text
stringlengths 14
5.77M
| meta
dict | __index_level_0__
int64 0
9.97k
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|---|---|---|
The New Harvest family would like to invite you to join us in fellowship and worship. While we are a small congregation we have a big heart and a welcoming spirit. Our fellowship is open to all who will come and join us.
Our worship style is comprised of a variety of music styles from traditional hymns to the latest contemporary Christian music. Our pastor's messages are directly from the Scriptures.
Our membership is comprised of people from diverse backgrounds, having moved here from places Maine, West Virginia, Jamaica, and Germany. While many others, like our pastor and his wife were born, raised and educated right here in South Cobb County.
You will find young people on occasion taking up offering, singing with the praise team and participating in many other areas of Worship.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,749
|
Vietnam Memorial FDC
Last one from Tom.... It's a Vietnam Memorial FDC cancelled on November 16 1984... Anyone has more info on this?
modern_who
This Vietnam Memorial FDC appears to be somebody's peace statement. The stamp featured is 2109, showing the memorial and visitors. The other add-on stamps are The Lions International issue, 1326 from 1967. The emblem shown is the MACV military insignia worn by members of the U.S. Army assigned to that unit. It stands for Military Assistance Command Vietnam, a non-combat unit whose purpose was to advise and support the military of the Republic of Vietnam or South Vietnam. The map shows the countries of the region, Vietnam, Laos, Cambodia, and Thailand. Together, Vietnam, Laos, and Cambodia were known a French Indo-China. Laos and Cambodia went their own way in 1951, and in 1954 the French were defeated in Vietnam but probably not without leaving their mark on that country forever. The wall has the name of every American who died or is still listed as missing in Vietnam inscribed on it.
Larry, APS Member
Modern-Vue Stamps on eBay
Thx modern, greatly appreciate your work her along with t360, laswabbie and others..
Mila_
Closer look.
Thx Mila...
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8
|
import type {Vocabulary} from "../../types"
import refKeyword from "./ref"
import typeKeyword, {JTDTypeError} from "./type"
import enumKeyword, {JTDEnumError} from "./enum"
import elements, {JTDElementsError} from "./elements"
import properties, {JTDPropertiesError} from "./properties"
import optionalProperties from "./optionalProperties"
import discriminator, {JTDDiscriminatorError} from "./discriminator"
import values, {JTDValuesError} from "./values"
import union from "./union"
import metadata from "./metadata"
const jtdVocabulary: Vocabulary = [
"definitions",
refKeyword,
typeKeyword,
enumKeyword,
elements,
properties,
optionalProperties,
discriminator,
values,
union,
metadata,
{keyword: "additionalProperties", schemaType: "boolean"},
{keyword: "nullable", schemaType: "boolean"},
]
export default jtdVocabulary
export type JTDErrorObject =
| JTDTypeError
| JTDEnumError
| JTDElementsError
| JTDPropertiesError
| JTDDiscriminatorError
| JTDValuesError
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,435
|
The 2020-21 Punjab State Super Football League was the 34th season of the Punjab State Super Football League, the top-tier football league in the Indian state of Punjab. Punjab Police are the champions. The league commenced from 19 December 2020.
Punjab Police FC, who were runners up last season, became champions this year.
Teams
A total number of 7 teams participated in the league. Last season champions Punjab FC didn't participated.
Standings
References
Punjab State Super Football League
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,143
|
{"url":"http:\/\/www.digitalfaq.com\/archives\/avisynth\/1705-avisynth-avisynth-dct.html","text":"Avisynth: New Avisynth DCT-Based Filter - digitalFAQ.com Forums [Archives]\n digitalFAQ.com Forums [Archives] Avisynth: New Avisynth DCT-Based Filter\n\n#1\n11-25-2002, 05:30 PM\n SansGrip Free Member Join Date: Nov 2002 Location: Ontario, Canada Posts: 1,135 Thanks: 0 Thanked 0 Times in 0 Posts\nTom Barry just released an experimental version of a filter that does a discrete cosine transform on the data before filtering. This should allow smoothing to be done only in certain frequency ranges.\n\nSounds very promising -- take a look at this Doom9 thread.\n\n@kwag:\n\nTheoretically this could target mosquito noise fairly precisely. Edit: Though possibly not as precisely as a full-blown FFT.\nSomeday, 12:01 PM\n Site Staff \/ Ad Manager Join Date: Dec 2002 Posts: 42 Thanks: \u221e Thanked 42 Times in 42 Posts\n#2\n11-25-2002, 08:14 PM\n black prince Free Member Join Date: Jul 2002 Posts: 1,224 Thanks: 0 Thanked 0 Times in 0 Posts\nHi SansGrip,\n\nSansGrip wrote:\nQuote:\n Tom Barry just released an experimental version of a filter that does a discrete cosine transform on the data before filtering. This should allow smoothing to be done only in certain frequency ranges. Sounds very promising -- take a look at this Doom9 thread. @kwag: Theoretically this could target mosquito noise fairly precisely. Edit: Though possibly not as precisely as a full-blown FFT.\nI saw this post earlier and found it interesting and realized you\nwere still working wavelets (GFR). This could further the knowledge\nto solve Gibbs effect. This filter is written using avisynth 2.5 experimental.\nThere is nothing available for avisynth 2.07. I don't know whether both\navisynths will work loaded at the same time. If these test filters were\nbeing created for avisynth 2.07, more testers could evaluation them.\n\n-black prince\n#3\n11-25-2002, 08:50 PM\n SansGrip Free Member Join Date: Nov 2002 Location: Ontario, Canada Posts: 1,135 Thanks: 0 Thanked 0 Times in 0 Posts\nQuote:\n Originally Posted by black prince I saw this post earlier and found it interesting and realized you were still working wavelets (GFR). This could further the knowledge to solve Gibbs effect.\nThey're really very similar concepts -- Tom's filter transforms the spatial data into the frequency domain, whereas a wavelet filter would transform it into the wavelet domain. Both approaches allow modification of the data in ways impossible (or exceptionally difficult) simply through manipulation of pixel values.\n\nQuote:\n Originally Posted by black prince This filter is written using avisynth 2.5 experimental. There is nothing available for avisynth 2.07.\nI know Tom used the DCT\/iDCT code from XViD, which means the transformation is probably done in YV12 space. It would be very hard to convert that part of the code to YUY2.\n\nQuote:\n Originally Posted by black prince I don't know whether both avisynths will work loaded at the same time.\nThey sure will. Simply download 2.5 and install it into a directory somewhere. DO NOT copy the new DLL into your system folder or you will overwrite 2.07.\n\nGet hold of the 2.5 filters you want to use, and copy those DLLs into the same directory that Avisynth 2.5 is in. Any .avs scripts you make IN THAT FOLDER will use 2.5 and the 2.5 filters instead of 2.07 and the 2.07 filters.\n\nYou can then use a script like:\n\nCode:\nLoadPlugin(\"mpeg2dec3.dll\")\nMpeg2Source(\"path\\to\\d2v\")\n# other filters\nDctFilter(...)\nIf you're sending this into XViD or DivX 5 via VirtualDub you can get specially modified versions of XViD and VirtualDub that will work with YV12 data. If you're not, you'll need to add a ConvertToYUY2() as the last line of the script. If you normally convert to RGB at the end, you'll need to do that AFTER the ConvertToYUY2() line, because currently ConvertToRGB() will not work with YV12 data.\n\nIt's not an ideal setup, but at least you can play with the new filters without harming your existing install .\n#4\n11-25-2002, 10:00 PM\n kwag Free Member Join Date: Apr 2002 Location: Puerto Rico, USA Posts: 13,537 Thanks: 0 Thanked 0 Times in 0 Posts\nQuote:\n Originally Posted by SansGrip @kwag: Theoretically this could target mosquito noise fairly precisely. Edit: Though possibly not as precisely as a full-blown FFT.\nThis looks good. Took a look at your sample .jpeg at doom9, and looks better than the original. BTW: Where's the link to AviSynth 2.5 ??\n\n-kwag\n#5\n11-26-2002, 09:02 AM\n KingTuk Free Member Join Date: Nov 2002 Posts: 107 Thanks: 0 Thanked 0 Times in 0 Posts\nhere is AviSynth 2.5\n\nhttp:\/\/cultact-server.novi.dk\/kpo\/av...nth_alpha.html\n#6\n11-26-2002, 09:18 AM\n kwag Free Member Join Date: Apr 2002 Location: Puerto Rico, USA Posts: 13,537 Thanks: 0 Thanked 0 Times in 0 Posts\nQuote:\n Originally Posted by KingTuk here is AviSynth 2.5 http:\/\/cultact-server.novi.dk\/kpo\/av...nth_alpha.html\nThanks KingTuk\n\n-kwag\n#7\n11-26-2002, 10:17 AM\n SansGrip Free Member Join Date: Nov 2002 Location: Ontario, Canada Posts: 1,135 Thanks: 0 Thanked 0 Times in 0 Posts\nQuote:\n Originally Posted by kwag This looks good. Took a look at your sample .jpeg at doom9, and looks better than the original.\nIf you read further into the thread you'll see that in 2.5 StackVertical -- for whatever reason -- reverses the order of the clips. What I thought was the sharpened, better-looking version is actually the original, and the blurred version is after DctFilter.\n\nThat said, iago is reporting compressibility gains of up to 30% without very noticible degredation in picture quality!\n#8\n11-26-2002, 11:37 AM\n kwag Free Member Join Date: Apr 2002 Location: Puerto Rico, USA Posts: 13,537 Thanks: 0 Thanked 0 Times in 0 Posts\nQuote:\nOriginally Posted by SansGrip\nQuote:\n Originally Posted by kwag This looks good. Took a look at your sample .jpeg at doom9, and looks better than the original.\nIf you read further into the thread you'll see that in 2.5 StackVertical -- for whatever reason -- reverses the order of the clips. What I thought was the sharpened, better-looking version is actually the original, and the blurred version is after DctFilter.\nIt was too good to be true\nQuote:\n That said, iago is reporting compressibility gains of up to 30% without very noticible degredation in picture quality!\nOh yeah! that sounds good, 234 minutes ( 180 + 30% ) on one CD-R (reduced quality) LBR and 156 minutes ( 120 + 30% ) for the regular 120 minute templates I have to do some test encodes\n\n-kwag\n#9\n11-26-2002, 02:41 PM\n Boulder Free Member Join Date: Sep 2002 Location: Lahti, Finland Posts: 1,652 Thanks: 0 Thanked 0 Times in 0 Posts\nI can't use v2.07 and 2.5a on the same computer. I did just as SansGrip said but no good. If 2.07 is in the system folder, it says \"there's no function called ConvertToYV12\" (I have to convert my YUY2 capture to YV12) .\n\nEdit: and I can't seem to be able to open the avs script in TMPGEnc with 2.5a. Despite the line ConvertToRGB24(), TMPGEnc just crashes.\n\nOS is WinXP, if this is of any use.\n#10\n11-26-2002, 03:24 PM\n SansGrip Free Member Join Date: Nov 2002 Location: Ontario, Canada Posts: 1,135 Thanks: 0 Thanked 0 Times in 0 Posts\nQuote:\n Originally Posted by Boulder I can't use v2.07 and 2.5a on the same computer. I did just as SansGrip said but no good. If 2.07 is in the system folder, it says \"there's no function called ConvertToYV12\" (I have to convert my YUY2 capture to YV12) .\nWhen Windows looks for \"avisynth.dll\" it looks first in the current directory, only moving to other places (such as your system folder) if it doesn't find it. This means only avs scripts in the same folder as the 2.5 dll will use that version. All others will use 2.07.\n\nQuote:\n Originally Posted by Boulder Edit: and I can't seem to be able to open the avs script in TMPGEnc with 2.5a. Despite the line ConvertToRGB24(), TMPGEnc just crashes.\nYou'll need to make sure that TMPGEnc is using the 2.5 directory as its working directory, otherwise it'll load the 2.07 dll and crash. To do this you could either copy TMPGEnc and all its files into that directory too, or you could make a new shortcut to it, and edit the working\/current directory entry in the shortcut properties.\n#11\n11-26-2002, 04:51 PM\n Boulder Free Member Join Date: Sep 2002 Location: Lahti, Finland Posts: 1,652 Thanks: 0 Thanked 0 Times in 0 Posts\nQuote:\n Originally Posted by SansGrip When Windows looks for \"avisynth.dll\" it looks first in the current directory, only moving to other places (such as your system folder) if it doesn't find it. This means only avs scripts in the same folder as the 2.5 dll will use that version. All others will use 2.07. You'll need to make sure that TMPGEnc is using the 2.5 directory as its working directory, otherwise it'll load the 2.07 dll and crash. To do this you could either copy TMPGEnc and all its files into that directory too, or you could make a new shortcut to it, and edit the working\/current directory entry in the shortcut properties.\n\nThe script was in the same directory.. VDubMod should probably be pointing to the very same directory just as TMPGEnc. That would make some sense. I'll have to experiment with this as I think that most plugin authors will end up writing only 2.5 plugins - and I can't live without your latest plugins!\n#12\n11-26-2002, 05:42 PM\n SansGrip Free Member Join Date: Nov 2002 Location: Ontario, Canada Posts: 1,135 Thanks: 0 Thanked 0 Times in 0 Posts\nQuote:\n Originally Posted by Boulder The script was in the same directory..\nThat's strange, then. If the script is in the same directory as the 2.5 dll and the 2.5 filters you want to use, you should be able to double-click the script and watch it in WMP. If that doesn't work then something strange is going on.\n\nQuote:\n Originally Posted by Boulder I'll have to experiment with this as I think that most plugin authors will end up writing only 2.5 plugins\nIt's not that there's much difference writing 2.5 filters as 2.07 filters, at least as far as YUY2 goes. In fact, pretty much all a plugin writer has to do is recompile the filter using the new 2.5 Avisynth header file.\n\nThe big difference is that 2.5 supports YV12, and it's generally much easier to make filters that operate on YV12 than on YUY2 (though there are some caveats). They also generally run faster, sometimes as much as 25-30%.\n\nSince it was decided that supporting YV12 would be a good idea (and it is), I look forward to the day when everyone is using 2.5 and YV12 and I won't have to write two versions of everything, which is a major pain in the neck .\n#13\n11-27-2002, 04:08 AM\n Boulder Free Member Join Date: Sep 2002 Location: Lahti, Finland Posts: 1,652 Thanks: 0 Thanked 0 Times in 0 Posts\nQuote:\n Originally Posted by SansGrip Since it was decided that supporting YV12 would be a good idea (and it is), I look forward to the day when everyone is using 2.5 and YV12 and I won't have to write two versions of everything, which is a major pain in the neck .\nI completely agree. As it's possible to use 2.5 with YUY2, there's really no need for the older one when 2.5 is considered non-alpha or -beta. The only thing that bugs me is that some 2.5 filters, for example Convolution3d, want YV12 data so I'll have to convert to YV12 if I want to use them.\n#14\n11-27-2002, 04:55 PM\n SansGrip Free Member Join Date: Nov 2002 Location: Ontario, Canada Posts: 1,135 Thanks: 0 Thanked 0 Times in 0 Posts\nHere's a present for y'all .\n#15\n11-29-2002, 12:49 PM\n SansGrip Free Member Join Date: Nov 2002 Location: Ontario, Canada Posts: 1,135 Thanks: 0 Thanked 0 Times in 0 Posts\nWhat, no-one's tried it yet??\n#16\n11-29-2002, 01:22 PM\n Boulder Free Member Join Date: Sep 2002 Location: Lahti, Finland Posts: 1,652 Thanks: 0 Thanked 0 Times in 0 Posts\nI experimented shortly with Tom's original AVS2.5 version. It seems that the filter messes up permanent subtitles quite easily. As we here in Finland have non-Finnish programs subtitled and I use permanent subs in VCDs, it is a problem I'll have to solve before taking it into everyday use. I could live without subtitles but my wife couldn't - now what can you say to that\n\nUnfortunately I haven't had the time to experiment with different parms, but I think I'll try to do that tomorrow. If you happen to have any tips, I'm all ears.\n#17\n12-02-2002, 04:49 AM\n Graal_CPM Free Member Join Date: Jun 2002 Location: Paris, France Posts: 54 Thanks: 0 Thanked 0 Times in 0 Posts\nHi Boulder!\n\nYou can try to put permanent subs command at the very end of your avisynth script .\n\nThanx SansGrip's AviSynth 2.0.X version, I have done few testings with dctfilter myself and it is very promising. Right now, here is the kinda \"production script\", I use to encode in PAL 352x288 with KVCD LBR :\n\nBilinearResize\nFluxSmooth()\nBB_Resolution = 352*288\nBB_StrengthConstant = 352 * 240 * 20 # Base strength\nStrengthValue = round(BB_StrengthConstant \/ BB_Resolution)\nBlockbuster( method=\"noise\", detail_min=1, detail_max=10, variance=1 )\nBlockbuster( method=\"sharpen\", detail_min=20, detail_max=90, strength=StrengthValue )\nDctFilter(1,1,1,1,1,1,.5,0)\nVobSub\nLegalClip()\n\nQuality is fine with Q around 35. I should try this with KVCDx3.\n\nCheers.\n\nEdit : maybe you were you talking about permanent subs *already* burned into the movie, like on some hudge trailers you can find around the internet?\n#18\n12-02-2002, 06:16 AM\n Boulder Free Member Join Date: Sep 2002 Location: Lahti, Finland Posts: 1,652 Thanks: 0 Thanked 0 Times in 0 Posts\nThanks for the tip, Graal_CPM. I was just about to report that in the AVS2.0x version of the filter, the permanent subs are not affected!However, it could be that I've done something differently, so I'll have to test the AVS2.5 version again.\n\nCompression seems to increase nicely without a loss of details with light settings\n#19\n12-06-2002, 06:47 AM\n jp Free Member Join Date: May 2002 Posts: 6 Thanks: 0 Thanked 0 Times in 0 Posts\nI can't access www.jungleweb.net from Portugal.\nWhere else can I get the DCT filter 2.0x ?\nI'm willing to try it because with the reduction of Max GOP in the last templates it's very dificult to put 1 film in 1 CD at 352*576.\n\nJP\n\ncaldas_lopes@clix.pt\n#20\n12-06-2002, 08:52 AM\n fozzieb Free Member Join Date: Sep 2002 Location: Scotland Posts: 46 Thanks: 0 Thanked 0 Times in 0 Posts\nI cant access it either, it must be down\n\n Similar Threads Thread Thread Starter Forum Replies Last Post Prodater64 Audio Conversion 2 12-27-2005 03:17 PM supermule Avisynth Scripting 1 10-23-2005 08:06 AM marcellus Avisynth Scripting 3 04-21-2004 01:00 PM nighthawk Avisynth Scripting 3 01-31-2004 12:24 PM black prince Avisynth Scripting 6 11-28-2002 08:51 AM","date":"2020-08-08 17:22:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.4899512529373169, \"perplexity\": 5303.138108114478}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439738015.38\/warc\/CC-MAIN-20200808165417-20200808195417-00247.warc.gz\"}"}
| null | null |
/* ###
* IP: GHIDRA
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package ghidra.trace.database.listing;
import ghidra.program.model.address.AddressRange;
import ghidra.program.model.lang.Register;
import ghidra.trace.model.Lifespan;
import ghidra.trace.model.guest.TracePlatform;
import ghidra.trace.model.listing.TraceBaseDefinedUnitsView;
import ghidra.trace.model.listing.TraceCodeUnit;
import ghidra.util.exception.CancelledException;
import ghidra.util.task.TaskMonitor;
public interface InternalTraceBaseDefinedUnitsView<T extends TraceCodeUnit>
extends TraceBaseDefinedUnitsView<T>, InternalBaseCodeUnitsView<T> {
@Override
default void clear(TracePlatform platform, Lifespan span, Register register,
TaskMonitor monitor) throws CancelledException {
AddressRange range = platform.getConventionalRegisterRange(getSpace(), register);
clear(span, range, true, monitor);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,898
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BJP women's wing will launch 'Kamal Diya' Campaign on October 12.
BJP women's wing will promote the plans of the Raje government and the central government through the bhajan mandalis. The women's wing is preparing the bhajans (hymns) of different schemes including Swachh Bharat Mission, Ujjwala Yojana, and Bhamashah Yojana.
According to the BJP women's wing president, Madhu Sharma, the workers of the BJP Mahila Morcha will campaign with the bhajan mandali and promote the plans across the state. The bhajan mandalis will go house-to-house and sing hymns based on the schemes of Raje government. 'Apart from this, we will also run the 'Akhand Navratra' and 'Kamal Diya (Lotus lamp)' campaign.' she added.
The BJP women's wing had launched 'Kamal Mehndi' Campaign during the previous assembly elections. Former state president of BJP's women wing and women commission, Suman Sharma said, 'Our Kamal Mehndi campaign was very successful. This time Kamal Diya campaign has been introduced and it too will be successful.' 'We have set a goal to reach 54 lakh families in the state.' She added.
Wing president Madhu Sharma said that the election campaign of Mahila Morcha will start on October 12, on the 100th birth anniversary of Rajmata Vijaya Raje Scindia. Hence, on October 12, women's wing will organize Vijay Sankal Nari Shakti Sammelan in all 200 constituencies of the state.
The Election Commission has announced the election dates in the state. December 7, will be the day when the future of Rajasthan will be written and the final counting will be on December 11.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,206
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\section{\label{sec:introduction} Introduction}
Radio frequency plasma sources such as capacitively or inductively coupled plasmas (CCPs/ICPs) are necessarily operated using external electric circuits to transfer the generated power into the discharge \cite{lieberman_principles_2005, chabert_physics_2011}. These circuits include generators, matching networks, filters and power lines among others. The voltage waveform at the driven electrodes and the current flowing through the plasma discharge depend decisively on the electrical properties of the network elements, which the plasma interacts with. Plasmas operated at radio frequency and low pressure also show a nonlinear behavior, making the interaction not easily predictable \cite{mussenbrock_nonlinear_2007,mussenbrock_enhancement_2008, czarnetzki_self-excitation_2006, lieberman_effects_2008, miller_electrical_1992, ziegler_temporal_2009, yamazawa_effect_2009, yamazawa_electrode_2015}.
Many plasma simulation techniques focus on the plasma dynamics itself and external circuits are often neglected or drastically simplified, e.g., to a simple bias capacitance. Electrical equivalent circuit models allow to comprisingly incorporate the plasma and complex external circuits in the solution algorithms \cite{schmidt_consistent_2018, schmidt_multi_2018}. In contrast, Particle-in-Cell (PIC) simulations have been coupled to external series circuits consisting of a resistance, an inductance and a capacitance via conservation of charge by Verboncoeur et al.\ \cite{verboncoeur_simultaneous_1993}. In their method, the differential equations following Kirchhoff's circuit laws have been discretized and solved, ensuring numerical stability of the whole simulation. While conceptually possible, an extension to more complex external networks involves a similarly elaborate procedure, which limits the practical applicability.
In this work, we propose a method for coupling an external circuit to any desired plasma simulation that gives a voltage-current relation based on the method of harmonic balance \cite{maas_nonlinear_2003, gilmore_nonlinear_1991}. Once implemented, the external circuit can be included via a simple netlist, hence, making changes in the setup easy to implement and investigate. The main idea is to split the whole system into a linear part -- the electric circuit -- and a nonlinear part -- the plasma. Notably, the nonlinear part may contain linear elements, whereas the linear part must not contain any nonlinear elements. The voltage drop between the interconnection(s) connecting the two is sought for, so that the currents nullify one another, i.e. Kirchhoff's nodal law is satisfied. This needs to be accomplished for every harmonic of interest, hence the term harmonic balance. The procedure is detailed subsequently.
The manuscript is organized as follows: In chapter \ref{sec:method} the principle and the algorithm of the method is discussed. Two different plasma descriptions are coupled to external circuits in this work. On the one hand, a global equivalent circuit is utilized, which can also be simulated using SPICE for reference. On the other hand, a 1-dimensional Particle-in-Cell (PIC) simulation is used and coupled to two variants of external circuits. These methods are proposed in detail in chapter \ref{sec:model} and the respective results discussed in comparison with reference methods in chapter \ref{sec:results}.
\section{\label{sec:method} Harmonic Balance Algorithm}
Harmonic balance is a common method for calculating the interactions of linear circuits with nonlinear elements such as diodes or transistors \cite{maas_nonlinear_2003, gilmore_nonlinear_1991}. The fundamental idea is to split the circuit into a linear and a nonlinear part and find the voltage in between those regimes for which each harmonic in the respective currents is the same as in the other circuit, i.e., balanced. Hence the term harmonic balance.
A detailed description of the implementation is provided by Maas \cite{maas_nonlinear_2003}. In this chapter, the basic concept is reviewed and details about the necessary changes in the method to adapt it for plasma--circuit simulations are discussed.
\begin{figure}[t!]
\centering
\resizebox{8cm}{!}{
\includegraphics[width=8cm]{Harmonic_balance2.pdf}
}
\caption{Harmonic balance setup}
\label{fig:harmonic_balance}
\end{figure}
The linear circuit consisting of resistances, inductances and capacitances on the one hand and the nonlinear plasma on the other hand. This is depicted in figure~\ref{fig:harmonic_balance}. The transient voltage $v(t)$ dropping between the interconnection and, respectively, the transient current $i(t)$ entering the nonlinear circuit part can be written in Fourier series representation as
\begin{eqnarray}
\label{eq:fourier_series1}
i(t) &=& \sum\nolimits_{k=-K}^K I_k e^{j k \omega t},\\
\label{eq:fourier_series2}
v(t) &=& \sum\nolimits_{l=-K}^K V_l e^{j l \omega t},
\end{eqnarray}
with $I_k = I_{-k}^*$ and $V_l = V_{-l}^*$. For the purpose of practicability $K$ is chosen as a finite number. In the algorithm described in the following paragraphs, only values for $k,l \geq 0$ are considered, which reduces the complexity of the procedure. Thereby (temporarily) complex values for $i(t)$ and $v(t)$ arise.
The Fourier coefficients $I_k$ and $V_l$ can be consistently defined as
\begin{eqnarray}
\label{eq:coefficients_complex}
I_k &=& \frac{1}{T} \int_{0}^T i(t) e^{- j k \omega t} dt, \\
V_l &=& \frac{1}{T} \int_{0}^T v(t) e^{- j k \omega t} dt.
\end{eqnarray}
These coefficients can be written in vector form $\boldsymbol{I}_\mathrm{NL}$ and $\boldsymbol{V}$ of dimension $K+1$, containing a (real valued) DC entry, a complex valued fundamental frequency component and $K-1$ complex valued harmonics of the fundamental frequency. The goal in harmonic balance is to find a voltage $\boldsymbol{V}$ for which the current flowing into the linear circuit $\boldsymbol{I}_\mathrm{L}$ and the current flowing into the plasma $\boldsymbol{I}_\mathrm{NL}$ satisfy Kirchhoff's nodal law. In other words, a current error vector
\begin{align}
\boldsymbol{F} = \boldsymbol{I}_\mathrm{L} + \boldsymbol{I}_\mathrm{NL}
\label{eq:ohms_law}
\end{align}
can be defined which is desired to vanish, $\boldsymbol{F} = 0$. Note that also the linear current $\boldsymbol{I}_\mathrm{L}$ is written as a vector of dimension $K+1$.
The transadmittance matrix $\underline{\boldsymbol{Y}}$ of a linear circuit can be calculated using nodal analysis, which has to be done for each frequency. This can easily be automated and the linear circuit information thereby included via a simple netlist as it is done for example in SPICE~\cite{nagel_spice_1973}. This leads to a linear current entering port 2
\begin{align}
\label{eq:linear}
\boldsymbol{I}_\mathrm{L} = \underline{\boldsymbol{Y}_{21}} \cdot \boldsymbol{V}_\mathrm{S} + \underline{\boldsymbol{Y}_{22}} \cdot \boldsymbol{V},
\end{align}
which entails current contributions due to the voltages $\boldsymbol{V}_\mathrm{S}$ and $\boldsymbol{V}$ at both ports 1 and 2 (cf.\ Figure~\ref{fig:harmonic_balance}). The linear current can be straightforwardly calculated in frequency space.
The plasma simulations used in this work are performed in time domain. Therefore, the nonlinear current $\boldsymbol{I}_\mathrm{NL}$ is calculated from the evolution $i(t)$, which is the result of a transient simulation of the plasma. The latter is subject to the voltage $v(t)$, obtained using the Fourier series representation of equation~\eqref{eq:fourier_series2} with coefficients $\boldsymbol{V}$.
\begin{figure}[t!]
\centering
\resizebox{8cm}{!}{
\includegraphics[width=8cm]{Harmonic_balance_algorithm.pdf}
}
\caption{Harmonic balance algorithm.}
\label{fig:algorithm}
\end{figure}
The algorithm of the simulation is depicted in figure~\ref{fig:algorithm}. Starting with an initial guess of $\boldsymbol{V}$ all described values can be calculated. The most complex part of the algorithm is to change $\boldsymbol{V}$ until $\boldsymbol{F}$ is satisfactory small, which is done by utilizing Newton's method: After the $p$-th iteration step, the voltage $\boldsymbol{V}^{p+1}$ can be calculated using the previous value $\boldsymbol{V}^{p}$ and the error $\boldsymbol{F}$ in the form of
\begin{align}
\label{eq:newton_next_step}
{\boldsymbol{V}}^{p+1} = \boldsymbol{V}^{p} - \underline{\boldsymbol{J}_\mathrm{F}}^{-1} {\boldsymbol{F}}({\boldsymbol{V}^p}),
\end{align}
with the Jacobian
\begin{eqnarray}
\underline{\boldsymbol{J}_\mathrm{F}} &=& \left.\frac{\partial {\boldsymbol{F}}({\boldsymbol{V}})}{\partial {\boldsymbol{V}}}\right|_{{\boldsymbol{V}} = {\boldsymbol{V}}^p} \nonumber \\
\label{eq:jacobian}
&=& \underline{\boldsymbol{Y}_{22}} + \frac{\partial \boldsymbol{I}_\mathrm{NL}}{\partial \boldsymbol{V}}.
\end{eqnarray}
The voltage source term $\underline{\boldsymbol{Y}_{21}} \cdot \boldsymbol{V}_\mathrm{S}$ is independent of $\boldsymbol{V}$ and vanishes.
The second term in equation~\eqref{eq:jacobian} can be written as
\begin{align}
\frac{\partial I_k}{\partial V_l} = \frac{1}{T} \int_{0}^T \frac{ \partial i(t) }{\partial V_l} e^{- j k \omega t} dt.
\end{align}
This can be expanded to
\begin{align}
\label{eq:jacobian_element}
\frac{\partial I_k}{\partial V_l} = \frac{1}{T} \int_0^T \frac{\partial i(t)}{\partial v(t)} e^{-j (k-l) \omega t} dt,
\end{align}
using $\partial v(t) / \partial V_l = e^{j l \omega t}$.~\cite{maas_nonlinear_2003}
For a time-varying voltage $v(t)$, the entries of the Jacobian depend on
\begin{eqnarray}
\frac{\partial i(t)}{\partial v(t)} &=& \frac{\partial i(t)}{\partial t} \left( \frac{\partial v(t)}{\partial t} \right)^{-1} \nonumber \\
\label{eq:partial1}
&=& \frac{\sum\nolimits_{k=1}^K I_k k e^{j k \omega t}}{\sum\nolimits_{l=1}^K V_l l e^{j l \omega t} },
\end{eqnarray}
using the definitions from equations~\eqref{eq:fourier_series1} and \eqref{eq:fourier_series2}. For DC excitation, the time derivative of $v(t)$ vanishes and equation~\eqref{eq:partial1} cannot be utilized. In this case, it is safe to assume a linear relation of the current components to the voltage
\begin{eqnarray}
\frac{\partial i(t)}{\partial v^{(0)}}
\label{eq:partial_DC}
&=& \frac{\sum\nolimits_{k=0}^K I_k e^{j k \omega t}}{V_0 }.
\end{eqnarray}
Equation \eqref{eq:partial1} and \eqref{eq:partial_DC} can be directly incorporated into equation~\eqref{eq:jacobian_element}.
By performing a plasma simulation with a specific voltage $\boldsymbol{V}$ (respectively $v(t)$, which may entail several non-zero components $V_l$), only the collective system response $\boldsymbol{I}_\mathrm{NL}$ to this particular excitation may be obtained. The system response is probed using $\frac{\partial i(t)}{\partial v(t)}$ for the specified work point only. Moreover, the integration kernel of equation~\eqref{eq:jacobian_element} solely depends on the index difference $k-l$ and not on the individual indices $k$ and $l$. Therefore, for a given response $\frac{\partial i(t)}{\partial v(t)}$, the matrix elements $G_{k,l} = \frac{\partial I_k}{\partial V_l}$ are the elements of a circulant matrix $\underline{\boldsymbol{G}}$ (fully specified by a vector with elements $g_k = G_{k,\,l=0}$). The corresponding Jacobian matrix, however, is insufficient as it does not entail the isolated influence of all specific frequency components $V_l$ of the voltage $\boldsymbol{V}$ on the system -- specifically, on the current $\boldsymbol{I}_\mathrm{NL}$.
To obtain the selective system response to the $m$-th harmonic of the voltage and to set up the corresponding parts of the Jacobian, the simulation needs to be performed not only with $\boldsymbol{V}$, but also with $K+1$ simulations using a voltage, which is disturbed at the $m$-th entry. We define a voltage $\boldsymbol{V_{\Delta}}$, which contains these disturbances for each frequency component. The latter should not be too large in order to not change the state of the system when performing the respectively different simulations, but large enough to be distinctive from noise. We found empirically that $\boldsymbol{V_{\Delta}} = 1/100 \, \boldsymbol{V}$ is a good choice. Especially for the DC value the disturbance may need to be chosen larger, which heavily depends on the specific plasma model.
The simulation is now performed $K+1$ times with a voltage $\boldsymbol{\hat{V}}$, which differs from $\boldsymbol{V}$ at the $m$-th entry by $V_{\Delta m}$, specifically $\hat{V}_l = V_l + V_{\Delta m} \delta_{l m}$ with $\delta_{lm}$ the Kronecker delta. The current resulting from the disturbed excitation is denoted by $\boldsymbol{\hat{I}}$. To now account for the variances in the current due to the specific difference in the excitation at the $m$-th entry in the voltage, equation~\eqref{eq:partial1} can be reformulated to
\begin{align}
\label{eq:partial2}
\frac{\partial i_{\Delta}^{(m)} (t)}{\partial v_{\Delta}^{(m)} (t)} = \frac{\sum\nolimits_{k=1}^K (I_k -\hat{I}_k ) k e^{j k \omega t}}{ \sum\nolimits_{l=1}^K (V_l -\hat{V}_l) l e^{j l \omega t} } = \frac{\sum\nolimits_{k=1}^K (I_k - \hat{I}_k) k e^{j k \omega t}}{- V_{\Delta m} m e^{j m \omega t} },
\end{align}
which includes the change of all harmonics in the current $k \in \left[1,K\right]$ due to the $m$-th probing harmonic in the voltage at frequency $m \omega$. The influence of the DC current component neglected in equation~\eqref{eq:partial2} is immaterial, as any contribution is integrated out in equation~\eqref{eq:jacobian_element}. Again, equation~\eqref{eq:partial2} is not defined for DC. In this case the evaluation needs to be based on equation~\eqref{eq:partial_DC} leading to
\begin{align}
\label{eq:partial_DC2}
\frac{\partial i_{\Delta}^{(0)} (t)}{\partial v_{\Delta}^{(0)} } = \frac{\sum\nolimits_{k=0}^K (I_k - \hat{I}_k) e^{j k \omega t}}{- V_{\Delta 0}}.
\end{align}
Plugging equation~\eqref{eq:partial2} and \eqref{eq:partial_DC2} into equation~\eqref{eq:jacobian_element}, the $m$-th column vector of the Jacobian matrix
\begin{align}
\label{eq:jacobian_element2}
G_{k,\,m} = \frac{\partial I_{\Delta k}}{\partial V_{\Delta m}} = \frac{1}{T} \int_0^T \frac{\partial i_{\Delta}^{(m)} (t)}{\partial v_{\Delta}^{(m)} (t)} e^{-j (k-m) \omega t} dt
\end{align}
may be evaluated. The latter can be computed directly in case of DC as a function of the response in the respective harmonic
\begin{align}
\label{eq:jacobian_element_dc}
G_{k,\,0} = \frac{I_k - \hat{I}_k}{- V_{\Delta 0}}.
\end{align}
Finally, the Jacobian matrix
\begin{align}
\underline{\boldsymbol{J}_F} = \underline{\boldsymbol{Y}_{22}} +
\begin{bmatrix}
G_{0,0} & G_{0,1} & \dots & G_{0,K}\\
G_{1,0} & G_{1,1} & \dots & G_{1,K}\\
\vdots & \vdots & \ddots & \\
G_{K,0} & G_{K,1} & \dots & G_{K,K}\\
\end{bmatrix}
\end{align}
is no longer circulant, but fully populated and dense.
\section{\label{sec:model} Plasma Simulation Models}
Harmonic balance can generally be applied to any plasma simulation that provides a voltage--current relation and operates in time-domain with steady-state. In this work, we use two different plasma models: a) A nonlinear global equivalent circuit model that can be coupled to a linear external circuit. b) A self-consistent 1-D PIC simulation to be coupled to a linear external circuit. Both models are used to simulate a capacitively coupled argon discharge at low pressure ($p < 10$~Pa).
In this work, only a single excitation frequency is considered. If the number of frequencies of interest need to be higher and these frequencies are harmonics of each other, the proposed method can be applied without any alterations. If this is not the case, the algorithm needs to be adjusted. A detailed discussion of this can be found in \cite{maas_nonlinear_2003}.
\subsection*{Global equivalent circuit model}
\begin{figure}[t!]
\centering
\resizebox{12cm}{!}{
\includegraphics[width=12cm]{Matching_1f.pdf}
}
\caption{Generator, matching network and stray elements attached to an equivalent circuit of the plasma.}
\label{fig:setup}
\end{figure}
The global plasma model utilized in this work is based on considerations introduced and discussed in~\cite{mussenbrock_enhancement_2008, mussenbrock_nonlinear_2006, lieberman_effects_2008, ziegler_temporal_2009}, while its interaction with an external electric circuit has been studied using ngSPICE~\cite{vogt_ngspice_2017} in previous works \cite{schmidt_consistent_2018, schmidt_multi_2018}. In the following, the model is only briefly reviewed. For a more detailed description the referenced literature is suggested.
\begin{table}
\begin{tabular}{l c}
\toprule
Parameter & Value \\
\colrule
$T_\mathrm{Ar}$ & 300 K \\
$n$ & $1.25 \times 10^{15}~\mathrm{m}^{-3}$ \\
$p$ & 0.66 Pa \\
$k_\mathrm{B} T_e$ & 4.73 eV \\
$A_\mathrm{E}$ & 100 cm$^2$\\
$A_\mathrm{G}$ & 300 cm$^2$ \\
$l_\mathrm{B}$ & 5.7 cm \\
$V_\mathrm{rf}$ & 100 V \\
$\omega$ & $2 \pi \times 13.56$~MHz \\
$R_\mathrm{rf}$ & $50~\Omega$ \\
$R_\mathrm{m}$ & $0.5~\Omega$ \\
$R_\mathrm{stray}$ & $0.5~\Omega$ \\
$C_\mathrm{stray}$ & $200$ pF \\
$C_\mathrm{m1}$ & $1550$ pF \\
$C_\mathrm{m2}$ & $175$ pF \\
$L_\mathrm{m2}$ & $1500$ nH \\
\botrule
\end{tabular}
\caption{Input parameters for the the global plasma model simulation.}
\label{table:parameter}
\end{table}
The model of the plasma is divided into two sheaths and a bulk. On the one hand, following a generalized Ohm's law, the bulk is modeled as an inductance $L_\mathrm{pl}=l_\mathrm{B} m_\mathrm{e}/e^2 n A_\mathrm{E}$ and a resistance $R_\mathrm{pl} = \nu_\mathrm{eff} L_\mathrm{pl}$, with the bulk length $l_\mathrm{B}$, the electron mass $m_\mathrm{e}$, the plasma density $n$, the electrode area $A_\mathrm{E}$, and the effective collision frequency $\nu_\mathrm{eff}$. The sheaths, on the other hand, consist of a nonlinear capacitance, a constant current source to account for the steady ion flux and a diode to model the electron dynamics in the sheath. The current source has a value of $I_\mathrm{i,1} = A_\mathrm{E} e n u_\mathrm{B}$ for the driven electrode and $I_\mathrm{i,2} = A_\mathrm{G} e n u_\mathrm{B}$ for the grounded electrode with the grounded area $A_\mathrm{G}$, the Bohm velocity $u_\mathrm{B} = \sqrt{k_\mathrm{B} T_e/m_\mathrm{i}}$, the electron temperature $T_e$, and the ion mass $m_\mathrm{i}$. The electron current depends on the sheath voltage and amounts to $I_\mathrm{e,1} = A_\mathrm{E} e n \bar{v}_\mathrm{e} \mathrm{exp}\left( -e V_\mathrm{S,1}/k_\mathrm{B} T_\mathrm{e} \right)$ for the driven electrode and $I_\mathrm{e,2} = A_\mathrm{G} e n \bar{v}_\mathrm{e} \mathrm{exp}\left( -e V_\mathrm{S,2}/k_\mathrm{B} T_\mathrm{e} \right)$ for the grounded electrode, with mean electron speed $\bar{v}_\mathrm{e} = \sqrt{8 k_\mathrm{B} T_e/\pi m_\mathrm{i}}$. Lastly, the nonlinear capacitances have a value of $C_\mathrm{S,1}= \left( e n \epsilon_0 A_\mathrm{E}^2/2 V_\mathrm{S,1} \right)^{\frac{1}{2}}$ and $C_\mathrm{S,2}= \left( e n \epsilon_0 A_\mathrm{G}^2/2 V_\mathrm{S,2} \right)^{\frac{1}{2}}$, respectively, resulting from a Matrix sheath model \cite{lieberman_principles_2005}. Again, a detailed discussion about this model coupled to an external electrical circuit can be found elsewhere \cite{schmidt_consistent_2018}. The resulting equivalent circuit is depicted in Figure~\ref{fig:setup} on the far right side attached to a generator, a matching network and stray elements. An obvious method to simulate such a model with an external network attached is to make use of a circuit simulation tool such as ngSPICE \cite{vogt_ngspice_2017}. This model therefore serves as a proof of concept for the harmonic balance analysis, since it provides a reference method to simulate the setup. The values chosen for the plasma parameters and the network elements are the same as published in \cite{schmidt_consistent_2018} and are listed in Table~\ref{table:parameter}.
\subsection*{Particle in Cell}
For many investigations, a global plasma model is not satisfactory since all spatial information on the plasma is integrated and not resolved. For a detailed study of, e.g., the electron dynamics inside the sheath or the ion energy distributions at the walls, a spatially resolved simulation such as PIC is useful. PIC incorporates the plasma dynamics self-consistently through a kinetic description of electrons and ions coupled to the electromagnetic fields. As a result it offers more insights than a global model at the cost of being computationally expensive. In this work the 1-dimensional PIC code \textit{yapic} is used \cite{turner_simulation_2013, trieschmann_particle--cell/test-particle_2017}.
Two different setups are investigated: A geometrical symmetric setup with an electrode area of $A_\mathrm{E} = 315~\mathrm{cm}^{2}$ and a resistance and capacitance in series (RC-unit) attached to it. The values of the external circuit elements are $V_\mathrm{S} = 100$~V, $R=10~\Omega$, and $C = 300$~pF. This simple circuit can be included by making use of harmonic balance, but alternatively also by solving the network's auxiliary differential equations simultaneously with the discharge as proposed by Verboncoeur et al. \cite{verboncoeur_simultaneous_1993}. Similar to the simulation of the global plasma model, this approach serves as a reference for a proof of concept, since two methods for solving the same case are available. The second setup adds a more realistic external circuit to the PIC simulation, namely, the one depicted in Figure~\ref{fig:setup} with the plasma being modeled by PIC instead of the equivalent circuit. A geometrically asymmetric discharge in spherical coordinates is simulated with a driven electrode area $A_\mathrm{E} = 50~\mathrm{cm}^{2}$ and a grounded electrode area $A_\mathrm{G}=1250~\mathrm{cm}^{2}$. For this network no reference simulation exists, making this a demonstration of the flexibility provided by harmonic balance.
Both cases use an argon discharge with a pressure $p=1$~Pa and a temperature of 650~K.
The simulations with respectively varied voltages are always initiated with the same steady-state solution, i.e. the same number of particles and their distribution in phase-space. The assumption is that the voltages and thereby the state of the system does not change significantly. In this case, convergence can be reached faster.
\section{\label{sec:results} Results and Discussion}
\subsection*{Global equivalent circuit model}
\begin{figure}[t!]
\centering
\resizebox{12cm}{!}{
\includegraphics[width=12cm]{SPICE_HB_ampl.png}
}
\caption{Voltage and current using the global plasma model. a) Transient solution. The straight lines indicate the results obtained using ngSPICE, while the dotted lines represent the results obtained using the harmonic balance algorithm. b) The absolute values of the different current harmonics. In black are the results obtained by ngSPICE, in grey are those which are obtained from the harmonic balance algorithm.}
\label{fig:result_global}
\end{figure}
The network depicted in Figure~\ref{fig:setup} is simulated with both ngSPICE and the harmonic balance algorithm. Within the latter, the whole global plasma model is treated as the nonlinear part, while the rest of the circuit (excluding the voltage source) is incorporated in the transadmittance matrix. Thereby, the solution of the plasma model can conceptually be obtained using any arbitrary method, such as ngSPICE or Mathematica. Performing the steps described in Section~\ref{sec:method}, a converged solution gives the steady-state voltages and currents in the system. The transient values of $v_\mathrm{pl}(t)$ and $i_\mathrm{pl}(t)$ (corresponding to Fourier components $V_\mathrm{pl}$ and $I_\mathrm{pl}$) are shown in Figure~\ref{fig:result_global}. In a) the result of the harmonic balance method is plotted together with a reference simulation using ngSPICE, while b) shows the corresponding Fourier components of the current $I_\mathrm{pl}$. It is obvious that both results are practically identical, which is the desired outcome. The differences that still arise can be explained with numerical inaccuracies and a finite number of $K=15$ considered harmonics in the Fourier series. While the voltage is almost completely sinusoidal at the fundamental excitation frequency, the current consists of a number of harmonics due to the strong interaction of the plasma bulk and the nonlinear sheaths at low pressure and given the asymmetry of the setup. More details on the discharge physics and the matching procedure are discussed in \cite{schmidt_consistent_2018}.
\subsection*{Particle in Cell}
\begin{figure}[t!]
\centering
\resizebox{12cm}{!}{
\includegraphics[width=12cm]{PIC_HB_ampl.png}
}
\caption{Plasma-voltage and current using PIC and an external RC-element. a) Transient solution. The straight lines indicate the results using the method proposed by Verboncoeur et al.\ \cite{verboncoeur_simultaneous_1993}, while the dotted plots show the results obtained using the harmonic balance algorithm, b) The absolute values of the different harmonics of the current. In black are the results obtained using the method proposed by Verboncoeur et al.\ \cite{verboncoeur_simultaneous_1993}, in grey are those obtained from the harmonic balance algorithm.}
\label{fig:result_PIC_RC}
\end{figure}
First, the symmetric setup with an RC-unit attached and solved using PIC simulations is considered. The current flowing through the discharge is expected to contain only a small amount of harmonics aside from the excitation frequency. Figure~\ref{fig:result_PIC_RC} shows the simulation result for the plasma current and voltage in which the external circuit is simulated using the harmonic balance approach and the method proposed by Verboncoeur et al.\ \cite{verboncoeur_simultaneous_1993}. Both cases lead to the same result of a sinusoidal voltage of 95~V amplitude and a current with 120~mA amplitude and small amounts of higher harmonics. It is worth noting that even in a symmetric arrangement of the discharge higher harmonics can be observed, since the nonlinear characteristics of the two opposing sheaths do not completely cancel. In terms of a Taylor expansion of the voltage charge characteristics of the sheaths, only the even series elements cancel. The odd elements remain. This is the reason why one can observe only odd harmonics in the current in the case of a perfectly symmetric discharge.
\begin{figure}[t!]
\centering
\resizebox{12cm}{!}{
\includegraphics[width=12cm]{PIC_HB_densities.png}
}
\caption{Densities of electrons and ions and flux of the particles resulting from a PIC simulation with an external RC-element. Straight lines indicate the solutions obtained from the method proposed by Verboncoeur et al.\ \cite{verboncoeur_simultaneous_1993}, dotted lines show the results from the harmonic balance algorithm. a) Density and flux of electrons. b) Density and flux of ions. }
\label{fig:result_PIC_RC_densities}
\end{figure}
Using these results, the plasma impedance can be calculated at the excitation frequency to $Z_\mathrm{pl} =(53-j 760)~\Omega$. The external circuit has an impedance of $Z_\mathrm{ext} = (10 - j 39)~\Omega$. Taking these impedances as a voltage divider and with the source having an amplitude of 100 V, the voltage drop at $Z_\mathrm{pl}$ is calculated to 95 V, which is consistent with the obtained simulation results.
The average ion and electron densities and fluxes depicted in Figure~\ref{fig:result_PIC_RC_densities} a) and b) are also identical within the level of statistical accuracy for both simulation methods. This proves that not only the global voltage and current are the same, but also the intrinsic plasma state, including the spatio-temporal dynamics.
\begin{figure}[t!]
\centering
\resizebox{12cm}{!}{
\includegraphics[width=12cm]{PIC_HB_ampl_matching.png}
}
\caption{Current and voltage resulting from a PIC simulation with an attached generator, matching network and stray elements. a) Transient solution b) Absolut values of different harmonics in the current}
\label{fig:result_PIC_match}
\end{figure}
The second case with an attached generator, the matching network and reactor losses was simulated accordingly. The voltage and current evolution is depicted in Figure~\ref{fig:result_PIC_match}. As expected for an asymmetric setup, the current is very nonlinear, containing multiple harmonics. At the same time, the voltage remains almost completely sinusoidal. The results shown are for a matched case, which means that the impedance seen by the generator at the fundamental frequency is $Z_\mathrm{TL}=V_\mathrm{TL}/I_\mathrm{TL} \approx 50~\Omega$. This is achieved by iteratively varying the capacitances $C_\mathrm{m1}$ and $C_\mathrm{m2}$ until matching is obtained \cite{schmidt_consistent_2018}. The resulting values are $C_\mathrm{m1}=1536$~pF and $C_\mathrm{m2}=185$~pF -- due to the matching to a different load -- while all other network elements have the same value as listed in Table~\ref{table:parameter}.
\section{\label{sec:conclusion} Conclusion}
A solution scheme is developed, which allows for the coupling of arbitrary lumped element circuits to radio-frequency plasma simulations. The approach is based on the harmonic balance method and incorporates the external circuit via a simple netlist, avoiding the deployment and the solution of Kirchhoff's differential equations by hand.
The validity of the proposed simulation approach is demonstrated using two different reference simulations. First, a global plasma model is established and attached to an external matching circuit. The results obtained by harmonic balance are identical to a simulation based on the electrical network analysis tool ngSPICE. Second, a geometrically symmetric 1-dimensional PIC simulation is coupled to a resistance and a capacitance in series (i) via conservation of charge at the driven electrode and a coupling of the circuit equations to the PIC simulation by hand and (ii) using harmonic balance. Also in this case, the results are practically indistinguishable.
Lastly, a more complicated electrical network consisting of a generator, a matching network and stray elements is connected to PIC using harmonic balance to demonstrate the versatility of the method also for cases where the circuit is not straightforwardly incorporated via auxiliary differential equations.
The presented method offers a fast and comfortable solution for the integration of complex external networks into plasma simulations.
Depending on the nonlinearity of the considered discharge or with voltage source contributions that are not harmonics of the fundamental frequency, a high number of harmonics in the current may result and, consequently, an equivalent number of simulations have to be considered.
This extensive necessity may ultimately lead to critical computational costs. In the future, an analysis of the latter and possibly an optimization might therefore be inevitable.
\section*{Acknowledgement}
This work is supported by the German Research Foundation (DFG) in the frame of Transregional Collaborative Research Centre TRR\,87, Collaborative Research Centre SFB\,1316, and DFG Research Grant MU2332/6-1.
\section*{ORCID IDs}
\noindent\href{https://orcid.org/0000-0001-6623-0464}{F. Schmidt: https://orcid.org/0000-0001-6623-0464}
\noindent\href{http://orcid.org/0000-0001-6445-4990}{T. Mussenbrock: http://orcid.org/0000-0001-6445-4990}
\noindent\href{http://orcid.org/0000-0001-9136-8019}{J. Trieschmann: http://orcid.org/0000-0001-9136-8019}
\noindent\href{http://orcid.org/0000-0001-5041-2941}{T. Gergs: http://orcid.org/0000-0001-5041-2941}
\bibliographystyle{aip}
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Q: How to reassign an object so that it changes the referenced object as well? Let's say I have an array of objects:
var people = [
{name: "Jack", height: 180},
{name: "Marry", height: 170},
{name: "Susan", height: 162},
]
If I take the first person and change the height like this:
var jack = people[0];
jack.height = 163;
Then this change is reflected in the object in the array as well like this:
people = [
{name: "Jack", height: 163},
{name: "Marry", height: 170},
{name: "Susan", height: 162},
]
However if I reassign the object like this
jack = {name: "Jack the developer", height: 163}
The array doesn't change:
people = [
{name: "Jack", height: 163},
{name: "Marry", height: 170},
{name: "Susan", height: 162},
]
How should I assign jack so that it changes the reference?
A: When you do this:
jack = {name: "Jack the developer", height: 163};
You're creating a new object and assigning it to jack, instead of changing the current object, E.g.:
jack.name = "John";
A: JS does have reference types, which is why this code works:
var jack = people[0];
jack.height = 163;
However, it still does assignments "by value" instead of "by reference", which means that jack holds the value of the reference type (the object) but if you reassign to jack, you're just reassigning a new reference value to a new object.
If JS had assignment "by reference", then your code would work, because jack would be referencing the original location of the object in the array, allowing you to work with jack as though you were working with that array index. That's not the case in JS.
A: As per your way you can do using Object.assign()
DEMO
const arr=[
{name: "Jack", height: 163},
{name: "Marry", height: 170},
{name: "Susan", height: 162},
] ,
jack = {name: "Jack the developer", height: 163};
Object.assign(arr[0],jack);
console.log(arr);
.as-console-wrapper {max-height: 100% !important;top: 0;}
You can also use find method of array and merge new value using Object.assign().
DEMO
const arr=[
{name: "Jack", height: 163},
{name: "Marry", height: 170},
{name: "Susan", height: 162},
] ,
jack = {name: "Jack the developer", height: 163};
let result = arr.find(({name})=>name=="Jack");
if(result){
Object.assign(result,jack);
};
console.log(arr);
.as-console-wrapper {max-height: 100% !important;top: 0;}
A: In JavaScript objects are passed and assigned by reference so
var jack = people[0];
here jack and people are both references to the same object.
So in first case,
When you modify the value like this jack.height = 163 then people object also get the reflected value.
in your second case,
With this line jack = {name: "Jack the developer", height: 163};
you're creating totally a new object jack with new value which will not reflect the people object that's why here you're not getting the re-assigned value.
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Dr. Yvonne Michelle Spicer (born June 30, 1962) is an American educator, democratic politician, and a former mayor from Framingham, Massachusetts. She was inaugurated on January 1, 2018, becoming the first African-American woman to be popularly elected mayor in Massachusetts, and served until January 1, 2022. She was previously the Vice President for Advocacy and Educational Partnerships at the Museum of Science in Boston, Massachusetts.
Early life and education
Yvonne Spicer grew up in Brooklyn, New York, the third of four children of Willie and Dorothy Spicer. When she was six years old, a visit to her class by Shirley Chisholm left a lasting impression of the importance of leadership and public service. Spicer's father died when she was ten. She was 13 years old when she got her first job running errands, peeling potatoes and stocking shelves for a Brooklyn restaurant, and her first official job was working for McDonald's. During the summers, she also helped her mother clean houses on the Upper East Side. Spicer attended Catholic middle school and graduated from Brooklyn Technical High School, then earned a B.S. in industrial arts & technology in 1984, followed by an M.S. in technology education in 1985, both from the State University of New York at Oswego, from which she was the first African-American woman to graduate. She earned her doctorate in Educational Leadership from the University of Massachusetts Boston in 2004.
Career
Education
After graduating from college, Spicer moved to Framingham, Massachusetts in 1985 for a job as a woodworking instructor. She worked in the Framingham Public Schools for 16 years, also teaching drafting, architecture, graphic arts, and photography, and eventually becoming Chair of Technology Education, the first woman to fill that position. During that period she also worked part-time as a realtor. She spent two years as Statewide Technology and Engineering Coordinator at the Massachusetts Department of Education, then five years as Director of Career and Technical Education in the Newton Public Schools. In 2006, Spicer was hired as Associate Director of the Museum of Science (Boston)'s National Center for Technological Literacy, where she rose to Vice President for Advocacy and Educational Partnerships, a division she created and led.
Politics
Spicer served on the Framingham Human Relations Commission and the Democratic Town Committee. She was elected to Framingham's representative Town Meeting in 2016, where she served as vice-chair for Precinct 6, and on the Standing Committee on Ways and Means. Spicer was elected Framingham's first mayor in November 2017, following town's decision to change the format of the local government from a Board of Selectmen to a mayor and City Council. Senator Elizabeth Warren held the bible for Spicer's swearing in on January 1, 2018.
She was appointed by Governor Deval Patrick to the Massachusetts Governor's STEM Advisory Council in 2010, and reappointed to the council in 2017 by Governor Charlie Baker.
On November 2, 2021, Dr. Spicer lost her bid for re-election. Her term ended on January 1, 2022 with the inauguration of new mayor Charlie Sisitsky.
Honors
Selected by the National Aeronautics & Space Administration in 2000 to participate in an aerospace engineering program for technology educators
Named Framingham State University's Global Education Teacher of the Year in 1993
Received the Anti-Defamation League's 1995 "A World of Difference" Teacher Incentive Award
Named one of 2009's ten "Women to Watch" by Mass High Tech: the Journal of New England Technology
2017 President-Elect to the International Technology Engineering Education Association (ITEEA)
References
Mayors of places in Massachusetts
African-American educators
African-American mayors in Massachusetts
People from Framingham, Massachusetts
Living people
1962 births
21st-century American politicians
21st-century American women politicians
Women mayors of places in Massachusetts
Educators from Massachusetts
American women educators
Massachusetts Democrats
21st-century African-American women
21st-century African-American politicians
20th-century African-American people
20th-century African-American women
African-American women mayors
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"redpajama_set_name": "RedPajamaWikipedia"
}
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The peaceful Holiday Inn Plymouth is surrounded by the grassy open spaces of Plymouth Hoe. Relax over dinner with views across Plymouth Sound from 11th-floor Elliot's restaurant and bar. Exeter International Airport is a 50-minute drive away and Plymouth Airport 6 miles from the hotel.
Leave your car in the car park and stroll to shops in Plymouth city centre. Yachts moor up next to cafes at the stylish Barbican marina, 5 minutes away, and sharks swim at the National Marine Aquarium.
Sea views are on the menu at breakfast and dinner from top-floor Elliott's restaurant, and you can order local Plymouth Gin in the laid-back, pub-style bar.
|
{
"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
Resistive random-access memory (ReRAM) is an emerging non-volatile memory technology that changes the resistance value of a memristor to represent two states of the binary user data: the High-Resistance State corresponding to logic 0 while the Low-Resistance State corresponding to logic 1. The memristor cell is positioned on each row-column intersection of a crossbar structure, which offers a huge density advantage for ReRAM systems \cite{Strukov}.
When a cell in a memory array is read, voltage is applied to the memristor cell, and the current flows through the memristor and senses the resistance value. If the memristor is detected with a High-Resistance State, the bit is decided to be a `0'; if it is detected with a Low-Resistance State, the bit is determined to be a `1'. A fundamental drawback of the ReRAM crossbar array is the sneak-path problem \cite{Zidan}. Sneak paths are undesirable paths in parallel to the selected cell for reading. The current goes through the sneak paths and degrades the measured resistance value. Sneak paths are detrimental when a cell with a High-Resistance State is read because the resistance degradation may lead to an erroneous decision.
In the literature, several works \cite{Yuval,Ben,CZH} tackled the sneak-path problem by using information and communication theoretical frameworks. In particular, Y. Cassuto \emph{et al.} \cite{Yuval} studied the maximum storage efficiency when the constrained codes are employed to completely avoid sneak paths. This method incurs a high code rate loss, especially when the array size is large. Y. Cassuto \emph{et al.} proved that as the array size approaches infinity, the storage information rate approaches 0 in order to achieve a sneak-path free channel. On the other hand, a commonly used method to eliminate the sneak paths is to introduce a cell selector in series to each array cell. However, these selectors are also prone to failure due to the imperfections of the memory fabrication and maintenance process, leading to the reoccurrence of the sneak paths \cite{Ben}\cite{CZH}. Y. Ben-Hur \cite{Ben} and Zehui \emph{et al.} \cite{CZH} considered ReRAM systems with imperfect selectors which fail with a certain probability. They built a probabilistic sneak-path model and developed the corresponding data detection schemes. A main challenge for the ReRAM channel is that the sneak-path induced interference to the channel is data-dependent. Moreover, the sneak-path interference is also correlated between different locations of the crossbar array. Previous work \cite{Ben} developed single-cell data detection schemes that detect the data for each memory cell independently. More sophisticated joint-cell data detection schemes were developed in \cite{CZH} by introducing pilot cells.
However, the probabilistic model in \cite{CZH} becomes too complex when the array size is getting large. No error correction code (ECC) was employed in previous research works \cite{Ben}\cite{CZH}.
According to the information theory, an efficient way to achieve reliable data storage is to apply ECC to the system. Such a design should not be a straightforward extension of the conventional ECC designed for the symbol-wise independent and identically distributed (i.i.d.) channels. Other than the channel noise, the code must overcome the sneak-path interference which is data-dependent and also correlated within the crossbar array.
In this paper, we propose efficient coding and decoding schemes for ReRAM channels. To reduce the correlation of the sneak-path interference within a codeword, we propose an across-array coding strategy which spreads a codeword to multiple independent memory
arrays, and also exploit a real-time channel estimation scheme to obtain the channel status of each memory array. Since the coded bits from different memory arrays experience independent channels, the overall channel will be averaged and a ``diversity" gain will be obtained during decoding. In this way, we can design the coding scheme over a symbol-wise i.i.d. channel. By further applying the treating-interference-as-noise (TIN) decoding, the ReRAM channel is equivalent to a block-varying channel whose status does not depend on the input data, based on which we derive the lower and upper bounds of its channel capacity and propose a coding scheme. The proposed coding scheme consists of an outer irregular repeat-accumulate (IRA) code being concatenated with an inner data shaper, which is used to change the input data distribution so as to achieve the maximum information rate. A low-complexity joint message-passing decoding for the IRA code and the data shaper is developed based on the state-of-the-art sparse-graph coding theory. With an optimized IRA code, our ReRAM system achieves a near capacity limit storage efficiency.
The rest of this paper is organized as follows. In Section~\ref{sec:model}, we present the ReRAM channel model and describe the data-dependent feature of the sneak-path interference.
The across-array coding strategy and the capacity bounds for the ReRAM channel are proposed in Section~\ref{sec:capacity}. In Section~\ref{sec:coding}, we propose a coding scheme for ReRAM channels and present both numerical and simulation results. In Section~\ref{sec:conclusion}, we conclude the paper.
\section{ReRAM Channel Model}\label{sec:model}
Consider an $m\times n$ crossbar memory array. The memristor that lies at the intersection of row $i$ and column $j$ denotes memory cell $(i, j)$.
Each array is able to store an $m\times n$ binary data matrix $\fat{X}=[x_{i,j}]_{m\times n}$, where bit $x_{i,j}\in\{0, 1\}$ is stored in memory cell $(i, j), i\in\{1,..., m\}, j\in\{1,..., n\}$. During the writing process, each bit is written into the memory cell by changing the resistance value of the memristor, i.e., for a logical ``0" bit, the cell is changed to a high resistance of $R_0$, referred to as the High-Resistance State, and for a logical ``1" bit, it is changed to a low resistance of $R_1$, referred to as the Low-Resistance State.
During the reading process, the data bit can be detected by measuring the resistance value of the corresponding cell. If it is in the High-Resistance State, the bit is identified as a `0'; if it is in the Low-Resistance State, the bit is identified as a `1'. However, due to the existence of the sneak-path interference, as well as a mix of other noises which can be modeled as an additive Gaussian noise, the memory reading becomes unreliable \cite{Yuval,Ben,CZH}. When the cell $(i, j)$ in a memory array is read, certain voltage is applied to the target cell to measure its resistance. A sneak path is defined as a closed path that originates from and returns to location $(i, j)$ while traversing logical-1 cells through alternating vertical and horizontal steps. An example is shown in Fig.~\ref{fig:SPmodel}, where $(3, 2)$ is a target cell for reading and the green line shows the desired path to measure the resistance. However, $(3, 2)\rightarrow(3, 4)\rightarrow(1, 4)\rightarrow(1, 2)\rightarrow(3, 2)$ forms a sneak path (red line) in parallel of the selected cell $(3, 2)$.
Since sneak paths always degrade the measured resistance value, they actually benefit the data detection when a cell in a Low-Resistance State (logic 1) is read. The detrimental effect only occurs when a High-Resistance State cell (logic 0) is read, making it more vulnerable to noise.
In this paper, we only consider the sneak path when a High-Resistance State cell is read.
\begin{figure}[t]
\includegraphics[width=
3.5 in]{Sneak-Path-model.pdf}
\centering
\caption{(a) Example of a $4\times 4$ memory array. (b) Corresponding logical values of memory array. $(3, 2)$ is a target cell for reading. Voltage is applied to memristor cell $(3, 2)$ and green line is the desired path for resistance measuring. However, $(3, 2)\rightarrow(3, 4)\rightarrow(1, 4)\rightarrow(1, 2)\rightarrow(3, 2)$ forms a sneak path (red line) in parallel of target cell $(3, 2)$ that degrades the measured resistance value. Arrows show current flow directions. Note that the sneak path produces a reverse current across cell $(1, 4)$.} \label{fig:SPmodel}
\end{figure}
The most popular method to mitigate the sneak-path interference is to introduce a cell selector in series to each array cell. A cell selector is an electrical device that allows current to flow only in one direction across the cell. Since sneak paths inherently produce reverse currents in at least one of the cells along the parallel path, the cell selector can completely eliminate sneak paths from the entire array. In this paper we follow the previous work \cite{Ben,CZH} and consider the 1D1R structure, even though our proposed approaches can be extended to other structures as well. Although cell selectors can effectively eliminate sneak paths, they are also prone to failures due to the imperfections in the production or the maintenance of memory array, leading to the reoccurrence of the sneak paths. Following previous work \cite{Ben,CZH}, we assume that the selectors in a memory array fail i.i.d. with probability $p_f$. However, our work is based on the fact that once a selector fails it will by no means recover, and hence the locations of the failed selectors are fixed during the reading of the whole array. This is different from the assumption made by the previous work \cite{Ben,CZH} that the locations of the failure cells vary randomly in the crossbar array during the reading of each cell. Although these two assumptions may not affect the detection performance significantly for the uncoded ReRAM systems, they are fundamentally different for the coded systems. The previous assumption \cite{Ben,CZH} actually leads to a near i.i.d. model for the sneak-path interference of each cell. In this work, the sneak-path events for the cells at different locations in the same array are highly correlated, which is much more difficult to be tackled by the channel coding scheme.
We define a sneak-path event indicator $e_{i, j}$ for cell $(i, j)$ to be a Boolean variable with the value $e_{i, j}=1$ if the cell $(i, j)$ is affected by sneak paths, otherwise, $e_{i, j}=0$. According to the previous work \cite{CZH}, sneak-path events occur during the reading of cell $(i, j)$ and lead to $e_{i, j}=1$ if and only if the following three conditions are satisfied:
\emph{[Sneak-Path Condition:]}
1) The cell $(i, j)$ is in a High-Resistance State, i.e., $x_{i, j}=0$.
2) There exists at least one combination of $k\in\{1,...,m\}, \ell\in\{1,...,n\}$ that induces a sneak path, defined by
\begin{equation}
x_{i, \ell}=x_{k, \ell}=x_{k, j}=1.
\end{equation}
3) The selector at the diagonal cell $(k, \ell)$ fails. Due to the circuit structure of the crossbar array, cells $(i,l)$ and $(k,j)$ will conduct current in the forward direction and not be affected by their selectors. Only when the selector of the cell $(k,l)$ is faulty, a sneak path will be formed.
The above Sneak-Path Condition definition limits the sneak path to length of 3, i.e., traversing three cells \cite{CZH}. More sophisticated cases of the sneak paths were considered in \cite{Ben}. The principle of our work can be extended to those cases.
Building on the above Sneak-Path Condition, we define a
\emph{[ReRAM Channel:]}
Let $\fat{X}=[x_{i,j}]_{m\times n}$ be the stored data array and $\fat{Y}=[y_{i,j}]_{m\times n}$ be the corresponding readback signal for the crossbar array. Let $\mathcal{R}$ be the set of real numbers.
An ReRAM channel is a channel with input $\fat{X}\in \{0, 1\}^{m\times n}$ and output $\fat{Y}\in \mathcal{R}^{m\times n}$:
\begin{equation}
y_{i,j}=\begin{cases} (\frac{1}{R_0}+\frac{e_{i,j}}{R_s})^{-1}+\eta_{i,j}&\mbox{if $x_{i,j}=0$}\\ R_1+\eta_{i,j}&\mbox{if $x_{i,j}=1$} \end{cases}
\label{eq:ReRAMchannel}
\end{equation}
where $R_s$ is the parasitic resistance value brought by sneak paths. Here $\eta_{i,j}\sim \mathcal{N}(0, \sigma^2), i=1,...,m, j=1,...,n$ is an additive white Gaussian noise (AWGN) with mean 0 and variance $\sigma^2$ \cite{Ben}. It is used to model a mix of various noises of the ReRAM system.
\begin{figure}[t]
\includegraphics[width=
3.2 in]{PDF_single.pdf}
\centering
\caption{PMFs of sneak-path rate within single memory array simulated for array sizes $m\times n=64\times64, 128\times128$ and input distributions with $q=0.25, 0.5$. The mean values of sneak-path rates for the four cases, indicated by the dashed lines, are $\epsilon_q=0.06, 0.3888, 0.2216$, and $0.8626$, respectively.} \label{fig:SP_rate}
\end{figure}
The fundamental problem of the ReRAM channel is to recover the stored data array $\fat{X}$ based on readback signal $\fat{Y}$ in the presence of sneak-path interference $[e_{i,j}]_{m\times n}$ and Gaussian noise $[\eta_{i,j}]_{m\times n}$. The ReRAM channel $\{0, 1\}^{m\times n}\rightarrow \mathcal{R}^{m\times n}$ with input and output size $m\times n$ actually consists of $mn$ symbol-wise channels with $\{0, 1\}\rightarrow \mathcal{R}$. Since the sneak-path indicator $e_{i,j}$ of each target cell is related to the entire data array, these $mn$ symbol-wise channels are data-dependent and correlated.
The ReRAM channel is also asymmetrical, whose channel status (sneak-path occurring probability) is highly related to the channel input distribution. We define the input distribution as i.i.d. Bernoulli $(q)$, i.e., $\textrm{Pr}(x_{i,j}=1)=q$ and $\textrm{Pr}(x_{i,j}=0)=1-q$ for $i=1,..., m, j=1,..., n$.
For a fixed input distribution, we investigate the fraction of sneak-path affected cells in the crossbar array and define a sneak-path rate over the array as $\frac{\sum_{i=1}^m\sum_{j=1}^ne_{i,j}}{mn(1-q)}$. Its mean value is derived as a function of $q$:
\begin{eqnarray}
\epsilon_q\!\!\!\!\!\!\!\!&&\overset{\Delta}{=}E\left[\frac{\sum_{i=1}^m\sum_{j=1}^ne_{i,j}}{mn(1-q)}\right]\label{eq:SP-rate}\\
\!\!\!\!\!\!\!\!&&=\textrm{Pr}(e_{i,j}=1| x_{i,j}=0)\nonumber\\
\!\!\!\!\!\!\!\!&&=1-
\sum_{u = 0}^{m - 1} \sum_{v = 0}^{n - 1}\binom{m-1}{u} \binom{n-1}{v}q^{u + v}( 1 - q )^{m - 1 - u + n - 1 - v}\nonumber\\
\!\!\!\!\!\!\!\!&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times(1 - p_fq)^{uv}.\label{eq:SP-rate-cal}
\end{eqnarray}
When $m$ or $n$ is large, (\ref{eq:SP-rate-cal}) is difficult to calculate.
However, when $p_fq$ is small, using the Taylor expansion $(1 - p_fq)^{uv}\approx 1 -uv p_fq+\alpha\binom{uv}{2}p_f^2q^2$, we can approximately calculate (\ref{eq:SP-rate-cal}):
\begin{eqnarray}
\epsilon_q\!\!\!\!\!\!\!\!&&\approx1-
\sum_{u = 0}^{m - 1} \sum_{v = 0}^{n - 1}\binom{m-1}{u} \binom{n-1}{v}q^{u + v}( 1 - q )^{m - 1 - u + n - 1 - v}\nonumber\\
\!\!\!\!\!\!\!\!&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\left(1 -uv p_fq+\alpha\binom{uv}{2}p_f^2q^2\right)\label{eq:talor}\\
\!\!\!\!\!\!\!\!&&=(m-1)(n-1)p_fq^3-\alpha\left(2q\binom{m-1}{2}\binom{n-1}{2}\right.\nonumber\\
\!\!\!\!\!\!\!\!&&\ \ \ \ \ \ \ \ \ \ \ \ \left.+(n-1)\binom{m-1}{2}+(m-1)\binom{n-1}{2}\right)p_f^2q^5
\end{eqnarray}
where $\alpha$ is a balance factor for the last term of the Taylor expansion. Here a good setting for $\alpha$ is $0.8$.
We define the probability mess function (PMF) of the sneak-path rate as
\begin{equation}
F(\epsilon)=\textrm{Pr}\left(\frac{\sum_{i=1}^m\sum_{j=1}^ne_{i,j}}{mn(1-q)}=\epsilon\right).
\end{equation}
For a memory array size of $m\times n=64\times64, 128\times128$ and input distributions with $q=0.25, 0.5$, we simulate $F(\epsilon)$ and the results are illustrated by Fig.~\ref{fig:SP_rate}. In particular, we generate a large amount of input data arrays, compute the sneak-path rate of each array, and obtain the PMF statistically. In the simulations as well as the numerical results of this paper, we assume the selectors fail i.i.d. with probability $p_f=10^{-3}$. Fig.~\ref{fig:SP_rate} shows that a larger value of $q$, or a larger array size, leads to higher sneak-path rates, i.e., worse channels. The values of the sneak-path rate are quite diverse for different input data patterns since the PMF spreads in a large range over the abscissa, which indicates that the channel varies significantly for different input data patterns. This is because the occurrence of sneak-path events depends on the input data pattern.
This creates a big challenge for designing the coding scheme for the ReRAM channels since the code directly designed based on the average sneak-path rate of $\epsilon_q=\sum_{\epsilon}\epsilon F(\epsilon)$ will be inadequate.
\begin{figure*}[t]
\includegraphics[width=
5.8 in]{Coding_system.pdf}
\centering
\caption{Across-array coding strategy for ReRAM.} \label{fig:Coding}
\end{figure*}
\section{Across-Array Coding Strategy and Channel Capacity Bounds}\label{sec:capacity}
To mitigate the variability of the ReRAM channel, we propose an across-array coding strategy that assigns a codeword to multiple crossbar arrays. Since the coded bits at different arrays experience independent channels, the sneak-path rate within one codeword will be close to its mean value and hence the channel is closer to an i.i.d. channel. Based on the across-array coding strategy, we further investigate the ReRAM channel capacity bounds.
As the sneak path is dependent on the data message, it is difficult to derive the exact capacity of the ReRAM channel.
By treating the sneak-path interference as the i.i.d. noise during decoding, the ReRAM channel over multiple memory arrays resembles a block-varying
channel that we will define in Section~\ref{sec:channel model}, whose status does not depend on the input data.
We then derive the capacity bounds of the block-varying channel, which can be regarded as an approximation of the ReRAM channel capacity.
\subsection{Across-Array Coding Strategy}
The proposed across-array coding strategy is illustrated in Fig.~\ref{fig:Coding}.
Consider the processing of the data vector $\fat{b}=(b_1, b_2,...,b_{NR})$, where $N$ is the code length and $R$ is the code rate. Here, $\fat{b}$ is encoded into codeword $\fat{x}=(x_1, x_2,...,x_N)$ which is assigned to $T$ memory arrays, where $N=sT$ for some integer $s$. Thus, we split $\fat{x}$ into $T$ equal-length vectors, each of which is assigned to an independent memory array. Without loss of generality, we assign $\fat{x}^t=(x_1^t, x_2^t, ...., x_{N/T}^t)$ with $x_i^t=x_{(t-1)N/T+i}, i=1,..., N/T$, to the $t$-th memory array for $t=1,..., T$. Since each memory array is of size $m\times n$, $mnT/N$ codewords can be stored by these $T$ memory arrays, where $mnT/N$ is assumed to be an integer. As the code rate is $R$, the storage efficiency is $R$ bits/cell.
Each codeword is decoded independently based on its readback signal. The codeword $\fat{x}=(\fat{x}^1,..., \fat{x}^T)$ is decoded based on its readback signal $\fat{y}=(\fat{y}^1,..., \fat{y}^T)$, where $\fat{y}^t$ is the readback signal of $\fat{x}^t$ from the $t$-th memory array, $t=1,..., T$, to obtain the estimated data $\hat{\fat{b}}$. Here we employ a suboptimal decoding scheme known as the TIN decoding \cite{Huang}. The TIN decoder ignores the correlation between the channel input and the sneak paths and treats the sneak-path interference as the i.i.d. noise.
To illustrate the advantage of across-array coding strategy more explicitly, in Fig~\ref{fig:PDF64_128}, we evaluate the PMF of the sneak-path rate over one codeword that is stored in $T$ memory arrays. In the figure, we employ the array sizes and the code lengths of $N=m\times n = 64\times64$ and $128\times128$. Since the coded bits are distributed in $T$ memory arrays, the sneak-path rate as well as its PMF are rewritten as $\sum_{t=1}^T\sum_{i=1}^{N/T}e_i^t/(mn(1-q))$ and $F^T_q(\epsilon)=\textrm{Pr}(\sum_{t=1}^T\sum_{i=1}^{N/T}e_i^t/(mn(1-q))=\epsilon)$, where $e_i^t$ is the sneak-path event indicator during the reading of the $i$-th bit that belongs to the $t$-th array. For each case of $m\times n= 64\times64$ and $128\times128$ and the given input distribution with $q=0.25$ and $0.5$, as $T$ increases, the spread of the PMF of the sneak-path rate gets smaller and concentrates closer to the mean value $\epsilon_q$ and the channel becomes more stable. The reason is that since the codeword is assigned to $T$ independent memory arrays, the sneak-path rate is averaged over the $T$ arrays. Based on the law of large numbers, as $T\rightarrow\infty$, the sneak-path rate converges exactly to the mean value with probability 1, and therefore, we can design a code based on this mean value to guarantee error free decoding. Note that the across-array coding strategy does not change the channel correlation. It actually reduces the correlation of the sneak-path interference within one codeword since the readback signals for coded bits from different memory arrays are independent with each other. As all the coded bits are encoded from the same message data, this resembles a ``code diversity" strategy for block fading channels \cite{Raymond}.
\begin{figure*}[t]
\includegraphics[width=
6.0 in]{PDF_128_64.pdf}
\centering
\caption{PMFs of sneak-path rates over one codeword distributed in $T$ memory arrays with $T=1, 2, 4, 8, 16$. Code length and memory array size are $m\times n= 64\times64$ (left) and $128\times128$ (right).} \label{fig:PDF64_128}
\end{figure*}
The drawback of a joint coding across $T$ arrays is a potential increase of the read/write latency. Note that such additional latency is unavoidable for many sneak-path mitigating approaches in the literature. For example, the multistage reading technique presented in \cite{Vontobel} requires three readings and three writings to get a better estimation of the sneak current; \cite{Ben} investigates the multiple-read detector and shows that it can achieve near-optimum performance with 10 reads. We remark that for our proposed across-array coding strategy, the additional read/write latency can be minimized if a parallel reading/writing circuit \cite{Zhou} is adopted across different crossbar arrays. Moreover, the additional latency incurred will become negligible if the two-dimensional crossbar arrays are stacked to form a 3D structure, which naturally enables the parallel reading/writing across different arrays \cite{Luo}.
\subsection{Channel Equivalence and Capacity Bounds}\label{sec:channel model}
We first define a block-varying $(\fat{\epsilon}, \sigma)$-channel and show how
the ReRAM channel capacity is related to that of the block varying
$(\fat{\epsilon}, \sigma)$-channel.
To begin, we first define an $(\epsilon, \sigma)$-channel.
As illustrated in Fig.~\ref{fig:ESchannel}, an $(\epsilon, \sigma)$-channel is a concatenation of an i.i.d. asymmetrical discrete channel and an i.i.d. additive Gaussian channel, and therefore, it is also an i.i.d. channel without channel correlation. The asymmetrical discrete channel is with binary-input $X\in\{0, 1\}$ and ternary-output from $\{R_0, R_0^\prime, R_1\}$ with $R_0^\prime=\left(\frac{1}{R_0}+\frac{1}{R_s}\right)^{-1}$, and the transition property is described by Pr$(R_0|0)=1-\epsilon$, Pr$(R_0^\prime|0)=\epsilon$, and Pr$(R_1|1)=1$. The additive Gaussian channel is with noise distribution $\eta\sim\mathcal{N}(0, \sigma^2)$, whose output $Y\in \mathcal{R}$ serves as the output of the $(\epsilon, \sigma)$-channel.
\begin{figure}[t]
\includegraphics[width=
3.3 in]{ESchannel.pdf}
\centering
\caption{$(\epsilon, \sigma)$-channel model with $R_0^\prime=\left(\frac{1}{R_0}+\frac{1}{R_s}\right)^{-1}$.} \label{fig:ESchannel}
\end{figure}
For given input distribution Pr$(X=0)=1-q$, Pr$(X=1)=q$, the $(\epsilon, \sigma)$-channel capacity can be derived as
\begin{eqnarray}
\!\!\!\!C_q(\epsilon, \sigma)&&\!\!\!\!\!\!\!\!=I(X;Y)\nonumber\\
&&\!\!\!\!\!\!\!\!=H(Y)-H(Y|X)\nonumber\\
&&\!\!\!\!\!\!\!\!=H(Y)-qH(Y|X=1)-(1-q)H(Y|X=0)\nonumber\\
&&\!\!\!\!\!\!\!\!=-\int_{-\infty}^{+\infty} p_Y(y)\log_2 p_Y(y)dy-q\log_2\sqrt{2\pi e\sigma^2}\nonumber\\
&&\!\!\!\!\!\!\!\! \ \ \ \ \ \ \ \ \ +(1-q)\int_{-\infty}^{+\infty} p_{Y|X=0}(y)\log_2 p_{Y|X=0}(y)dy\nonumber
\end{eqnarray}
where
\begin{eqnarray}
p_Y(y)=\!\!\!\!\!\!\!\!&&(1-q)\left(\epsilon \phi(y, R_0^\prime)+(1-\epsilon)\phi(y, R_0)\right)+q\phi(y, R_1)\nonumber\\
p_{Y|X=0}(y)=\!\!\!\!\!\!\!\!&&\epsilon \phi(y, R_0^\prime)+(1-\epsilon)\phi(y, R_0)\nonumber\\
\phi(y, m)=\!\!\!\!\!\!\!\!&&{1}/{(\sqrt{2\pi}\sigma)}e^{-\frac{(y-m)^2}{2\sigma^2}}.\nonumber
\end{eqnarray}
The $(0, \sigma)$- and $(1, \sigma)$-channels are asymmetrical binary-input AWGN channels, which are two special cases of an $(\epsilon, \sigma)$-channel. Obviously, the $(\epsilon, \sigma)$-channel capacity decreases as $\epsilon$ increases leading to $C_q(1, \sigma)<C_q(\epsilon, \sigma)<C_q(0, \sigma)$.
A $T$-block block-varying $(\fat{\epsilon}, \sigma)$-channel with parameters $\fat{\epsilon}=(\epsilon^1,\epsilon^2,...,\epsilon^T)$ varies from data block to data block, while within the $t$-th data block, the channel is a symbol-wise i.i.d. $(\epsilon^t, \sigma)$-channel, $t=1, 2,...,T$. The block-varying $(\fat{\epsilon}, \sigma)$-channel is a type of channel with block interference as proposed by \cite{Mceliece}.
The ReRAM channel over $T$ memory arrays resembles the $T$-block block-varying $(\fat{\epsilon}, \sigma)$-channel. In particular, the sneak-path rate of the ReRAM channel varies from memory array to memory array resembles the block-varying property (the channel varies from block to block) of the block-varying $(\fat{\epsilon}, \sigma)$-channel. The channel parameter $\epsilon^t=\textrm{Pr}(R_0^\prime|X=0)$ resembles the instantaneous sneak-path rate of the $t$-th memory array. The main difference is that in the ReRAM channel, the sneak-path interference is dependent on the input data and this data-dependency leads to the channel correlation, while the parameter $\fat{\epsilon}$ of block-varying $(\fat{\epsilon}, \sigma)$-channel is data-independent and the channel within each block is i.i.d. However, if we adopt TIN decoding where the decoder ignores this data-dependency and regards the sneak paths as the i.i.d. noise, an ReRAM channel is equivalent to a block-varying $(\fat{\epsilon}, \sigma)$-channel. The memory block length, denoted by $M$, of the block-varying $(\fat{\epsilon}, \sigma)$-channel should be identical to the data array size of the ReRAM channel, i.e., $M=mn$. The channel parameters $\epsilon^t, t=1, 2, ..., T$ should be i.i.d. generated based on PMF of the sneak-path rate of the ReRAM channel. Therefore, the maximum achievable coding rate over the ReRAM channel under TIN decoding can be approximated by the block-varying $(\fat{\epsilon}, \sigma)$-channel capacity.
In preparation to give the capacity limit, we define an $(\epsilon, \sigma)$-channel code:
\begin{definition}\label{def:code}
An $(\epsilon, \sigma)$-channel code includes a sequences of codes with rate $C_q(\epsilon, \sigma)$ and different code lengths $n$, which achieve asymptotical error free decoding over the i.i.d. $(\epsilon, \sigma)$-channel as the code length approaches infinity.
\end{definition}
The existence of the $(\epsilon, \sigma)$-channel code ensemble is guaranteed by the conventional channel coding theorem.
Consider a block-varying $(\fat{\epsilon}, \sigma)$-channel with memory block size $M$. Parameter $\epsilon^t$ has the PMF $F(\epsilon^t), t=1, 2, ..., T$, and mean value $\bar{\epsilon}=\sum_{\epsilon}\epsilon F(\epsilon)$, as defined in (\ref{eq:SP-rate}). Since the channel varies from block to block, we consider joint $T$-block encoding and decoding. The code length is therefore $MT$. Let $R$ be the encoding rate, and we then have the following theorem:
\begin{theorem}\label{thm:thm}
For fixed input distribution of Bernoulli $(q)$, as $T\rightarrow \infty$, the maximum achievable code rate $R$ with joint $T$-block encoding and decoding is bounded by
\begin{equation}
C_q(\bar{\epsilon}, \sigma)\leq R\leq \overline{C}_q(\sigma)
\end{equation}
where $\overline{C}_q(\sigma)=\sum_{\epsilon} F(\epsilon)C_q(\epsilon, \sigma)$.
\end{theorem}
\emph{Proof:}
We first show that for a fixed input distribution of Bernoulli $(q)$, $C_q(\bar{\epsilon}, \sigma)$ is achievable. Let $\fat{x}=(\fat{x}^1,..., \fat{x}^T)$ be the joint $T$-block codeword, where $t$-th block $\fat{x}^t=(x^t_1, ..., x^t_M)$ experiences an $(\epsilon^t, \sigma)$-channel, i.e., each symbol $x^t_j, j=1,..., M$, experiences an i.i.d. $(\epsilon^t, \sigma)$-channel. We assume that the codeword is encoded in the way that the $i$-th bits, located at different data blocks, i.e., $(x_i^1, x_i^2, ..., x_i^T)$, belong to a codeword of a length-$T$ $(\bar{\epsilon}, \sigma)$-channel code. In this way, the original length-$MT$ codeword $\fat{x}$ can be considered as a vector consisting $M$ length-$T$ $(\bar{\epsilon}, \sigma)$-channel codewords. This is possible because we can always split the uncoded data vector into $M$ equal-length sub-vectors, and encode each of them independently using an $(\bar{\epsilon}, \sigma)$-channel code. As the encoding rate of each $(\bar{\epsilon}, \sigma)$-channel code is $C_q(\bar{\epsilon}, \sigma)$, the overall code rate is $R=C_q(\bar{\epsilon}, \sigma)$.
During decoding, for each $i=1,..., M$, $(x_i^1, x_i^2, ..., x_i^T)$ is decoded based on its channel output $(y_i^1, y_i^2, ..., y_i^T)$, where $y_i^t$ is a channel observation of $x_i^t$. Since coded bit $x_i^t$ experiences an $(\epsilon^t, \sigma)$-channel, where $\epsilon^t, t=1, 2,..., T$, are i.i.d. generated based on the PMF of $F(\epsilon)$, the overall codeword experiences an $(\bar{\epsilon}, \sigma)$-channel where $\bar{\epsilon}=\sum_{\epsilon}\epsilon F(\epsilon)$. Since $(x_i^1, x_i^2, ..., x_i^T)$ is an $(\bar{\epsilon}, \sigma)$-channel codeword, the decoding error probability approaches 0 as $T\rightarrow\infty$ according to Definition~\ref{def:code}.
In \cite{Mceliece}, an upper bound of the block interference channel is proposed. That is, given the channel parameters $\fat{\epsilon}$, the channel becomes memoryless, leading to $R\leq\frac{1}{MT} I(\fat{x};\fat{y})\leq\frac{1}{MT} I(\fat{x};\fat{x}|\fat{\epsilon})\leq I(x_1^1;y_1^1|\epsilon^1) =\sum_{\epsilon}F(\epsilon) C_q(\epsilon, \sigma)=\overline{C}_q(\sigma)$.
\myQED
It was also shown in \cite{Mceliece} that if the channel state is finite, i.e., $\epsilon$ is from a finite set, the upper bound is
tight when the block size $M\rightarrow\infty$.
\begin{figure*}[t]
\includegraphics[width=
6.2 in]{Capacity.pdf}
\centering
\caption{Upper bound $\sum_\epsilon F_q(\epsilon)C_q(\epsilon, \sigma)$ and lower bound $C(\epsilon_q, \sigma)$ of the capacity as $T\rightarrow\infty$, with $R_1=100\ \Omega, R_0=1000\ \Omega, R_s=250\ \Omega$, and $p_f=10^{-3}$.} \label{fig:capacity}
\end{figure*}
Based on our channel equivalence, $\max_qC_q(\epsilon_q, \sigma)$ is an approximate lower bound of the ReRAM channel capacity with TIN decoding, and $\max_q\sum_\epsilon F_q(\epsilon)C_q(\epsilon, \sigma)$ is an upper bound. Since we have the explicit formula (\ref{eq:SP-rate-cal}) for $\epsilon_q$, the lower bound is much easier to be calculated than the upper bound, which requires $F_q(\epsilon)$.
Fortunately, we can show that when the array size $N$ is large, the two bounds are numerically very close with each other, and hence the lower bound $\max_qC(\epsilon_q, \sigma)$ can be a good approximation of the ReRAM channel capacity. Fig.~\ref{fig:capacity} illustrates the capacity upper bound $\sum_\epsilon F_q(\epsilon)C_q(\epsilon, \sigma)$ and lower bound $C(\epsilon_q, \sigma)$ as functions of $q$, for different memory array sizes $m\times n= 32\times32, 64\times64, 128\times128, 256\times 256$ and different noise values of $\sigma=30, 50, 100$. The resistance parameters are fixed with $R_1=100\ \Omega, R_0=1000\ \Omega$, and $R_s=250\ \Omega$. Observe that when the memory array size $N$ is large, the two bounds are very close with each other.
Fig.~\ref{fig:capacity} also indicates that the ReRAM channel capacity bounds decrease as the data size increases due to the increase of the sneak-path rate, i.e., the larger the data array size the lower the average storage efficiency for each cell, and vice versa. For a very low noise level of $\sigma=30$, the capacity bounds are maximized at about $q=0.5$, and for $\sigma=50, 100$, they are typically maximized when $q<0.5$. The optimal value of $q$ that maximizes the channel capacity bounds decreases as noise level increases. This is because noise amplifies the detrimental effect of sneak paths, while reducing $q$ effectively reduces the sneak-path rate.
\section{Design of the Coding Scheme}\label{sec:coding}
In this section, we present the design of a coding scheme for the ReRAM channel. We also propose a real-time maximum likelihood channel estimation, based on which the message-passing decoding is performed. We utilize the state-of-the-art sparse-graph code and message-passing decoding theories to design the coding scheme, which is essentially an $(\epsilon_q, \sigma)$-channel code design. A major difference between the $(\epsilon_q, \sigma)$-channel code and the classical ECC is that since the former works over an asymmetrical channel, the coded data should follow the desired distribution to approach the channel capacity (Fig.~\ref{fig:capacity}). We propose a coding scheme, which is a serial concatenation between a classical ECC and a data shaper that shapes the desired data distribution. Bit error rate (BER) simulations and performance comparisons are also presented in this section for our proposed coding scheme.
\begin{figure*}[t]
\includegraphics[width=
4.8 in]{Code_shape.pdf}
\centering
\caption{Proposed coding scheme for the ReRAM channel.} \label{fig:Code_shape}
\end{figure*}
\subsection{Coding and Decoding Model}
A system model for the proposed coding scheme is illustrated in Fig.~\ref{fig:Code_shape}. The encoder includes an ECC encoder and a data shaper. The ECC encoder encodes data vector $\fat{b}=(b_1, b_2,...,b_{NR})$ into codeword $\fat{c}=(c_1, c_2,...,c_N)$ whose entries are uniformly distributed on $\{0, 1\}$. The data shaper reforms the data distribution into Bernoulli $(q)$. Its output is $\fat{x}=(x_1, x_2,...,x_N)$. Here the data shaper in our system has rate-1, and therefore, the overall code rate is still $R$.
The decoder involves a real-time channel estimator, elementary signal estimator (ESE), a de-shaper, and an ECC decoder. Since the decoding is actually a block-varying $(\fat{\epsilon}, \sigma)$-channel decoding, the channel estimator first estimates the channel parameters $\fat{\epsilon}=(\epsilon^1,...,\epsilon^T)$ over the $T$ memory arrays based on readback signal $\fat{y}=(\fat{y}^1,..., \fat{y}^T)$. Based on $\hat{\fat{\epsilon}}$ and $\fat{y}$, the ESE calculates a soft estimation $\{L(x_i^t|y_i^t, \hat{\epsilon}^t)\}$, i.e., the log-likelihood ratio (LLR), for each coded bit $x_i^t$ that is used as the decoder input. The decoder consists of a de-shaper and an ECC decoder, both of which use soft-in soft-out (SISO) processings and perform iteratively to improve the decoding reliability. The corresponding decoding is standard message-passing decoding. Specifically, the de-shaper calculates soft LLR $\{L^e(c_i)\}$ for each ECC coded bit, based on which an ECC decoding refines the estimation and feds back an updated LLR $\{L^a(c_i)\}$ to the de-shaper for the next round of decoding iterations. After a fixed maximum number of iterations, a hard decision is performed at the ECC decoder to produce data estimation $\hat{\fat{b}}$.
\begin{figure}[t]
\includegraphics[width=
3.5 in]{Shaper_deshaper.pdf}
\centering
\caption{Data shaper and its factor graph illustration.} \label{fig:Shaper}
\end{figure}
\subsection{Data Shaper}\label{sec:shaper}
The data shaper consists of a length-$L$ repeater, a length-$NL$ bit interleaver $\pi$, and an $L$-to-1 mapper (Fig.~\ref{fig:Shaper}). The repeater duplicates each ECC coded bit $L$ times. The bit interleaver permutes the repeater output to relocate the bits. The $L$-to-1 mapper maps every $L$ bits to one bit, i.e., $\mathcal{M}: \{0, 1\}^L\rightarrow \{0, 1\}$. Therefore, the data shaper's overall rate is 1. For the $L$-to-1 mapper, since there are $2^L$ patterns for the mapping inputs, by mapping $i$ of them to 1 and $2^L-i$ of them to 0, we obtain the data output with a distribution of Bernoulli $(\frac{i}{2^L})$. By choosing $i=1, 2,..., 2^L-1$, we achieve data distributions of Bernoulli $(q)$ with $q=\frac{1}{2^L}, \frac{2}{2^L}, ..., \frac{2^L-1}{2^L}$.
\begin{figure}[t]
\includegraphics[width=
1.3 in]{Mapping3.pdf}
\centering
\caption{3-to-1 mapping for output data distribution Pr$(\tilde{x}=0)=\frac{3}{4}$, Pr$(\tilde{x}=1)=\frac{1}{4}$.} \label{fig:map3}
\end{figure}
The interleaver inside the data shaper is crucial in our scheme. Rather than adopting random interleaving, we propose a structured interleaving scheme, as shown in the data shaper's factor graph in Fig.~\ref{fig:Shaper}. The interleaver $\pi$ consists of $L$ sub-interleavers $\pi_i, i=1,..., L$, each of which can be random. The data-shaping process can be described using a factor graph. Each variable node is associated with an ECC coded bit, where the $i$-th variable node is associated with $c_i$. There are $L$ edges from a variable node to the interleavers corresponding to the $L$ repetitions of the ECC coded bit. Each mapping node has $L$ edges from the interleavers corresponding to the $L$ mapping inputs. The $i$-th mapping node is associated with the mapping output $x_i$. By using our structured interleaver, the $i$-th repetitions of the ECC coded bits enter a sub-interleaver $\pi_i$ whose outputs are used as the $i$-th inputs of the mapping nodes. By doing so, each ECC coded bit has exactly one repetition that occupies the $i$-th input of a mapping node for $i=1, ..., L$.
The advantage of employing this structured interleaver can be explained using an example. Consider a data shaper with an $(L=3)$-repeater and a 3-to-1 mapper $\mathcal{M}(\tilde{c}_1,\tilde{c}_2,\tilde{c}_3)=\tilde{x}$ (Fig.~\ref{fig:map3}). There are eight patterns for three binary inputs $\tilde{c}_1,\tilde{c}_2,\tilde{c}_3$, where only two of them, 110 and 111, are mapped to 1, and the other six are mapped to 0. If $\tilde{c}_1,\tilde{c}_2,\tilde{c}_3$ are i.i.d. with Pr$(\tilde{c}_i=0)$=Pr$(\tilde{c}_i=1)=\frac{1}{2}, i=1, 2, 3$, the mapping can realize output data distribution with Pr$(\tilde{x}=0)=\frac{3}{4}$, Pr$(\tilde{x}=1)=\frac{1}{4}$. Next we address the de-mapping. Mapping output $\tilde{x}$ actually contains a different quantity of information about the three input bits $\tilde{c}_1,\tilde{c}_2,\tilde{c}_3$. By formulating the mapping rule as $\tilde{x}=\mathcal{M}(\tilde{c}_1,\tilde{c}_2,\tilde{c}_3)=\tilde{c}_1\cdot\tilde{c}_2$, where $\cdot$ is a multiply operation, we evaluate the mutual information between $\tilde{x}$ and each input bit as $I(\tilde{x};\tilde{c}_1)=I(\tilde{x};\tilde{c}_2)=\frac{3}{4}\log_2\frac{4}{3}$ and $I(\tilde{x};\tilde{c}_3)=0$.
Therefore, if the de-mapper is sufficiently near-optimal, during de-mapping we can obtain information about the first and second bits $\tilde{c}_1, \tilde{c}_2$. Unfortunately, we cannot get any information about the third bit $\tilde{c}_3$ since $\tilde{x}$ does not contain any information about $\tilde{c}_3$. In other words, one-third of the bits are erased after de-mapping. If random interleaving is employed, with probability $\frac{1}{27}$, all the three repetitions of an ECC coded bit will be assigned as the third input of a mapping node and erased. In other words, with probability $\frac{1}{27}$, an ECC coded bit will be erased after de-shaping, which leads to a poor ECC decoding performance.
Our structured interleaving guarantees that all the ECC coded bits can obtain a positive and statistically equal quantity of information from the de-shaper to benefit the ECC decoding.
\subsection{Channel Estimation and ESE}
We propose a maximum likelihood channel estimation to obtain parameters $\fat{\epsilon}=(\epsilon^1,...,\epsilon^T)$.
Since the decoder assumes the channel created by the $t$-th memory array as an i.i.d. $(\epsilon^t, \sigma)$-channel, with the channel observation of $\fat{y}^t=(y^t_1,...,y^t_{N/T})$, the log-likelihood function of $\epsilon^t$ is written as:
\begin{eqnarray}
\Lambda(\epsilon^t;\fat{y}^t)
&&\!\!\!\!\!\!\!\!\!\!=\log \prod_{i=1}^{N/T}\textrm{Pr}\left(y^t_i|\epsilon^t\right)\\
&&\!\!\!\!\!\!\!\!\!\!=\sum_{i=1}^{N/T}\log\left[ (1-q)\left(\epsilon^t \phi(y^t_i, R_0^\prime)+(1-\epsilon^t)\phi(y^t_i, R_0)\right)\right.\nonumber\\
&&\!\!\!\!\!\!\!\!\!\!\ \ \ \ \ \ \left.+q\phi(y^t_i, R_1)\right]. \label{eq:logsum}
\end{eqnarray}
Next we consider the approximation for (\ref{eq:logsum}). Let $\bar{y}^t_i=\arg\min_{x\in\{R_0, R_0^\prime, R_1\}}|y^t_i-x|$ be the hard decision value of $y^t_i$. Since each term of (\ref{eq:logsum}) is in the form of $\log\sum_{x\in\{R_0, R_0^\prime, R_1\}}p_x\phi(y^t_i, x)$, when the channel noise level is low, it is dominated by the term of $x=\bar{y}^t_i$. We thereby apply
\begin{eqnarray}
\log\sum_{x\in\{R_0, R_0^\prime, R_1\}}p_x\phi(y^t_i, x)&&\!\!\!\!\!\!\!\!\!\!\approx \log \left(p_{\bar{y}^t_i}\phi(y^t_i, \bar{y}^t_i)\right)\nonumber\\
&&\!\!\!\!\!\!\!\!\!\!=\log p_{\bar{y}^t_i}-\frac{(y^t_i-\bar{y}^t_i)^2}{2\sigma^2}-\frac{1}{2}\log(2\pi\sigma^2),\nonumber
\end{eqnarray}
and hence have the following approximation:
\begin{eqnarray}
\Lambda(\epsilon^t;\fat{y}^t)\approx n_{R_0^\prime}\log\epsilon^t+n_{R_0}\log(1-\epsilon^t)-\sum_{i=1}^{N/T}\frac{(y^t_i-\bar{y}^t_i)^2}{2\sigma^2}\!+\!c\nonumber
\end{eqnarray}
where $n_x\overset{\Delta}{=}\sum_{i=1}^{N/T}1\{\bar{y}^t_i=x\}$ is the total number of $y^t_i$ whose hard decision is $x$ and $c$ is a constant term independent of $\epsilon^t$. By maximizing $\Lambda(\epsilon^t;\fat{y}^t)$ we obtain the estimation of $\epsilon^t$:
\begin{eqnarray}
\hat{\epsilon}^t=\arg\max_{\epsilon^t}\Lambda(\epsilon^t;\fat{y}^t)\approx \frac{n_{R_0^\prime}}{n_{R_0^\prime}+n_{R_0}}.\label{eq:channel_est}
\end{eqnarray}
Next, the ESE calculates the LLR for each coded bit of $(\fat{x}^1,..., \fat{x}^T)$ based on the readback signal $(\fat{y}^1,..., \fat{y}^T)$ and the estimated channel parameters:
\begin{eqnarray}
L(x_i^t|y_i^t, \hat{\epsilon}^t) &&\!\!\!\!\!\!\!\!\!\!=\log\frac{\textrm{Pr}(y_i^t|x_i^t=0, \hat{\epsilon}^t)}{\textrm{Pr}(y_i^t|x_i^t=1, \hat{\epsilon}^t)}\\
&&\!\!\!\!\!\!\!\!\!\!=\log\frac{\hat{\epsilon}^t \phi(y_i^t, R_0^\prime)+(1-\hat{\epsilon}^t)\phi(y_i^t, R_0)}{\phi(y_i^t, R_1, \sigma^2)}
\end{eqnarray}
for $i=1,..., N/T, t=1,...,T$.
Note that the implementation of ESE (\ref{eq:ESE}) ignores the correlation between cells in a memory array and regards the sneak-path interference as the i.i.d. noise.
A more sophisticated decoding scheme can be developed by utilizing the cell correlation and performing joint data and sneak path detection. It is left as our future work.
\begin{figure}[t]
\includegraphics[width=
3.5 in]{Channel_est_mse.pdf}
\centering
\caption{MSE: $E[(\hat{\epsilon}-\epsilon)^2]$ of our proposed channel estimation (solid lines) and MSE: $E[(\epsilon_q-\epsilon)^2]$ of the average channel parameter (dashed lines). } \label{fig:Channel_mse}
\end{figure}
To demonstrate the accuracy of the proposed channel estimation, in Fig.~\ref{fig:Channel_mse}, we illustrate the mean squared error (MSE) between the estimated and the actual sneak-path rates. We obtain the MSE: $E[(\hat{\epsilon}-\epsilon)^2]$ by simulation for $T=1$ and memory array sizes $m\times n=64\times64, 128\times128$, where $\epsilon$ is the actual sneak-path occurrence rate, and $\hat{\epsilon}$ is the estimated value obtained by (\ref{eq:channel_est}). In general, the MSE is below $10^{-2}$ and decreases as the channel noise level decreases. For comparison, we also illustrate the MSE: $E[(\epsilon_q-\epsilon)^2]$ between the average and the actual sneak-path rates, where the average sneak-path rate is employed by the decoder when channel estimation is unavailable. Our proposed channel estimation is much more accurate to predict the channel than the average channel parameter, especially for large array size.
\subsection{De-Shaper}
The SISO de-shaper can also be realized by message-passing processing over the factor graph shown in Fig.~\ref{fig:Shaper}. During the de-shaping, each node performs as a local processor and the edges pass LLR messages. The message passing on the edges is bi-directional. The overall processing is performed iteratively. In each iteration, each node in the factor graph acts once. A mapping node performs de-mapping processing based on the $L$ priori LLRs from its neighboring variable nodes and the LLR from ESE and outputs an extrinsic LLR for each mapping input. The extrinsic LLR is used as an a priori LLR for variable node processings. A variable node combines the $L$ priori LLRs from its neighboring mapping nodes and feeds back an extrinsic LLR to each of its neighboring mapping nodes. After a certain number of iterations the variable nodes output a more reliable LLR for each ECC coded bit as the de-shaper output.
\subsubsection{Mapping Node Processing}
A mapping node represents a mapping constraint, i.e., the $L$ edges on the left side link to the $L$ variable nodes that are the $L$ mapping input, and the edge on the right side links to a mapping output. Therefore, the $i$-th mapping node represents mapping constraint $\mathcal{M}(c_{i,1}^\prime, c_{i,2}^\prime, \cdots, c_{i,L}^\prime)=x_i$, where $c_{i,1}^\prime, c_{i,2}^\prime, \cdots, c_{i,L}^\prime$ are the mapping inputs and $x_i$ is the mapping output. Thus, the edges from the left of a mapping node should pass the LLRs for $c_{i,1}^\prime, c_{i,2}^\prime, \cdots, c_{i,L}^\prime$ and the edge from the right side should pass LLR for $x_i$.
Let $L(x_i)=\log\frac{\textrm{Pr}(y_i|x_i=0)}{\textrm{Pr}(y_i|x_i=1)}$ be the LLR about $x_i, i=1,...,N$, obtained from the ESE.
Let $L^a(c_{i,j}^\prime)=\log\frac{\textrm{Pr}(c_{i,j}^\prime=0)}{\textrm{Pr}(c_{i,j}^\prime=1)}$ be an a priori LLR of $c_{i,j}^\prime$ from a variable node.
The $i$-th mapping node calculates an extrinsic LLR for $c_{i,k}^\prime, k=1,2,...,L$, given by
\begin{eqnarray}
L^e(c_{i,k}^\prime)&&\!\!\!\!\!\!\!\!=\log\frac{\textrm{Pr}(y_i|c_{i,k}^\prime=0)}{\textrm{Pr}(y_i|c_{i,k}^\prime=1)}\nonumber\\
&&\!\!\!\!\!\!\!\!=\log\frac{\sum_{c_{i,j}^\prime, j\neq k}\textrm{Pr}(y_i|c_{i,k}^\prime=0, c_{i,j}^\prime, j\neq k)\prod_{j\neq k }\textrm{Pr}(c_{i,j}^\prime)}{\sum_{c_{i,j}^\prime, j\neq k}\textrm{Pr}(y_i|c_{i,k}^\prime=1, c_{i,j}^\prime, j\neq k)\prod_{j\neq k }\textrm{Pr}(c_{i,j}^\prime)}\nonumber\\
&&\!\!\!\!\!\!\!\!=\log\frac{\sum_{c_{i,1}^\prime, c_{i,2}^\prime, \cdots, c_{i,L}^\prime}(1-c_{i,k}^\prime)\textrm{Pr}(y_i|x_i)\prod_{j\neq k }\textrm{Pr}(c_{i,j}^\prime)}{\sum_{c_{i,1}^\prime, c_{i,2}^\prime, \cdots, c_{i,L}^\prime}c_{i,k}^\prime\textrm{Pr}(y_i|x_i)\prod_{j\neq k }\textrm{Pr}(c_{i,j}^\prime)}\nonumber\\
&&\!\!\!\!\!\!\!\!=\log\frac{\sum_{c_{i,1}^\prime, c_{i,2}^\prime, \cdots, c_{i,L}^\prime}(1-c_{i,k}^\prime)e^{(1-x_i)L(x_i)+\sum_{j\neq k}(1-c_{i,j}^\prime)L^a(c_{i,j}^\prime)}}{\sum_{c_{i,1}^\prime, c_{i,2}^\prime, \cdots, c_{i,L}^\prime}c_{i,k}^\prime e^{(1-x_i)L(x_i)+\sum_{j\neq k}(1-c_{i,j}^\prime)L^a(c_{i,j}^\prime)}}\nonumber
\end{eqnarray}
where $x_i=\mathcal{M}(c_{i,1}^\prime, c_{i,2}^\prime, \cdots, c_{i,L}^\prime)$.
\subsubsection{Variable Node Processing}
Since a variable node is associated with an ECC coded bit, i.e., the $i$-th node is associated with $c_i$, the edges connected to it should pass LLR messages for the same bit, i.e., $c_{i,1}=c_{i,2}=\cdots=c_{i,L}=c_i$, where $c_{i,j}, j=1,..., L$, are $L$ repetitions of $c_i$.
Consider the processing at the $i$-th variable node. Let $L^a(c_i)$ be the priori LLR about $c_i$ from the ECC decoder and $L^a(c_{i,j})$ be the priori LLR about $c_{i,j}, j=1,...,L$, from the neighboring mapping nodes. Since $c_{i,1}=c_{i,2}= \cdots = c_{i,L}=c_i$,
the variable node calculates an extrinsic LLR for each $c_{i,k}$, given by
\begin{eqnarray}
L^e(c_{i,k}) =L^a(c_i)+\sum_{j\neq k}L^a(c_{i,k}).
\end{eqnarray}
After a certain number of processing iterations, the variable node outputs an extrinsic LLR about $c_i$ to the ECC decoder
\begin{eqnarray}
L^e(c_i) =\sum_{k=1}^LL^a(c_{i,k}).
\end{eqnarray}
\subsection{ECC Optimization and BER Simulations}
In this section, we present the
ECC optimization and BER simulation results for ReRAM systems. With our proposed ECC, we achieve a high storage efficiency with a gap of less than 0.1 bit/cell from the ReRAM channel capacity.
\begin{figure}[t]
\includegraphics[width=
3.3 in]{Mapping4.pdf}
\centering
\caption{Mappings A and B that achieved data distributions with $q=5/16$ and $q=3/16$.} \label{fig:Mapping4}
\end{figure}
\begin{table}[t]
\caption{Code parameters for $m\times n=64\times64$ and $128\times128$ ReRAM systems. Parameters of IRA code involves its variable node degree distribution: $\{\lambda_i\}$ and combiner factor: $a$ \cite{RA,IRA}.}
\label{tab:code}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline\hline
Array size&\multirow{2}{*}{$64\times64$} &\multirow{2}{*}{$128\times128$}\\
$m\times n$ & & \\
\hline
\multirow{3}{*}{Data shaper}& $L=4$ & $L=4$\\
& Mapping A & Mapping B\\
& $q=5/16$ & $q=3/16$\\
\hline
\multirow{3}{*}{IRA code (ECC)} & $\lambda_3=0.567736$&$\lambda_3=0.501564$\\
& $\lambda_{50}=0.432264$&$\lambda_{50}=0.498436$\\
&Combiner: $a=6$&Combiner: $a=4$\\
\hline
Code rate &$R=0.542824$&$R=0.414735$\\
\hline
$\max_q(C_q(\epsilon_q, \sigma=100))$ &0.660 &0.494\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[t]
\includegraphics[width=
3.5 in]{BER_64.pdf}
\centering
\caption{BERs for IRA-coded ReRAM channel (solid line) and IRA-coded block-varying $(\fat{\epsilon}, \sigma)$-channel (labeled by $\circ$) with across-array coding strategy where channel parameters are set as $m\times n=64\times64$, $R_1=100\ \Omega, R_0=1000\ \Omega, R_s=250\ \Omega$, and $p_f=10^{-3}$.} \label{fig:BER64}
\end{figure}
\begin{figure}[t]
\includegraphics[width=
3.5 in]{BER_128.pdf}
\centering
\caption{BERs for IRA-coded ReRAM channel (solid line) and IRA-coded block-varying $(\fat{\epsilon}, \sigma)$-channel (labeled by $\circ$) with across-array coding strategy where channel parameters are set as $m\times n=64\times64$, $R_1=100\ \Omega, R_0=1000\ \Omega, R_s=250\ \Omega$, and $p_f=10^{-3}$.} \label{fig:BER128}
\end{figure}
\begin{figure}[t]
\includegraphics[width=
3.5 in]{BER_channel_est.pdf}
\centering
\caption{BER comparison between the IRA-coded ReRAM channel with and without channel estimation (CE), where memory array size is $m\times n=64\times64, 128\times128$ and joint-coding array number is $T=1, 16$.} \label{fig:BER_channel_est}
\end{figure}
\begin{figure}[t]
\includegraphics[width=
3.5 in]{BER_interleave.pdf}
\centering
\caption{BER comparison between our proposed structured interleaving and random interleaving over the i.i.d. $(\epsilon_q, \sigma)$-channel.} \label{fig:interleave}
\end{figure}
For the ECC, we adopt an IRA code which is a type of irregular low-density parity-check (LDPC) code that is able to approach the i.i.d. channel capacity with low encoding and decoding complexity \cite{RA,IRA}. We consider the code design for two ReRAM systems with a memory array sizes of $m\times n=64\times64$ and $128\times128$, and at a noise level of $\sigma=100$. We adopt the data shapers with a $(L=4)$-repeater and 4-to-1 mappers with Mappings A and B (Fig.~\ref{fig:Mapping4}). Mappings A and B produce data distributions of $q=\frac{5}{16}$ and $q=\frac{3}{16}$, respectively, which approach the maximum storage efficiency for the considered ReRAM systems (Fig.~\ref{fig:capacity}). Here the maximum storage efficiencies for $m\times n=64\times64$ and $128\times128$ ReRAM systems are $\max_qC_q(\epsilon_q, \sigma=100)=0.660$ and $0.494$ bit/cell, respectively. We optimize the IRA code for these two cases over an i.i.d. $(\epsilon_q, \sigma)$-channel using a density evolution method \cite{LDPC_DE}. The code parameters for these two ReRAM systems are listed in TABLE~\ref{tab:code}. Our codes achieve $R=0.542824$ and $0.414735$ bit/cell, which are close to the capacity bound with gaps of about $0.12$ and $0.08$ bit/cell.
In Figs.~\ref{fig:BER64} and \ref{fig:BER128}, we simulate the BER of the two coded ReRAM systems in TABLE~\ref{tab:code} with the across-array coding strategy over both the ReRAM channels (solid lines) and the equivalent block-varying $(\fat{\epsilon}, \sigma)$-channels (labeled by $\circ$). Message-passing decoding were employed for both the de-shaper and the ECC decoder, where the decoding of IRA codes can be found \cite{IRA}. For the $m\times n=64\times 64$ system, we employ a code length $N=64\times64=4096$ and for $128\times128$, we adopt $N=128\times128=16384$. In both figures, the BERs over the ReRAM channels are almost the same as those over the block-varying $(\fat{\epsilon}, \sigma)$-channels, thus demonstrating the proposed channel equivalence. The BERs improve as $T$ increases and approach the decoding performance over the i.i.d. $(\epsilon_q, \sigma)$-channel, which is the performance limit for ReRAM channels as $T\rightarrow\infty$. This performance improvement as $T$ increases can be regarded as a ``diversity" gain by assigning the codeword to multiple memory arrays. For comparison, we also illustrate the BER performances of the same IRA codes without data shaping $(q=1/2)$ over i.i.d. $(\epsilon_q, \sigma)$-channel. The decoding performance deteriorates significantly due to the lack of data shaping.
To emphasize the importance of the proposed real-time channel estimation, in Fig.~\ref{fig:BER_channel_est}, we compared the BERs between the IRA-coded ReRAM channels with and without channel estimation for memory array sizes $m\times n=64\times64, 128\times128$ and joint-coding array numbers $T=1, 16$. We employ the same code parameters listed in TABLE~\ref{tab:code} for both cases. For the IRA-coded ReRAM channel without channel estimation, we employ the average channel parameter $\epsilon_q$ for the decoder. We observe that by applying the channel estimation, the BER improvement is obvious for each comparison pair.
Note that the codes in TABLE~\ref{tab:code} are designed at $\sigma=100$ for the i.i.d. $(\epsilon_q, \sigma)$-channel, which means that theoretically the code can be decoded without errors at $\sigma=100$. However, the actual decoding performances shown in Figs.~\ref{fig:BER64} and \ref{fig:BER128} are much worse. This is because with density evolution, the code is designed under an infinite code length assumption, while in our simulations finite-length codes were employed. In other words, the codes are asymptotically decodable at $\sigma=100$ as the code length approaches infinity. For the same reason, the $128\times128$ ReRAM system achieves a lower BER performance than the $64\times64$ system since a much longer code is employed in the $128\times128$ system although the codes in both systems are designed at the same noise level of $\sigma=100$.
In Fig.~\ref{fig:interleave}, we provide BER comparisons between our proposed structured interleaving scheme and the random interleaving for the codes in TABLE~\ref{tab:code} over the i.i.d. $(\epsilon_q, \sigma)$-channel. For both codes, our proposed structured interleaving scheme outperforms the random interleaving. The BER curves are steeper with structured interleaving than that with random interleaving. This verifies our analysis in Section~\ref{sec:shaper}. Note that similar performance gain can also be observed from the ReRAM channels, which are the block-varying forms of the $(\epsilon, \sigma)$-channel.
\section{Conclusion} \label{sec:conclusion}
We have considered the design of effective channel coding
schemes to tackle both the sneak-path interference and the
additive noise for the ReRAM channels. We have proposed
an across-array coding strategy to mitigate the channel instability. It also enables
a ``diversity" gain during decoding. By employing TIN decoding, the ReRAM channel is equivalent to a block-varying channel whose status is not data-dependent, based on which, we proposed the capacity limit as well as a coding scheme. We have also proposed a
real-time channel estimation scheme to obtain the sneak-path
rates of the $T$ arrays, based on which an ESE calculates the
LLR for each coded bit for decoding.
To deal with the channel asymmetry, we proposed an ECC concatenated with a data shaper, where the later forms the desired input data distribution to achieve the maximum information rate. With an optimal ECC design, the ReRAM system achieved a high storage efficiency with a gap of less than 0.1 bit/cell from the ReRAM channel capacity limit.
We would also like to point out some possible extensions that lead to our future work. Although we only considered the AWGN noise in this paper, our work can be easily extended to other types of noises, such as the lognormal noise, through reformulating the channel capacity and the LLR formula in the ESE. Moreover, to consider more general sneak-path models that involve multiple sneak paths affecting a read cell, the channel model in Fig.~\ref{fig:ESchannel} should be modified as a binary input multi-level output channel, where the types of the output signal levels depend on the corresponding sneak-path combinations \cite{Ben}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,538
|
Garage Door Bottom Weatherstrip have 40 picture of decorating ideas, it's including Garage Door Bottom Weatherstrip Awesome Weatherseals For Doors Accessories Seals Decorating Ideas 1. Garage Door Bottom Weatherstrip Improbable Weather Seal Replacement Kit Fits Up To 18 Decorating Ideas 2. Garage Door Bottom Weatherstrip Far Fetched Rubber Weather Strip Overhead Seals Decorating Ideas 3. Garage Door Bottom Weatherstrip Shock Shop Frost King G16H Weatherstripping 2 1 4 X 16 Decorating Ideas. Garage Door Bottom Weatherstrip Stupefy Seals Decorating Ideas 5.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,453
|
{"url":"http:\/\/physics.stackexchange.com\/tags\/statistical-mechanics\/new","text":"# Tag Info\n\n2\n\nThere are two definitions of entropy, which physicists believe to be the same (modulo the dimensional Boltzman scaling constant) and a postulate of their sameness has so far yielded agreement between what is theoretically foretold and what is experimentally observed. There are theoretical grounds, namely most of the subject of statistical mechanics, for our ...\n\n0\n\nWhat is entropy really? I want to answer(!) this question from a different point of view. First off, I focus on your title and the phrase \u201creally\u201d. We don\u2019t know what entropy is really. We don\u2019t know what energy is really also, and any thing or concept else too. Entropy, like all other concepts created by humans, is a convention between some people to ...\n\n1\n\nI read in the documentation that came with this simulator that when you change the number of particles \"N\", the total energy of the system \"E\" or the number of dimensions, the field will turn yellow (which indicates that the simulator has not incorporated the changes) until you hit enter. I made changes to these values, hit enter with the cursor still in ...\n\n1\n\nWhy does a critical point even exist? I think this question is equal to this one: \"Why the width of the two phase region is bigger at lower temperatures and pressures?\" Specific volume of liquids mostly depends on the temperature of them in comparing with their pressure. This means, for a well-defined increment of the pressure, we can neglect its effect ...\n\n2\n\nAttempting to answer the \"why\" question intuitively: In a liquid, the molecules experience significant intermolecular force - so much so, that the average energy of the molecules is insufficient to escape the attractive force of the surrounding materials. The result is that it energetically favorable for them to remain close together, even if that means ...\n\n14\n\nI will try to answer these questions from different views. Macroscopic view The \"quantitative\" rather than qualitative difference in a liquid-gas phase transition is due to the fact that the molecules arrangement does not change so much (there is no qualitative difference) but the value of the compressibility changes a lot (quantitative difference). This ...\n\n8\n\nFor a pure substance that can exist in the solid, liquid, and vapor states (i.e., wood is not in this category), let's assume that a closed container is half full of liquid and half full of vapor. As the temperature rises, the liquid expands and the liquid density falls. Also, as the temperature rises, the pressure in the container rises due to the vapor ...\n\n2\n\nGood question. I don't have my Widom around, but I'll try to answer from memory. I think the consensus is to say a substance is at its gas state if it could be a liquid at the same temperature. This, as opposed to same pressure, same volume, etc. If the temperature is supercritical, there is no transition between liquid and gas, and the generic term \"fluid\"...\n\n1\n\nI'll try to explain why there could be a critical line and not just a critical point, and hopefully that will answer your question. If you think about the Ising model, we have the standard Hamiltonian: $$-\\beta H = J_1\\sum_{<i,j>}s_i s_j + h\\sum_{i}s_i$$ where $\\sum_{<i,j>}$ is a sum over nearest neighbors. This model ...\n\n2\n\nThe dimension issue is solved easily by defining the probability density function(PDF) as $$P(\\{q,p\\})=\\frac{E_0}{h^{3N}} \\ \\delta (H(\\{q,p\\})-E)$$ where $E_0$ is an arbitrary constant which will not affect any thermodynamic quantity or equilibrium property. Actually, this definition is incomplete. We have to take into account the indistinguishability of ...\n\n0\n\nI think the key here is that you're misunderstanding these integrals. Let's look at the following integral you wrote: $$\\Omega(E,V,N)=\\frac{1}{h^{3N}}\\cdot\\int_{H=E} d\\Gamma$$ And let's look at a common example when H is a function of p and q. Now this definition becomes: $$\\Omega(E,V,N)=\\frac{1}{h^{3N}}\\cdot\\int_{H(p,q)=E} d\\Gamma(p,q)$$ Can you see ...\n\n2\n\nParticle on a rotating ring For further discussion purpose, let's considere the dynamics of a quantum particle on a $r_0$ radius rotating circle at a constant angular velocity $\\mathbf{\\Omega}=\\Omega\\,\\hat{e}_z$ . In cylindrical coordinates, we fix $z=0$, and we have the azimuthal angle $\\theta$, which is the \"good\" degree of freedom describing the ...\n\n0\n\nIn CFT, we are interested in the continuum limit, where we can classify classes of models at their critical points. By means of the Jordan-Wigner transformation one can construct a fermion operator out of the spin operators of the usual 2D Ising model. Then, the continuum critical Ising model is described by a massless real fermion: $$S=\\frac 12\\int d^2 z\\... 0 That calculation has restrictions, but, one in particular should be mentioned, that master equation is supposed to be connected to this entropy, but is not necessarily, the master equation can be connected to general entropic form, and that is a fundamental idea for a more complete proff. 9 Preliminaries: How do we define 'localized?' For a single particle, or for multiple non-entangled particles, it is easy to tell from the expressions for the wavefunctions whether they are localized or delocalized. For example, you might say that if the wavefunction is falling off exponentially or faster for large x, that is with a form like \\psi(x)\\sim e^... 2 In the context of solid-state physics, a closely related question has been an area of active research in the past few years. Most interacting systems do indeed thermalize (and thus delocalize) over long time scales. However, certain systems whose disorder is much stronger than their interactions experience \"Many-Body Localization,\" in which the individual ... 3 To solve your problem exactly, you would have to solve the Schr\u00f6dinger equation$$i \\frac{\\partial}{\\partial t} \\Psi (\\vec r_1 \\dots \\vec r_N,t)= H \\ \\Psi(\\vec r_1 \\dots \\vec r_N,t)$$where \\Psi (\\vec r_1 \\dots \\vec r_N,t) is the wave function of the N particles and$$H=\\sum_i^N \\frac{p_i^2}{2 m} + \\sum_{i<j}^N u_{ij}+V_{\\text{ext}}$$where u_{ij}... 2 I agree that the language is very confusing - I'm a native English speaker, and it also took me a while to understand what they were saying. When they talk about the dimension of \"the statistical system itself,\" they mean the spacetime dimension. So if a system has two spatial dimensions, then it has three dimensions total (including time), and the ... 0 I suppose you mean that the gas is contained in a magic box. Otherwise the walls become part of the system, exchanging momentum\/energy with the 'particles'. I have no answer for you; I don't know. What I do know is that none of the particle-particle collisions can be characterized other than by using a probability distribution. Common sense demands that ... 2 Quantum effects appear if the concentration of particles satisfies,$$\\frac{N}{V} \\ge n_q$$where N is the number of particles, V is the volume, and n_q is the quantum concentration, for which the interparticle distance is equal to the thermal de Broglie wavelength, so that the wavefunctions of the particles are barely overlapping. As the quantum ... 3 It really depends on the boundary conditions. For boundary conditions like a 3D box with reflecting walls, the initial quantum state \\Psi will stay a quantum state with the unique wave function depending on variables of each particle:$$\\Psi({\\bf{r}}_1,...,{\\bf{r}}_n, t).$$If the boundary conditions are such that allow exchange with the environment, then ... 0 The behaviour of the molecules in your gedanken experiment can be approached by using decoherence. But I do not believe you can get a definitive answer until somebody makes a full scale simulation (or until some expert's answer can make a formal proof of what really happens, but I am not skilled to do that). The decoherence effects can be argued ... 2 As @valerio92 points out, your mistake is that S = k \\ln (\\omega\\, \\delta E), not \\delta S. To get \\delta S, you differentiate the right-hand expression to get \\delta S = k \\frac{\\delta \\omega}{\\omega}, and the \\delta E drops out and you get an expression with the right dimensions. The notation is a bit misleading, because the \\delta in the \\... 5$$S=k \\ln [\\Omega(E)] = k \\ln [\\omega (E) \\delta E] = k \\ln [\\omega(E)] +k \\ln (\\delta E)$$Last term is an arbitrary constant, so that we can set$$S = k \\ln[\\omega (E)]$$from which$$\\delta S = k \\frac{\\delta \\omega}{\\omega}$$If we can ignore the power contribution and set \\omega (E) \\simeq e^{\\beta E}, we get$$\\delta S = k \\frac{\\delta(e^{\\beta ...\n\n4\n\nYour equation (2) is trivially a solution of (1), because $v$ and $T$ are constant. This is a disappointing answer, because it leaves unanswered the question what makes the Boltzmann distribution unique. The answer is that you only wrote down the collision-less Boltzmann equation, but in the real world collisions are always present (and indeed, systems ...\n\n2\n\nThere is no single point where this becomes true - it is a very gradual change. The buzzwords are microscopic $\\to$ mesoscopic $\\to$ macroscopic. There is no special kind of mathematics involved; in the mesoscopic domain one uses a mix of quantum mechanics and statistical mechanics. See https:\/\/en.wikipedia.org\/wiki\/Mesoscopic_physics\n\n1\n\nDensity is the average amount of mass per unit volume $\\rho(\\vec{r},t) = \\frac{M}{V_r}$. Distribution function is defined as a number of particles per unit phase space volume $f(\\vec{r}, \\vec{c},t) = \\frac{N}{V_r V_c}$ (which is space volume times velocity volume). Each particle has mass $M$, so to get the total mass density, we need to sum distribution ...\n\n0\n\nThe information paradox is 40-45 years old. AdS\/CFT is not even 20 years old, modern string theory (after D-branes) in probably 25 years old. This is to say that the information paradox doesn't need any of these, even though of course you can try to solve it in the context of string theory. In a nutshell, the information paradox is a sharp theoretical ...\n\n1\n\nThis has been open for a while so I will bite. The information paradox has two versions or iterations. The previous one is that information is demolished by black holes by the entropy of its event horizon and that Hawking radiation that is emitted is in a pure blackbody distribution. A blackbody distribution of radiation is maximally random. If you make a ...\n\n1\n\nClausius' statement about heat not being able to flow spontaneously from a cold body to a warm body is sufficient to prove that no engine can have an efficiency greater than that of a perfectly reversible engine. But it's not enough to prove that the Carnot engine is the only reversible engine. For example, there could be a perfectly reversible engine where ...\n\n1\n\nBut for an ideal gas, internal energy is only a function of temperature and so internal energy remains constant here,no change in average kinetic energy of gas particles takes place, so where does the chaos come from to increase entropy of the system. 'Chaos' is not a very well defined term in context of statistical physics. It is not necessary to use it ...\n\n0\n\nThe chaos comes from by changing of volume or pressure of the system. The average kinetic energy doesn't change, but number of collisions increases (if pressure increase) or length of paths increases (if volume increases).\n\n3\n\nThe Maxwell-Boltzmann distribution and the Boltzmann distributions are probability distributions, i.e. functions $\\rho(\\vec x,\\vec v)$ of velocity and position of a particle, that say what is the probability density that the velocity and position belong to the small cube around the given value of them. The Boltzmann distribution is the more general one, \\$\\...\n\n0\n\nGreiner's Thermodynamics and Statistical Mechanics is pretty good from a few short readings I did. Also, it has better reviews from almost all of the other popular textbooks on the subject in goodreads.com\n\n0\n\nYou said it yourself the molecules have direction rather than randomly moving about. Picture a wide spot in a river, slow water, maybe eddy currents, random water flow. River narrows, water is directed through the slot. I agree with you, speed before pressure differential. What do you think about this, molecules entering Venturi creating vacuum which sucks ...\n\nTop 50 recent answers are included","date":"2016-07-24 01:10:15","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\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.772691547870636, \"perplexity\": 360.24827228935874}, \"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-2016-30\/segments\/1469257823805.20\/warc\/CC-MAIN-20160723071023-00318-ip-10-185-27-174.ec2.internal.warc.gz\"}"}
| null | null |
package proxy
import (
"fmt"
"google.golang.org/grpc"
"google.golang.org/protobuf/proto"
)
// Codec returns a proxying grpc.Codec with the default protobuf codec as parent.
//
// See CodecWithParent.
//
// Deprecated: No longer necessary.
func Codec() grpc.Codec {
return CodecWithParent(&protoCodec{})
}
// CodecWithParent returns a proxying grpc.Codec with a user provided codec as parent.
//
// Deprecated: No longer necessary.
func CodecWithParent(fallback grpc.Codec) grpc.Codec {
return &rawCodec{fallback}
}
type rawCodec struct {
parentCodec grpc.Codec
}
type frame struct {
payload []byte
}
func (c *rawCodec) Marshal(v interface{}) ([]byte, error) {
out, ok := v.(*frame)
if !ok {
return c.parentCodec.Marshal(v)
}
return out.payload, nil
}
func (c *rawCodec) Unmarshal(data []byte, v interface{}) error {
dst, ok := v.(*frame)
if !ok {
return c.parentCodec.Unmarshal(data, v)
}
dst.payload = data
return nil
}
func (c *rawCodec) String() string {
return fmt.Sprintf("proxy>%s", c.parentCodec.String())
}
// protoCodec is a Codec implementation with protobuf. It is the default rawCodec for gRPC.
type protoCodec struct{}
func (protoCodec) Marshal(v interface{}) ([]byte, error) {
return proto.Marshal(v.(proto.Message))
}
func (protoCodec) Unmarshal(data []byte, v interface{}) error {
return proto.Unmarshal(data, v.(proto.Message))
}
func (protoCodec) String() string {
return "proto"
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,554
|
<?php
class LoginForm extends CFormModel {
public $password;
public $remember_me = true;
public function rules() {
return array(
array('password', 'required'),
array('password', 'authenticate'),
array('remember_me', 'boolean')
);
}
public function attributeLabels() {
return array(
'password' => 'Пароль:',
'remember_me' => 'Запомнить',
);
}
public function authenticate($attribute, $params) {
if (!$this->hasErrors()) {
$this->identity = new UserIdentity($this->password);
if (!$this->identity->authenticate()) {
$this->addError('password', 'Неверный пароль.');
}
}
}
public function login() {
if(is_null($this->identity)) {
$this->identity = new UserIdentity($this->password);
$this->identity->authenticate();
}
if ($this->identity->errorCode === UserIdentity::ERROR_NONE) {
Yii::app()->user->login($this->identity, $this->remember_me ?
Constants::REMEMBER_DURATION_IN_S : 0);
return true;
} else {
return false;
}
}
private $identity;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,902
|
Defining Papillary Carcinoma of the Thyroid: A Short Review and Analysis.
Objectives:To review how changes in the pathologic definitions for papillary tumors of the thyroid during recent decades have affected outcomes for patients with these tumors. Methods:Forty-nine previous reports or studies involving collectively 53,606 patients were reviewed, and new analyses were performed on the data to include analyses of agreement, incidence, survival, and diagnostic categories. Results:The past emphasis on cytologic features to define papillary tumors has not resulted in ideal pairwise agreement between pathologists and has produced incidence and survival data suggesting overdetection and overdiagnosis. Most recently, tissue patterns have been reemphasized. Conclusions:With the recent reemphasis on diagnostic tissue patterns (over cytologic criteria), agreements between pathologists for the diagnosis of papillary tumors should improve, and the incidence of papillary carcinoma should decline. Nevertheless, updated survival analyses demonstrate excellent long-term survival for most of those diagnosed with papillary carcinomas.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,006
|
Q: getting an error while executing the code In the below code i want to print the lines which are between "first", and on those line searching for "new.txt" line.. when i am running i get an error :
if "first" in lines[i+n]:
IndexError: list index out of range
my code:
def find_path(self):
f = open("/output",'w')
for line in self.logs:
f.write(line)
f = open('/output','rb')
lines = f.readlines()
for i,line in enumerate(lines):
if "first" in line:
pattern = line
for n in range(1,len(lines)):
if "first" in lines[i+n]:
break
else:
if "new.txt" in line:
print line
print lines[i+n]
f.close()
A: It's because i+n can be and will be greater than length of lines list.
for i,line in enumerate(lines):
That enumeration creates values for i from 0 to len(lines) - 1, so the maximum value of i is len(lines) - 1.
The following lines tell us that values of n can be from 1 to len(lines) - 1, so maximum value of n is len(lines) - 1 again:
for n in range(1,len(lines)):
if "first" in lines[i+n]:
break
So, value of i + n can be from 1 to 2 * (len(lines) - 1) - that's why you're getting IndexError.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,811
|
Moreover, if i use null for inputDate i get Exception during the query!
when it's false using our Boolean Branching technology so the query plan will not contain table scans for expressions like above.
What about the latter exception?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,288
|
Suunto has really pushed the envelope with the 9 in creating the best battery life. "I've worn these boots on rainy days in Portland and hikes in Big Cottonwood.
Telephoto lenses are able to bring far away objects to us, and they offer amazing compression which can. In this article w.
Experts say the best of these destinations offer co-working. Primely situated between the United States and Japan, Hawaii.
A trend that's on the rise right now is finding some of the best dog booties to protect canine's paws from snow, ice, heat, cold and salt.
. north shore. Hawaii's premier zipline tours waterfall hikes, swims, & horseback rides. Come Adventure With Us! Get away. We Are A Top Rated Company.
[Kentucky.com] Wonder what the story is behind the FC Dallas trade of Kellyn Acosta. He's going to be in the mix for the US National team at the 2022 World Cup. [Dallas News] Why you'd have to pay me.
Jun 29, 2017. So, here we are providing you a list of the top best Hiking Trails in. This hiking trail situated in Colorado, USA is quite a short distance trail.
When away from home (especially if hiking or camping) I use one almost every day of my. Have something you think we should.
After all, you want the best deals–especially Prime Day outdoors deals. Prime Day camping deals have brought us these incredible Etekcity Portable LED Lanterns. For the price, you're receiving thre.
When most of us abandon our heated living rooms and voice. to do if you encounter a cougar in the wild. — Go in a group "H.
So, should you carry a gun when you go hiking, camping, or other stuff outdoors. rather than risk passing near us. They pr.
Here, we pack all four seasons with the best of the outdoors, the indoors, and everything in between. outdoor playground complete with Gold Medal fly- fishing rivers, stellar mountain biking trails, and majestic alpine hikes. You with us?
In Michigan, fall colors are expected to peak around the week of Oct. 22 and if you're like us, you're ready to take. loca.
As a dad who has encouraged his kids to spend time outside, pored over naturalists guides to be able to identify plants and w.
Best Western Plus Landmark Inn Your Park City Vacation Starts Here. For places to stay in Canyons Village Park City Park City Utah, our Best Western Landmark Inn is the perfect spot! Close to popular tourist attractions, you and your family are guaranteed to experience everything this unique town has to offer.
Jul 19, 2018. He said he's "not thrilled" with rate hikes and worries that the work the. But at the same time I'm letting them do what they feel is best. that it will put the U.S. at a "disadvantage" while the Fed's counterparts like the European.
Fed officials, including Powell, have long maintained that the best way they can help global financial stability is by keeping U.S. growth on track, while communicating clearly about where policy is h.
We scoured hiking resources to present you with several spectacular hikes that you can get to via public transportation. And for those who want to keep things closer to home, we included one fantastic.
"For many of us if we used this system would actually be travelling farther. "Paying more will never prove popular, but we.
(KNWA) — Time Magazine's website has ranked Rogers as one of the top 20 best cities to live in the United States. Time.com's.
For the record, Bacon is also a big fan of hiking—even on a rainy day. "It was a really big honor," Bacon said. "When I ma.
In a fall season that has given us involving open-world adventures and storylines that. though the first episode highlight.
Apr 1, 2012. Across the U.S., you'll find lush forests, stunning plains and dusty deserts, so it's no wonder the country has some of the most beautiful hiking.
Many of us who've made that trek have returned with. have some steep sections and uneven passages over rocks. It's best to.
You just show up and experience the best your backyard has to offer. But don't take it from us. It's the best thing you'll do all week. "This was my first hiking experience and Marc was super friendly and answered all my question about.
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(Photo by Silver Screen Collection/Getty Images)
Photo: Silver Screen Collection
Four amazing things about "Willy Wonka"
Rare.us
We learned Monday of the loss of iconic American actor and comic Gene Wilder at the age of 83 due to complications from Alzheimer's Disease, his nephew revealed in a letter you should take the time to read.
Wilder may not have known it at the time he played the role of Willy Wonka in "Willy Wonka and the Chocolate Factory," but he would learn over the years of the incredible impact the film had, especially on children.
>>PHOTOS: Gene Wilder through the years
There are three aspects about the classic film that exhibited the brilliance of Wilder as an actor and of the care he took to make every role stand the test of time.
Everyone surely remembers Willy Wonka's grand entrance in the film, where the mysterious candy man walks out slowly with a cane and a limp only to somersault into a flourish to the crowd's confusion and delight.
>>MORE: Gene Wilder dead at 83
It turns out that Wilder saw this as critical for the overall effect and impact of the story before anyone else.
(At the 8:41 mark in video)
Before he committed to the role, Wilder told director Mel Stuart that he thought the script was very good, but he insisted make his grand entrance with a cane and all of the other quirks described.
"What do you want to do that for?" the director replied.
"From that moment on, no one would know if I was lying or telling the truth," Wilder responded.
"Okay, I'll give it a try," Stuart said.
The rest is history.
The Spooky Boat Ride Through the Chocolate Tunnel
According to Movie Pilot, Wilder was the only one who knew what was going to happen on that boat ride, so if the passengers looked scared, it's because they actually were.
You can see the terror on the faces of the passengers, both kids and adults alike, but it wasn't just good acting — their terror was real. All the actors had been set up for was a simple boat ride; only Gene Wilder himself knew what was really in store.
The Chocolate Room
Wilder used the same technique of keeping actors in the dark for the famous chocolate room scene that gave us "Pure Imagination."
It was first time that the child actors had seen the room, ensuring that they would react like kids in a candy store — the candy store of their dreams.
The Lasting Impact on Kids and Adults Alike
All you have to do is search "Willy Wonka" on social media right now and you'll understand what Wilder meant to so many.
But Wilder himself was asked in a sit-down interview with film buff of repute Robert Osborne in 2013 about the impact of Willy Wonka.
(At the 11:40 mark in video)
"Do people mention it a lot when they meet you?" Osborne asked.
"I get five mails, I would say I get five a day — not Saturday or Sunday," Wilder replied. "'Would you sign this for me?'"
"And I do it, but it piles up," he said. "It's all because they saw Willy Wonka."
"Sometimes it's someone who's 12 years old, sometimes it's 21 years old and sometimes it's 34 years old," he continued. "But they want to have it signed — a picture and a signature."
May Gene Wilder rest in peace.
|
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| 7,902
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A 21-year Houston Police Department veteran was found dead Friday morning after he apparently committed suicide with a single gunshot to the head at a Houston police patrol station, Chief Art Acevedo said.
The sergeant was found around 8:35 a.m. inside of a stairwell on the fourth floor at the Westside Patrol Station, where he worked, Acevedo said. The fourth floor is not currently in use, so no one heard the gunshot, reports the Houston Chronicle.
Officers working in the station decided to search the facility after they discovered at 7 a.m. the sergeant had not come to work.
The sergeant, whose name has not been released, was married and had two children, ages 10 and 12.
"You can't explain these things," Acevedo said. "We ask that people please just pray for the family, pray for those young children."
The investigation continues. HPD psychological services and chaplain services were on site at the Westside Patrol Station on Friday morning.
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Happy Friday! I want to first say I'm sorry to all of you for not replying to your comments and questions as quickly as I usually do. I'm moving from San Francisco to Boston this weekend, so it's been an absolutely crazy week! I'm looking forward to next week when I'll be all settled and back to my regular schedule.
We took a break from packing and had a light lunch downtown. On the way we stopped by this lovely neighborhood and snapped a few pics. It was a cool sunny day, and this sweater kept me nice and cozy, I really love how comfortable it is. It's on sale right now with an extra 40% off if you use the code SALETIME.
I really loved how these pictures came out, I hope you guys do as well. What are you all doing this weekend? Are you going to watch the big game and which team are you rooting for?!
Next time I post, I'll be in Boston! Have a wonderful weekend and thanks for reading.
|
{
"redpajama_set_name": "RedPajamaC4"
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Q: recall the same fragment I'm in fragment A and I want that after pressing the switch button, present in fragment A, this redirects me to the same fragment A (I want to start onStart () of fragment A, every time I press the switch button). My code is:
public class A extends Fragment implements View.OnClickListener {
private TextView line;
private TextView direction;
private Adapter_post a;
private String departure;
private String sid_user;
public static RequestQueue requestQueue;
@Override
public View onCreateView(LayoutInflater inflater, ViewGroup container,
Bundle savedInstanceState) {
Log.d("main_activity", "ok eseguito e parte main activity post");
return inflater.inflate(R.layout.A_fragment, container, false);
}
@Override
public void onStart() {
super.onStart();
Log.d("main_activity","this don't start");
}
@Override
public void onClick(View view) {
switch (view.getId()) {
case R.id.switch:
appCompatActivity.getSupportFragmentManager().beginTransaction().
replace(R.id.fragment_container, this).
addToBackStack(null).commit();
break;
}
}
after having pressed the switch button from fragment A, how can I make the same fragment A suffer (I want to start the onStart () of fragment A). How can i do?
|
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"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/physics.lnu.edu.ua\/jps\/2019\/1\/abs\/a1903-5.html","text":"Journal of Physical Studies 23(1), Article 1903 [5 pages] (2019)\nDOI: https:\/\/doi.org\/10.30970\/jps.23.1903\n\n## DYNAMICS OF THE FINE STRUCTURE OF THE 22-YEAR SOLAR ACTIVITY MAGNETIC CYCLE\n\nM. M. Koval'chuk1, R. Ye. Rykaluk2, M. I. Stodilka1, O. A. Baran1, M. B. Hirnyak1\n\n1Astronomical Observatory of the Ivan Franko National University of Lviv,\n8, Kyryla i Mefodia St., Lviv, UA--79005, Ukraine\n2Faculty of Applied Mathematics and Informatics of the Ivan Franko National University of Lviv,\n1, Universytetska St., Lviv, UA--79000, Ukraine\nMyroslav.Stodilka@lnu.edu.ua\n\nAn analysis of the dynamics of the fine structure of the 22-year cycle of solar activity (1964-1986) was carried out. The complete homogeneous data series for sunspot data such as Wolf numbers, squares of groups and individual spots, their coordinates, polarity and magnetic fluxes were retrieved from the electronic data base of Daily Solar Data.\n\nDepending on the logarithm of the maximum areas of the spot groups, we divided them into two populations -- the large and small ones (hypothesis by Nagovitsyn, Pevtsov's {\\em et al.}). They intersect in the domain lg\\,$S=1.8-1.9$, where $S=60-90$ m.f.h. In this domain, the two populations of the spot groups are separated by the average lifetime takes place: less than 3-4 days -- for small and more than 4-5 days -- for large groups of spots. The two sunspot populations differ in the dynamics of the magnetic field distribution: the fields with a weak field strength (about 1600\\,G) are localized in small spots, while the fields with a powerful flow -- in large spots, where magnetic field reaches 3000\\,G. The analysis of the fine structure of the 22-year cycle confirmed the existence of the so-called points of the Waldmeier's fracture points on the growth branches, after which the sunspot production slows before the solar maximum. Comparison of the spot groups in the northern and southern hemispheres of the Sun shows the predominance of spottedness on the branches of growth in the northern hemisphere. So, a positive asymmetry dominates there, and, vice versa, on the recession branches, the negative asymmetry of the southern hemisphere starts to dominate. The asymmetry of the sunspots' magnetic field in the two hemispheres behaves in a similar way. The indexes of the magnetic asymmetry are maximal on the branches of growth till the maxima of the magnetic activity cycle, namely, during the periods of the sunspots' magnetic field polarity reversal.\n\nPACS number(s): 96.60.qd\n\nReferences\n1. A.\u00a0A. Pevtsov et al., Solar Phys. 289, 593 (2014);\nCrossRef\n2. Yu. Nagovitsyn et al., Astrophys. J. Lett. 758, L20 (2012);\nCrossRef\n3. \u042e.\u00a0\u0410.\u00a0\u041d\u0430\u0433\u043e\u0432\u0438\u0446\u044b\u043d, \u0410.\u00a0\u0410.\u00a0\u041f\u0435\u0432\u0446\u043e\u0432, \u0410.\u00a0\u0410.\u00a0\u041e\u0441\u0438\u043f\u043e\u0432\u0430, \u0432 \u0422\u0440\u0443\u0434\u044b XIX \u0412\u0441\u0435\u0440\u043e\u0441\u0441\u0438\u0439\u0441\u043a\u043e\u0439 \u0435\u0436\u0435\u0433\u043e\u0434\u043d\u043e\u0439 \u043a\u043e\u043d\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0438 \u043f\u043e \u0444\u0438\u0437\u0438\u043a\u0435 \u0421\u043e\u043b\u043d\u0446\u0430 \u0421\u043e\u043b\u043d\u0435\u0447\u043d\u0430\u044f \u0438 \u0441\u043e\u043b\u043d\u0435\u0447\u043d\u043e-\u0437\u0435\u043c\u043d\u0430\u044f \u0444\u0438\u0437\u0438\u043a\u0430 --- 2015'' (\u0421\u0430\u043d\u043a\u0442-\u041f\u0435\u0442\u0435\u0440\u0431\u0443\u0440\u0433, 2015), \u0441.\u00a0293.\n4. \u0418.\u00a0\u0413.\u00a0\u041a\u043e\u0441\u0442\u044e\u0447\u0435\u043d\u043a\u043e, \u0432 \u0422\u0440\u0443\u0434\u044b XIX \u0412\u0441\u0435\u0440\u043e\u0441\u0441\u0438\u0439\u0441\u043a\u043e\u0439 \u0435\u0436\u0435\u0433\u043e\u0434\u043d\u043e\u0439 \u043a\u043e\u043d\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0438 \u043f\u043e \u0444\u0438\u0437\u0438\u043a\u0435 \u0421\u043e\u043b\u043d\u0446\u0430 \u0421\u043e\u043b\u043d\u0435\u0447\u043d\u0430\u044f \u0438 \u0441\u043e\u043b\u043d\u0435\u0447\u043d\u043e-\u0437\u0435\u043c\u043d\u0430\u044f \u0444\u0438\u0437\u0438\u043a\u0430 --- 2016'', c.\u00a0171.\n5. A.\u00a0A.\u00a0Pevtsov et al., Astrophys. J. Lett. 742, L36 (2011);\nCrossRef\n6. V.\u00a0M.\u00a0Efimenko, V.\u00a0G. Lozitsky, Adv. Space Res. 61, 2820 (2018);\nCrossRef\n7. N.\u00a0I. Lozitska et al., Adv. Space Res. 55, 897 (2015);\nCrossRef\n8. \u042e.\u00a0\u0418.\u00a0\u0412\u0438\u0442\u0438\u043d\u0441\u043a\u0438\u0439, \u0426\u0438\u043a\u043b\u0438\u0447\u043d\u043e\u0441\u0442\u044c \u0438 \u043f\u0440\u043e\u0433\u043d\u043e\u0437\u044b \u0441\u043e\u043b\u043d\u0435\u0447\u043d\u043e\u0439 \u0430\u043a\u0442\u0438\u0432\u043d\u043e\u0441\u0442\u0438 (\u041d\u0430\u0443\u043a\u0430, \u041c\u043e\u0441\u043a\u0432\u0430, 1973).","date":"2022-01-25 13:53:40","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\": 0.7162875533103943, \"perplexity\": 5228.160077218326}, \"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\/1642320304835.96\/warc\/CC-MAIN-20220125130117-20220125160117-00574.warc.gz\"}"}
| null | null |
Q: Force and Restrain on same Regex I'm trying to set a password input that will force at least one of each lowercase, uppercase, and numbers, but at the same time refuse any other input like special characters. so far im able to force one of each but haven't been able to deny special characters..
Thank You
pattern="(?=.*\d)(?=.*[a-z])(?=.*[A-Z]).{5,25}"
A: You need to replace the consuming . with [a-zA-Z0-9] character class, or whatever chars/ranges you allow in your regex:
pattern="(?=.*\d)(?=.*[a-z])(?=.*[A-Z])[a-zA-Z0-9]{5,25}"
^^^^^^^^^^^
The reason is that . matches any character (other than line break chars).
Note: the length limit works here because HTML5 pattern regex is anchored by default.
|
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{"url":"http:\/\/pballew.blogspot.com\/2014\/11\/a-23rd-and-very-first-known-way-to.html","text":"## Thursday, 13 November 2014\n\n### A 23rd (and the very first known?) way to solve a quadratic\n\nA very long time ago (2007) in a land far, far away (Lakenheath, England) I completed a somewhat longer than average paper (for me) with the somewhat tongue-in-cheek title of \"There must be Twenty Ways to Solve a Quadratic,\" with reference to an even older song called \"Fifty Ways to Leave your Lover.\"\nAmong them were several nice geometric approaches, and much later (May 2014) I blogged here about two additional geometric approaches I found quite lovely.\nToday while looking at a Kindle sample of Taming the Unknown by Victor Katz and Karen Parshall, I discovered yet another that I had overlooked, and it might well be the oldest quadratic problem ever found, and thus the earliest approach ever printed. The authors refer to it as \"the only example extant in the Egyptian Papyri...\".\u00a0 The approach was well known, but only it's application to quadratics had escaped my attention.\nI will describe this 23rd solution to a quadratic with one other very early quadratic problem from the same source to follow.\nIn the Berlin Papyrus (c 1900 BC) a problem is given to divide a square of 100 sq units into two squares where the ratio of the sides of the squares is 1: 3\/4.\nThis would be written in modern notation as $x^2 + (\\frac{3x}{4})^2 = 100$\nThe scribe's approach was to use the rule of false position. He begins by assuming one side of the larger square has sides of 1 unit, and the other would then be 3\/4. From this he obtains a total sum of squares to be $\\frac{25}{16}$ The scribe then chose to take the square root of this ratio since it represents a ratio between areas, not the side lengths he was comparing, to obtain 5\/4. Knowing that the sides of the original square were ten units, he computed the number of multiples of 5\/4 to achieve ten by dividing, and realized that his final sides must be 8 times as large as his false trial, or 8 units for one square and six for the other, producing squares with areas of 64 sq units and 36 sq units to make the desired total of 100 sq units.\nNot that the problem solved in this example is of the simple form Ax2 = k.\u00a0 I have not worked out for myself yet whether such an approach is feasible for a common trinomial quadratic $Ax^2 + Bx = C$ .\n YBC4663\n\nIn the same book they describe a problem from a Mesopotamian Tablet (C 1700 BC (see bottom of post)) that is the first known quadratic to be solved by the method of completing the square. I think this is still my favorite of all the ways to solve quadratics because of it's geometric elements. The table is YBC4663 and the eighth problem describes in part, the problem states that in rectangle the length exceeds the width by 3 1\/2 units and the area is 7 1\/2 square units. The problem is to find the lengths of the two sides of the rectangle. The scribe uses the method of least squares in purely geometric methods to solve the challenge.\n from Math is Fun.com (Let b= 3 1\/2)\nHe first divides the rectangle into a square and another rectangle by subtracting 3 1\/2 from the longer side and dropping a perpendicular to the short side, thus creating a square of x2 units and another that is 3 1\/2 x square units.\nThen he divides the rectangle in half (a process of geometric manipulation by cut and paste that seems to be unique to the Mesopotamian mathematics) and matching the two pieces against the x by x square to make an l-shaped gnomen (square with a square cut out of one corner). Knowing the edges of the two small rectangles forming this \"missing square\" is \\( \\frac{7}{4})^2= \\frac{49}{16}, the scribe realizes that if he added that area to the original 7 1\/2, he would have an area of 10 9\/16 or 169\/16 .\nHe takes the square root to determine that the completed square has sides of 3 1\/4 units, and that that if he adds the side of the small square to the large one, (3 1\/4\u00a0+ 1 3\/4 = 5) he will get the length of the original rectangle, and by subtracting it, (3 1\/4 - 1 3\/4 =1 1\/2) he gets the width.\nMy search continues, so if you know of an approach unique to all these, be assured, I'm looking for solution method number 24 .\nMy 2007 document has a statement that completing the square was later in Mesopotamian Math, around 400 BCE. This seems to be contradicted by several assertions that the above Tablet was from the Old Babylonian period (c.1792 - 1595 BC)","date":"2017-02-28 03:34:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7306398749351501, \"perplexity\": 675.8732169438878}, \"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-2017-09\/segments\/1487501174124.9\/warc\/CC-MAIN-20170219104614-00079-ip-10-171-10-108.ec2.internal.warc.gz\"}"}
| null | null |
<?php
namespace CloudDevStudio\EAV;
class Decimal extends Value
{
/**
* @param $format
* @return bool|string
*/
public function getFormatedValue()
{
return number_format($this->getValue(), 2);
}
}
|
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"redpajama_set_name": "RedPajamaGithub"
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Chicago VI je peti studijski album chichaške rock zasedbe Chicago, ki je izšel leta 1973 pri založbi Columbia Records. Gre za prvi album skupine, pri katerem je sodeloval tolkalist Laudir de Oliveira, ki je pri albumu Chicago VIII postal polnopravni član skupine.
Ozadje
Potem, ko je skupina posnela vseh prvih pet studijskih albumov v New Yorku (razen delov drugega albuma v Los Angelesu), je producent James William Guercio leta 1972 v Nederlandu, Kolorado, postavil Caribou Studios. Dela so bila končana tako, da je skupina lahko februarja 1973 začela s snemanjem albuma. Studio je ostal snemalna baza skupine za naslednja štiri leta.
Robert Lamm je napisal polovico skladb, vključno s svojim odgovorom nekaterim negativnim kritikom Chicaga v skladbi »Critics' Choice«. James Pankow je napisal dva hita z albuma, »Just You 'n' Me« (4. mesto) in »Feelin' Stronger Every Day« (10. mesto), katerega soavtor je bil Peter Cetera, ki je prispeval še skladbo »In Terms of Two« ter odpel vse tri.
Chicago VI je bil komercialno uspešen in je pet tednov preživel na vrhu ameriške lestvice Billboard 200. V Združenem kraljestvu se skupina ni uvrstila na lestvico albumov vse do leta 1976, ko je izšel album Chicago X.
Album je bil miksan in izdan tako v stereu kot v kvadrofoniji. Originalno izdajo je za zgoščenko masteriziral Joe Gastwirt. Leta 2002 je bil album remasteriziran in ponovno izdan pri založbi Rhino Records, dodani pa sta bili še dve skladbi: demo Terryja Katha »Beyond All Our Sorrows« in posnetek Al Greenove »Tired of Being Alone«, ki je bil vzet iz TV oddaje Chicago in the Rockies. Leta 2013 je podjetje Mobile Fidelity Sound Lab remasteriziralo album in ga izdalo v formatu Hybrid SACD.
Seznam skladb
Osebje
Chicago
Peter Cetera – bas, glavni vokal, spremljevalni vokal, orglice pri »In Terms of Two«
Terry Kath – električne kitare, akustične kitare, slide kitara, glavni vokal, spremljevalni vokal
Robert Lamm – klavir, Hammond orgle, clavinet, električni klavir Wurlitzer, Fender Rhodes, sintetizator ARP, Hohner Pianet, glavni vokal, spremljevalni vokal
Lee Loughnane – trobenta, spremljevalni vokal, tolkala
James Pankow – trombon, trobilni aranžmaji
Walter Parazaider – saksofoni, flavta
Danny Seraphine – bobni, tolkala
Dodatni glasbeniki
Laudir de Oliveira – konge
Joe Lala – konge
J. G. O'Rafferty – pedal steel
Produkcija
Producent: James William Guercio
Inženir: Wayne Tarnowski
Asistent inženirja: Jeff Guercio
Miks: Phil Ramone
Asistent: Richard Blakin
Oblikovanje ovitka: John Berg, Nick Fasciano
Fotografija: Barry Feinstein
Lestvice
Tedenske lestvice
Singli
Certifikati
Sklici
Albumi leta 1973
Albumi skupine Chicago
Albumi založbe Columbia Records
|
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\section*{Abstract}
{\bf
We study a generalized quantum spin ladder with staggered long range interactions that decay as a power-law with exponent $\alpha$. Using large scale quantum Monte Carlo (QMC) and density matrix renormalization group (DMRG) simulations, we show that this model undergoes a transition from a rung-dimer phase characterized by a non-local string order parameter, to a symmetry broken N\'eel phase. We find evidence that the transition is second order. In the magnetically ordered phase, the spectrum exhibits gapless modes, while excitations in the gapped phase are well described in terms of triplons -- bound states of spinons across the legs. We obtain the momentum resolved spin dynamic structure factor numerically and find a well defined triplon band that evolves into a gapless magnon dispersion across the transition. We further discuss the possibility of deconfined criticality in this model.
}
\vspace{10pt}
\noindent\rule{\textwidth}{1pt}
\tableofcontents\thispagestyle{fancy}
\noindent\rule{\textwidth}{1pt}
\vspace{10pt}
\section{Introduction}
The study of exotic phases of matter of quantum origin is one of the cornerstones of modern condensed matter physics, motivating a quest for materials and models that could exhibit novel unconventional properties, such as fractionalized excitations that cannot be described as Landau quasiparticles, topological states that do not admit a local order parameter, and quantum phase transitions that defy the Landau-Ginzburg paradigm.
Quantum magnets exhibit a vast and varied phenomenology and offer a relatively simple and intuitive playground where to test and verify these ideas. A prototypical example of phase transition that has been extensively studied is the one between a disordered dimer phase and a N\'eel ordered antiferromagnet(AFM) \cite{Lohofer2015, Qin2017, Lohofer2017}. On both sides of the critical point, excitations carry spin $S=1$: triplons in the magnetically disordered gapped phase; gapless magnons in the ordered phase.
At the transition, besides two gapless Goldstone modes, a massive amplitude mode (also referred-to as the Higgs mode) is expected.
Remarkably, this behavior has has been experimentally observed under pressure in TlCuCl$_3$ \cite{Matsumoto2004, Ruegg2008,Kra2004,Hong2017}.
A crucial reason explaining why the theoretical study of these phenomena has been limited to two and three spatial dimensions is justified by the Mermin-Wagner theorem\cite{Mermin1966}, that establishes that quantum Hamiltonians with short range interactions cannot spontaneously break a continuous symmetry in dimensions lower than $D=2$. Even in 2D systems, this can only occur at zero temperature $T=0$. In this work, we circumvent these restrictions by introducing long range non-frustrating interactions to the problem. We can thus conceive a ladder Hamiltonian that exhibits true long range N\'eel order and apply numerical techniques that are well suited for studying low-dimensional problems. Explicitly, the model of interest is a conventional Heisenberg ladder with additional algebraically decaying all-to-all couplings:
\begin{equation}
H= - J \sum_{i>j} \frac{(-1)^{\vert x_i+y_i-x_j-y_j\vert}}{\vert\vec{r}_i-\vec{r}_j\vert^\alpha} \vec{S}_i\cdot\vec{S}_j.
\label{rkky}
\end{equation}
where the spin operators $\vec{S}_i$ are localized at positions $\vec{r_i}=(x_i,y_i)$ on a two leg $2\times L$ ladder with $y_i=1,2$. The alternating sign on the interactions ensures that they will be AFM between spins on opposite sublattices, and ferromagnetic otherwise (See Fig. \ref{fig:J_ladder} for a graphical representation). One could in principle envision such interactions emerging from a proximity coupling with a higher dimensional antiferromagnet or other ladders in a perturbative sense. The only free parameter in the problem is the exponent $\alpha$; for large $\alpha$ we expect the ground state to be in the same phase as the conventional Heisenberg ladder and the physics is well understood: the correlation length is short, of a few lattice spaces, and the gap is of the order of the coupling $J$\cite{Barnes1993,Barnes1994,White1994,Gopalan1994,Troyer1994,Troyer1996,Shelton1996,Fouet2006,Giamarchi2011,Normand2011,Giamarchi2012,Giamarchi2013}.
This ground state is adiabatically connected to the trivial limit of the conventional Heisenberg ladder corresponding to anisotropic couplings along the legs and rungs $J_{rung} \gg J_{leg}$. In this ``strong rung coupling limit'' the ground state is a product of rung dimers, the single-triplet gap is of order $\mathcal{O}(J_{rung})$ and excitations are rung triplets that can propagate coherently along the ladder.
\begin{figure}[t]
\centering
\includegraphics[width=0.55\textwidth]{J_ladder.png}
\caption{Exchange interaction between two spins at distance $x$ along the same and opposite leg, for a value of $\alpha=1.8$. }
\label{fig:J_ladder}
\end{figure}
Notoriously, unlike the case of dimerized chains, this ``rung singlet phase'' does not break any lattice symmetry, and even though it is adiabatically connected to a product state in the limit of $J_{rung}\rightarrow \infty$, it is characterized by a broken ``hidden'' symmetry \cite{Kennedy1992,Oshikawa1992,Takada1992,Nishiyama1995,Pollmann2012} described in terms of a ``string'' order parameter\cite{White1996,Kim2000,Gibson2011}:
\begin{gather}
O = \lim_{l \to \infty} O_l,\label{eq:O} \\
O_l = -\left\langle \tilde{S}^z_0\left(\prod_{j=1}^{l-1} e^{i\pi\tilde{S}^z_j}\right)\tilde{S}^z_l \right\rangle,\label{eq:Ol}
\end{gather}
with $\tilde{S}^z_j=S^z_{j+1,1}+S^z_{j,2}$ connects spins along one of the diagonals between two rungs. The connection between ladders and the topological aspects of the Haldane chain has been established for some time now \cite{White1996}.
On the other hand, in the model described by Eq.(\ref{rkky}) we anticipate that the all-to-all unfrustrating interactions will yield a ground state with long-range AFM (N\'eel) order and gapless excitations for relatively small $\alpha$. These expectations are based on previous studies of Hamiltonian (\ref{rkky}) in 1D chains\cite{Yusuf2004,Laflorencie2005,Sandvik2010,Tang2015,Yang2021,Yang2020deconfined}, where a transition between a gapless spin-liquid and a gapless ordered phase was revealed.
In this work, we focus on identifying and characterizing the quantum critical point, as well as understanding the excitation spectrum at and away from the transition. Given that the model described by Eq.(\ref{rkky}) does not have a sign problem we use QMC to study the properties of the transition while using the time-dependent density matrix renormalization group method (tDMRG) \cite{white2004a,daley2004,vietri,Paeckel2019} to understand the low-energy excitations. The behavior of the gap and order parameters is discussed in Sec.~\ref{sec:qpt}, offering compelling evidence for a continuous quantum phase transition between the N\'eel and rung-dimer phases at $\alpha_c=2.519(1)$. In Sec.~\ref{sec:dynamics} we present results for the dynamic spin structure factor $S^z(q,\omega)$. We finally close with a summary and discussion of our findings.
\section{Quantum critical point}
\label{sec:qpt}
In this section, we study the transition between the N\'eel phase and the rung-dimer phase. In order to exclude the possibility of an intermediate phase, we use both order parameters to calculate the critical point $\alpha_c$ and the correlation length exponent $\nu$. We shall show that both order parameters give the same $\alpha_c$ and $\nu$ providing evidence of a direct order-to-order transition.
To perform these calculations, we use standard finite-size scaling (FSS) methods~\cite{campostrini14}. We calculate the quantities of interest using projector QMC applied to Eq.(\ref{rkky}) with periodic boundary conditions\footnote{We enforce the boundary conditions in the definition of the distance between the sites, $\vert\vec{r}_i-\vec{r}_j\vert$, being the minimum of the two possible distances.}. We start the projector QMC with a initial state expressed as an amplitude product state in the valance bond basis~\cite{Sandvik2010b,Beach2006,Tang2015}. We use the same initial state for all values of $\alpha$. We choose the amplitudes such that the initial state has long-range N\'eel order to reduce the number of projector steps needed to reach the ground state in the gapless N\'eel phase. The gap opens up in the rung-dimer phase; as such, the initial state has little effect on the number of projector steps required to reach the ground state.
To calculate the critical point using the N\'eel order parameter we use the Binder Cumulant defined as:
\begin{equation}
B = \frac{3}{2}\left(1-\frac{1}{3}\frac{\langle M_s^2\rangle}{\langle\vert M_s\vert\rangle^2}\right),
\end{equation}
where $M_s=\sum_{i}(-1)^{x_i+y_i}S^z_i$. In our QMC simulations we
exploit the full $SU(2)$ symmetry of the ground state~\cite{Sandvik2010b,Beach2006}.
On the other hand, the rung-dimer phase characterized by a string order parameter $O$, defined in Eq.(\ref{eq:O}). Much like a correlation function, we observe that in a finite size system with finite $l$, $O_l$ has a non-vanishing value even in the N\'eel phase. We suspect this is due to finite-size effects. In the same spirit as the correlation length discussed in Ref.~\cite{campostrini14} we study the log-ratio between $O_l$ measured at two values of $l$:
\begin{equation}
R = \log\left(\frac{O_{L/2}}{O_{L/4}}\right).
\end{equation}
In order to calculate $O_l$ in our QMC simulations we use the standard estimator calculated in the $S^z$ basis. Because of the non-local nature of this order parameter, the results have more noise compared to $B$.
Before we show the results from our simulations, we discuss how to interpret $R$ as a function of system size $L$. There are few possible outcomes for $R$ in the thermodynamic limit depending on the asymptotic behavior of $O_l$. If $O_l$ tends to a constant as $l \rightarrow \infty$, by definition $R \rightarrow 0$. Otherwise, because of the $\log$, if $O_l$ decays exponentially or faster, then $R$ diverges as $L\rightarrow\infty$. If $O_l$ decays slower than exponential, $R\rightarrow {\rm const}\le 0$, {\it i.e.} for a power-law decay this constant will be negative.
\begin{figure}[t]
\centering
\includegraphics[width=0.55\textwidth]{critical_crossing.pdf}
\caption{Binder cumulant $B$ (solid lines) and the log string-order parameter ratio $R$ (dotted lines) as a function of $\alpha$ for different system sizes. Note the scale for $B$ and $R$ are on the left and right side of the frame respectively. The error bars for both quantities are smaller than the symbols. }
\label{fig:crossing}
\end{figure}
In Fig.~\ref{fig:crossing} we show both $B$ and $R$ as a function of $\alpha$ for various system sizes. We observe that $B$ monotonically increases for decreasing $\alpha$. This behavior indicates the onset of long-range N\'eel order for small values of $\alpha$. On the other hand, $R$, which is always negative by definition, is growing in absolute value as $\alpha$ decreases. For larger values of $\alpha$, in the gapped phase, $R$ tends to $0$ with increasing system size, implying that $O>0$. In the N\'eel phase (small $\alpha$) $R$ converges to a finite value, indicating that $O=0$. The ``steepness'' of the $B$ and $R$ curves increases with increasing system size. This observation is consistent with the finite-size behavior one would expect from a phase transition~\cite{campostrini14}.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{crossing_QMC.pdf}
\caption{Crossing points for both the Binder cumulant $B$ (circles) and the log string-order parameter ratio $R$ (triangles) as a function of $1/L$ for different system sizes. The dashed lines show the extrapolation to the thermodynamic limit and the legend indicates the extrapolated value along with the normalized $\chi^2$ value for the fit.}
\label{fig:critical_alpha}
\end{figure}
To determine the location of the critical point, we look at the crossing points between two curves (either $B$ or $R$) corresponding to different system sizes. Because of FSS corrections, the crossing points will drift closer to the critical point as the system sizes increase~\cite{campostrini14}. Specifically, we look at the crossing points between curves corresponding to system sizes $L$ and $2L$, which we denote as $\alpha^*(L)$.
By construction, both $B$ and $R$ have no multiplicative scaling factor as a function of $L$ in their respective FSS form. As such, it is possible to use them to extract the correlation length exponent, $\nu$, from the following limit:
\begin{gather}
\nu = \lim_{L\rightarrow\infty} \nu^*(L) \\ \nu^*(L) = \left[\log_2\left(\frac{\partial_\alpha Y(\alpha^*(L),2L)}{\partial_\alpha Y(\alpha^*(L),L)}\right)\right]^{-1}\label{eq:nu_estimate}
\end{gather}
where $Y(\alpha,L)$ corresponds to either $B$ or $R$~\cite{campostrini14}. In practice one can only study a finite number of values of $\alpha$, we interpolate those points with a polynomial. Using the interpolation for $L$ and $2L$ we can calculate both $\alpha^*(L)$ and $\nu^*(L)$. To account for the statistical errors coming from the QMC calculations we use the bootstrapping method. This involves drawing a new set of values for $B$ and $R$ from a normal distribution with a mean and standard deviation given by the QMC mean and standard error for each point respectively. After drawing the new points, a polynomial is fitted from which $\alpha^*(L)$ and $\nu^*(L)$ are obtained. This procedure is repeated for many realizations of the data points. From this set of values the mean and standard deviation are calculated. In this work we use $10000$ realizations to generate the mean and standard deviation corresponding to the points and error bars shown in Figs.~\ref{fig:critical_alpha} and \ref{fig:critical_nu}.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{nu_QMC.pdf}
\caption{Extrapolation of the exponent $\nu^*(L)$ for both the Binder cumulant $B$ (circles) and the log string-order parameter ratio $R$ (triangles) as a function of $1/L$ for different system sizes. The legend indicates the extrapolated values for $\nu$ along with the normalized $\chi^2$ values for each fit.}
\label{fig:critical_nu}
\end{figure}
In Fig.~\ref{fig:critical_alpha} we show $\alpha^*(L)$ for both $B$ and $R$ as a function of $1/L$. The curves going through the points depict a power-law fit ($\alpha^*(L) = \alpha_c + b/L^\gamma$) of the finite-size data extrapolated to the thermodynamic limit. We obtain a critical point of $2.520$ (within error bars) for both order parameters. Turning to the correlation exponent $\nu$, in Fig.~\ref{fig:critical_nu} we show $\nu^*(L)$ versus $1/L$ for both $B$ and $R$. The lines in the figure depict the linear fit of the finite-size data extrapolated to the thermodynamic limit. Much like the critical point, both order parameters give a value of $\nu=1.79$. Our numerical results show that both order parameters point to the same critical point with the same correlation length exponent providing strong evidence of a direct transition between the rung-dimer phase and the N\'eel phase. Our next goal is to establish if the transition is continuous or first order. \\
\section{Gap and dynamic exponent $z$}
The dynamic exponent $z$ can provide useful information about the behavior of excitations as well as whether or not the transition is continuous. In order to obtain $z$ we use finite size extrapolations of the spin-triplet gap, calculated using the DMRG method~\cite{White1992,White1993}.
What can make this problem particularly challenging is the possibility of a volume law entanglement law due to the presence of all-to-all interactions. However, in the gapped phase, the correlation length remains finite and the entanglement remains under control. Surprisingly, the entanglement entropy does not grow dramatically in the gapless phase and across the transition. This may appear to be a general feature of one-dimensional models with long-range interactions as has been observed in quantum spin chains, which display a $log(L)$ behavior\cite{Ren2020, Gong2017, Lerose2020,Yang2021}. The main numerical cost lies on the fact that the number of terms in the Hamiltonian grows quadratically with system size. In the calculations presented here we have studied ladders of size $L \times 2$ sites with $L$ up to 48, with open boundary conditions and adjusting the bond dimension such that the truncation error is kept under $10^{-7}$, translating into up to 1000 states.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{gap_close.png}
\includegraphics[width=0.45\textwidth]{z_vs_al_new.png}
\caption{(left) Gap extrapolated to the thermodynamic limit as a function of $\alpha$. The inset shows the finite-size gaps, obtained with DMRG, used in the extrapolation of $\Delta_0$ with second order polynomial fit. (right) Dynamic critical exponent $z(L)$ as a function of $\alpha$ for various system sizes. The circles correspond to $z(L)$ calculated using the gaps obtained from DMRG. The curves are the best fit using fourth order polynomials.}
\label{fig:fig_gap_close}\label{fig:fig_z_alpha}
\end{figure}
As discussed in the previous subsection, in the limit $\alpha \rightarrow \infty$ the problem reduces to the conventional Heisenberg ladder Hamiltonian with nearest neighbor interactions. As the value of $\alpha$ is decreased, the antiferromagnetic correlations are enhanced and the gap is reduced. In the left panel of Fig.~\ref{fig:fig_gap_close} we show the behavior of both the gap extrapolated to the thermodynamic limit as a function of $\alpha$. To carry out the extrapolation to the thermodynamic limit we use a second order polynomial fit of finite-size data as a function of $1/L$, as shown in the inset. We do not see a closing of the gap at $\alpha=2.520$ due to the strong sublinear scaling behavior that introduces corrections that require larger system sizes (as we describe below, a power-law extrapolation is ill-conditioned for this small dataset). We point out that the upturn of the curve for small $\alpha$ is likely an artifact of the extrapolation that becomes less reliable as the spectrum becomes more singular at the ordering wave vector. Even though, once the system orders, the spectrum is expected to remain gapless, we do not discard the possibility of a gap reopening for small $1<\alpha<2$, since the long range interactions violate Goldstone's theorem hypotheses and symmetry breaking could be accompanied by a gap \cite{Morchio1985,Strocchi_book, Auerbach_book} (we discuss this point in more detail in the Conclusions).
Given the power-law nature of the interactions, the finite-size effects are much stronger compared to a local Hamiltonian making the dynamic exponent difficult to extract when fitting the gap directly. We can account for these corrections by expressing the gap as:
\begin{equation}
\Delta (L) = aL^{-z}(1+f_\Delta (L)).
\end{equation}
Here we include all finite-size corrections in $f_\Delta (L)$ such that, in the thermodynamic limit, $f_\Delta(L)\rightarrow 0$. Instead of fitting the gap directly, we can define an approximation of the dynamic exponent for a finite-size system by calculating the log of the ratio of the gap between system sizes $L$ and $2L$,
\begin{equation}
z(L)\equiv\log_2\left(\cfrac{\Delta (L)}{\Delta (2L)}\right) = z + \log_2\left(\cfrac{1+f_\Delta(L)}{1+f_\Delta(2L)}\right). \label{eq:z_L}
\end{equation}
When $L\rightarrow\infty$ the second term on the right side will vanish. Using $z(L)$ allows one to directly extrapolate the dynamic exponent removing any bias in trying to guess the functional form of finite-size corrections.
At a transition between a gapless and gapped phase, $z$ will have a discontinuous jump at the critical point from $0$ to a finite value, much like the Binder cumulant for an order parameter. For finite-size systems the non-analytic behavior becomes smooth but the evidence of this discontinuity becomes more pronounced as system sizes become larger. As a result, the crossing points between any two finite-size curves will converge to the exact critical point in the limit where both system sizes go to infinity, giving us another independent method to calculate the location of the critical point. The behavior of the finite-size curves in the right panel of Fig.~\ref{fig:fig_z_alpha} indicate that the dynamic exponent is going to $0$ above $\alpha_c$, while approaching a value larger than $0$ below the transition.
We can also use $z(L)$ to determine whether this transition is first or second order. In the continuous case, $z(L)$ at the critical point will tend towards a finite value in the thermodynamic limit while for a first order transition $z(L)$ will tend to infinity. There is no indication in our results of a divergence in the dynamic exponent for any of the values of $\alpha$ we looked at, thus providing evidence for a second order transition.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{alpha_c.pdf}
\includegraphics[width=0.48\textwidth]{z_plots.pdf}
\caption{(left) Extrapolation of crossing point of the curves in the right panel of Fig.~\ref{fig:fig_z_alpha} to the thermodynamic limit using power-law ({\it e.g.} $\alpha_{\rm cross}(L)=\alpha_{c}+b/L^\beta$) and quadratic fits shown in the plot as orange and green lines respective. (right) Extrapolation of $z(L)$ values at the crossing points with a power-law and quadratic fits shown in the plot as orange and green lines respective. The error bars for each point are calculated from the co-variance matrix of the polynomial interpolations and the error bar for the critical point and dynamic exponent in the figure have been calculated the bootstrap method. The $\chi^2$ value for the fit is shown in the figures. We have normalized it by the number of left-over degree's of freedom, in this case is one. }
\label{fig:alpha_crit}\label{fig:z_crit}
\end{figure}
Using the crossing points between system sizes $L$ and $L+4$, we can estimate the critical point by extrapolating them to the thermodynamic limit as a function of $1/L$. We show the results for our extrapolation in the left panel of Fig.~\ref{fig:alpha_crit}. To fit the finite-size data we use a power-law ({\it e.g.} $y=y_0+b/L^\beta$), and a quadratic fit. We find that the critical point is around $2.5$ based on the two different extrapolation methods.
It is worth noting that a value of $z=1$ would indicate the possibility of an underlying conformal invariance and, consequently, a deconfined quantum critical point. To determine the value of $z$ at the critical point one can extrapolate the values of $z(L)$ at the crossing points just as we did for the Binder cumulant~\cite{campostrini14}. The results are shown in the left panel of Fig.~\ref{fig:z_crit}. Unlike in the critical point estimate, the extrapolated values differ significantly between the power-law and the quadratic extrapolations indicating there are larger finite-size corrections to this quantity. In this case, we cannot provide an accurate estimate for the critical exponent.
\section{Spin dynamics}
\label{sec:dynamics}
In order to calculate the spin dynamic structure factor we used the time-dependent DMRG method (tDMRG) \cite{white2004a,daley2004}, a well established technique described in detail in the original work Refs.\cite{white2004a,Feiguin2005} and reviews Refs.\cite{vietri,Paeckel2019}. The longitudinal two-time spin-spin correlation function is defined as:
\begin{equation}
\langle S^z_r(t)S^z_0(0) \rangle = \langle \psi_0\vert e^{iHt}S^z_r e^{-iHt}S^z_0\vert\psi_0 \rangle,
\label{sqw}
\end{equation}
where we take $S^z_0$ at the center of the one of the legs of the ladder, and $r$ is the distance from the middle. The spectral function is obtained by Fourier transforming from real space and time to momentum space and frequency. This procedure is carried out over a finite time range (in our case $t_{max}=10$). For this reason, the poles in the spectral function will not be well defined deltas, but will display artifacts such as artificial ringing that can be attenuated by means of standard windowing techniques also used in signal processing. As a consequence, the width of the spectral features will be inversely proportional to the width of our time window.
Due to the long-range nature of the terms in the Hamiltonian, we employ a time-step targeting procedure with a Krylov expansion of the time-evolution operator \cite{Feiguin2005}. We fixed the time step $\delta t=0.05$ (measured in units of $J^{-1}$). We fixed the maximum truncation error to $10^{-7}$ in the time range considered. All results shown here are for a relatively small ladder of length $L=20$. Unlike the ground-state calculations, the entanglement entropy grows rapidly in time, together with the number of states required to keep the error under bounds which can be as large as $m=1500$. In addition, as mentioned before, the number of terms in the Hamiltonian makes the time evolution very time consuming.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{sk_dmrg_4.png}
\caption{Longitudinal dynamic spin structure factor $S(k,\omega)$ for a $20\times 2$ ladder with long range interactions and different exponent $\alpha$ across the quantum critical point, obtained with tDMRG. Upper(lower) row shows the symmetric (antisymmetric) channel. Ringing at high energies is due to the finite time integration window (see text). Also shown are the linear spin-wave dispersion and results for the conventional ladder with only nearest-neighbor terms, $\alpha=\infty$.}
\label{fig:sqw}
\end{figure}
Our results for the longitudinal spin dynamic structure factor are shown in Fig.~\ref{fig:sqw}, for both the symmetric ($k_y=0$) and antisymmetric ($k_y=\pi$) channels, together with the linear spin-wave (SW) dispersion. We show a similar color density plot in Fig.~\ref{fig:sqw_color} focusing on the antisymmetric sector with $k_y=\pi$. Notice that the SW results agree very well with the DMRG data in the gapless phase, but as the gap open, the differences become more obvious, since spin-wave theory cannot describe the gapped phase of the Heisenberg ladder\cite{Barnes1993}. Elementary excitations on a two leg ladder are conventionally understood as rung triplons: a spin will pair with another one on the opposite leg forming a triplet excitation that costs an energy $\Delta \sim J$. The energy is lowered by propagating the triplet via spin-flips, in what can be qualitatively interpreted as a hard-core boson moving in a vacuum of rung-singlets. For $\alpha > \alpha_c$ we observe a gapped coherent band in the symmetric channel with vanishing spectral weight around $k_x=0$, since $S^z_{total}=0$. The antisymmetric channel presents coherent features at high energies, but the spectrum broadens as the momentum approaches $\vec{k}=(\pi,\pi)$ (this is more clearly seen in Fig.~\ref{fig:sqw}(g)).
As the value of $\alpha$ is reduced and approaches the quantum critical point, the two dispersive branches condense at $\vec{k}=(0,0)$ and $(\pi,\pi)$, respectively. The excitations display a sharp elastic peak at the ordering vector $(\pi,\pi)$, and we can observe how the bandwidth increases.
Interestingly, the width of the continuum in the symmetric $k_y=0$ channel seems to get smaller as we approach $\vec{k}=(0,\pi)$ and both the magnon band and the spinon continuum seem to merge into a single sharp coherent dispersion. We also notice that the high energy features near the center of the Brillouin zone evolve adiabatically and are insensitive to the phase transition. It is thus reasonable to assume that in this region, magnons and triplons do not differ qualitatively. In fact, the same could be said about the symmetric branch, and the main distinction becomes question of semantics: in one case they are gapless, and in the other gapped, but otherwise, they are both interpreted as bound states of spinons.
In Fig.~\ref{fig:sqw_color} we observe a very sharp peak at the ordering vector and a range of intermediate values of energies with little spectral weight below what looks like a separate branch. Since spin-wave theory is expected to work in the ordered phase, there is in principle no reason to expect two dispersive branches. Another, more reasonable possibility, is that in reality the space between the upper coherent band and the large elastic peak is occupied by an incoherent continuum with very small spectral weight, but finite size effects should not be discarded. Unfortunately, our limited resolution and the sublinear dispersion with a large slope near $\omega=0$ prevent us from fully answering this issue.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{skw2.png}
\caption{Longitudinal dynamic spin structure factor $S(k,\omega)$ with $k_y=\pi$, for a $20\times 2$ ladder with long range interactions and different exponent $\alpha$ across the quantum critical point, obtained with tDMRG. }
\label{fig:sqw_color}
\end{figure}
\section{Summary and Conclusions}
\label{sec:conclusions}
Our numerical evidence points at a second order phase transition at $\alpha_c \sim 2.5$ from a gapped, magnetically disordered rung dimer phase with triplon excitations, to an antiferromagnetic phase with long range order and magnon excitations. However, the possibility of a weak first order transition should not be discarded. Our results in Fig.~\ref{fig:fig_gap_close} are conspicuous enough to grant the question: is there a gap opening for $\alpha < \alpha_c$? If we trust that our extrapolation to the thermodynamic limit is indeed within error bars, this is definitely possible. In the quantum magnetism folklore, it is assumed that symmetry breaking is directly associated with the presence of gapless Goldstone modes\cite{Anderson1952,Anderson1984,LhuillierLectures,Zhitomirsky2013}. However, it is easy to see that in the case of $\alpha=0$ our model would realize symmetry breaking, but also that the energy would become superextensive, with a huge gap to the first excitation proportional to the system size\cite{Yusuf2004}. The presence of a gap in systems with long range interactions should not come as a surprise; after all, Goldstone's theorem relies on the condition that the Hamiltonian is relatively local, with short range interactions (rigorously speaking, the soft modes should no longer be referred-to as ``Goldstone modes'' in the presence of non-local interactions). In addition, the assumption that the spin-wave dispersion should be linear is no longer valid in our case.
While triplons are intuitively easy to visualize as rung triplet excitations that propagate coherently, spin-waves are rather understood as fluctuations of the order parameter around a symmetry broken ordered state\cite{Anderson1952,Anderson1984}. In the case of the pressure induced transition in TlCuCl$_3$, the triplon excitations condense at the transition and become gapless spin waves on the ordered phase \cite{Lohofer2015} and excitations remain coherent throughout the transition. However, we notice that the ``disordered'' phase of our ladder system realizes hidden topological order characterized by a non-local string order parameter.
Thus, one question that emerges from our studies is whether deconfined criticality can be realized or not \cite{Senthil2004a,Senthil2004,Ashvin2004,Senthil2005,Sachdev2008}: while Landau's arguments forbid a direct second order phase transition between phases with order parameters that describe different symmetries, it is possible that in certain cases the transition could be continuous and that, when this occurs, quasiparticle excitations would not be well defined at the critical point, with the spectrum displaying a broad incoherent continuum. While these arguments rely on a direct transition between two ordered phases with incompatible local ordered parameters, in our case one of the phases has topological order.
In our results, the peculiar features observed in the spectrum around $\vec{k}=(\pi,\pi)$ offers suggestive evidence of deconfined excitations, possibly in terms of spinons that carry spin $S=1/2$\cite{Ma2018}. A deconfined critical point would also be characterized by a critical exponent $z=1$, but our results are not conclusive. It is natural to ask whether the critical point can be identified with a conformal field theory or a new kind of criticality, but we do not have enough information to answer this question, since the algebra is not well defined in a finite volume because the theory is non-local. Quantum criticality connecting a topological ordered phase and a conventional Landau ordered phase could represent a new paradigm in our understanding of quantum phase transitions.
In systems with long range interactions one typically finds sublinear dispersion with $z < 1$ \cite{Laflorencie2005,Hauke2013,Nicolo17,Frerot2018,Kuwahara2020,Tran2020,Chen2019,Cevolani2015,Cevolani2018,schneider2020spreading}. However, the anti-ferromagnetic transverse field Ising chain with long-range interactions shows critical exponents that correspond to the standard 1D transverse field Ising chain indicating the possibility of a CFT critical point in a long-range interacting model~\cite{Puebla2019}.
More work needs to be done in order to establish the universality class of the transition.
\section{Acknowledgments}
LY and AEF acknowledge support from the National Science Foundation under grant No. DMR-1807814. The authors thank S. Sachdev, A. W. Sandvik, E. Katz, L. Manuel, and A. Trumper for useful discussions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,071
|
jest.dontMock("../sprint");
describe('before', function() {
it('should do something', function() {
var $ = require("../sprint.js");
// code...
});
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,025
|
package org.kie.workbench.common.stunner.core.client.components.palette.model.definition.impl;
import java.util.List;
import org.kie.workbench.common.stunner.core.client.components.palette.model.AbstractPaletteDefinition;
import org.kie.workbench.common.stunner.core.client.components.palette.model.definition.DefinitionPaletteItem;
import org.kie.workbench.common.stunner.core.client.components.palette.model.definition.DefinitionsPalette;
public final class DefinitionsPaletteImpl
extends AbstractPaletteDefinition<DefinitionPaletteItem>
implements DefinitionsPalette {
protected DefinitionsPaletteImpl(final List<DefinitionPaletteItem> items,
final String defSetId) {
super(items,
defSetId);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,191
|
Nonprofit Spotlight | The LEAD Project and "A Toast to LEAD" Benefit
Nonprofit Spotlight| The LEAD Project and "A Toast to LEAD" Benefit
By YNPN Twin Cities Cities
An Interview with Peter Wagner, LEAD Board Member
What is The LEAD Project?
LEAD was founded in 2006 and stands for the Leadership Emergence and Development Project. Our goal is to engage young professionals in the charitable and philanthropic communities of the Twin Cities.
How do you achieve your goal?
LEAD's goal, and our mission, is to engage young professionals in the Twin Cities nonprofit community. We host various events that are all aimed at creating relationships between the young professional community and the nonprofit community through skills-based volunteer opportunities. We host several different types of events including Board Bootcamps, PhilanthroFairs and large scale charitable events. To learn more about our events click here.
What has been your impact on our local community?
We estimate that LEAD has helped connect over 250 young professionals with Board and other skills-based volunteer positions at local nonprofits. We've helped educate over 1000 young professionals on the expectations of being a nonprofit Board of Director through our Board Boot Camps. We have held 13 large charitable events and as a result have donated over $130,000 back to nonprofits. We have also partnered with over 50 nonprofits in the past two years through our PhilanthroFairs. But we believe our true impact is making philanthropy accessible to young professionals and maximizing the amount that they contribute by donating their time in meaningful ways.
Tell us about the upcoming LEAD Event.
A Toast to LEAD celebrates our five year anniversary and the 50+ nonprofits we have partnered with.
When: Saturday Feb. 18 (8 pm- 12 am)
Where: Le Meridien Chambers
An all-night open bar with a signature drink
Separate party rooms/floors with two DJs
Access to Le Meridien Chamber's Ice Bar
Premier silent auction
A special tribute to all the nonprofits we have partnered with over the past five years
The chance to vote for which nonprofit the money we raise should go towards including; Project Success, World without Genocide, People Serving People, Make-a-Wish Foundation and Bolder Options.
Cost: $75 per ticket through Thursday, February 16th (so don't delay!), then $95 per ticket
Tickets: www.theleadproject.org
For more information about The LEAD Project visit: www.theleadproject.org
Tags: News Interview Nonprofit Spotlight Peter Wagner
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,540
|
{"url":"https:\/\/zbmath.org\/?q=an%3A0930.34025","text":"## Oscillation and nonoscillation criteria for two-dimensional systems of first-order linear ordinary differential equations.(English)Zbl\u00a00930.34025\n\nThis article contains oscillation and nonoscillation criteria for the linear system $u'= p(x)v,\\quad v'=- q(x)u,\\quad x\\in\\mathbb{R}_+,\\tag{1}$ with $$p,q\\in L^1_{\\text{loc}}(\\mathbb{R}_+, \\mathbb{R}_+)$$, $$\\int^\\infty p=+\\infty$$, and $$\\int^\\infty q<\\infty$$. The two main theorems establish sufficient conditions of Hille\u2019s type for all nontrivial solutions to (1) to be oscillatory at $$\\infty$$ (componentwise). Additional results are nonoscillation criteria for (1), in which $$q(x)$$ is permitted to change sign. The theorems generalize classic ones of E. Hille [Trans. Am. Math. Soc. 64, 234-252 (1948; Zbl\u00a00031.35402)] and Z. Nehari [Trans. Am. Math. Soc. 85, 428-445 (1957; Zbl\u00a00078.07602)]. Analogues for linear and semilinear scalar differential equations were obtained by A. Lomtatidze [Arch. Math., Brno, 32, No. 3, 181-193 (1996; Zbl\u00a00908.34023); Georgian Math. J. 4, No. 2, 129-138 (1997; Zbl\u00a00877.34029)].\n\n### MSC:\n\n 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34A30 Linear ordinary differential equations and systems\n\n### Keywords:\n\noscillation; nonoscillation; linear system\n\n### Citations:\n\nZbl 0031.35402; Zbl 0078.07602; Zbl 0908.34023; Zbl 0877.34029\nFull Text:","date":"2022-06-30 07:36:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.4137912392616272, \"perplexity\": 2549.6241674147414}, \"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\/1656103669266.42\/warc\/CC-MAIN-20220630062154-20220630092154-00126.warc.gz\"}"}
| null | null |
Q: Accidentally created another root user. How to delete or change permissions? I need a non-root user to do certain (safe) things. I didn't pay attention, and used useradd -ou 0 -g 0 new_user_name, which gave root permissions to the user. It also has the same UID of 0.
Before I do something stupid, can anyone help me on how to delete this user, or simply change the permissions/UID? I already discovered what happens with pkill. Hahaha.
A: userdel has an option -f to force removal. userdel would probably just delete the passwd entry and home directory, without affecting the actual root account.
To be safer, I might be inclined to hand-edit the password file to remove the entry for your new root user, then hand-remove new root user's home directory. You may have a command on your system named nano, which lets you safely edit /etc/passwd in a text editor.
Source: https://superuser.com/questions/804210/how-can-i-delete-root-users
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,235
|
Q: Mathematical Induction proof classical logic tautologies I need to prove through mathematical induction that a logic L (Łukasiewicz) does not contain all tautologies of classical logic. I get that to start you show the contrapositive, and then start with base cases propositions with connectives. But I have no idea how to show why mathematical induction does not work for Łukasiewicz logic...
A: To prove the claim you only need to present some tautology of classical logic, which is not a tautology of Łukasiewicz. Such an example can be the law of excluded middle $A\vee\neg A$ or the contraction axiom $(A\to(A\to B))\to (A\to B)$. Why would someone want to use an inductive argument here?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,626
|
package com.sectong.config;
/**
* 常量
* @author ScienJus
* @date 2015/7/31.
*/
public class Constants {
/**
* 存储当前登录用户id的字段名
*/
public static final String CURRENT_USER_ID = "CURRENT_USER_ID";
/**
* token有效期(小时)
*/
public static final int TOKEN_EXPIRES_HOUR = 72;
/**
* 存放Authorization的header字段
*/
public static final String AUTHORIZATION = "authorization";
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,003
|
A Jobs Marketplace App for Employees and Employers
Industry : Jobs Marketplace
Use2Connect is an on-demand service app that allows service providers to find relevant work opportunities (according to their skills) by connecting with various employers. As of now, Use2Connect has helped many users utilize their skills to find new projects and jobs that relate to their niche. However, there was a time when Use2Connect was just a simple idea and Sascha Reinhard, the CEO of Use2Connect, was looking for a reliable development team to turn this idea into a fully functional app.
Before joining hands with RV Technologies, Reinhard interviewed many development teams, but none of them met his requirements. Our development team, on the other hand, was highly intrigued by the idea and asked many queries related to the app's scope. Reinhard was quite impressed by the curiosity of our developers and instantly decided to hire RV Technologies as his development partner.
With Use2Connect, the primary objective was to build a cross-platform mobile app that would allow users to list their projects/job requirements and find skilled service providers accordingly. Similarly, a service provider (be it a developer or designer) could see the projects (listed by other users) and apply for different projects according to their skills. The idea was to make job hunting much easier for independent workers while allowing project owners to find skilled individuals for their projects.
As soon as our development team jumped in and analyzed Reinhard's vision, they could foresee a few challenges with the development process. First of all, we knew that the mobile app market was stacked with similar on-demand service apps. So, our designers and developers needed to follow a completely new approach to easily build a distinctive app that would easily attract the target audience.
Secondly, Reinhard wanted to integrate a subscription model to Use2Connect, which meant we had to implement the right security mechanisms to completely secure the monetary transactions.
Did RV Technologies Succeed in Developing an On-Demand Service Platform for Sascha Reinhard?
After analyzing the challenges, our designing team took over the reins and designed the initial wireframes for different modules of the app. We chose Flutter as the front-end framework for Use2Connect as it would allow us to deploy the same codebase to create native apps for both Android and iOS.
Meanwhile, our development team was working on the backend. Since security was one of the major requirements of Use2Connect, we used Laravel to build and manage the backend for the app.
So, after the UI was ready, we integrated the following features to add the desired functionality to Use2Connect.
Single User Account
AWS Integration
Tile-Based Home UI
Project Listings
1 Single User Account
Even though Use2Connect was targeted towards both service providers and project owners, we designed a single-user login to keep the log-in process as convenient as possible. we designed a static UI that would allow users to switch between consumer and service provider profiles easily.
2 AWS Integration
To help Use2Connect interact with the registered users more effectively, we also integrated AWS that would allow the admin to configure automatic emails and send them to all the users easily.
3 Tile-Based Home UI
Since Reinhard wanted to keep the "profile switch" as smooth as possible, we opted for a tile-based UI for Use2Connect's home. We added three titles - News, Job Inquiries, and Your Projects. By tapping the "Job Inquiries" tile, the user would be prompted to all the inquiries that he/she has received from companies/employers.
4 Chat Support
With Use2Connect, we also integrated Flutter's built-in Chat API and customized it to make the app more usable. With the "Chat" menu, both employers and service providers could easily communicate with each other and discuss the project status whenever they want.
5 Project Listings
As soon as the employers would list a dedicated project on their profile, all the service providers would see it on their profile. The project owners could add projects under different categories (that are pre-defined by the admin himself). Based on their skills and past experience, the service providers would be able to apply for a dedicated job.
The project owner would receive all the job requests from different service providers. To enhance the overall user-friendliness of Use2Connect, we integrated a simple "Swipe Left/Right" feature via which the project owners could select or reject a candidate.
6 Push Notifications
For any on-demand service app, push notifications are an integral feature. Our developers designed a dedicated "Notification Panel" where the users (both employers and service providers) would see the pending notifications. Here the user could also turn off the push notification feature by toggling the switch if they don't want to get disturbed for a while.
How Use2Connect Became One of the Top On-Demand Service Apps for Employers & Service Providers?
Thanks to the continuous efforts of our developers, we were able to develop a feature-packed job marketplace for Sascha Reinhard within the 10 weeks timeframe.
As of now, Use2Connect has a 4.5-star rating on both Google Play Store and Apple's App Store. The app has already helped thousands of users find relevant work opportunities based on their skills.
Even today, RV Technologies continues to offer support and maintenance services to the Use2Connect mobile app. Our vision is to keep integrating new features to make job-hunting a hassle-free task for the users.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,876
|
\section{Introduction
\label{sec: Introduction}
}
\noindent
Nuclear reactions mediated by weak interactions are central to a variety of frontier problems in nuclear and astrophysics as well as high-energy physics. Single-weak-current processes like $pp$-fusion and (anti)neutrino-deuteron scattering are two prominent examples. The former is a critical process in understanding the energy production mechanism in a range of stars~\cite{Adelberger:2010qa}, and the latter is used to probe properties of neutrinos in several neutrino experiments~\cite{Aharmim:2011vm,Fukuda:2001nj,Fukuda:2002pe}. At the next order in weak currents, double-$\beta$ decay transitions are of major importance. Two important modes of this transition are two-neutrino double-$\beta$ ($2\nu\beta\beta$) decay and neutrinoless double-$\beta$ ($0\nu\beta\beta$) decay. The former process conserves the total lepton number~\cite{GoeppertMayer:1935qp}, and is the rarest Standard Model (SM) process that has been measured~\cite{Barabash:2020nck}. Besides providing insights into the SM weak interactions and nuclear structure, $2\nu\beta\beta$ decay can also shed light on potential beyond SM scenarios~\cite{Deppisch:2020mxv}. The $0\nu\beta\beta$ mode is forbidden in the SM as it changes the lepton number by two units, and if observed, would indicate that neutrinos are of Majorana type~\cite{Schechter:1981bd}. An extensive experimental program continues to seek evidence for $0\nu\beta\beta$ decays~\cite{Bilenky:2014uka, DellOro:2016tmg,Biassoni:2020byh,Dolinski:2019nrj,Cappuzzello:2018wek,Cappuzzello:2016zlj, Bilenky:2014uka,Bilenky:2020wjn}. However, the new-physics implications of the current and the future double-$\beta$ decay measurements are limited by the uncertainties in the theoretical predictions of their decay rates.\par
A major source of uncertainty in calculating the decay rate of weak processes is the matrix elements (MEs) of weak currents between the initial and final nuclear states. For energies well below the pion mass, $m_{\pi}$, that is often relevant for single-weak processes in the few-nucleon sector, pionless EFT~\cite{Kaplan:1998tg,Kaplan:1998we,vanKolck:1998bw,Bedaque:1997qi,Bedaque:1998mb,Bedaque:1999vb,Chen:1999tn} accurately describes the dynamics, see Ref.~\cite{Hammer:2019poc} for a review. For the double-$\beta$ decays that naturally occur in large nuclear isotopes, the corresponding nuclear-ME calculations suffer from uncertainties that stem from both approximations in quantum many-body methods as well as uncertainties in (multi)nucleon interactions and weak currents~\cite{Engel:2016xgb,Giuliani:2012zu}. The latter can be mitigated by improving the accuracy of MEs in the two-nucleon (NN) sector using an effective Lagrangian along with a power-counting scheme, and then using them as an input in an \textit{ab initio} framework to calculate the many-body MEs for larger nuclei~\cite{Coraggio:2020iht,Engel:2016xgb}. The NN transitions between the two-neutron initial state, $nn$, and the two-proton final state, $pp$, are not observed in free space, but they occur as off-shell subprocesses in transitions of larger nuclei. The typical Fermi momentum of nucleons in these nuclei is comparable to $m_{\pi}$, but at a first approximation, the pionless EFT is expected to provide a good description. Subsequently, the effect of pions can be included systematically using pionfull EFT~\cite{Kaplan:1998tg,Kaplan:1998we} or chiral nuclear EFTs~\cite{Weinberg:1990rz, Weinberg:1991um, Machleidt:2011zz}.\par
For SM processes involving more than two nucleons, the nuclear MEs of isovector axial-vector currents corresponding to Gamow-Teller transitions are not constrained precisely in pionless EFT. This is in part due to a large uncertainty on the renormalization-scale ($\mu$) dependent LEC $L_{1,A}$ that contributes at the next-to-leading order (NLO) and determines the strength of the momentum-independent isovector axial-vector two-body current~\cite{Kong:1999tw,Butler:1999sv,Butler:2000zp}. While constituting only a few percent of the total amplitude, the contribution to the Gamow-Teller transitions from the $L_{1,A}$ term remains the dominant source of uncertainty in determining the decay rate of processes such as $pp$ fusion in Sun and similar stars~\cite{Adelberger:2010qa}. The value of $L_{1,A}$ determined from experimental data has improved over the years~\cite{Chen:2002pv, Butler:2002cw, Butler:2000zp, Chen:2005ak, De-Leon:2016wyu}, with the most recent constraint given by\footnote{Throughout this work, values of $\mu$-dependent LECs are given at $\mu=m_{\pi}$.} $L_{1,A}= 4.9 ^{+1.9}_{-1.5}\text{ fm}^3$~\cite{Acharya:2019fij}, which has a significant uncertainty. On the other hand, no experimental constraint exists on the nuclear ME of $0\nu\beta\beta$ decay transition due to lack of observation. Furthermore, recent analyses in the light neutrino exchange scenario of the $0\nu\beta\beta$ decay transition in the two-nucleon sector, i.e. $nn \to pp e^- e^-$, have shown that the corresponding nuclear ME is unknown even at the leading order (LO) in pionless EFT~\cite{Cirigliano:2017tvr,Cirigliano:2018hja,Cirigliano:2019vdj}. In fact, a new $\mu$-dependent LEC, $g_\nu^{NN}$, is needed at LO for the decay amplitude to be manifestly renormalizable. Recently, an indirect estimate of $g_\nu^{NN}$ was obtained in Refs.~\cite{Cirigliano:2020dmx,Cirigliano:2021qko}: $\widetilde{g}_{\nu}^{NN} = 1.3 \pm 0.6$,
where $\widetilde{g}_{\nu}^{NN}$ is a dimensionless parameter related to $g_{\nu}^{NN}$. Subsequent analyses using this value showed that the missing short-range contribution to the nuclear ME of various candidate nuclei is comparable to the rest of the contributions~\cite{Wirth:2021pij,Jokiniemi:2021qqv}. This indicates the importance of improving the constraint on $g_\nu^{NN}$, preferably using a first-principles approach such as LQCD.\par
A direct way of constraining nuclear MEs is to solve the underlying short-distance theory of quantum chromodynamics (QCD) numerically using the technique of LQCD~\cite{Davoudi:2020ngi,Briceno:2014tqa,Cirigliano:2019jig,Kronfeld:2019nfb,Drischler:2019xuo,Cirigliano:2020yhp}. LQCD was in fact used in Ref.~\cite{Savage:2016kon} to constrain $L_{1,A}$ from the relevant LQCD three-point correlation functions albeit at unphysical quark masses corresponding to $m_{\pi}\approx 806$ MeV, see also Ref.~\cite{Detmold:2021oro}. The obtained value of $L_{1,A}=3.9(1.4) \text{ fm}^3$ at the physical quark masses required an uncertain quark-mass extrapolations but found to be comparable to experimental constraints with similar uncertainties. On the other hand, no LQCD determination of the $g_\nu^{NN}$ coupling is yet reported although progress in simpler $0\nu\beta\beta$ processes in the pion sector is being made in recent years~\cite{Feng:2018pdq, Tuo:2019bue, Detmold:2020jqv, Nicholson:2018mwc}. In LQCD, the QCD action is defined on a finite spacetime grid with a Euclidean time, and the $n$-point correlation functions are computed using Monte Carlo methods. A formalism for obtaining two-hadron scattering amplitudes from finite-volume (FV) Euclidean correlation functions was introduced by L\"uscher~\cite{Luscher:1986pf,Luscher:1990ux} and extended to other systems in Refs.~\cite{Rummukainen:1995vs, Beane:2003da, Kim:2005gf, He:2005ey, Davoudi:2011md, Leskovec:2012gb, Briceno:2012yi, Hansen:2012tf, Gockeler:2012yj, Briceno:2013lba, Feng:2004ua, Lee:2017igf, Bedaque:2004kc, Luu:2011ep, Briceno:2013hya, Briceno:2013bda, Briceno:2014oea,Polejaeva:2012ut,Briceno:2012rv, Hansen:2014eka, Hansen:2015zga, Hammer:2017uqm,Hammer:2017kms,Guo:2017ism,Mai:2017bge}. The formalism for obtaining transition amplitudes of processes involving external currents was first developed by Lellouch and L\"uscher~\cite{Lellouch:2000pv} and later generalized in Refs.~\cite{Briceno:2012yi,Christ:2005gi,Meyer:2011um,Bernard:2012bi,Beane:2014qha,Detmold:2004qn, Meyer:2011um, Briceno:2012yi, Bernard:2012bi, Briceno:2014uqa, Feng:2014gba, Briceno:2015csa, Briceno:2015tza, Hansen:2021ofl}. For the hadronic MEs involving long-range processes, the generalization of the above mappings resolves an additional complexity arising from the relative time between the two hadronic currents~\cite{Shanahan:2017bgi,Tiburzi:2017iux,Feng:2018pdq,Tuo:2019bue,Briceno:2019opb,Detmold:2020jqv,Feng:2020nqj,Davoudi:2020xdv,Davoudi:2020gxs,Christ:2012se,Christ:2020hwe}. Recently, we applied this formalism to single- and double-$\beta$ decays in the NN sector to obtain the needed matching relations that constrain the $L_{1,A}$ and $g_\nu^{NN}$ LECs from the LQCD output~\cite{Davoudi:2020xdv,Davoudi:2020gxs}.\par
Given the complexity of the matching relations involved, it is not immediately obvious what the precision requirement of the upcoming LQCD studies at the physical quark masses should be to reach the precision goal of the LECs, that is to be compatible or superior to phenomenological constraints. In particular, it is important to ask if anticipated uncertainties on the lowest-lying FV energies and on the MEs, as well as achievable physical volumes in LQCD, will guarantee precise determinations of LECs such as $L_{1,A}$ and $g_\nu^{NN}$. As a result, in this paper we embark on an investigation based on synthetic data to determine the sensitivity of the output of the matching relations (hence the LECs) along with their uncertainties on the values and uncertainties of the input to these relations, namely the LQCD energies and MEs. This also allows determining the range of volumes which leads to better constraints, hence guiding future LQCD calculations on their resource planning. This follows the spirit of Ref.~\cite{Briceno:2013bda} which demonstrated that a precise determination of the small S-D mixing parameter in the deuteron channel from LQCD is achievable in future LQCD calculations of the lowest-lying spectra of NN systems in boosted frames. This investigation, furthermore, aligns with recent valuable analyses of the sensitivity of nuclear spectra and MEs to the uncertainties in the input LECs of interactions and currents, when those uncertainties are propagated through \emph{ab initio} many-body calculations~\cite{Ekstrom:2019lss}.
\begin{figure}[t]
\centering
\includegraphics[scale=0.9]{flowchart.pdf}
\caption{The procedure used to perform the sensitivity analysis of $L_{1,A}$ and $g_\nu^{NN}$. The sequence of steps followed is indicated by the numbers enclosed in the circles. The LECs $L_{1,A}$ and $g_\nu^{NN}$ are represented by a crossed circle and a solid diamond, respectively. The wavy line denotes external leptons from a single weak-current insertion. A nucleon is denoted by the small solid circle in the diagrams for NN processes in infinite volume. Dotted lines in the NN energy spectrum in a finite volume are the excited-state energies, and the ground state energy, $E_0$ ($\widetilde{E}_0$), in the spin-singlet (spin-triplet) channel is denoted by the solid line. The FV nuclear MEs for the decay transitions are represented by large solid circles enclosed in dotted cubes with one and two weak-current insertions, respectively. The solid line denotes the FV nucleon state. The simulation of LQCD uncertainties using Gaussian fluctuations and uncertainty analysis of LECs from the synthetic data is discussed in Secs.~\ref{sec: L1A} and~\ref{sec: gvNN}. \label{fig: flowchart}}
\end{figure}
Explicitly, we consider the single-$\beta$ decay (Sec.~\ref{sec: L1A}) and $0\nu\beta\beta$ decay (Sec.~\ref{sec: gvNN}) transitions in the two-nucleon sector: $nn \to np e^-\bar{\nu}_e$ and $nn \to pp e^-e^-$, respectively. First using the L\"uscher's quantization condition (QC), the low-energy spectra of NN systems in a range of spatial cubic volumes with periodic boundary conditions (PBCs) are calculated using the phase shifts reported in the experimental NN scattering database~\cite{NNonline}. These spectra are expected to be the same as those calculated from the two-point function with LQCD at the physical quark masses, up to exponential corrections in $m_\pi$. Second, the central values of $L_{1,A}$ and $g_{\nu}^{NN}$ from Ref.~\cite{Acharya:2019fij} and Refs.~\cite{Cirigliano:2020dmx,Cirigliano:2021qko} are used to evaluate the physical transition amplitudes for single- and (neutrinoless) double-$\beta$ decay processes with initial and final energies set to the lowest energy eigenvalues obtained in the first step. These scattering amplitudes are then used in matching relations of Refs.~\cite{Davoudi:2020xdv} and \cite{Davoudi:2020gxs}, respectively, to obtain a reasonable guess for the central values of the corresponding FV three- and four-point functions. Next, Gaussian fluctuations are introduced to the quantities that are expected to be extracted from LQCD, namely the FV energy eigenvalues and the three- and four-point functions, to generate a set of synthetic data for performing the sensitivity analysis. This introduces uncertainties in the supposedly LQCD ingredients. Finally, matching relations are used once again to obtain $L_{1,A}$ and $g_{\nu}^{NN}$ from the synthetic dataset, along with their uncertainties. Figure~\ref{fig: flowchart} summarizes the procedure used for performing the sensitivity analysis of this work. \par
A detailed account of our findings is provided in Sec.~\ref{sec: conclusions}. To summarize, achieving small uncertainties in $L_{1,A}$ is found to be more challenging than $g_{\nu}^{NN}$, and demands (sub)percent-level precision in the two-nucleon spectra and the ME to supersede the current phenomenological constraints. On the other hand, the short-distance coupling of the neutrinoless double-$\beta$ decay, $g_{\nu}^{NN}$, turns out to be less sensitive to uncertainties on both LQCD energies and the ME, and promises competitive precision compared with the current indirect estimates, even with few-percent uncertainties on LQCD energies and MEs. The volume requirements are moderate and for ground-state to ground-state transitions, smaller volumes are shown to lead to more precise extractions.
\section{Formalism
\label{sec: Formalism}
}
\noindent
In this section, we present a brief overview of the pionless EFT~\cite{Kaplan:1998tg,Kaplan:1998we,vanKolck:1998bw,Bedaque:1997qi,Bedaque:1998mb,Bedaque:1999vb,Chen:1999tn} employed to evaluate the hadronic scattering amplitudes of $nn \to np e^-\bar{\nu}_e$ and $nn \to pp e^-e^-$ transitions. Furthermore, the FV ingredients required to perform the sensitivity analyses of Secs.~\ref{sec: L1A} and~\ref{sec: gvNN} are obtained via the application of L\"uscher's quantization condition that relates the FV energy eigenvalues to the physical two-hadron scattering amplitudes. Our notation follows that used in Ref.~\cite{Davoudi:2020xdv}.
\subsection{Pionless EFT
\label{subec: Pionless EFT}}
In the pionless EFT, the hadronic Lagrangian is arranged according to the number of nucleons. The relativistic corrections and the isospin-breaking effects contribute at higher orders than considered in this work. The single-nucleon Lagrangian is given by
\begin{equation}
{\cal L}_{(1)}=N^{\dagger }\bigg(i\partial_{t}+\frac{{\nabla}^2}{2M}\bigg)N+\cdots,
\label{eq: EFT 1 nucleon Lagrangian}
\end{equation}
where ellipsis denotes relativistic corrections. Here, $\partial_t$ is the time derivative and ${\bf \nabla}$ is the spatial gradient operator. $N=(p,n)^T$ is an isospin doublet composed of the proton, $p$, and the neutron, $n$, fields, each with mass $M$. The NN contact interactions are governed by the Lagrangian,
\begin{align}
{\cal L}_{(2)}=&-C_{0}(N^{T}\mathcal{P}_iN)^{\dagger }(N^{T}\mathcal{P}_iN) \nonumber -\widetilde{C}_{0}(N^{T}\widetilde{\mathcal{P}}_iN)^{\dagger }(N^{T}\widetilde{\mathcal{P}}_iN)\,+ \nonumber \\
&~~~{\frac{C_{2}}{8}}\left[ (N^{T}\mathcal{P}_i%
N)^{\dagger }(N^{T}(\overleftarrow{{\bf\nabla}}^{2}\mathcal{P}_i-2%
\overleftarrow{{\bf\nabla}}\cdot \mathcal{P}_i\overrightarrow{{\bf\nabla}}+%
\mathcal{P}_i\overrightarrow{{\bf\nabla}}^{2})N)+{\rm H.c.}\right]+
\nonumber\\
&~~~{\frac{\widetilde{C}_{2}}{8}}\left[ (N^{T}\widetilde{\mathcal{P}}_iN)^{\dagger }(N^{T}(%
\overleftarrow{{\bf\nabla}}^{2}\widetilde{\mathcal{P}}_i-2\overleftarrow{{\bf\nabla}}\cdot \widetilde{\mathcal{P}}_i%
\overrightarrow{{\bf\nabla}}+\widetilde{\mathcal{P}}_i\overrightarrow{{\bf\nabla}}^{2})N)+{\rm H.c.}\right]
+ \cdots ,
\label{eq: Nucleon EFT Lagrangian}
\end{align}
The overhead arrow indicates which nucleon field is acted by the derivative operator, and ellipsis denotes higher-derivative operators that will not contribute to the order at which the analysis of this work is performed. Index $i=1,2,3$ is summed over. $\mathcal{P}_i$ and $\widetilde{\mathcal{P}}_i$ are the spin-isospin projection operators for the spin-singlet $(^1S_0)$ and spin-triplet $(^3S_1)$ channels, respectively.\footnote{The spin-triplet channel couples S and D partial waves. Since partial-wave mixing both in infinite and finite volumes is neglected in this work, the spin-triplet channel will be denoted by $^3S_1$ instead of $^3S_1-{^3}D_1$.} Strong-interaction LECs for these channels are distinguished by an overhead tilde for the $^3S_1$ channel.\par
For NN systems in the $^1S_0$ channel at a low center-of-mass (CM) energy, $E$, the scattering amplitude, $\mathcal{M}$ is described by an $S$-wave scattering phase shift, $\delta$,
\begin{equation}
\mathcal{M} = \frac{4\pi}{M} \frac{1}{p\cot{\delta}-i p} ,
\label{eq: scattering amplitude}
\end{equation}
where $p=\sqrt{ME}$ and higher partial-wave contributions are ignored. Below the t-channel cut, the effective-range function $p\cot{\delta}$ can be expansed in $p^2$ near $p^2=0$, resulting in an effective-range expansion,
\begin{equation}
p\cot{\delta} = -\frac{1}{a} + \frac{1}{2} r p^2 + \cdots,
\label{eq: ERE}
\end{equation}
where $a$ is the scattering length, $r$ is the effective range, and ellipsis denotes higher-order terms that will be neglected in this analysis. In the pionless EFT with the Kaplan-Savage-Wise power counting~\cite{Kaplan:1998tg,Kaplan:1998we}, the S-wave scattering amplitude is expanded to LO and NLO amplitudes:
\begin{align}
\mathcal{M}^{\rm (LO)} &=
-\frac{4\pi}{M} \frac{1}{(1/a + ip)},
\label{eq: MLO NN scattering}
\\
\mathcal{M}^{\rm (NLO)} &=
-\frac{2\pi}{M} \frac{r\,p^2}{(1/a + ip)^2}.
\label{eq: MNLO NN scattering}
\end{align}
The LO amplitude, $\mathcal{M}^{\rm (LO)}$, is given by the tree-level NN contact interaction, $C_0$, plus any number of $C_0$ vertices connected by the s-channel two-nucleon loops. The NLO amplitude, $\mathcal{M}^{\rm (NLO)}$, involves one insertion of the NN derivative coupling, $C_2$, dressed by the NN propagator and the LO amplitude from both sides. The NN s-channel loop is an ultraviolet (UV) divergent integral that is regularized with the power-divergence subtraction scheme introduced in Ref.~\cite{Kaplan:1998tg}. By comparing these amplitudes with Eqs.~\eqref{eq: MLO NN scattering} and~\eqref{eq: MNLO NN scattering} for the $^1S_0$ channel, the NN contact interactions at a given renormalization scale, $\mu$, can be expressed in terms of the effective-range expansion parameters defined in Eq.~\eqref{eq: ERE}:
\begin{align}
C_0(\mu) & = \frac{4\pi}{M} \frac{1}{(-\mu + 1/a)},
\label{eq: C0 scale relation}\\
C_2(\mu) & = \frac{2\pi}{M} \frac{r}{(-\mu + 1/a)^2}.
\label{eq: C2 scale relation}
\end{align}
Equations~\eqref{eq: scattering amplitude}-\eqref{eq: C2 scale relation} are valid for the $^3S_1$ channel too upon replacements, $\delta \to \widetilde{\delta}$,\footnote{$\tilde{\delta}$ is the $\alpha$-wave phase shift in the Blatt-Biedenharn parametrization of the coupled $^3S_1-{^3}D_1$ channel~\cite{blatt1952neutron}, but here it will be referred to as an S-wave phase shift for simplicity.} $a \to \widetilde{a}$, $r \to \widetilde{r}$, $\mathcal{M}^{\rm (LO)} \to \widetilde{\mathcal{M}}^{\rm (LO)}$, $\mathcal{M}^{\rm (NLO)} \to \widetilde{\mathcal{M}}^{\rm (NLO)}$, $C_0 \to \widetilde{C}_0$ and $C_2 \to \widetilde{C}_2$, where the overhead tilde denotes the analogous quantity in the $^3S_1$ channel.\footnote{Overhead tilde is used throughout to denote two-nucleon quantities in the $^3S_1$ channel. The only exceptions to this rule are $\widetilde{L}_{1,A}$ and $\widetilde{g}_{\nu}^{NN}$ that denote renormalization-scale independent LECs in Secs.~\ref{sec: L1A} and~\ref{sec: gvNN}. The convention for these LECs is maintained to be consistent with the literature.} The NN scattering amplitudes and contact LECs introduced in this section are needed in matching relations for $L_{1,A}$ and $g_{\nu}^{NN}$ in Eq.~\eqref{eq: matching relation for L1A} and~\eqref{eq: matching relation gvNN}, respectively.\par
\subsection{L\"uscher's method
\label{subec: FV formalism}}
In LQCD, the $n$-point correlation functions are computed on a finite Euclidean spacetime lattice. Assuming the continuum limit for a hypercubic lattice with periodic boundary conditions, L\"uscher's quantization condition gives a direct relation between the FV energy eigenvalues of two hadrons obtained from LQCD and the corresponding scattering amplitudes. The mapping is valid up to exponentially suppressed corrections governed by the range of the interactions. For the low-energy NN systems, the interaction range is set by the Compton wavelength of the pion. The quantization conditions are then valid up to $\mathcal{O}(e^{-m_\pi L})$ corrections, where $L$ denotes the spatial extent of the volume. \par
\begin{figure}[t]
\centering
\includegraphics[scale=1]{QCplot.pdf}
\caption{The effective-range function (solid lines) and L\"uscher's function (dotted lines) in Eq.~\eqref{eq: quantization condition} are plotted independently against the CM energy of NN systems. Equation~\eqref{eq: ERE} is used for the effective-range function with the effective-range expansion parameters given in Eq.~\eqref{eq: ERE values} for the two channels, $^1S_0$ (cyan) and $^3S_1$ (magenta). The function $4\pi c_{00}(p^2,L)$ is plotted for three different volumes with $L=8\;{\rm fm}$ (red), $L=12\;{\rm fm}$ (blue) and $L=16\;{\rm fm}$ (green). The diamonds, circles, triangles, and stars denote, respectively, the location of energy eigenvalues of the ground, first, second, and third excited states in each volume, and satisfy the quantization condition in Eq.~\eqref{eq: quantization condition} (and its counterpart for the $^3S_1$ channel). The numerical values associated with this figure are provided in Appendix~\ref{app:detail}.
\label{fig: QC plot}}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[scale=1]{rplots-abs-combined.pdf}
\caption{The absolute values of the LL residue function in the $^1S_0$ (left) and $^3S_1$ (right) channels is plotted against the CM energy for three different volumes with $L=8\;{\rm fm}$ (red), $L=12\;{\rm fm}$ (blue), and $L=16\;{\rm fm}$ (green). Dashed lines indicate energy eigenvalues in the respective volumes. The numerical values of $|\mathcal{R}|$ and $|\widetilde{\mathcal{R}}|$ evaluated at the FV ground- and first excited-state energies in the corresponding volumes are provided in Appendix \ref{app:detail}.
\label{fig: R plot}}
\end{figure}
The cubic volume does not respect the rotational symmetry, and as a result, the FV quantization conditions mix scattering amplitudes in all partial waves. However, at low energies the scattering amplitude is expected to be dominated by the $S$-wave interaction. Ignoring the contribution from all higher-order partial waves, the FV quantization condition relates the $S$-wave phase shifts to a discrete set of FV energy eigenvalues, $E_n$. For NN systems in the $^1S_0$ channel, the quantization condition is given by
\begin{equation}
p_n\cot{\delta} = 4\pi c_{00}(p_n^2,L).
\label{eq: quantization condition}
\end{equation}
Here, $p_n=\sqrt{ME_n}$, and $\delta$ is the corresponding $S$-wave scattering phase shift. The FV function $c_{00}(p_n^2,L)$ is given by~\cite{Luscher:1986pf,Luscher:1990ux,Kim:2005gf}
\begin{align}
c_{00}({p}_n^2,L)=\frac{1}{L\sqrt{4\pi^3}}\; \mathcal{Z}_{00}\left[1;(p_{n}L/2\pi)^2\right],~\text{with}~\mathcal{Z}_{00}[s;x^2]=\frac{1}{\sqrt{4\pi}}\sum_{\bm n \in \mathbb{Z}^3}\frac{1}{\left(|\bm{n}|^2-x^2\right)^s}.
\label{eq: Luscher's function}
\end{align}
Here, $\bm{n}$ is a Cartesian vector with integer components. The quantization condition in Eq.~\eqref{eq: quantization condition} is obtained by locating the singularities of the two-point correlation function of two nucleons, which is governed by the FV function $\mathcal{F}$,
\begin{align}
\mathcal{F}(E) & = \frac{1}{F^{-1}_0(E)+\mathcal{M}(E)},
\label{eq: definition of mathcal F}
\end{align}
where $F_0$ is another FV function related to the $c_{00}$ function defined above,
\begin{align}
F_0(E) =\frac{M}{4\pi}\left[-4\pi \; c_{00}({p}^2,L) +i p\right],
\label{eq: F0 expression}
\end{align}
and $\mathcal{M}$ is the NN scattering amplitude defined in Eq.~\eqref{eq: scattering amplitude}. Another useful quantity, which appears in the matching relations in Eqs.~\eqref{eq: matching relation for L1A} and~\eqref{eq: matching relation gvNN}, is the generalized Lellouch-L\"uscher (LL) residue matrix, $\mathcal{R}$, which is the residue of the FV function $\mathcal{F}$ at FV energies $E_n$, and is given by
\begin{align}
\mathcal{R}(E_n) = \lim_{E \to E_n} (E-E_n)\;\mathcal{F}(E) = \bigg[\frac{d\mathcal{F}^{-1}}{dE}\biggr|_{E=E_n}\bigg]^{-1} .
\label{eq: LL residue}
\end{align}
In the limit where higher partial waves are ignored, Eq.~\eqref{eq: quantization condition} is also valid for the $^3S_1$ channel after replacing $\delta$ with $\widetilde{\delta}$. Similarly, the replacement $\mathcal{M} \to \widetilde{\mathcal{M}}$ in Eq.~\eqref{eq: definition of mathcal F} defines $\mathcal{F}\to\widetilde{\mathcal{F}}$, which leads to the FV residue function $\widetilde{\mathcal{R}}$ for the $^3S_1$ channel.\par
\begin{figure}[t]
\centering
\includegraphics[scale=1]{erecombined.pdf}
\caption{The inverse scattering length (left column) and the effective range (left column) for the $^1S_0$ (top row) and $^3S_1$ (bottom row) channels obtained from synthetic data with $\Delta_{E},\Delta_{\widetilde{E}}=10\%$, $5\%$, and $1\%$, from lighter to darker bands, respectively, are shown as a function of $L$. The bands indicate mid-$68\%$ uncertainty on the parameters from synthetic data, whereas gray thin bands denote the corresponding experimental values. Selected numerical values associated with this figure are provided in Appendix~\ref{app:detail}.
\label{fig: band ERE}}
\end{figure}
In the following sections, we investigate the sensitivity of constraining LECs $L_{1,A}$ and $g^{NN}_{\nu}$ to LQCD inputs from future LQCD calculations of the corresponding three- and four-point correlation functions at physical quark masses. The lowest-lying FV energy eigenvalues in each of the NN channels enter the necessary matching relations and these energies will be evaluated \emph{ab initio} from LQCD FV two-point correlation functions. As no LQCD determination of the FV spectrum at the physical quark masses exist to date, one can estimate the expected energies for given volumes by solving L\"uscher's quantization condition in Eq.~\eqref{eq: quantization condition} using experimental input for scattering amplitudes, as illustrated in Fig.~\ref{fig: QC plot}. Here, the function $p\cot \delta$ ($p\cot \widetilde{\delta}$) on the left-hand side of Eq.~\eqref{eq: quantization condition} is given by the effective-range expansion defined in Eq.~\eqref{eq: ERE} (and its counterpart for the $^3S_1$ channel). The effective-range expansion parameters
\begin{equation}
\begin{split}
a & =-23.5\;[\text{fm}], \hspace{2cm} r = 2.75\;[\text{fm}],
\\
\widetilde{a} & =5.42\;[\text{fm}], \hspace{2.35cm} \widetilde{r} = 1.75\;[\text{fm}],
\end{split}
\label{eq: ERE values}
\end{equation}
are obtained using NN phase shifts for $S$-wave scattering generated by the Nijmegen phenomenological NN potential~\cite{Stoks:1994wp}, that are the result of fits to NN scattering data in Ref.~\cite{NNonline}. The ground-state energies of the NN systems in the $^1S_0$ ($^3S_1$) channel with the CM energy $E_0$ ($\widetilde{E}_0$) for the volumes shown are negatively shifted compared with the threshold as noted in Fig.~\ref{fig: QC plot}, and asymptote polynomially (exponentially) to zero (to -2.2245 MeV) in the infinite-volume limit. Additionally, the absolute values of the LL residue functions are plotted in Fig.~\ref{fig: R plot} as a function of energy for $L=8,12$ and $16$~fm. Note that only the absolute values of these functions appear in the matching relations for the matrix elements.
The small uncertainties on the scattering parameters from experiment are ignored as the goal is to obtain central values of FV energies. For the sensitivity analyses of the upcoming sections, uncertainties need to be artificially introduced on these energies in generating synthetic data to mimic the expected LQCD uncertainties on energy extractions. This indicates that the scattering parameters associated with these energies will become uncertain too. Since the scattering parameters enter the LO and NLO NN scattering amplitudes, and hence impact the matching relations of the next sections, the subsequent uncertainty on scattering parameters must be taken into account. Uncertainties on the first two lowest-lying energies in each channel (which is a minimal set in a single volume to constrain the scattering length and effective range) can be introduced through a randomly-generated Gaussian distribution of energies with central values equal to $E_0$ and $E_1$ ($\widetilde{E}_0$ and $\widetilde{E}_1$) and the width equal to $\Delta_{E_0} \times |E_0|$ and $\Delta_{E_1} \times |E_1|$ ($\Delta_{\widetilde{E}_0} \times |\widetilde{E}_0|$ and $\Delta_{\widetilde{E}_1} \times |\widetilde{E}_1|$) for the ground- and first excited-state energies of the NN systems in the $^1S_0$ ($^3S_1$) channels, respectively. The scattering length and effective range corresponding to each channel for the choices of $\Delta_{E}\equiv\Delta_{E_0}=\Delta_{E_1}=10\%,~5\%$, and $1\%$ and $\Delta_{\widetilde{E}}\equiv\Delta_{\widetilde{E}_0}=\Delta_{\widetilde{E}_1}=10\%,~5\%$, and $1\%$ are then obtained by solving the quantization condition in Eq.~\eqref{eq: quantization condition}, resulting in uncertainties in the scattering parameters as shown in Fig.~\ref{fig: band ERE}. A similar analysis was performed in Ref.~\cite{Briceno:2013bda} in the isosinglet channel to study the viability of the extraction of the S-D mixing parameter from the upcoming LQCD calculations.
Constraints on more than two energies, including in more than one volume and with various different boost vectors, will improve uncertainties on the extracted scattering parameters, possibilities that are not considered in this initial analysis. More radically, one may attempt to input the experimental determination of the scattering parameters (and hence the energy eigenvalues derived using quantization conditions) to avoid an uncertainty introduced in both quantities in costly LQCD calculations. This can reduce the uncertainty on the extracted LECs, as the only LQCD input will be matrix elements that are unknown experimentally. Nonetheless, the upcoming LQCD calculations will first evaluate these matrix elements at the isospin-symmetric limit where quantum electrodynamics (QED) effects and the non-vanishing mass difference among the light quarks are ignored. This means that for consistency, one needs to input the scattering parameters associated with the $^1S_0$ and $^3S_1$ channels in such a limit. As obtaining the isospin-symmetric parameters from experimental data involves model/EFT uncertainties, it is preferred that all inputs to the quantization and matching conditions are evaluated from first-principles LQCD calculations consistently. That is the strategy adopted in this synthetic data analysis. While the experimental parameters are used to obtain the central values of the FV energies, the subsequent analysis assumes energies and hence the scattering parameters are obtained directly from LQCD and hence involve likely sizable uncertainties in early calculations.\footnote{The inaccuracy in the central values of the FV energies compared to what is expected at the isospin symmetric limit will have minimal impact in the conclusions reached in the upcoming sections, as we have verified by slightly changing the central values of the synthetic data in our analysis and observed no significant sensitivity in achieved uncertainties on the LECs.}
\section{Sensitivity analysis for $L_{1,A}$
\label{sec: L1A}}
\noindent
At the NLO in the pionless EFT, the two-body axial-vector current contributes to single- and double-weak processes, including $pp$ fusion, neutrino(antineutrino)-induced disintegration of the deuteron, and muon capture on the deuteron~\cite{Butler:1999sv,Butler:2000zp,Davoudi:2020xdv}, and its strength is characterized by the LEC $L_{1,A}$. Constraints on $L_{1,A}$ were obtained using elastic and inelastic (anti)neutrino-deuteron scattering data from nuclear reactors: $L_{1,A}=3.6 \pm 5.5\text{ fm}^3$~\cite{Chen:1999tn}, as well as from Sudbury Neutrino Observatory~\cite{Bellerive:2016byv} and Super-K~\cite{Fukuda:2001nj,Fukuda:2002pe} experiments: $L_{1,A}=4.0 \pm 6.3\text{ fm}^3$~\cite{Butler:2002cw}. A more precise constraint was obtained in Ref.~\cite{Acharya:2019fij} where improved low-energy chiral EFT results of inelastic (anti)neutrino-deuteron scattering amplitude were matched to those of pionless EFT, resulting in: $L_{1,A}= 4.9 ^{+1.9}_{-1.5}\text{ fm}^3$. It is expected that the uncertainty in $L_{1,A}$ will be reduced to $\sim 1.25 \text{ fm}^3$ from the precise measurement of reaction rate of muon capture on the deuteron that is underway in the MuSun experiment~\cite{Andreev:2010wd}.
Furthermore, a constraint on $L_{1,A}$ has been obtained from a LQCD study of the $pp$-fusion process in Ref.~\cite{Savage:2016kon} giving the value $L_{1,A}=3.9(0.2)(1.4) \text{ fm}^3$. Even though the statistical uncertainty shown in the first parentheses is small, the overall uncertainty is similar to the experimental constraints due to the large systematic uncertainty indicated in the second parentheses. The major source of uncertainty is the extrapolation to the physical quark masses as the correlation function for the $pp$-fusion process was calculated at larger quark masses corresponding to $m_\pi\approx 806$ MeV. Thus, it is expected that this uncertainty will improve in future LQCD calculations at lighter quark masses. However, the extraction of this LEC at such a large pion mass did not require the involved matching relation that will be presented shortly, as the NN states appeared deeply bound. Furthermore, achieving the quoted statistical uncertainty with quark masses near the physical values will be challenging. The question that will be addressed here is whether these features will limit the precise extraction of $L_{1,A}$ at the physical values of the quark masses.
\par
In this section, we investigate the accuracy with which $L_{1,A}$ can be obtained from future LQCD calculations performed at the physical pion mass. In Sec.~\ref{subsec: matching relation L1A}, the matching relation provided in Ref.~\cite{Davoudi:2020xdv} will be reviewed, relating the hadronic scattering amplitude for the $nn \to npe^-\bar{\nu}_e$ decay (or alternatively the $pp$ fusion process $pp \to npe^+\nu_e$) to the corresponding nuclear ME calculated using LQCD. The matching relation is then used to perform a sensitivity analysis on $L_{1,A}$ in Sec~\ref{subsec: sensitivity analysis L1A} through studying the effects of the LQCD inputs and their uncertainty on the $L_{1,A}$ extraction.
\subsection{Matching Relation
\label{subsec: matching relation L1A}}
Consider the single-$\beta$ decay transition $nn \to npe^-\bar{\nu}_e$ with the kinematics chosen such that the total three-momentum of the electron and anti-neutrino is zero, and the NN systems are unboosted in the initial and final states. The hadronic amplitude receives non-vanishing contribution from the Gamow-Teller-type transitions mediated by one-body ($n=1$) and two-body ($n=2$) axial-current operators, $A_{k(n)}^{i}$, where $k$ and $i$ denote spin and isospin indices, respectively. In the spin-isospin symmetric limit, the amplitude is independent of the azimuthal spin quantum number of the final state. Thus, one can consider the hadronic transition $nn \to np$ $(k=3)$ in the pionless EFT. The LO contribution is characterized by the corresponding LO NN contact interactions in Eq.~\eqref{eq: Nucleon EFT Lagrangian} for each channel and the one-body axial-vector current operator corresponding to the nucleon axial charge, $g_A$. At the NLO, the hadronic amplitude receives a contribution from the two-body axial-vector current operator corresponding to the LEC $L_{1,A}$:
\begin{equation}
A_{3(2)}^{+}=L_{1,A}\big( N^{T}\widetilde{\mathcal{P}}_{3}N \big) ^{\dagger }\big( N^{T}\mathcal{P}_{+}N \big),
\label{eq: isovector current : two body}
\end{equation}
where $\mathcal{P}_{+} = (\mathcal{P}_{1}+i \mathcal{P}_{2})/{\sqrt{2}}$.\par
The hadronic amplitude is related to the FV nuclear ME of the weak current between the ground states of the NN system with energies $E_0$ and $\widetilde{E}_0$ corresponding to the $^1S_0$ and $^3S_1$ channels, respectively, via the matching relation~\cite{Davoudi:2020xdv, Briceno:2012yi}
\begin{align}
L^6&\left|\left[\vphantom{B^\dagger}\langle E_{0},L|\,\mathcal{J}({0})\,|\widetilde{E}_{0},L\rangle \right]_L\right|^2 = \left|\widetilde{\mathcal{R}}(\widetilde{E}_{0})\right| \left|\mathcal{M}^{\rm{DF},V}_{nn\to np} (E_{0},\widetilde{E}_{0})\right|^2 \left|\mathcal{R}(E_{0}) \vphantom{\widetilde{\mathcal{R}}(\widetilde{E}_{0})} \right|,
\label{eq: matching relation for L1A}
\end{align}
where the equality is up to exponentially suppressed corrections in $L$. Here, $|\cdot|$ denotes the absolute value, and $\mathcal{J}$ denotes the hadronic part of the weak current placed at the origin, see Ref.~\cite{Davoudi:2020xdv}. The FV $S$-wave states, $|E,L\rangle$, are labeled with the CM energy, $E$, and the spatial extent of the cubic volume $L$, and the FV nature of the ME is emphasized by the subscript $L$. The quantity, $\mathcal{M}^{\rm{DF},V}_{nn\to np}$, is related to infinite-volume amplitude via
\begin{align}
i \mathcal{M}^{\rm{DF},V}_{nn\to np} (E_0,\widetilde{E}_{0})=
i\mathcal{M}^{\rm DF}_{nn\to np} & (E_0,\widetilde{E}_{0})-ig_A\,F_1(\widetilde{E}_{0},E_0)
\left[\mathcal{M}^{\rm (LO)}({E}_{0})\widetilde{\mathcal{M}}^{\rm (LO)}(\widetilde{E}_{0})+ \right.
\nonumber
\\
& \left.\mathcal{M}^{\rm (LO)}({E}_{0})\widetilde{\mathcal{M}}^{\rm (NLO)}(\widetilde{E}_{0})+\mathcal{M}^{\rm (NLO)}({E}_{0})\widetilde{\mathcal{M}}^{\rm (LO)}(\widetilde{E}_{0})\right],
\label{eq: L1A DF amplitude: finite V}
\end{align}
where $\mathcal{M}^{\rm DF}_{nn\to np}$ is the divergence-free infinite-volume amplitude, which is obtained after removing from the full amplitude the contributions from the Feynman diagrams with the weak current on the external nucleon legs. The LO and NLO NN scattering amplitudes in the $^1S_0$ channel, $\mathcal{M}^{\rm (LO)}$ and $\mathcal{M}^{\rm (NLO)}$, are defined in Eqs.~\eqref{eq: MLO NN scattering} and~\eqref{eq: MNLO NN scattering}, respectively. $\mathcal{R}$ is the LL residue function defined in Eq.~\eqref{eq: LL residue} for the $^1S_0$ channel. The corresponding quantities for the $^3S_1$ channel are denoted with an overhead tilde. $F_1$ is a FV function originating from the s-channel loop diagram with three nucleon propagators. It is related to the $F_0$ function defined in Eq.~\eqref{eq: F0 expression},
\begin{equation}
F_1(\widetilde{E}_{0},E_0) =\frac{1}{E_0-\widetilde{E}_{0}}\,\left[F_0(\widetilde{E}_{0})-F_0(E_0)\right].
\label{eq:F1def}
\end{equation}
Finally, the amplitude $\mathcal{M}^{\rm DF}_{nn\to np}$ depends on the LEC $L_{1,A}$,
\begin{align}
i\mathcal{M}^{\rm DF}_{nn\to np} = -i\widetilde{L}_{1,A} \,\widetilde{\mathcal{M}}^{\rm LO}(\widetilde{E}_{0}) & \mathcal{M}^{\rm LO}(E_0)-ig_A\,I_1(\widetilde{E}_{0},E_0)
\left[\mathcal{M}^{\rm (LO)}({E}_{0})\widetilde{\mathcal{M}}^{\rm (LO)}(\widetilde{E}_{0}) \right.
\nonumber\\
& \left.+\mathcal{M}^{\rm (LO)}({E}_{0})\widetilde{\mathcal{M}}^{\rm (NLO)}(\widetilde{E}_{0})+\mathcal{M}^{\rm (NLO)}({E}_{0})\widetilde{\mathcal{M}}^{\rm (LO)}(\widetilde{E}_{0})\right],
\label{eq: L1A DF amplitude: inf volume}
\end{align}
where $\widetilde{L}_{1,A}$ is the renormalization-scale-independent combination of $L_{1,A}$ and the LO and NLO NN LECs introduced in Sec.~\ref{subec: Pionless EFT}:
\begin{equation}
\widetilde{L}_{1,A} =
\frac{L_{1,A}}{C_0\,\widetilde{C}_0}-\frac{g_A\,M}{2}\frac{(C_2+\widetilde{C}_2)}{C_0\,\widetilde{C}_0},
\label{eq: L1A scale independent}
\end{equation}
and $I_1$ in is defined as
\begin{equation}
I_1(\widetilde{E}_{0},E_0) =\frac{iM^{3/2}}{4\pi}\frac{1}{\sqrt{\widetilde{E}_0}+\sqrt{E_0 \vphantom{\widetilde{E}_0}} }.
\label{eq: I1 definition}
\end{equation}
\subsection{Sensitivity analysis
\label{subsec: sensitivity analysis L1A}}
Future constraints on $L_{1,A}$ from LQCD calculations at the physical quark masses will depend on LQCD determinations of the low-lying FV energy eigenvalues of the NN systems in the $^1S_0$ and $^3S_1$ channels, as well as the nuclear MEs of the axial-vector current between these states, as is clear from the ingredients of Eq.~\eqref{eq: matching relation for L1A}. Furthermore, the matching relation depends upon the LO and NLO NN scattering amplitudes in both the $^1S_0$ and $^3S_1$ channels, as well as the derivative of scattering amplitudes with respect to energy that enters the LL residue function in Eq.~(\ref{eq: LL residue}), requiring the values of the scattering length and effective range in the two NN channels. These are obtained from the knowledge of at least two energy levels in the spectrum, i.e., the ground and the first excited states, as outlined in Sec.~\ref{subec: FV formalism}. The precision with which $L_{1,A}$ can be obtained depends on the precision and correlation of these ingredients. In order to quantify the uncertainty on $L_{1,A}$ extracted from a future LQCD calculation performed in a given volume, one can introduce percent precision with which the nuclear ME of a single axial-vector current and the NN ground- (and first excited-) state energies are expected to reach, to be denoted by $\Delta_{\beta}$ and $\Delta_{E(\widetilde{E})}$, respectively. A sample set of these ingredients is then generated from a Gaussian distribution with the mean represented by the value obtained from (the central values of) the phenomenological constraint for the quantity, and the precision level multiplied by the mean for its standard deviation.
The mean values for the expected ground- and first excited-state energies of the NN channels are obtained using the quantization condition in Eq.~\eqref{eq: quantization condition} with the NN phase shifts for the $S$-channel from Ref.~\cite{NNonline}, as was already discussed in Sec.~\ref{subec: FV formalism} and demonstrated in Fig.~\ref{fig: QC plot}. The mean value of the expected FV ME is obtained by using the matching relation in Eq.~\eqref{eq: matching relation for L1A} with the FV energies being the mean values discussed above, the experimental value of $g_A\approx1.27$, and the central value of the $L_{1,A}$ (or and its scale-independent counterpart) from a recent phenomenological determination~\cite{Acharya:2019fij}
\begin{equation}
L_{1,A}= 4.9 ^{+1.9}_{-1.5}\;\text{ fm}^3~~\text{or}~~\widetilde{L}_{1,A}= -449.7^{+19.5}_{-15.4}\;\text{ fm}^3.
\label{eq: L1A value}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[scale=0.99]{L1A.pdf}
\caption{The value of $\widetilde{L}_{1,A}$ as a function of $L$ for the $^1S_0 \to {^3}S_1$ transition obtained from synthetic data with various combinations of $\Delta_{E}=\Delta_{\widetilde{E}}$ and $\Delta_{\beta}$ values. The gray horizontal band denotes the experimental value, whereas the colored bands indicate mid-$68\%$ uncertainty on extracted $L_{1,A}$ for the ground-state to ground-state (purple) and first excited-state to first excited-state (green) transitions. Note the smaller range of the $\widetilde{L}_{1,A}$-axes in the most-left plots compared to the rest. Selected numerical values associated with this figure are provided in Appendix~\ref{app:detail}.
\label{fig: band L1A}}
\end{figure}
The expected mean values are then used to generate the Gaussian samples for the FV energies and FV ME. With the samples generated, the matching relation in Eq.~\eqref{eq: matching relation for L1A} is used once again to solve for the $L_{1,A}$ values associated with each set of energies and MEs, leading to a distribution for the expected $L_{1,A}$ values. In the following, the scale-dependent quantity $\widetilde{L}_{1,A}$ is used but it can be converted to $L_{1,A}$ values give the values of the NN LECs evaluated at the corresponding values of the scattering length and effective range. Note that since the scattering parameters are obtained \emph{ab initio} from LQCD, the uncertainties in energies impact their precision, as discussed in Sec.~\ref{subec: FV formalism}.\par
The effect of $\Delta_{\beta}$ and $\Delta_E$ on determining $\widetilde{L}_{1,A}$ is illustrated in Fig.~\ref{fig: band L1A}, where the volume dependence of $L_{1,A}$ values obtained from the sample sets for various combinations of $\Delta_E$ and $\Delta_{\beta}$ values is shown. In all cases, the uncertainty on $L_{1,A}$ (determined from the mid-$68\%$ of the sample) increases with increasing $\Delta_E$, $\Delta_{\widetilde{E}}$, and $\Delta_\beta$. Only the most precisely determined sample set and at volumes with $L \approx 8\;{\rm fm}$, constraints on $L_{1,A}$ become comparable in precision to that in Eq.~\eqref{eq: L1A value}. Thus, future LQCD calculations at the physical quark masses need to determine the NN ground and first excited-state energies and the FV MEs with below percent-level precision to supersede the current phenomenological constraints. The situation is likely alleviated in the actual LQCD calculations where energy and ME extractions are partially correlated, and where the NN scattering amplitude can be determined more precisely with a larger set of precise FV energies.
Since LQCD can, in principle, obtain FV MEs for transitions involving excited states, one may wonder if constraining $L_{1,A}$ through the first excited-state to the first excited-state transition will be more beneficial and relaxes the precision requirements on the FV energies and ME above. The green bands in Fig.~\ref{fig: band L1A} denote the $\widetilde{L}_{1,A}$ values and uncertainties obtained from the first excited-state to the first excited-state transition. It is clear that the ground-state to ground-state transition leads to better constraints at smaller volumes---volumes that are more readily accessibly to upcoming LQCD calculations at the physical pion mass, but for larger volumes with $L \gtrsim 14\;{\rm fm}$, the constraints from the first excited-state to the first excited-state transition become comparable or more precise. The reverse trend in uncertainties as a function of volume between the two cases is a consequence of different behavior of the LL residue functions near negative and positive CM energies, as illustrated in Fig.~\ref{fig: R plot}. One cautionary note is the loss of accuracy in using the effective-range expansion and the associated LO and NLO NN scattering amplitudes in the pionless EFT near the first excited-state energies. However, at large volume where the $\widetilde{L}_{1,A}$ constraints from excited-state transition become more precise, the FV energies tend to their asymptotic value of zero and are therefore near or within the t-channel cut. On the other hand, at such large volumes, the density of states in the spectrum increases, and the identification of excited states with current methods may present a challenge. Variational techniques such as those developed in Refs.~\cite{Horz:2020zvv, Green:2021qol, Amarasinghe:2021lqa} will likely constrain the lowest-lying levels with comparable precision to the ground state.
\section{Sensitivity analysis for $g_{\nu}^{NN}$
\label{sec: gvNN}}
\noindent
In the light neutrino exchange model of the low-energy $nn \to pp e^-e^-$ decay, there exists an undetermined LEC, $g_{\nu}^{NN}$, at the LO the pionless EFT, which is introduced to absorb the UV scale dependence of the amplitude through renormalization group~\cite{Cirigliano:2017tvr,Cirigliano:2018hja,Cirigliano:2019vdj}. The Lagrangian density corresponding to this short-distance contribution consists of a four-nucleon-two-electron contact interaction:
\begin{equation}
\mathcal{L}_{N}^{\Delta L =2} = \left(\frac{4V_{ud}G_{F}}{2\sqrt{2}}\right)^2 \, m_{\beta\beta } \, g_\nu^{NN} \left[\overline{e}_L\, C\, \bar{e}^T_L \right] \left[ (N^T\mathcal{P}_-N)^\dagger N^T\mathcal{P}_+N)\right] + {\rm H.c.}
\label{eq: gvNN operator}
\end{equation}
Here, $G_f$ is Fermi's constant, $V_{ud}$ is a Cabibbo-Kobayashi-Maskawa (CKM) matrix element~\cite{Cabibbo:1963yz, Kobayashi:1973fv}, $m_{\beta\beta }$ is the effective Majorana mass, $m_{\beta\beta} = \sum_i U^2_{ei} m_i$, where $U_{ei}$ are the elements of the Pontecorvo-Mako-Nakagawa-Sato (PMNS) matrix~\cite{Pontecorvo:1957qd, Maki:1962mu}, with $m_i$ being the mass of the neutrino-mass eigenstate $i$. $C$ is the charge-conjugation matrix, and $e_L$ is a left-handed electron field.\par
As shown in Refs.~\cite{Cirigliano:2020dmx,Cirigliano:2021qko}, a constraint on the $g_{\nu}^{NN}$ value can be obtained by expressing the $nn \to pp e^-e^-$ decay amplitude as a product of momentum integral of the Majorana neutrino propagator and the generalized forward Compton scattering amplitude, in analogy to the
Cottingham formula~\cite{Cottingham:1963zz,PhysRevLett.17.1303} for the electromagnetic contribution to hadron masses. A model-independent representation of the integrand using the chiral EFT and operator product expansion can then be obtained. The missing parts of the full amplitude can be filled by interpolating between the known regions using nucleon form factors for the weak current and information on NN scattering. The constraint on $g_{\nu}^{NN}$ via this method is:
\begin{equation}
\widetilde{g}_{\nu}^{NN} = 1.3 \pm 0.6,
\label{eq: gvNNtilde value}
\end{equation}
where $\widetilde{g}_{\nu}^{NN}$ is a dimensionless parameter related to $g_{\nu}^{NN}$ and the momentum-independent NN LEC in Eq.~\eqref{eq: Nucleon EFT Lagrangian}:
\begin{equation}
\widetilde{g}_{\nu}^{NN}= \bigg(\frac{4\pi}{M C_0}\bigg)^2 g_{\nu}^{NN}\,.
\label{eq: gvNNtilde}
\end{equation}
The value in Eq.~\eqref{eq: gvNNtilde value} has a large uncertainty, and a more precise and direct constraint on $g_{\nu}^{NN}$ using LQCD will be desired. As shown in Ref.~\cite{Davoudi:2020gxs}, a prescription exists for obtaining the $g_{\nu}^{NN}$ (or equivalently the $\widetilde{g}_{\nu}^{NN}$) value from a Euclidean four-point correlation function calculated using LQCD. With LQCD calculations of these correlation functions underway, it would be useful to know the precision with which one can constrain the $\widetilde{g}_{\nu}^{NN}$ value for a given LQCD setup. In this section, we perform the sensitivity analysis of constraining $\widetilde{g}_{\nu}^{NN}$ by estimating the uncertainty on $\widetilde{g}_{\nu}^{NN}$ from a synthetic data representing a future LQCD calculation of the four-point correlation function at the physical quark masses.\par
\subsection{Matching Relation
\label{subsec: matching relation gvNN}}
Consider the transition $nn \to pp e^-e^-$ in the spin-isospin symmetric limit with simple kinematics, where the currents carry zero energy and momentum such that the initial CM energy, $E_i \equiv E_{CM}$, remains unchanged. The Euclidean four-point function for this process, which is accessible via LQCD methods, can be analytically continued to Minkowski spacetime to obtain
\begin{eqnarray}
\mathcal{T}^{(\rm M)}_L \equiv \int dz_0\,\int_L d^3z \,
\big[\langle E_0,L|\, T[\mathcal{J}(z_0,\bm{z})\,S_\nu(z_0,\bm{z})
\mathcal{J}(0)]\, |E_0,L\rangle\big]_L,
\label{eq: TML definition}
\end{eqnarray}
using the procedure described in Ref.~\cite{Davoudi:2020gxs}. In Eq.~\eqref{eq: TML definition}, $T$ denotes time ordering, the superscript $(\rm M)$ denotes a Minkowski time signature, the subscript $L$ on the spatial integral indicates that the integral is performed over a finite cubic volume (with PBCs), and $z_0$ is the Minkowski time coordinate. $S_\nu$ is the Minkowski propagator of a Majorana neutrino in a finite volume that is given by
\begin{equation}
S_\nu(z_0,\bm{z}) = \frac{1}{L^3}\sum_{\bm{k} \in \frac{2\pi}{L}\mathbb{Z}^3\neq \bm{0}} \int \frac{dk_0}{2\pi}e^{i\bm{k} \cdot \bm{z}-ik_0z_0} \frac{-i\,m_{\beta\beta}}{k_0^2-|\bm{k}|^2+i\epsilon},
\label{eq: Neutrino propagator Minkowski}
\end{equation}
where the neutrino four-momentum is given by $(k_0,\bm{k})$ with quantized spatial momenta $\bm{k}$. Contributions from the small non-zero neutrino mass in the denominator of the neutrino propagator can be ignored at the LO in the EFT power counting, and the infrared divergence is regulated by removing the zero-momentum mode of the neutrino. The remaining notation in Eq.~\eqref{eq: TML definition} is the same as in Eq.~\eqref{eq: matching relation for L1A}. It is important to note that for a ground-state to ground-state transitions at low energies corresponding to the FV energy eigenvalues in the range of volumes studied, no intermediate single-neutrino-two-nucleon state can go on shell and the analytic continuation from the Euclidean correlation function of LQCD to the Minkowski counterpart in Eq.~(\ref{eq: TML definition}) is straightforward. With on-shell intermediate states, the complete formalism of Ref.~\cite{Davoudi:2020gxs} needs to be implemented but this will not be necessary in the upcoming LQCD calculations given realistic volumes and energies.
$\mathcal{T}^{(\rm M)}_L$ is related to the physical decay amplitude through the following matching relation:
\begin{equation}
L^6\;\bigg|\mathcal{T}^{(\rm M)}_L(E_i,E_f) \bigg|^2=\bigg|\mathcal{R}(E_i)\bigg | \, \bigg|\mathcal{M}^{0\nu,V}_{nn\to pp} (E_i,E_f) \bigg|^2 \bigg|\mathcal{R}(E_f)\bigg |,
\label{eq: matching relation gvNN}
\end{equation}
where
\begin{equation}
\mathcal{M}^{0\nu,V}_{nn\to pp} (E_{i},E_{f}) = \mathcal{M}^{(\rm{Int.})}_{nn\to pp} (E_{i},E_{f})-m_{\beta\beta}(1+3g_A^2)\mathcal{M}^{(\rm LO)}(E_i)\delta J^V(E_{i},E_{f}) \mathcal{M}^{(\rm LO)}(E_f).
\label{eq: Finite volume amplitude 0vbb}
\end{equation}
The right-hand side of Eq.~\eqref{eq: matching relation gvNN} contains the LL residue matrix, $\mathcal{R}$, defined in Eq.~\eqref{eq: LL residue}, and the FV quantity $\mathcal{M}^{0\nu,V}_{nn\to pp}$ which is related to the physical scattering amplitude of the $0\nu\beta\beta$ decay with the initial (final) CM energy $E_i$ $(E_f)$, as defined in Eq.~\eqref{eq: Finite volume amplitude 0vbb}. Here, $\mathcal{M}^{({\rm LO})}$ is the LO NN scattering amplitude defined in Eq.~\eqref{eq: MLO NN scattering}, $\mathcal{M}^{(\rm{Int.})}$ is the infinite-volume decay amplitude evaluated in the pionless EFT after removing the contributions from the diagrams in which the neutrino propagates between two external nucleons. The full scattering amplitude is evaluated assuming that the amplitude is approximated by the s-wave interactions of the nucleons and only receives contributions from a static neutrino potential. Moreover, contributions to the full infinite-volume amplitude from radiative neutrinos are ignored. With these assumptions, $\mathcal{M}^{(\rm{Int.})}$ is given by~\cite{Cirigliano:2017tvr,Cirigliano:2018hja,Cirigliano:2019vdj}
\begin{eqnarray}
\mathcal{M}^{(\rm{Int.})}_{nn\to pp} (E_i,E_f)=m_{\beta\beta}\;\mathcal{M}^{(\rm LO)}(E_i) \bigg [ -(1+3g_A^2) J^{\infty}(E_i,E_f;\mu)+\frac{2g_\nu^{NN}}{C_0^2}\bigg]
\mathcal{M}^{(\rm LO)}(E_f).
\label{eq: M^int}
\end{eqnarray}
The first term denotes contributions from the diagrams in which the neutrino propagates between two nucleons dressed by strong interactions on both sides. $J^\infty$ is a known function given by
\begin{equation}
J^{\infty}(E_i,E_f;\mu)=\frac{M^2}{32\pi^2} \bigg[-\gamma_E+\ln(4\pi)+\ln\left(\tfrac{\mu^2/M}{-(\sqrt{E_i}+\sqrt{E_f})^2-i\epsilon}\right)+1\bigg],
\label{eq: Jinf}
\end{equation}
with $\gamma_E$ being Euler's constant. This arises from evaluating the s-channel two-loop diagram with an exchanged Majorana neutrino. The UV divergence is regularized in the dimensional-regularization scheme, introducing the scale $\mu$. The second term in square brackets Eq.~\eqref{eq: M^int} denotes contributions from diagrams with the NN short-range operator in Eq.~\eqref{eq: gvNN operator} dressed by the NN propagator and the LO NN amplitude on both sides. Finally, $\delta J^V(E_i,E_f)$ in Eq.~\eqref{eq: matching relation gvNN} is a FV function corresponding to the FV two-loop diagram with the exchanged neutrino propagator, that is defined by
\begin{eqnarray}
\delta J^V(E_i,E_f)=
\bigg[\frac{1}{{L^6}}\sum_{\substack{\bm{k}_1,\bm{k}_2 \in \frac{2\pi}{L}\mathbb{Z}^3\\ \bm{k}_1 \neq \bm{k}_2}}-
\int \frac{d^3k_1}{(2\pi)^3}\frac{d^3k_2}{(2\pi)^3}
\bigg]
\frac{1}{E_i-\tfrac{|\bm{k}_1^2|}{M}+i\epsilon} \frac{1}{E_f-\tfrac{|\bm{k}_2^2|}{M}+i\epsilon}\frac{1}{|\bm{k}_1-\bm{k}_2|^2}.
\label{eq: deltaJV}
\end{eqnarray}
\begin{figure}[t]
\centering
\includegraphics[scale=1]{delJ_plottedReandIm.pdf}
\caption{The real (solid cyan) and imaginary (dotted dashed cyan) parts of $J^{\infty}(E_i,E_f;\mu=m_\pi)$ defined in Eq.~(\ref{eq: Jinf}), as well as real (solid magenta) and imaginary (dotted dashed magenta) parts of $\delta J^V(E_i,E_f)$ defined in Eq.~(\ref{eq: deltaJV}), both evaluated at $E_i=E_f\equiv E$. The red, blue, and green dashed lines denote the FV ground-state energy eigenvalues with $L=8$, $12$, and $16\;{\rm fm}$, respectively, obtained from the quantization condition in Eq.~(\ref{eq: quantization condition}) (as plotted in Fig.~\ref{fig: QC plot}). These are the values at which the LQCD four-point function will be evaluated in the future studies at the physical quark masses. Selected numerical values for the functions shown are provided in Appendix~\ref{app:detail}.
\label{fig:delJ}
}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[scale=1]{TLM-M0vv-combo.pdf}
\caption{$\big| \mathcal{M}^{0\nu,V({\rm Int.})}_{nn\to pp}\big|$ (left) and $\big|\mathcal{T}_L^{(M)}\big|$ (right) functions defined in Eqs.~(\ref{eq: M^int})-(\ref{eq: matching relation gvNN}), with $L=8\;{\rm fm}$ (red), $L=12\;{\rm fm}$ (blue), and $L=16\;{\rm fm}$ (green) are plotted against the CM energy of the NN state, considering the kinematics $E_i=E_f \equiv E$. The effective neutrino mass $m_{\beta\beta}$ is set to $1\;{\rm MeV}$. The dashed lines in both panels denote the ground-state energy eigenvalues in the corresponding volumes obtained from the quantization condition in Eq.~(\ref{eq: quantization condition}) (as plotted in Fig.~\ref{fig: QC plot}). Selected numerical values for the functions shown are provided in Appendix~\ref{app:detail}.
\label{fig: TLMMV vs ECM}
}
\end{figure}
This sum-integral difference is calculated numerically using the technique presented in the supplemental material of Ref.~\cite{Davoudi:2020gxs}. The real and imaginary parts of $J^{\infty}$ and $\delta J^V$ are depicted in Fig.~\ref{fig:delJ} for a range of negative and positive $E_i=E_f\equiv E$ values.
The absolute value of the FV amplitude $\mathcal{M}^{0\nu,V}_{nn\to pp}$ for the kinematics $E_i=E_f \equiv E$ is plotted against the CM energy in the left panel of Fig.~\ref{fig: TLMMV vs ECM} along with $|\mathcal{M}^{\rm (Int.)}_{nn \to pp}|$, using the value of $g_{\nu}^{NN}$ obtained from the central value of the constraint in Eq.~\eqref{eq: gvNNtilde value}. The dependence of the $|\mathcal{T}_L^{(M)}|$ on the CM energy of the NN system in different volumes is shown in the right panel of Fig.~\ref{fig: TLMMV vs ECM}(b) using the matching relation in Eq.~\eqref{eq: matching relation gvNN}.
\subsection{Sensitivity Analysis
\label{subsec: sensitivity analysis gvNN}}
Equation~\eqref{eq: matching relation gvNN} indicates that the precision with which $g_{\nu}^{NN}$, and thus $\widetilde{g}_{\nu}^{NN}$, can be obtained from LQCD depends on the precision with which the FV ground-state energy in a given volume, $E_0$, and the FV ME are obtained from the LQCD calculations of the corresponding two- and four-point functions, respectively. Furthermore, the matching relation depends upon the LO NN scattering amplitude in the $^1S_0$ channel as well as the derivative of the NLO+LO scattering amplitude with respect to energy that enters the LL residue function in Eq.~(\ref{eq: LL residue}), requiring the values of the scattering length and effective range in the $^1S_0$ channel. These depend on the central value and the uncertainty of at least two energy levels in the spectrum, e.g., the ground and the first excited states, as outlined in Sec.~\ref{subec: FV formalism}. In this section, we investigate the uncertainty on $\widetilde{g}_{\nu}^{NN}$ from the precision levels with which these LQCD inputs are obtained in future LQCD calculations at the physical quark masses.
The expected value of $E_0$ for a given volume is calculated using L\"uscher's quantization condition in Eq.~\eqref{eq: quantization condition} and NN phase shifts in the $^1S_0$ channel obtained from Ref.~\cite{NNonline}. This expected value of $E_0$ and the central value of the constraint on $\widetilde{g}_{\nu}^{NN}$ given in Eq.~\eqref{eq: gvNNtilde value} are then used to obtain an estimate on the expected value of $\mathcal{T}^{(\rm M)}_L$ with the use of Eqs.~\eqref{eq: matching relation gvNN} and~\eqref{eq: M^int}. Note that even though the expected value of $\mathcal{T}^{(\rm M)}_L$ from Eqs.~\eqref{eq: TML definition}-\eqref{eq: M^int} is dependent on $m_{\beta\beta}$, the mean value and the uncertainty on $\widetilde{g}_{\nu}^{NN}$ obtained from synthetic data using Eq.~\eqref{eq: matching relation gvNN} is independent of $m_{\beta\beta}$.
\begin{figure}[t]
\centering
\includegraphics[scale=1]{gvNN.pdf}
\caption{ The value of $\widetilde{g}_{\nu}^{NN}$ obtained from the synthetic data is plotted against $L$ for different combinations of $\Delta_{\beta\beta}$ and $\Delta_E$. The gray band denotes the uncertainty in the value of $\widetilde{g}_{\nu}^{NN}$ from Eq.~\eqref{eq: gvNNtilde value} from the indirect determination of Ref.~\cite{Cirigliano:2020dmx}. The corresponding central value is used to obtain the expected values of $\mathcal{T}^{(\rm M)}_L$, which enables this sensitivity analysis. The purple band is the mid-$68\%$ uncertainty band corresponding to the sample sets with uncorrelated fluctuations. Selected numerical values associated with this figure are provided in Appendix~\ref{app:detail}.
\label{fig: band gvNN}}
\end{figure}
The percent precision on $E_0$ (and $E_1$) is denoted by $\Delta_E$, whereas the percent precision on $\mathcal{T}^{(\rm M)}_L$ is denoted by $\Delta_{\beta\beta}$. Similar to Sec.~\ref{subsec: sensitivity analysis L1A}, the uncertainty on $\widetilde{g}_{\nu}^{NN}$ is taken as the mid-$68\%$ of the ensemble of $\widetilde{g}_{\nu}^{NN}$ values obtained from synthetic data that incorporates uncertainties on $E_{0(1)}$ and $\mathcal{T}^{(\rm M)}_L$ as Gaussian fluctuations. The precision levels, $\Delta_E$ and $\Delta_{\beta\beta}$, are incorporated in this synthetic data by making the standard deviation of the fluctuations equal to the expected values of the quantities multiplied by the corresponding percent precision. The scattering length and effective range in the $^1S_0$ channel are obtained by solving L\"uscher's quantization condition in Eq.~\eqref{eq: quantization condition} for the generated ensembles of the ground- and the first excited-state energies, as outlined in sec.~\ref{subec: FV formalism}.
The $\widetilde{g}_{\nu}^{NN}$ values obtained for various combinations of $\Delta_{\beta\beta}$ and $\Delta_E$ are plotted against $L$ in Fig.~\ref{fig: band gvNN}. The LQCD constraints on $\widetilde{g}_{\nu}^{NN}$ are almost always more precise than the constraint of Ref.~\cite{Cirigliano:2020dmx} for input uncertainties below $\sim10\%$ level, which indicates that future LQCD calculations can confidently improve the current constraint, especially for smaller volumes, provided that $\Delta_{\beta\beta}$ and $\Delta_E$ are a few percents. This situation is more promising than the case of $L_{1,A}$, where (sub)precent-level uncertainties appear to be the requirement. As the LQCD input for energies and the ME will be partially correlated, the constraint on $\widetilde{g}_{\nu}^{NN}$ will likely be further improved.
\section{Conclusions
\label{sec: conclusions}}
\noindent
This paper presents an analysis of the effect of uncertainties in the future lattice quantum chromodynamics calculations at the physical quark masses on the accuracy with which the hadronic amplitudes of $\beta$ decays can be constrained in the two-nucleon sector. The nuclear matrix elements of the single-$\beta$ decay and the neutrinoless double-$\beta$ decay within the light neutrino exchange scenario are studied for this purpose, and the precision with which the low-energy constants $L_{1,A}$ and $\tilde{g}_{\nu}^{NN}$, corresponding to the respective two-body isovector and isotensor operators, can be obtained from future calculations was deduced from a synthetic data analysis.
\par
For processes that are studied here, matching relations exist that relate the three- and four-point functions of LQCD evaluated in a finite Euclidean spacetime to their respective physical scattering amplitudes~\cite{Briceno:2012yi,Detmold:2004qn,Briceno:2015tza,Davoudi:2020xdv,Davoudi:2020gxs}. The LQCD inputs that go into these matching relations involve the lowest-lying two-nucleon energy spectra for a given volume, the matrix elements of a single axial-vector weak current (for the single-$\beta$ decay), and of two axial-vector weak currents along with a Majorana neutrino propagator (for the $0\nu\beta\beta$ decay) between appropriate two-nucleon states. Using these matching relations, constraints were obtained on $L_{1,A}$ and $\tilde{g}_{\nu}^{NN}$ from the synthetic data of the relevant LQCD ingredients. In order to synthesize this data to represent the underlying LQCD uncertainties, Gaussian fluctuations were introduced on the supposedly LQCD ingredients that go into these matching relations.
The precision with which $L_{1,A}$ and $\tilde{g}_{\nu}^{NN}$ can be obtained from the synthetic data was obtained for a range of input uncertainties at or below $\sim10\%$ level. The uncertainty on the LECs grows with volume in both cases assuming ground-state to ground-state transitions, and so smaller volumes that are more feasible computationally appear to be more advantageous. The constraints from LQCD studies on $L_{1,A}$ will likely be worse than the current experimental constraints for the range of volumes and plausible input uncertainties considered here, and may require (sub)percent-level precision on the finite-volume energies and matrix element. The situation may be alleviated in actual LQCD calculations where the uncertainties in the inputs to the matching relations are (partially) correlated. Furthermore, one may imagine inputting the precise experimental parameters and associated FV energies in those analyses, rather than obtaining them directly from LQCD calculations, to decrease the uncertainty in the extraction of the unknown LECs. Nonetheless, such an approach will not be \emph{ab initio}, particularly since the early calculations will take place at the isospin-symmetric limit and excluding QED, and for consistency and model independency, scattering parameters need to be evaluated directly from LQCD.
Finally, for precision levels on the LQCD energies and the ME below $10\%$, the constraint on $\tilde{g}_{\nu}^{NN}$ will likely improve the existing constraint, and will therefore provide a direct precise determination arising from first-principles calculations rooted in QCD. As a result, the present study further motivates future studies of the $nn \to ppee$ process within the light Majorana exchange scenario from LQCD at or near the physical values of the quark masses.
\section*{Acknowledgment}
\noindent
We acknowledge valuable discussions on a range of topics at the Institute for Nuclear Theory's virtual program on ``Nuclear Forces for Precision Nuclear Physics'' (INT-21-1b held in Spring 2021) which inspired the need for the analysis of this work. ZD and SVK are supported by the Alfred P. Sloan fellowship and by the Maryland Center for Fundamental Physics at the University of Maryland, College Park.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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L'Oracle de Delphes és un curtmetratge mut francès del 1903 dirigit per Georges Méliès. Va ser estrenat per la Star Film Company de Méliès i té el número 476 als seus catàlegs.
Argument
A l'antic Egipte, dos sacerdots dipositen una caixa ornamentada dins d'un temple, tancant les portes darrere d'ells. Després de la seva marxa, un lladre irromp al temple i roba la caixa, però és atrapat per una figura amb barba que apareix misteriosament. L'home barbut recupera la capsa i dóna vida a les dues estàtues d'esfinx col·locades a les portes. Les esfinxs apareixen com a dones vives i ataquen al lladre, el cap del qual es converteix de seguida en cap d'ase. Les esfinxs tornen a convertir-se en estàtues, l'home barbut desapareix i el lladre de rucs queda assegut astorat a terra.
Producció
Méliès interpreta el lladre a la pel·lícula. Tot i que el títol es refereix a l'oracle de Delfos, una gran sacerdotessa de la religió grega antiga, els decorats i els vestits indiquen un ambient de l'antic Egipte. El palanquí i els accessoris de caixa, al seu torn, es van reutilitzar més tard per a un altre escenari, l'orientalisme eclèctic de la pel·lícula de Méliès Le Palais des mille et une nuits (1905). Les tècniques d'efectes especials de la pel·lícula inclouen escamoteigs així com fosa i exposició múltiple.
Versions
Pel·lícules anteriors a 1903 de Méliès, especialment la popular Le Voyage dans la Lune, eren freqüentment pirata per productors estatunidencs com Siegmund Lubin. Per combatre la pirateria, Méliès va obrir una sucursal nord-americana de la seva Star Film Company i va començar a produir dos negatius de cada pel·lícula que realitzava: una per als mercats nacionals i una altra per a l'estranger. Per produir els dos negatius separats, Méliès va construir una càmera especial que utilitzava dues lents i dues bobines de pel·lícula simultàniament.
A la dècada del 2000, els investigadors de la companyia cinematogràfica francesa Lobster Films es van adonar que el sistema de dues lents de Méliès era en efecte una càmera de pel·lícula estèreo no intencionada, però totalment funcional, i per tant que versions 3D de les pel·lícules de Méliès es podria fer simplement combinant les impressions nacionals i estrangeres de la pel·lícula. Serge Bromberg, el fundador de Lobster Films, va presentar versions en 3D de L'Oracle de Delphes i un altre pel·lícula de Méliès de 1903, Le Chaudron infernal, en una presentació el gener de 2010 a la Cinémathèque Française. Segons la crítica de cinema Kristin Thompson, "l'efecte del 3D era deliciós... les pel·lícules sincronitzades per Lobster semblaven exactament com si Méliès les hagués dissenyat per a 3D." Bromberg va tornar a projectar ambdues pel·lícules —així com la pel·lícula de Méliès de 1906 Alchimiste Parafaragaramus ou la Cornue infernale, preparada de manera similar per a 3D—en una presentació de setembre de 2011 a l'Acadèmia d'Arts i Ciències Cinematogràfiques.
Referències
Enllaços externs
Curtmetratges de França
Pel·lícules de França del 1903
Pel·lícules de França en blanc i negre
Pel·lícules dirigides per Georges Méliès
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{
"redpajama_set_name": "RedPajamaWikipedia"
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static bool g_is_crash_reporting_enabled = false;
static Lumix::DefaultAllocator stack_node_allocator;
namespace Lumix
{
namespace debug
{
void enableFloatingPointTraps(bool enable)
{
unsigned int cw = _control87(0, 0) & MCW_EM;
if (enable)
{
cw &= ~(_EM_OVERFLOW | _EM_ZERODIVIDE | _EM_INVALID | _EM_DENORMAL); // can not enable _EM_INEXACT because it is common in QT
}
else
{
cw |= _EM_OVERFLOW | _EM_INVALID | _EM_DENORMAL; // can not enable _EM_INEXACT because it is common in QT
}
_control87(cw, MCW_EM);
}
void debugOutput(const char* message)
{
OutputDebugString(message);
}
void debugBreak()
{
DebugBreak();
}
int StackTree::s_instances = 0;
struct StackNode
{
~StackNode()
{
LUMIX_DELETE(stack_node_allocator, m_next);
LUMIX_DELETE(stack_node_allocator, m_first_child);
}
void* m_instruction;
StackNode* m_next = nullptr;
StackNode* m_first_child = nullptr;
StackNode* m_parent;
};
StackTree::StackTree()
{
m_root = nullptr;
if (atomicIncrement(&s_instances) == 1)
{
HANDLE process = GetCurrentProcess();
SymInitialize(process, nullptr, TRUE);
}
}
StackTree::~StackTree()
{
LUMIX_DELETE(stack_node_allocator, m_root);
if (atomicDecrement(&s_instances) == 0)
{
HANDLE process = GetCurrentProcess();
SymCleanup(process);
}
}
void StackTree::refreshModuleList()
{
ASSERT(s_instances > 0);
SymRefreshModuleList(GetCurrentProcess());
}
int StackTree::getPath(StackNode* node, Span<StackNode*> output)
{
u32 i = 0;
while (i < output.length() && node)
{
output[i] = node;
i++;
node = node->m_parent;
}
return i;
}
StackNode* StackTree::getParent(StackNode* node)
{
return node ? node->m_parent : nullptr;
}
bool StackTree::getFunction(StackNode* node, Span<char> out, int& line)
{
HANDLE process = GetCurrentProcess();
alignas(SYMBOL_INFO) u8 symbol_mem[sizeof(SYMBOL_INFO) + 256 * sizeof(char)] = {};
SYMBOL_INFO* symbol = reinterpret_cast<SYMBOL_INFO*>(symbol_mem);
symbol->MaxNameLen = 255;
symbol->SizeOfStruct = sizeof(SYMBOL_INFO);
BOOL success = SymFromAddr(process, (DWORD64)(node->m_instruction), 0, symbol);
IMAGEHLP_LINE64 line_info;
DWORD displacement;
if (SymGetLineFromAddr64(process, (DWORD64)(node->m_instruction), &displacement, &line_info))
{
line = line_info.LineNumber;
}
else
{
line = -1;
}
if (success) copyString(out, symbol->Name);
return success != FALSE;
}
void StackTree::printCallstack(StackNode* node)
{
while (node)
{
HANDLE process = GetCurrentProcess();
alignas(SYMBOL_INFO) u8 symbol_mem[sizeof(SYMBOL_INFO) + 256 * sizeof(char)];
SYMBOL_INFO* symbol = reinterpret_cast<SYMBOL_INFO*>(symbol_mem);
memset(symbol_mem, 0, sizeof(symbol_mem));
symbol->MaxNameLen = 255;
symbol->SizeOfStruct = sizeof(SYMBOL_INFO);
BOOL success = SymFromAddr(process, (DWORD64)(node->m_instruction), 0, symbol);
if (success)
{
IMAGEHLP_LINE line;
DWORD offset;
if (SymGetLineFromAddr(process, (DWORD64)(node->m_instruction), &offset, &line))
{
OutputDebugString("\t");
OutputDebugString(line.FileName);
OutputDebugString("(");
char tmp[20];
toCString((u32)line.LineNumber, Span(tmp));
OutputDebugString(tmp);
OutputDebugString("):");
}
OutputDebugString("\t");
OutputDebugString(symbol->Name);
OutputDebugString("\n");
}
else
{
OutputDebugString("\tN/A\n");
}
node = node->m_parent;
}
}
StackNode* StackTree::insertChildren(StackNode* root_node, void** instruction, void** stack)
{
StackNode* node = root_node;
while (instruction >= stack)
{
StackNode* new_node = LUMIX_NEW(stack_node_allocator, StackNode)();
node->m_first_child = new_node;
new_node->m_parent = node;
new_node->m_next = nullptr;
new_node->m_first_child = nullptr;
new_node->m_instruction = *instruction;
node = new_node;
--instruction;
}
return node;
}
StackNode* StackTree::record()
{
static const int frames_to_capture = 256;
void* stack[frames_to_capture];
USHORT captured_frames_count = CaptureStackBackTrace(2, frames_to_capture, stack, 0);
void** ptr = stack + captured_frames_count - 1;
if (!m_root) {
m_root = LUMIX_NEW(stack_node_allocator, StackNode)();
m_root->m_instruction = *ptr;
m_root->m_first_child = nullptr;
m_root->m_next = nullptr;
m_root->m_parent = nullptr;
--ptr;
return insertChildren(m_root, ptr, stack);
}
StackNode* node = m_root;
while (ptr >= stack)
{
while (node->m_instruction != *ptr && node->m_next)
{
node = node->m_next;
}
if (node->m_instruction != *ptr)
{
node->m_next = LUMIX_NEW(stack_node_allocator, StackNode);
node->m_next->m_parent = node->m_parent;
node->m_next->m_instruction = *ptr;
node->m_next->m_next = nullptr;
node->m_next->m_first_child = nullptr;
--ptr;
return insertChildren(node->m_next, ptr, stack);
}
if (node->m_first_child)
{
--ptr;
node = node->m_first_child;
}
else if (ptr != stack)
{
--ptr;
return insertChildren(node, ptr, stack);
}
else
{
return node;
}
}
return node;
}
static const u32 UNINITIALIZED_MEMORY_PATTERN = 0xCD;
static const u32 FREED_MEMORY_PATTERN = 0xDD;
static const u32 ALLOCATION_GUARD = 0xFDFDFDFD;
void* GuardAllocator::allocate_aligned(size_t size, size_t align) {
const size_t pages = 1 + ((size + 4095) >> 12);
void* mem = VirtualAlloc(nullptr, pages * 4096, MEM_RESERVE, PAGE_READWRITE);
VirtualAlloc(mem, (pages - 1) * 4096, MEM_COMMIT, PAGE_READWRITE);
if (align == 4096) return mem;
u8* ptr = (u8*)mem;
return (void*)(uintptr_t(ptr + (pages - 1) * 4096 - size) & ~size_t(align - 1));
}
void GuardAllocator::deallocate_aligned(void* ptr) {
VirtualFree((void*)((uintptr_t)ptr & ~(size_t)4095), 0, MEM_RELEASE);
}
Allocator::Allocator(IAllocator& source)
: m_source(source)
, m_root(nullptr)
, m_total_size(0)
, m_is_fill_enabled(true)
, m_are_guards_enabled(true)
{
m_sentinels[0].next = &m_sentinels[1];
m_sentinels[0].previous = nullptr;
m_sentinels[0].stack_leaf = nullptr;
m_sentinels[0].size = 0;
m_sentinels[0].align = 0;
m_sentinels[1].next = nullptr;
m_sentinels[1].previous = &m_sentinels[0];
m_sentinels[1].stack_leaf = nullptr;
m_sentinels[1].size = 0;
m_sentinels[1].align = 0;
m_root = &m_sentinels[1];
}
void Allocator::checkLeaks()
{
AllocationInfo* last_sentinel = &m_sentinels[1];
if (m_root != last_sentinel)
{
OutputDebugString("Memory leaks detected!\n");
AllocationInfo* info = m_root;
while (info != last_sentinel)
{
StaticString<2048> tmp("\nAllocation size : ", info->size, " , memory ", (u64)(info + sizeof(info)), "\n"); //-V568
OutputDebugString(tmp);
m_stack_tree.printCallstack(info->stack_leaf);
info = info->next;
}
debugBreak();
}
}
Allocator::~Allocator()
{
checkLeaks();
}
void Allocator::lock()
{
m_mutex.enter();
}
void Allocator::unlock()
{
m_mutex.exit();
}
void Allocator::checkGuards()
{
if (m_are_guards_enabled) return;
auto* info = m_root;
while (info)
{
auto user_ptr = getUserPtrFromAllocationInfo(info);
void* system_ptr = getSystemFromUser(user_ptr);
ASSERT(*(u32*)system_ptr == ALLOCATION_GUARD);
ASSERT(*(u32*)((u8*)user_ptr + info->size) == ALLOCATION_GUARD);
info = info->next;
}
}
size_t Allocator::getAllocationOffset()
{
return sizeof(AllocationInfo) + (m_are_guards_enabled ? sizeof(ALLOCATION_GUARD) : 0);
}
size_t Allocator::getNeededMemory(size_t size)
{
return size + sizeof(AllocationInfo) + (m_are_guards_enabled ? sizeof(ALLOCATION_GUARD) << 1 : 0);
}
size_t Allocator::getNeededMemory(size_t size, size_t align)
{
return size + sizeof(AllocationInfo) + (m_are_guards_enabled ? sizeof(ALLOCATION_GUARD) << 1 : 0) +
align;
}
Allocator::AllocationInfo* Allocator::getAllocationInfoFromSystem(void* system_ptr)
{
return (AllocationInfo*)(m_are_guards_enabled ? (u8*)system_ptr + sizeof(ALLOCATION_GUARD)
: system_ptr);
}
void* Allocator::getUserPtrFromAllocationInfo(AllocationInfo* info)
{
return ((u8*)info + sizeof(AllocationInfo));
}
Allocator::AllocationInfo* Allocator::getAllocationInfoFromUser(void* user_ptr)
{
return (AllocationInfo*)((u8*)user_ptr - sizeof(AllocationInfo));
}
u8* Allocator::getUserFromSystem(void* system_ptr, size_t align)
{
size_t diff = (m_are_guards_enabled ? sizeof(ALLOCATION_GUARD) : 0) + sizeof(AllocationInfo);
if (align) diff += (align - diff % align) % align;
return (u8*)system_ptr + diff;
}
u8* Allocator::getSystemFromUser(void* user_ptr)
{
AllocationInfo* info = getAllocationInfoFromUser(user_ptr);
size_t diff = (m_are_guards_enabled ? sizeof(ALLOCATION_GUARD) : 0) + sizeof(AllocationInfo);
if (info->align) diff += (info->align - diff % info->align) % info->align;
return (u8*)user_ptr - diff;
}
void* Allocator::reallocate(void* user_ptr, size_t size)
{
if (user_ptr == nullptr) return allocate(size);
if (size == 0) return nullptr;
void* new_data = allocate(size);
if (!new_data) return nullptr;
AllocationInfo* info = getAllocationInfoFromUser(user_ptr);
memcpy(new_data, user_ptr, info->size < size ? info->size : size);
deallocate(user_ptr);
return new_data;
}
void* Allocator::allocate_aligned(size_t size, size_t align)
{
void* system_ptr;
AllocationInfo* info;
u8* user_ptr;
size_t system_size = getNeededMemory(size, align);
m_mutex.enter();
system_ptr = m_source.allocate_aligned(system_size, align);
user_ptr = getUserFromSystem(system_ptr, align);
info = new (NewPlaceholder(), getAllocationInfoFromUser(user_ptr)) AllocationInfo();
info->previous = m_root->previous;
m_root->previous->next = info;
info->next = m_root;
m_root->previous = info;
m_root = info;
m_total_size += size;
m_mutex.exit();
info->align = u16(align);
info->stack_leaf = m_stack_tree.record();
info->size = size;
if (m_is_fill_enabled)
{
memset(user_ptr, UNINITIALIZED_MEMORY_PATTERN, size);
}
if (m_are_guards_enabled)
{
*(u32*)system_ptr = ALLOCATION_GUARD;
*(u32*)((u8*)system_ptr + system_size - sizeof(ALLOCATION_GUARD)) = ALLOCATION_GUARD;
}
return user_ptr;
}
void Allocator::deallocate_aligned(void* user_ptr)
{
if (user_ptr)
{
AllocationInfo* info = getAllocationInfoFromUser(user_ptr);
void* system_ptr = getSystemFromUser(user_ptr);
if (m_is_fill_enabled)
{
memset(user_ptr, FREED_MEMORY_PATTERN, info->size);
}
if (m_are_guards_enabled)
{
ASSERT(*(u32*)system_ptr == ALLOCATION_GUARD);
size_t system_size = getNeededMemory(info->size, info->align);
ASSERT(*(u32*)((u8*)system_ptr + system_size - sizeof(ALLOCATION_GUARD)) == ALLOCATION_GUARD);
}
{
MutexGuard lock(m_mutex);
if (info == m_root)
{
m_root = info->next;
}
info->previous->next = info->next;
info->next->previous = info->previous;
m_total_size -= info->size;
} // because of the lock
info->~AllocationInfo();
m_source.deallocate_aligned((void*)system_ptr);
}
}
void* Allocator::reallocate_aligned(void* user_ptr, size_t size, size_t align)
{
if (user_ptr == nullptr) return allocate_aligned(size, align);
if (size == 0) return nullptr;
void* new_data = allocate_aligned(size, align);
if (!new_data) return nullptr;
AllocationInfo* info = getAllocationInfoFromUser(user_ptr);
memcpy(new_data, user_ptr, info->size < size ? info->size : size);
deallocate_aligned(user_ptr);
return new_data;
}
void* Allocator::allocate(size_t size)
{
void* system_ptr;
AllocationInfo* info;
size_t system_size = getNeededMemory(size);
{
MutexGuard lock(m_mutex);
system_ptr = m_source.allocate(system_size);
info = new (NewPlaceholder(), getAllocationInfoFromSystem(system_ptr)) AllocationInfo();
info->previous = m_root->previous;
m_root->previous->next = info;
info->next = m_root;
m_root->previous = info;
m_root = info;
m_total_size += size;
} // because of the lock
void* user_ptr = getUserFromSystem(system_ptr, 0);
info->stack_leaf = m_stack_tree.record();
info->size = size;
info->align = 0;
if (m_is_fill_enabled)
{
memset(user_ptr, UNINITIALIZED_MEMORY_PATTERN, size);
}
if (m_are_guards_enabled)
{
*(u32*)system_ptr = ALLOCATION_GUARD;
*(u32*)((u8*)system_ptr + system_size - sizeof(ALLOCATION_GUARD)) = ALLOCATION_GUARD;
}
return user_ptr;
}
void Allocator::deallocate(void* user_ptr)
{
if (user_ptr)
{
AllocationInfo* info = getAllocationInfoFromUser(user_ptr);
void* system_ptr = getSystemFromUser(user_ptr);
if (m_is_fill_enabled)
{
memset(user_ptr, FREED_MEMORY_PATTERN, info->size);
}
if (m_are_guards_enabled)
{
ASSERT(*(u32*)system_ptr == ALLOCATION_GUARD);
size_t system_size = getNeededMemory(info->size);
ASSERT(*(u32*)((u8*)system_ptr + system_size - sizeof(ALLOCATION_GUARD)) == ALLOCATION_GUARD);
}
{
MutexGuard lock(m_mutex);
if (info == m_root)
{
m_root = info->next;
}
info->previous->next = info->next;
info->next->previous = info->previous;
m_total_size -= info->size;
} // because of the lock
info->~AllocationInfo();
m_source.deallocate((void*)system_ptr);
}
}
} // namespace Debug
BOOL SendFile(LPCSTR lpszSubject,
LPCSTR lpszTo,
LPCSTR lpszName,
LPCSTR lpszText,
LPCSTR lpszFullFileName)
{
HINSTANCE hMAPI = ::LoadLibrary("mapi32.dll");
if (!hMAPI) return FALSE;
LPMAPISENDMAIL lpfnMAPISendMail = (LPMAPISENDMAIL)::GetProcAddress(hMAPI, "MAPISendMail");
PathInfo fi(lpszFullFileName);
char szFileName[MAX_PATH] = {0};
strcat_s(szFileName, fi.m_basename);
strcat_s(szFileName, ".");
strcat_s(szFileName, fi.m_extension);
char szFullFileName[MAX_PATH] = {0};
strcat_s(szFullFileName, lpszFullFileName);
MapiFileDesc MAPIfile = {0};
ZeroMemory(&MAPIfile, sizeof(MapiFileDesc));
MAPIfile.nPosition = 0xFFFFFFFF;
MAPIfile.lpszPathName = szFullFileName;
MAPIfile.lpszFileName = szFileName;
char szTo[MAX_PATH] = {0};
strcat_s(szTo, lpszTo);
char szNameTo[MAX_PATH] = {0};
strcat_s(szNameTo, lpszName);
MapiRecipDesc recipient = {0};
recipient.ulRecipClass = MAPI_TO;
recipient.lpszAddress = szTo;
recipient.lpszName = szNameTo;
char szSubject[MAX_PATH] = {0};
strcat_s(szSubject, lpszSubject);
char szText[MAX_PATH] = {0};
strcat_s(szText, lpszText);
MapiMessage MAPImsg = {0};
MAPImsg.lpszSubject = szSubject;
MAPImsg.lpRecips = &recipient;
MAPImsg.nRecipCount = 1;
MAPImsg.lpszNoteText = szText;
MAPImsg.nFileCount = 1;
MAPImsg.lpFiles = &MAPIfile;
ULONG nSent = lpfnMAPISendMail(0, 0, &MAPImsg, 0, 0);
FreeLibrary(hMAPI);
return (nSent == SUCCESS_SUCCESS || nSent == MAPI_E_USER_ABORT);
}
static void getStack(CONTEXT& context, Span<char> out)
{
BOOL result;
STACKFRAME64 stack;
alignas(IMAGEHLP_SYMBOL64) char symbol_mem[sizeof(IMAGEHLP_SYMBOL64) + 256];
IMAGEHLP_SYMBOL64* symbol = (IMAGEHLP_SYMBOL64*)symbol_mem;
char name[256];
copyString(out, "Crash callstack:\n");
memset(&stack, 0, sizeof(STACKFRAME64));
HANDLE process = GetCurrentProcess();
HANDLE thread = GetCurrentThread();
DWORD64 displacement = 0;
DWORD machineType;
#ifdef _M_X64
machineType = IMAGE_FILE_MACHINE_AMD64;
stack.AddrPC.Offset = context.Rip;
stack.AddrPC.Mode = AddrModeFlat;
stack.AddrStack.Offset = context.Rsp;
stack.AddrStack.Mode = AddrModeFlat;
stack.AddrFrame.Offset = context.Rbp;
stack.AddrFrame.Mode = AddrModeFlat;
#elif defined _M_IA64
#error not supported
machineType = IMAGE_FILE_MACHINE_IA64;
stack.AddrPC.Offset = context.StIIP;
stack.AddrPC.Mode = AddrModeFlat;
stack.AddrFrame.Offset = context.IntSp;
stack.AddrFrame.Mode = AddrModeFlat;
stack.AddrBStore.Offset = context.RsBSP;
stack.AddrBStore.Mode = AddrModeFlat;
stack.AddrStack.Offset = context.IntSp;
stack.AddrStack.Mode = AddrModeFlat;
#else
#error not supported
machineType = IMAGE_FILE_MACHINE_I386;
stack.AddrPC.Offset = context.Eip;
stack.AddrPC.Mode = AddrModeFlat;
stack.AddrStack.Offset = context.Esp;
stack.AddrStack.Mode = AddrModeFlat;
stack.AddrFrame.Offset = context.Ebp;
stack.AddrFrame.Mode = AddrModeFlat;
#endif
do
{
result = StackWalk64(machineType,
process,
thread,
&stack,
&context,
NULL,
SymFunctionTableAccess64,
SymGetModuleBase64,
NULL);
symbol->SizeOfStruct = sizeof(IMAGEHLP_SYMBOL64);
symbol->MaxNameLength = 255;
BOOL sybol_valid = SymGetSymFromAddr64(process, (ULONG64)stack.AddrPC.Offset, &displacement, symbol);
auto err = GetLastError();
DWORD num_char = UnDecorateSymbolName(symbol->Name, (PSTR)name, 256, UNDNAME_COMPLETE);
if (!catString(out, symbol->Name)) return;
if (!catString(out, "\n")) return;
} while (result);
}
static LONG WINAPI unhandledExceptionHandler(LPEXCEPTION_POINTERS info)
{
if (!g_is_crash_reporting_enabled) return EXCEPTION_CONTINUE_SEARCH;
HANDLE process = GetCurrentProcess();
SymInitialize(process, nullptr, TRUE);
debug::StackTree::refreshModuleList();
struct CrashInfo
{
LPEXCEPTION_POINTERS info;
DWORD thread_id;
};
auto dumper = [](void* data) -> DWORD {
LPEXCEPTION_POINTERS info = ((CrashInfo*)data)->info;
uintptr base = (uintptr)GetModuleHandle(NULL);
StaticString<4096> message;
if(info)
{
getStack(*info->ContextRecord, Span(message.data));
message << "\nCode: " << (u32)info->ExceptionRecord->ExceptionCode;
message << "\nAddress: " << (uintptr)info->ExceptionRecord->ExceptionAddress;
message << "\nBase: " << (uintptr)base;
os::messageBox(message);
}
else
{
message.data[0] = '\0';
}
char minidump_path[LUMIX_MAX_PATH];
GetCurrentDirectory(sizeof(minidump_path), minidump_path);
catString(minidump_path, "\\minidump.dmp");
HANDLE process = GetCurrentProcess();
DWORD process_id = GetProcessId(process);
HANDLE file = CreateFile(
minidump_path, GENERIC_WRITE, 0, nullptr, CREATE_ALWAYS, FILE_ATTRIBUTE_NORMAL, nullptr);
MINIDUMP_TYPE minidump_type = (MINIDUMP_TYPE)(
MiniDumpWithFullMemoryInfo | MiniDumpFilterMemory | MiniDumpWithHandleData |
MiniDumpWithThreadInfo | MiniDumpWithUnloadedModules);
MINIDUMP_EXCEPTION_INFORMATION minidump_exception_info;
minidump_exception_info.ThreadId = ((CrashInfo*)data)->thread_id;
minidump_exception_info.ExceptionPointers = info;
minidump_exception_info.ClientPointers = FALSE;
MiniDumpWriteDump(process,
process_id,
file,
minidump_type,
info ? &minidump_exception_info : nullptr,
nullptr,
nullptr);
CloseHandle(file);
minidump_type = (MINIDUMP_TYPE)(MiniDumpWithFullMemory | MiniDumpWithFullMemoryInfo |
MiniDumpFilterMemory | MiniDumpWithHandleData |
MiniDumpWithThreadInfo | MiniDumpWithUnloadedModules);
file = CreateFile(
"fulldump.dmp", GENERIC_WRITE, 0, nullptr, CREATE_ALWAYS, FILE_ATTRIBUTE_NORMAL, nullptr);
MiniDumpWriteDump(process,
process_id,
file,
minidump_type,
info ? &minidump_exception_info : nullptr,
nullptr,
nullptr);
CloseHandle(file);
SendFile("Lumix Studio crash",
"SMTP:mikulas.florek@gamedev.sk",
"Lumix Studio",
message,
minidump_path);
return 0;
};
DWORD thread_id;
CrashInfo crash_info = { info, GetCurrentThreadId() };
auto handle = CreateThread(0, 0x8000, dumper, &crash_info, 0, &thread_id);
WaitForSingleObject(handle, INFINITE);
StaticString<4096> message;
getStack(*info->ContextRecord, Span(message.data));
logError(message);
return EXCEPTION_CONTINUE_SEARCH;
}
void enableCrashReporting(bool enable)
{
g_is_crash_reporting_enabled = enable;
}
void installUnhandledExceptionHandler()
{
SetUnhandledExceptionFilter(unhandledExceptionHandler);
}
} // namespace Lumix
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,584
|
Produced by Ted Garvin, Dave Morgan and PG Distributed Proofreaders
LA FIAMMETTA
BY
GIOVANNI BOCCACCIO
TRANSLATED BY JAMES C. BROGAN
1907.
INTRODUCTION
Youth, beauty, and love, wit, gayety and laughter, are the component
parts of the delightful picture conjured up by the mere name of Giovanni
Boccaccio, the prince of story-tellers for all generations of men. This
creator of a real literary epoch was born in Paris, in 1313, (in the
eleventh year of Dante's exile), of an Italian father and a French-woman
of good family. His father was a merchant of Florence, whither he
returned with his son when the child was seven years old. The boy
received some education, but was placed in a counting-house when he was
only thirteen, and at seventeen he was sent by his father to Naples to
enter another commercial establishment. But he disliked commerce, and
finally persuaded his father to allow him to study law for two years at
the University of Naples, during which period the lively and attractive
youth made brisk use of his leisure time in that gay and romantic city,
where he made his way into the highest circles of society, and
unconsciously gleaned the material for the rich harvest of song and
story that came with his later years. At this time he was present at the
coronation of the poet Petrarch in the Capitol, and was fired with
admiration for the second greatest poet of that day. He chose Petrarch
for his model and guide, and in riper manhood became his most intimate
friend.
By the time he was twenty-five, Boccaccio had fallen in love with the
Lady Maria, a natural daughter of King Robert of Naples, who had caused
her to be adopted as a member of the family of the Count d'Aquino, and
to be married when very young to a Neapolitan nobleman. Boccaccio first
saw her in the Church of San Lorenzo on the morning of Easter eve, in
1338, and their ensuing friendship was no secret to their world. For the
entertainment of this youthful beauty he wrote his _Filicopo_, and the
fair Maria is undoubtedly the heroine of several of his stories and
poems. His father insisted upon his return to Florence in 1340, and
after he had settled in that city he occupied himself seriously with
literary work, producing, between the years 1343 and 1355, the _Teseide_
(familiar to English readers as "The Knight's Tale" in Chaucer,
modernized by Dryden as "Palamon and Arcite"), _Ameto, Amorosa Visione,
La Fiammetta, Ninfale Fiesolona_, and his most famous work, the
_Decameron_, a collection of stories written, it is said, to amuse Queen
Joanna of Naples and her court, during the period when one of the
world's greatest plagues swept over Europe in 1348. In these years he
rose from the vivid but confused and exaggerated manner of _Filocopo_ to
the perfection of polished literary style. The _Decameron_ fully
revealed his genius, his ability to weave the tales of all lands and all
ages into one harmonious whole; from the confused mass of legends of the
Middle Ages, he evolved a world of human interest and dazzling beauty,
fixed the kaleidoscopic picture of Italian society, and set it in the
richest frame of romance.
While he had the _Decameron_ still in hand, he paused in that great
work, with heart full of passionate longing for the lady of his love,
far away in Naples, to pour out his very soul in _La Fiammetta_, the
name by which he always called the Lady Maria. Of the real character of
this lady, so famous in literature, and her true relations with
Boccaccio, little that is certain is known. In several of his poems and
in the _Decameron_ he alludes to her as being cold as a marble statue,
which no fire can ever warm; and there is no proof, notwithstanding the
ardor of Fiammetta as portrayed by her lover--who no doubt wished her to
become the reality of his glowing picture--that he ever really received
from the charmer whose name was always on his lips anything more than
the friendship that was apparent to all the world. But she certainly
inspired him in the writing of his best works.
The best critics agree in pronouncing _La Fiammetta_ a marvelous
performance. John Addington Symonds says: "It is the first attempt in
any literature to portray subjective emotion exterior to the writer;
since the days of Virgil and Ovid, nothing had been essayed in this
region of mental analysis. The author of this extraordinary work proved
himself a profound anatomist of feeling by the subtlety with which he
dissected a woman's heart." The story is full of exquisite passages, and
it exercised a widespread and lasting influence over all the narrative
literature that followed it. It is so rich in material that it furnished
the motives of many tales, and the novelists of the sixteenth century
availed themselves freely of its suggestions.
After Boccaccio had taken up a permanent residence in Florence, he
showed a lively interest in her political affairs, and fulfilled all
the duties of a good citizen. In 1350 he was chosen to visit the lords
of various towns of Romagna, in order to engage their cooperation in a
league against the Visconti family, who, already lords of the great and
powerful city of Milan, desired to extend their domains beyond the
Apennines. In 1351 Boccaccio had the pleasure of bearing to the poet
Petrarch the news of the restoration of his rights of citizenship and of
his patrimony, both of which he had lost in the troubles of 1323, and
during this visit the two geniuses became friends for life. They delved
together into the literature of the ancients, and Boccaccio determined,
through the medium of translation, to make the work of the great Greek
writers a part of the liberal education of his countrymen. A knowledge
of Greek at that time was an exceedingly rare accomplishment, since the
serious study of living literatures was only just beginning, and the
Greek of Homer had been almost forgotten. Even Petrarch, whose erudition
was marvelous, could not read a copy of the _Iliad_ that he possessed.
Boccaccio asked permission of the Florentine Government to establish a
Greek professorship in the University of Florence, and persuaded a
learned Calabrian, Leonzio Pilato, who had a perfect knowledge of
ancient Greek, to leave Venice and accept the professorship at Florence,
and lodged him in his own house. Together the Calabrian and the author
of _La Fiammetta_ and the _Decameron_ made a Latin translation of the
_Iliad_, which Boccaccio transcribed with his own hand. But his literary
enthusiasm was not confined to his own work and that of the ancients.
His soul was filled with a generous ardor of admiration for Dante;
through his efforts the Florentines were awakened to a true sense of the
merits of the sublime poet, so long exiled from his native city, and the
younger genius succeeded in persuading them to establish a professorship
in the University for the sole study of the _Divine Comedy_, he himself
being the first to occupy the chair, and writing a _Life of Dante_,
besides commentaries on the _Comedy_ itself.
Mainly through his intimacy with the spiritual mind of Petrarch,
Boccaccio's moral character gradually underwent a change from the
reckless freedom and unbridled love of pleasure into which he had easily
fallen among his associates in the court life at Naples. He admired the
delicacy and high standard of honor of his friend, and became awakened
to a sense of man's duty to the world and to himself. During the decade
following the year 1365 he occupied himself at his home in Certaldo,
near Florence, with various literary labors, often entertaining there
the great men of the world.
Petrarch's death occurred in 1374, and Boccaccio survived him but one
year, dying on the twenty-first of December, 1375. He was buried in
Certaldo, in the Church of San Michele e Giacomo.
That one city should have produced three such men as the great
triumvirate of the fourteenth century--Dante, Petrarch, Boccaccio--and
that one half-century should have witnessed their successive triumphs,
is the greatest glory of Florence, and is one of the most notable facts
in the history of genius.
We quote once more from Symonds: "Dante brought the universe into his
_Divine Comedy_. 'But the soul of man, too, is a universe', and of this
inner microcosm Petrarch was the poet and genius. It remained for
Boccaccio to treat of daily life with an art as distinct and dazzling as
theirs. From Dante's Beatrice, through Petrarch's Laura, to Boccaccio's
La Fiammetta--from woman as an allegory of the noblest thoughts and
purest stirrings of the soul, through woman as the symbol of all beauty
worshiped at a distance, to woman as man's lover, kindling and
reciprocating the most ardent passion; from mystic, stately periods to
Protean prose; from verse built up into cathedral-like dignity, through
lyrics light as arabesques and pointed with the steely touch of polished
style, to that free form of speech which takes all moods and lends
itself alike to low or lofty things--such was the rapid movement of
Italian genius within the brief space of fifty years. So quickly did the
Renaissance emerge from the Middle Ages; and when the voices of that
august trio were silenced in the grave, their echoes ever widened and
grew louder through the spacious time to come."
No translation into English of _La Fiammetta_ has been made since
Shakespeare's time--when a small edition was published, which is now so
rare as to be practically unattainable--until the appearance of the
present Scholarly and poetic rendering, which places within the reach of
all one of the world's greatest masterpieces of literature.
D.K.R.
PROLOGUE
_Beginneth the Book called Elegy of Madonna Fiammetta, sent by her to
Ladies in Love._
When the wretched perceive or feel that their woes arouse compassion,
their longing to give vent to their anguish is thereby increased. And
so, since, from long usance, the cause of my anguish, instead of growing
less, has become greater, the wish has come to me, noble ladies--in
whose hearts, mayhap, abides a love more fortunate than mine--to win
your pity, if I may, by telling the tale of my sorrows. Nor is it at all
my intent that these my words should come to the ears of men. Nay,
rather would I, so far as lies in my power, withhold my complaints from
them; for, such bitterness has the discovery of the unkindness of one
man stirred in me, that, imagining all other men to be like him,
methinks I should be a witness of their mocking laughter rather than of
their pitying tears. You alone do I entreat to peruse my story, knowing
full well that you will feel with me, and that you have a pious concern
for others' pangs. Here you will not find Grecian fables adorned with
many lies, nor Trojan battles, foul with blood and gore, but amorous
sentiments fed with torturing desires. Here will appear before your very
eyes the dolorous tears, the impetuous sighs, the heart-breaking words,
the stormy thoughts, which have harrowed me with an ever-recurring goad,
and have torn away from me sleep and appetite and the pleasant times of
old, and my much-loved beauty. When you behold these things, and behold
them with the ardent feelings which ladies are wont to have, sure I am
that the cheeks of each separately, and of all when brought together,
will be bathed in tears, because of those ills which are alone the
occasion of my never-ending misery. Do not, I beseech you, refuse me
these tears, reflecting that your estate is unstable as well as mine,
and that, should it ever come to resemble mine (the which may God
forfend!), the tears that others shed for you will be pleasing to you in
return. And that the time may pass more rapidly in speaking than in,
weeping, I will do my best to fulfil my promise briefly, beginning with
that love which was more happy than lasting, so that, by comparing that
happiness with my present case, you may learn that I am now more unhappy
than any woman ever has been. And afterward I will trace with mournful
pen, as best I can, all the agonies which are justly the source of my
lamentations. But first, if the prayers of the wretched are heard, if
there is in Heaven any Deity whose holy mind can be touched with
compassion for me, afflicted as I am, bathed in my own tears, Him I
beseech to aid my despondent memory and support my trembling hand in its
present task. So may the tortures which I have felt and still feel in my
soul become fruitful, and the memory will suggest the words for them,
and the hand, more eager than apt for such duty, will write them down.
Chapter I
_Wherein the lady describes who she was, and by what signs her
misfortunes were foreshadowed, and at what time, and where, and in what
manner, and of whom she became enamored, with the description of the
ensuing delight._
In the time when the newly-vestured earth appears more lovely than
during all the rest of the year came I into the world, begotten of noble
parents and born amid the unstinted gifts of benignant fortune. Accursed
be the day, to me more hateful than any other, on which I was born! Oh,
how far more befitting would it have been had I never been born, or had
I been carried from that luckless womb to my grave, or had I possessed a
life not longer than that of the teeth sown by Cadmus, or had Atropos
cut the thread of my existence at the very hour when it had begun! Then,
in earliest childhood would have been entombed the limitless woes that
are the melancholy occasion of that which I am writing. But what boots
it to complain of this now? I am here, beyond doubt; and it has pleased
and even now pleases God that I should be here. Born and reared, then,
amid boundless affluence, I learned under a venerable mistress whatever
manners and refinements it beseems a demoiselle of high rank to know.
And as my person grew and developed with my increasing years, so also
grew and developed my beauty. Alas! even while a child, on hearing that
beauty acclaimed of many, I gloried therein, and cultivated it by
ingenious care and art. And when I had bidden farewell to childhood, and
had attained a riper age, I soon discovered that this, my beauty
--ill-fated gift for one who desires to live virtuously!--had power to
kindle amorous sparks in youths of my own age, and other noble persons
as well, being instructed thereupon by nature, and feeling that love can
be quickened in young men by beauteous ladies. And by divers looks and
actions, the sense of which I did but dimly discern at the time, did
these youths endeavor in numberless ways to kindle in my heart the fire
wherewith their own hearts glowed--fire that was destined, not to warm,
but rather to consume me also in the future more than it ever has burned
another woman; and by many of these young men was I sought in marriage
with most fervid and passionate entreaty. But after I had chosen among
them one who was in every respect congenial to me, this importunate
crowd of suitors, being now almost hopeless, ceased to trouble me with
their looks and attentions. I, therefore, being satisfied, as was meet,
with such a husband, lived most happily, so long as fervid love, lighted
by flames hitherto unfelt, found no entrance into my young soul. Alas! I
had no wish unsatisfied; nothing that could please me or any other lady
ever was denied me, even for a moment. I was the sole delight, the
peculiar felicity of a youthful spouse, and, just as he loved me, so did
I equally love him. Oh, how much happier should I have been than all
other women, if the love for him that was then in my heart had endured!
It was, then, while I was living in sweet content, amid every kind of
enjoyment, that Fortune, who quickly changes all things earthly,
becoming envious of the very gifts which she herself had bestowed,
withdrew her protecting hand. At first uncertain in what manner she
could succeed in poisoning my happiness, she at length managed, with
subtle craft, to make mine own very eyes traitors and so guide me into
the path that led to disaster. But the gods were still propitious to me,
nay, were even more concerned for my fate than I myself. Having seen
through her veiled malice, they wished to supply me with weapons, had I
but known how to avail me thereof, wherewith I might fend my breast,
and not go unarmed to the battle wherein I was destined to fall. Yea, on
the very night that preceded the day which was the beginning of all my
woes, they revealed to me the future in my sleep by means of a clear and
distinct vision, in such wise as follows:
While lying on my spacious couch, with all my limbs relaxed in deepest
slumber, I seemed to be filled with greater joy than I had ever felt
before, and wherefore I knew not. And the day whereon this happened was
the brightest and loveliest of days. I was standing alone in verdant
grass, when, with the joy whereof I spoke, came the thought to me that
it might be well for me to repose in a meadow that appeared to be
shielded from the fervid rays of the sun by the shadows cast by various
trees newly garbed in their glossy foliage. But first, gathering divers
flowers, wherewith the whole sward was bejeweled, I placed them, with my
white hands, in a corner of my robe, and then, sitting down and choosing
flower after flower, I wove therefrom a fair garland, and adorned my
head with it. And, being so adorned, I arose, and, like unto Proserpine
at what time Pluto ravished her from her mother, I went along singing in
this new springtime. Then, being perchance weary, I laid me down in a
spot where the verdure was deepest and softest. But, just as the tender
foot of Eurydice was pierced by the concealed viper, so meseemed that a
hidden serpent came upon me, as I lay stretched on the grass, and
pierced me under the left breast. The bite of the sharp fang, when it
first entered, seemed to burn me. But afterward, feeling somewhat
reassured, and yet afraid of something worse ensuing, I thought I
clasped the cold serpent to my bosom, fancying that by communicating to
it the warmth of that bosom, I should thereby render it more kindly
disposed in my regard in return for such a service. But the viper, made
bolder and more obdurate by that very favor, laid his hideous mouth on
the wound he had given me, and after a long space, and after it had
drunk much of my blood, methought that, despite my resistance, it drew
forth my soul; and then, leaving my breast, departed with it. And at the
very moment of the serpent's departure the day lost its brightness, and
a thick shadow came behind me and covered me all over, and the farther
the serpent crept, the more lowering grew the heavens, and it seemed
almost as if the reptile dragged after it in its course the masses of
thick, black clouds that appeared to follow in its wake, Not long
afterward, just as a white stone flung into deep water gradually
vanishes from the eyes of the beholder, so it, too, vanished from my
sight. Then the heavens became darker and darker, and I thought that the
sun had suddenly withdrawn and night had surely returned, as it had
erstwhile returned to the _Greeks_ because of the crime of Atrcus. Next,
flashes of lightning sped swiftly along the skies, and peals of crashing
thunder appalled the earth and me likewise. And through all, the wound
made in my breast by the bite of the serpent remained with me still, and
full of viperous poison; for no medicinal help was within my reach, so
that my entire body appeared to have swollen in a most foul and
disgusting manner. Whereupon I, who before this seemed to be without
life or motion--why, I do not know--feeling that the force of the venom
was seeking to reach my heart in divers subtle ways, now tossed and
rolled upon the cool grass, expecting death at any moment. But methought
that when the hour of my doom arrived, I was struck with terror at its
approach, and the anguish of my heart was so appalling, while looking
forward to its coming, that my inert body was convulsed with horror, and
so my deep slumber was suddenly broken. No sooner was I fully awake
than, being still alarmed by the things I had seen, I felt with my right
hand for the wound in my breast, searching at the present moment for
that which was already being prepared for my future misery. Finding that
no wound was there, I began to feel quite safe and even merry, and I
made a mock of the folly of dreams and of those who believe in them,
and so I rendered the work of the gods useless. Ah, wretched me! if I
mocked them then, I had good reason to believe in them afterward, to my
bitter sorrow and with the shedding of useless tears; good reason had I
also to complain of the gods, who reveal their secrets to mortals in
such mystic guise that the things that are to happen in the future can
hardly be said to be revealed at all. Being then fully awake, I raised
my drowsy head, and, as soon as I saw the light of the new-risen sun
enter my chamber, laying aside every other thought directly, I at once
left my couch.
That day, too, was a day of the utmost solemnity for almost everyone.
Therefore, attiring myself carefully in glittering cloth of gold, and
adorning every part of my person with deft and cunning hand, I made
ready to go to the August festival, appareled like unto the goddesses
seen by Paris in the vale of Ida. And, while I was lost in admiration of
myself, just as the peacock is of his plumage, imagining that the
delight which I took in my own appearance would surely be shared by all
who saw me, a flower from my wreath fell on the ground near the curtain
of my bed, I know not wherefore--perhaps plucked from my head by a
celestial hand by me unseen. But I, careless of the occult signs by
which the gods forewarn mortals, picked it up, replaced it on my head,
and, as if nothing portentous had happened, I passed out from my abode.
Alas! what clearer token of what was to befall me could the gods have
given me? This should have served to prefigure to me that my soul, once
free and sovereign of itself, was on that day to lay aside its
sovereignty and become a slave, as it betided. Oh, if my mind had not
been distempered, I should have surely known that to me that day would
be the blackest and direst of days, and I should have let it pass
without ever crossing the threshold of my home! But although the gods
usually hold forth signs whereby those against whom they are incensed
may be warned, they often deprive them of due understanding; and thus,
while pointing out the path they ought to follow, they at the same time
sate their own anger. My ill fortune, then, thrust me forth from my
house, vain and careless that I was; and, accompanied by several ladies,
I moved with slow step to the sacred temple, in which the solemn
function required by the day was already celebrating. Ancient custom, as
well as my noble estate, had reserved for me a prominent place among the
other ladies. When I was seated, my eyes, as was my habit of old,
quickly wandered around the temple, and I saw that it was crowded with
men and women, who were divided into separate groups. And no sooner was
it observed that I was in the temple than (even while the sacred office
was going on) that happened which had always happened at other times,
and not only did the men turn their eyes to gaze upon me, but the women
did the same, as if Venus or Minerva had newly descended from the skies,
and would never again be seen by them in that spot where I was seated.
Oh, how often I laughed within my own breast, being enraptured with
myself, and taking glory unto myself because of such things, just as if
I were a real goddess! And so, nearly all the young gentlemen left off
admiring the other ladies, and took their station around me, and
straightway encompassed me almost in the form of a complete circle; and,
while speaking in divers ways of my beauty, each finished his praises
thereof with well-nigh the same sentences. But I who, by turning my eyes
in another direction, showed that my mind was intent on other cares,
kept my ears attentive to their discourse and received therefrom much
delectable sweetness; and, as it seemed to me that I was beholden to
them for such pleasure, I sometimes let my eyes rest on them more kindly
and benignantly. And not once, but many times, did I perceive that some
of them, puffed up with vain hopes because of this, boasted foolishly of
it to their companions.
While I, then, in this way looked at a few, and that sparingly, I was
myself looked at by many, and that exceedingly, and while I believed
that my beauty was dazzling others, it came to pass that the beauty of
another dazzled me, to my great tribulation. And now, being already
close on the dolorous moment, which was fated to be the occasion either
of a most assured death or of a life of such anguish that none before me
has ever endured the like, prompted by I know not what spirit, I raised
my eyes with decent gravity, and surveyed with penetrating look the
crowds of young men who were standing near me. And I discerned, more
plainly than I saw any of the others, a youth who stood directly in
front of me, all alone, leaning against a marble column; and, being
moved thereto by irresistible fate, I began to take thought within my
mind of his bearing and manners, the which I had never before done in
the case of anyone else. I say, then, that, according to my judgment,
which was not at that time biased by love, he was most beautiful in
form, most pleasing in deportment, and apparently of an honorable
disposition. The soft and silky locks that fell in graceful curls beside
his cheeks afforded manifest proof of his youthfulness. The look
wherewith he eyed me seemed to beg for pity, and yet it was marked by
the wariness and circumspection usual between man and man. Sure I am
that I had still strength enough to turn away my eyes from his gaze, at
least for a time; but no other occurrence had power to divert my
attention from the things already mentioned, and upon which I had deeply
pondered. And the image of his form, which was already in my mind,
remained there, and this image I dwelt upon with silent delight,
affirming within myself that those things were true which seemed to me
to be true; and, pleased that he should look at me, I raised my eyes
betimes to see whether he was still looking at me. But anon I gazed at
him more steadily, making no attempt to avoid amorous snares. And when I
had fixed my eyes on his more intently than was my wont, methought I
could read in his eyes words which might be uttered in this wise:
"O lady, thou alone art mine only bliss!"
Certainly, if I should say that this idea was not pleasing to me, I
should surely lie, for it drew forth a gentle sigh from my bosom,
accompanied by these words: "And thou art mine!" unless, perchance, the
words were but the echo of his, caught by my mind and remaining within
it. But what availed it whether such words were spoken or not? The heart
had good understanding within itself of that which was not expressed by
the lips, and kept, too, within itself that which, if it had escaped
outside, might, mayhap, have left me still free. And so, from that time
forward, I gave more absolute liberty to my foolish eyes than ever they
had possessed before, and they were well content withal. And surely, if
the gods, who guide all things to a definite issue, had not deprived me
of understanding, I could still have been mistress of myself. But,
postponing every consideration to the last one that swayed me, I took
delight in following my unruly passion, and having made myself meet, all
at once, for such slavery, I became its thrall. For the fire that leaped
forth from his eyes encountered the light in mine, flashing thereunto a
most subtle ray. It did not remain content therewith, but, by what
hidden ways I know not, penetrated directly into the deepest recesses of
my heart; the which, affrighted by the sudden advent of this flame,
recalled to its center its exterior forces and left me as pale as
death, and also with the chill of death upon me. But not for long did
this continue, rather it happened contrariwise; and I felt my heart not
only glow with sudden beat, but its forces speeded back swiftly to their
places, bringing with them a throbbing warmth that chased away my pallor
and flushed my cheeks deeply; and, marveling wherefore this should
betide, I sighed heavily; nor thereafter was there other thought in my
soul than how I might please him.
In like fashion, he, without changing his place, continued to scrutinize
my features, but with the greatest caution; and, perhaps, having had
much practice in amorous warfare, and knowing by what devices the
longed-for prey might be captured, he showed himself every moment more
humble, more desperate, and more fraught with tender yearning. Alas! how
much guile did that seeming desperation hide, which, as the result has
now shown, though it may have come from the heart, never afterward
returned to the same, and made manifest later that its revealment on the
face was only a lure and a delusion! And, not to mention all his deeds,
each of which was full of most artful deception, he so wrought upon me
by his own craft, or else the fates willed it should so happen, that I
straightway found myself enmeshed in the snares of sudden and
unthought-of love, in a manner beyond all my powers of telling, and so I
remain unto this very hour.
It was this one alone, therefore, most pitiful ladies, that my heart, in
it mad infatuation, chose, not only among so many high-born, handsome
and valiant youths then present, but even among all of the same degree
having their abode in my own Parthenope, as first and last and sole lord
of my life. It was this one alone that I loved, and loved more than any
other. It was this one alone that was destined to be the beginning and
source of my by any pleasure, although often tempted, being at last
vanquished, have burned and now burn in the fire which then first caught
me. Omitting many thoughts that came into my mind, and many things that
were told me, I will only say that, intoxicated by a new passion, I
returned with a soul enslaved to that spot whence I had gone forth in
freedom.
When I was in my chamber, alone and unoccupied, inflamed with various
wild wishes, filled with new sensations and throbbing with many
anxieties, all of which were concentrated on the image of the youth who
pleased me, I argued within myself that if I could not banish love from
my luckless bosom, I might at least be able to keep cautious and secret
control of it therein; and how hard it is to do such a thing, no one can
discover who does not make trial of the same. Surely do I believe that
not even Love himself can cause so great anguish as such an attempt is
certain to produce. Furthermore, I was arrested in my purpose by the
fact that I had no acquaintance with him of whom I professed myself
enamored. To relate all the thoughts that were engendered in me by this
love, and of what nature they were, would take altogether too much time.
But some few I must perforce declare, as well as certain things that
were beginning to delight me more than usual. I say, then, that,
everything else being neglected, the only thing that was dear to me was
the thought of my beloved, and, when it occurred to my mind that, by
persevering in this course, I might, mayhap, give occasion to some one
to discover that which I wished to conceal, I often upbraided myself for
my folly. But what availed it all? My upbraidings had to give way to my
inordinate yearning for him, and dissolved uselessly into thin air.
For several days I longed exceedingly to learn who was the youth I
loved, toward whom my thoughts were ever clearly leading me; and this I
craftily learned, the which filled me with great content. In like
manner, the ornaments for which I had before this in no way cared, as
having but little need thereof, began to be dear to me, thinking that
the more I was adorned the better should I please. Wherefore I prized
more than hitherto my garments, gold, pearls, and my other precious
things. Until the present moment it had been my custom to frequent
churches, gardens, festivals, and seaside resorts, without other wish
than the companionship of young friends of my own sex; now, I sought the
aforesaid places with a new desire, believing that both to see and be
seen would bring me great delectation. But, in sooth, the trust which I
was wont to place in my beauty had deserted me, and now I never left my
chamber, without first seeking the faithful counsel of my mirror: and my
hands, newly instructed thereunto by I know not what cunning master,
discovering each day some more elegant mode of adornment than the day
before, and deftly adding artificial charms to my natural loveliness,
thereby caused me to outshine all the other ladies in my surpassing
splendor. Furthermore, I began to wish for the honors usually paid to me
by ladies, because of their gracious courtesy, though, perhaps, they
were rather the guerdon of my noble birth, being due to me therefor,
thinking that if I appeared so magnificent to my beloved's eyes, he
would take the more delight in beholding me. Avarice, too, which is
inborn in women, fled from me, so that I became free and openhanded, and
regarded my own possessions almost as if they were not my own. The
sedateness that beseems a woman fell away from me somewhat, and I grew
bolder in my ways; and, in addition to all this, my eyes, which until
that day looked out on the world simply and naturally, entirely changed
their manner of looking, and became so artful in their office that it
was a marvel. And many other alterations appeared in me over and above
these, all of which I do not care to relate, for besides that the
report thereof would be too tedious, I ween full well that you, like me,
also have been, or are, in love, and know what changes take place in
those who are in such sad case.
He was a most wary and circumspect youth, whereunto my experience was
able to bear witness frequently. Going very rarely, and always in the
most decorous manner, to the places where I happened to be, he used to
observe me, but ever with a cautious eye, so that it seemed as if he had
planned as well as I to hide the tender flames that glowed in the
breasts of both. Certainly, if I denied that love, although it had
clutched every corner of my heart and taken violent possession of every
recess of my soul, grew even more intense whenever it happened that my
eyes encountered his, I should deny the truth; he added further fuel to
the fires that consumed me, and rekindled such as might be expiring, if,
mayhap, there were any such. But the beginning of all this was by no
means so cheerful as the ending was joyless, as soon as I was deprived
of the sight of this, my beloved, inasmuch as the eyes, being thus
robbed of their delight, gave woful occasion of lamentation to the
heart, the sighs whereof grew greater in quality as well as in quantity,
and desire, as if seizing my every feeling, took me away from myself,
and, as if I were not where I was, I frequently gave him who saw me
cause for amazement by affording numberless pretexts for such
happenings, being taught by love itself. In addition to this, the quiet
of the night and the thoughts on which my fancy fed continuously, by
taking me out of myself, sometimes moved me to actions more frantic than
passionate and to the employment of unusual words.
But it happened that while my excess of ornaments, heartfelt sighs, lost
rest, strange actions, frantic movements, and other effects of my recent
love, attracted the notice of the other domestics of the household, they
especially struck with wonder a nurse of mine, old in years and
experienced, and of sound judgment, who, though well aware of the flames
that tortured my breast, yet making show of not knowing thereof,
frequently chided me for my altered manners. One day in particular,
finding me lying disconsolate on my couch, seeing that my brow was
charged with doleful thoughts, and believing that we were not likely to
be interrupted by other company, she began to speak as follows:
"My dearest daughter, whom I love as my very self, tell me, I pray you,
what are the sorrows that have for some time past been harassing you?
You who were wont to be so gay formerly, you whom I have never seen
before with a mournful countenance, seem to me now to be the prey of
grief and to let no moment pass without a sigh."
Then, having at first feigned to be asleep and not to have heard her, I
heaved a deep sigh, and, my face, at one time flushing, at another
turning pale, I tossed about on the couch, seeking what answer I should
make, though, indeed, in my agitation, my tongue could hardly shape a
perfect sentence. But, at length, I answered:
"Indeed, dear nurse, no fresh sorrows harass me; nor do I feel that I am
in any way different from what I am wont to be. Perhaps some troubles I
may have, but they are such as are incidental to all women."
"Most certainly, you are trying to deceive me, my child," returned the
aged nurse, "and you seem not to reflect how serious a matter it is to
attempt to lead persons of experience to believe one thing because it is
couched in words and to disbelieve the opposite, although it is made
plainly evident by deeds. There is no reason why you should hide from me
a fact whereof I have had perfect knowledge since several days ago."
Alas! when I heard her speak thus, provoked and stung by her words, I
said:
"If, then, thou wittest of all this, wherefore dost thou question me?
All that thou hast to do now is to keep secret that which thou hast
discovered."
"In good truth," she replied, "I will conceal all that which it is not
meet that another should know, and may the earth open and engulf me in
its bowels before I ever reveal aught that might turn to thy open shame!
Therefore, do thou live assured of this, and guard thyself carefully
from letting another know that which I, without either thyself or anyone
else telling me, have learned from observing thy looks. As for myself,
it is not now, but long ere now, that I have learned to keep hidden that
which should not be disclosed. Therefore, do thou continue to feel
secure as to this matter, and watch most carefully that thou lettest not
another know that which I, not witting it from thee or from another,
most surely have discovered from thine own face and from its changeful
seeming. But, if thou art still the victim of that folly by which I know
thou hast been enslaved, if thou art as prone now as erewhile to indulge
that feeling to which thou hast already given way, then know I right
well that I must leave thee to thy own devices, for bootless will be my
teachings and my warnings. Still, although this cruel tyrant, to whom in
thy youthful simplicity being taken by surprise thou hast yielded thy
freedom, appears to have deprived thee of understanding as well as of
liberty, I will put thee in mind of many things, and entreat thee to
fling off and banish wicked thoughts from thy chaste bosom, to quench
that unholy fire, and not to make thyself the thrall of unworthy hopes.
Now is the time to be strong in resistance; for whoso makes a stout
fight in the beginning roots out an unhallowed affection, and bears
securely the palm of victory; but whoso, with long and wishful fancies,
fosters it, will try too late to resist a yoke that has been submitted
to almost unresistingly."
"Alas!" I replied, "how far easier it is to say such things than to
lead them to any good result."
"Albeit they be not easy of fulfilment," she said, "yet are they
possible, and they are things that it beseems you to do. Take thou
thought whether it would be fitting that for such a thing as this thou
shouldst lose the luster of thy exalted parentage, the great fame of thy
virtue, the flower of thy beauty, the honor in which thou art now held,
and, above all, the favor of the spouse whom thou hast loved and by whom
thou art loved: certainly, thou shouldst not wish for this; nor do I
believe thou wouldst wish it, if thou didst but weigh the matter
seriously in thine own mind. Wherefore, in the name of God, forbear, and
drive from thy heart the false delights promised by a guilty hope, and,
with them, the madness that has seized thee. By this aged breast, long
harassed by many cares, from which thou didst take thy first nutriment,
I humbly beseech thee to have the courage to aid thyself, to have a
concern for thine own honor, and not to disdain my warnings. Bethink
thee that the very desire to be healed is itself often productive of
health."
Whereto I thus made answer:
"Only too well do I know, dear nurse, the truth of that which thou
sayest. But a furious madness constrains me to follow the worse course;
vainly does my heart, insatiable in its desires, long for strength to
enable it to adopt thy advice; what reason enjoins is rendered of no
avail by this soul-subduing passion. My mind is wholly possessed by
Love, who rules every part thereof, in virtue of his all-embracing
deity; and surely thou art aware that his power is absolute, and 'twere
useless to attempt to resist it."
Having said these words, I became almost unconscious, and fell into her
arms. But she, now more agitated than before, in austere and rebuking
tones, said:
"Yes, forsooth, well am I aware that you and a number of fond young
women, inflamed and instigated thereunto by vain thoughts, have
discovered Love to be a god, whereas a juster name for him would be that
of demon; and you and they call him the son of Venus, and say that his
strength has come to him from the third heaven, wishing, seemingly, to
offer necessity as an excuse for your foolishness. Oh, was ever woman so
misled as thou? Truly, thou must be bereft entirely of understanding!
What a thing thou sayest! Love a deity! Love is a madness, thrust forth
from hell by some fury. He speeds across the earth in hasty flight, and
they whom he visits soon discover that he brings no deity with him, but
frenzy rather; yet none will he visit except those abounding overmuch in
earthly felicity; for they, he knows, in their overweening conceit, are
ready to afford him lodgment and shelter. This has been proven to us by
many facts. Do we not see that Venus, the true, the heavenly Venus,
often dwells in the humblest cot, her sole concern being the
perpetuation of our race? But this god, whom some in their folly name
Love, always hankering after things unholy, ministers only to those
whose fortunes are prosperous. This one, recoiling from those whose food
and raiment suffice to meet the demands of nature, uses his best efforts
to win over the pampered and the splendidly attired, and with their food
and their habiliments he mixes his poisons, and so gains the lordship of
their wicked souls; and, for this reason, he gladly seeks a harborage in
lofty palaces, and seldom, or rather never, enters the houses of the
lowly, because this horrible plague always resorts by choice to scenes
of elegance and refinement, well knowing that such places are best
fitted for the achievement of his fell purposes. It is easy for us to
see that among the humble the affections are sane and well ordered; but
the rich, on the other hand, everywhere pluming themselves on their
riches, and being insatiable in their pursuit of other things as well as
of wealth, always show more eagerness therein than is becoming; and they
who can do much desire furthermore to have the power of doing that which
they must not do: among whom I feel that thou hast placed thyself, O
most hapless of women, seeing that thou hast already entered and
traveled far on a path that will surely lead to guilt and misery."
After hearing which, I said:
"Be silent, old woman, and provoke not the wrath of the gods by thy
speech. Now that thou art incapacitated from love by age and rejected by
all the gods, thou railest against this one, blaspheming him in whom
thou didst erstwhile take delight. If other ladies, far more puissant,
famous, and wise than I, have formerly called him by that name, it is
not in my power to give him a name anew. By him am I now truly enslaved;
whatever be the cause of this, and whether it be the occasion of my
happiness or misery, I am helpless. The strength wherewith I once
opposed him has been vanquished and has abandoned me. Therefore either
death or the youth for whom I languish can alone end my tortures. If
thou art, then, as wise as I hold thee to be, bestow such counsel and
help on me as may lighten my anguish, or, at least, abstain from
exasperating it by censuring that to which my soul, unable to act
differently, is inclined with all its energy."
Thereupon, she, being angry, and not without reason, making no answer,
but muttering to herself, passed out of the chamber and left me alone.
When my dear nurse had departed without making further discourse, and I
was again alone, I felt that I had acted ill in despising her advice. I
revolved her sayings within my restless breast; and, albeit my
understanding was blinded, I perceived that what she had said was
replete with wisdom, and, almost repenting of what I had uttered and of
the course which I had declared I purposed taking, I was wavering in my
mind. And, already beginning to have thoughts of abandoning that course
which was sure to be in every way most harmful, I was about to call her
back to give me encouragement, when a new and unforeseen event suddenly
changed my intention. For a most beautiful lady, come to my private
chamber I know not whence, presented herself before my eyes, enveloped
in such dazzling light that scarcely could my sight endure the
brightness thereof. But while she stood still and silent before me, the
effulgent radiance that had almost blinded my vision, after a time left
it unobscured, and I was able so to portray her every aspect to my mind,
as her whole beauteous figure was impressed on my memory. I saw that she
was nude, except for a thin and delicate drapery of purple, which,
albeit in some parts it covered the milk-white body, yet no more
concealed it from my ravished eyes than does the transparent glass
conceal the portrait beneath it. Her head, the hair whereof as much
surpassed gold in its luster as gold surpasses the yellowest tresses to
be found among mortals, was garlanded with a wreath of green myrtle,
beneath whose shadow I beheld two eyes of peerless splendor, so
enchanting that I could have gazed on them forever; they flashed forth
such luminous beams that it was a marvel; and all the rest of her
countenance had such transcendent loveliness that the like never was
seen here below. At first she spake no word, perchance content that I
should look upon her, or perchance seeing me so content to look upon
her. Then gradually through the translucent radiance, she revealed more
clearly every hidden grace, for she was aware that I could not believe
such beauty possible except I beheld it with my eyes, and that even then
words would fail me to picture it to mortals with my tongue. At last,
when she observed that I had sated my eyes with gazing on her, and when
she saw that her coming hither was as wondrous to me as her loveliness,
with smiling face, and in a voice sweeter than can be conceived by minds
like ours, she thus addressed me:
"Prithee, young woman, what art thou, the most fickle of thy sex,
preparing to do in obedience to the late counsels of thy aged nurse?
Knowest thou not that such counsels are far harder to follow than that
very love which thou desirest to flee? Hast thou reflected on the dire
and unendurable torments which compliance with them will entail on thee?
O most insensate one! dost thou then, who only a few hours ago wert my
willing vassal, now wish to break away from my gentle rule, because,
forsooth, of the words of an old woman, who is no longer vassal of mine,
as if, like her, thou art now unwitting of what delights I am the
source? O most witless of women! forbear, and reflect whether thou
shouldst not find befitting happiness in that which makes the happiness
of Heaven and earth. All things that Phoebus beholds during the bright
day, from what time he emerges from Ganges, until he plunges with his
tired steeds into the Hesperian waves, to seek due repose after his
wearisome pilgrimage; all things that are confined between cold Arcturus
and the red-hot pole, all own the absolute and authentic lordship of my
winged son; and in Heaven not only is he esteemed a god, like the other
deities, but he is so much more puissant than them all that not one
remains who has not heretofore been vanquished by his darts. He, flying
on golden plumage throughout his realms, with such swiftness that his
passage can hardly be discerned, visits them all in turn, and, bending
his strong bow, to the drawn string he fits the arrows forged by me and
tempered in the fountains sacred to my divinity. And when he elects
anyone to his service, as being more worthy than others, that one he
rules as it likes him. He kindles raging fires in the hearts of the
young, fans the flames that are almost dead in the old, awakens the
fever of passion in the chaste bosoms of virgins and instils a genial
warmth into the breasts of wives and widows equally. He has even
aforetime forced the gods, wrought up to a frenzy by his blazing torch,
to forsake the heavens and dwell on earth under false appearances.
Whereof the proofs are many. Was not Phoebus, though victor over huge
Python and creator of the celestial strains that sound from the lyres of
Parnassus, by him made the thrall, now of Daphne, now of Clymene, and
again of Leucothea, and of many others withal? Certainly, this was so.
And, finally, hiding his brightness under the form of a shepherd, did
not Apollo tend the flocks of Admetus? Even Jove himself, who rules the
skies, by this god coerced, molded his greatness into forms inferior to
his own. Sometimes, in shape of a snow-white fowl, he gave voice to
sounds sweeter than those of the dying swan, and anon, changing to a
young bull and fitting horns to his brow, he bellowed along the plains,
and humbled his proud flanks to the touch of a virgin's knees, and,
compelling his tired hoofs to do the office of oars, he breasted the
waves of his brother's kingdom, yet sank not in its depths, but joyously
bore away his prize. I shall not discourse unto you of his pursuit of
Semele under his proper form, or of Alcmena, in guise of Amphitryon, or
of Callisto, under the semblance of Diana, or of Danae for whose sake he
became a shower of gold, seeing that in the telling thereof I should
waste too much time. Nay, even the savage god of war, whose strength
appalls the giants, repressed his wrathful bluster, being forced to such
submission by this my son, and became gentle and loving. And the forger
of Jupiter, and artificer of his three-pronged thunderbolts, though
trained to handle fire, was smitten by a shaft more potent than he
himself had ever wrought. Nay I, though I be his mother, have not been
able to fend off his arrows: Witness the tears I have shed for the death
of Adonis! But why weary myself and thee with the utterance of so many
words? There is no deity in heaven who has passed unscathed from his
assaults; except, perhaps, Diana only, who may have escaped him by
fleeing to the woods; though some there be who tell that she did not
flee, but rather concealed the wound. If haply, however, thou, in the
hardness of thy unbelief, rejectest the testimony of heaven, and
searchest rather for examples of those in this nether world who have
felt his power, I affirm them to be so multitudinous that where to begin
I know not. Yet this much may I tell thee truly: all who have confessed
his sway have been men of might and valor. Consider attentively, in the
first place, that undaunted son of Alcmena, who, laying aside his arrows
and the formidable skin of the huge lion, was fain to adorn his fingers
with green emeralds, and to smooth and adjust his bristling and
rebellions hair. Nay, that hand which aforetime had wielded the terrific
club, and slain therewith Antaeus, and dragged the hound of hell from the
lower world, was now content to draw the woolen threads spun from
Omphale's distaff; and the shoulders whereon had rested the pillars of
the heavens, from which he had for a time freed Atlas, were now clasped
in Omphale's arms, and afterward, to do her pleasure, covered with a
diaphanous raiment of purple. Need I relate what Paris did in obedience
to the great deity? or Helen? or Clytemnestra? or AEgisthus? These are
things that are well known to all the world. Nor do I care to speak of
Achilles, or of Scylla, of Ariadne or Leander, of Dido, or of many
others, of whom the same tale could be told, were there need to tell it.
Believe me when I affirm that this fire is holy, and most potent as
well. Thou hast heard that heaven and earth are subject to my son
because of his lordship over gods and men. But what shall I say of the
power that he exercises over irrational animals, whether celestial or
terrene? It is through him that the turtle is fain to follow her mate;
it is through him that my pigeons have learned to caress his ringdoves
with fondest endearments. And there is no creeping or living creature
that has ever at any time attempted to escape from his puissance: in the
woods the timid stag, made fierce by his touch, becomes brave for sake
of the coveted hind and by bellowing and fighting, they prove how strong
are the witcheries of Love. The ferocious boars are made by Love to
froth at the mouth and sharpen their ivory tusks; the African lions,
when Love quickens them, shake their manes in fury. But leaving the
groves and forests, I assert that even in the chilly waters the
numberless divinities of the sea and of the flowing rivers are not safe
from the bolts of my son. Neither can I for a moment believe that thou
art ignorant of the testimony thereof which has been rendered by
Neptune, Glaucus, Alpheus, and others too numerous to mention: not only
were they unable to quench the flame with their dank waters, but they
could not even moderate its fury, which, when it had made its might
felt, both on the earth and in the waters, continued its onward course,
and rested not until it had penetrated into the gloomy realms of Dis.
Therefore Heaven and Earth and Ocean and Hell itself have had experience
of the potency of his weapons. And, in order that thou mayest understand
in a few words the power of the deity, I tell thee that, while
everything succumbs to nature, and nothing can ever be emancipated from
her dominion, Nature herself is but the servant of Love. When he
commands, ancient hatreds perish, and angry moods, be they old or new,
give place to his fires; and lastly, his sway has such far-reaching
influence that even stepmothers become gracious to their stepchildren, a
thing which it is a marvel to behold. Therefore what seekest thou? Why
dost thou hesitate? Why dost thou rashly avoid him? When so many gods,
when so many men, when so many animals, have been vanquished by him, art
ashamed to be vanquished by him also? In good sooth, thou weenest not
what thou art doing. If thou fearest to be blamed for thy obedience to
him, a blame so unmerited never can be thy portion. Greater sins than
thou canst commit have been committed by thousands far greater than
thou, and these sins would plead as thy excuse, shouldst thou pursue
that course which others have pursued--others who far excel thee. Thou
wilt have sinned but a little, seeing that thou hadst far less power of
resistance than those aforementioned. But if my words move thee not, and
thou wouldst still wish to withstand the god, bethink thee that thy
power falls far short of that of Jove, and that in judgment thou canst
not equal Phoebus, nor in wealth Juno, nor me in beauty; and yet, we all
have been conquered. Thou art greatly deceived, and I fear me that thou
must perish in the end, if thou persist in thy changed purpose. Let that
which has erstwhile sufficed for the whole world, suffice for thee, nor
try to render thyself cold-hearted, by saying: 'I have a husband, and
the holy laws and the vowed faith forbid me this'; for bootless are such
reasonings against the puissance of this god. He discards the laws of
others scornfully, as thinking them of no account, and ordains his own.
Pasiphae? had a husband, and Phaedra, and I, too, even though I have
loved. And it is these same husbands who most frequently fall in love
with others, albeit they have wives of their own: witness Jason and
Theseus and valiant Hector and Ulysses. Therefore to men we do no wrong
if we apply to them the same laws that they apply to others; for to
them no privilege has been granted which is not accorded to us withal.
Banish, then, thy foolish thoughts, and, in all security, go on loving
him whom thou hadst already begun to love. In good sooth, if thou
refusest to own the power of mighty Love, it behooves thee to fly; but
whither canst thou fly? Knowest thou of any retreat where he will not
follow and overtake thee? He has in all places equal puissance. Go
wheresoever thou wilt, never canst thou pass across the borders of his
realms, and within these realms vain it is for mortals to try to hide
themselves when he would smite them. But let it comfort thee to know,
young woman, that no such odious passion shall trouble thee as erstwhile
was the scourge of Myrrha, Semiramis, Byblis, Canace, and Cleopatra.
Nothing strange or new will be wrought by my son in thy regard. He has,
as have the other gods, his own special laws, which thou art not the
first to obey, and shouldst not be the last to entertain hopes
therefrom. If haply thou believest that thou art without companions in
this, foolish is thy belief. Let us pass by the other world, which is
fraught with such happenings; but observe attentively only thine own
city! What an infinite number of ladies it can show who are in the same
case with thyself! And remember that what is done by so many cannot be
deemed unseemly. Therefore, be thou of our following, and return thanks
to our beauty, which thou hast so closely examined. But return special
thanks to our deity, which has sundered thee from the ranks of the
simple, and persuaded thee to become acquainted with the delights that
our gifts bestow."
Alas! alas! ye tender and compassionate ladies, if Love has been
propitious to your desires, say what could I, what should I, answer to
such and so great words uttered by so great a goddess, if not: "Be it
done unto me according to thy pleasure"? And so, I affirm that as soon
as she had closed her lips, having already harvested within my
understanding all her words, and feeling that every word was charged
with ample excuse for what I might do, and knowing now how mighty she
was and how resistless, I resolved at once to submit to her guidance;
and instantly rising from my couch, and kneeling on the ground, with
humbled heart, I thus began, in abashed and tremulous accents:
"O peerless and eternal loveliness! O divinest of deities! O sole
mistress of all my thoughts! whose power is felt to be most invincible
by those who dare to try to withstand it, forgive the ill-timed
obstinacy wherewith I, in my great folly, attempted to ward off from my
breast the weapons of thy son, who was then to me an unknown divinity.
Now, I repeat, be it done unto me according to thy pleasure, and
according to thy promises withal. Surely, my faith merits a due reward
in time and space, seeing that I, taking delight in thee more than do
all other women, wish to see the number of thy subjects increase forever
and ever."
Hardly had I made an end of speaking these words, when she moved from
the place where she was standing, and came toward me. Then, her face
glowing with the most fervent expression of affection and sympathy, she
embraced me, and touched my forehead with her divine lips. Next, just as
the false Ascanius, when panting in the arms of Dido, breathed on her
mouth, and thereby kindled the latent flame, so did she breathe on my
mouth, and, in that wise, rendered the divine fire that slumbered in my
heart more uncontrollable than ever, and this I felt at that very
moment. Thereafter, opening a little her purple robe, she showed me,
clasped in her arms against her ravishing breast, the very counterpart
of the youth I loved, wrapped in the transparent folds of a Grecian
mantle, and revealing in the lineaments of his countenance pangs that
were not unlike those I suffered.
"O damsel," she said, "rivet thy gaze on the youth before thee: we have
not given thee for lover a Lissa, a Geta, or a Birria, or anyone
resembling them, but a person in every way worthy of being loved by
every goddess in the heavens. Thee he loves more than himself, as we
have ordained, and thee will he ever love; therefore do thou, joyfully
and securely, abandon thyself to his love. Thy prayers have moved us to
pity, as it is meet that prayers so deserving should, and so, be of good
hope, and fear not that thou shalt be without the reward due thee in the
future."
And thereafter she suddenly vanished from my eyes. _Oime!_ wretched me!
I do not for a moment doubt now, after considering the things which
followed, that this one who appeared unto me was not Venus, but rather
Tisiphone, who, doffing from her head the horrid snakes that served it
for hair, and assuming for the while the splendid form of the Goddess of
Love, in this manner lured me with deceitful counsels to that disaster
which at length overwhelmed me. Thus did Juno, but in different fashion,
veiling the radiance of her deity and transforming herself for the
occasion into the exact likeness of her aged nurse, persuaded Semele to
her undoing. Woe is me! my resolve to be so advised was the cause--O
hallowed Modesty! O Chastity, most sacred of all the virtues! sole and
most precious treasure of righteous women!--was the cause, I repeat,
wherefore I drove ye from my bosom. Yet do I venture to pray unto ye for
pardon, and surely the sinner who repents and perseveres in repentance
should in due season obtain your forgiveness.
Although the goddess had disappeared from my sight, my whole soul,
nevertheless, continued to crave her promised delights; and, albeit the
ardor of the passion that vexed my soul deprived me of every other
feeling, one piece of good fortune, for what deserving of mine I know
not, remained to me out of so many that had been lost--namely, the power
of knowing that seldom if ever has a smooth and happy ending been
granted to love, if that love be divulged and blazed abroad. And for
this reason, when influenced by my highest thoughts, I resolved,
although it was a most serious thing to do so, not to set will above
reason in carrying this my desire unto an ending. And assuredly,
although I have often been most violently constrained by divers
accidents to follow certain courses, yet so much grace was conceded to
me that, sustained by my own firmness, I passed through these agonies
without revealing the pangs that tortured me. And in sooth, I have still
resolution enough to continue to follow out this my purpose; so that,
although the things I write are most true, I have so disposed them that
no one, however keen his sagacity, can ever discover who I am, except
him who is as well acquainted with these matters as I, being, indeed,
the occasion of them all. And I implore him, should this little book
ever come into his hands, in the name of that love which he once bore
me, to conceal that which, if disclosed, would turn neither to his
profit nor honor. And, albeit he has deprived me of himself, and that
through no fault of mine, let him not take it upon himself to deprive me
of that honor which I still possess, although, perchance, undeservedly;
for should he do so, he could never again give it back to me, any more
than he can now give me back himself.
Having, therefore, formed my plans in this wise, I showed the most
long-suffering patience in manifesting my keenest and most covetous
yearnings, and I used my best efforts, but only in secret ways and when
opportunities were afforded me, to light in this young man's soul the
same flames wherewith my own soul glowed, and to make him as
circumspect as myself withal. Nor, in truth, was this for me a task of
great difficulty; for, inasmuch as the lineaments of the face always
bear most true witness to the qualities of the heart, it was not long
before I became aware that my desire would have its full fruition. I
perceived that, not only was he throbbing with amorous enthusiasm, but
that he was also imbued with most perfect discretion, and this was
exceedingly pleasing to me. He, being at once wishful to preserve my
honor in all its luster, and, at the same time, to arrange convenient
times and places for our meetings, employed many ingenious stratagems,
which, methinks, must have cost him much toil and trouble. He used every
subtle art to win the friendship of all who were related to me, and, at
last, of my husband; and not only did he enjoy their friendship, but he
possessed it in such a supreme degree that no pleasure was agreeable to
them unless he shared it. How much all this delighted me you will
understand without its being needful to me to set it down in words. And
is there anyone so dull of wit as not to conclude that from the
aforesaid friendship arose many opportunities for him and me of holding
discourse together in public? But already had he bethought himself of
acting in more subtle ways; and now he would speak to this one, now to
that one, words whereby I, being most eager for such enlightenment,
discovered that whatever he said to these was fraught with figurative
and hidden meanings, intended to show forth his ardent affection for
myself. When he was sensible that I had a clear perception of the occult
significance of his questions and answers, he went still further, and by
gestures, and mobile changes in the expression of his features, he would
make known to me his thoughts and the various phases of his passion,
which was to me a source of much delectation; and I strove so hard to
comprehend it all and to make fitting response thereunto, that neither
could he shadow forth anything to me, nor I to him, that either of us
did not at once understand.
Nay, not satisfied even with this, he employed other symbols and
metaphors, and labored earnestly to discipline me in such manner of
speech; and, to render me the more assured of his unalterable love, he
named me Fiammetta, and himself Panfilo. Woe is me! How often, when
warmed with love and wine, did we tell tales, in the presence of our
dearest friends, of Fiammetta and Panfilo, feigning that they were
Greeks of the days of old, I at one time, he at another; and the tales
were all of ourselves; how we were first caught in the snares of Love,
and of what tribulations we were long the victims, giving suitable names
to the places and persons connected with the story! Certainly, I
frequently laughed at it all, being made merry by the simplicity of the
bystanders, as well as by his astuteness and sagacity. Yet betimes I
dreaded that in the flush of his excitement he might thoughtlessly let
his tongue wander in directions wherein it was not befitting it should
venture. But he, being ever far wiser than I imagined, guarded himself
craftily from any such blundering awkwardness.
_Oime!_ most compassionate ladies, what is there that Love will not
teach to his subjects? and what is there that he is not able to render
them skilful in learning? I, who of all young women was the most
simple-minded, and ordinarily with barely power to loose my tongue, when
among my companions, concerning the most trivial and ordinary affairs,
now, because of this my affection, mastered so speedily all his modes of
speech that, in a brief space, my aptness at feigning and inventing
surpassed that of any poet! And there were few questions put to me in
response to which, after meditating on their main points, I could not
make up a pleasing tale: a thing, in my opinion, exceedingly difficult
for a young woman to begin, and still more difficult to finish and
relate afterward. But, if my actual situation required it, I might set
down numerous details which might, perhaps, seem to you of little or no
moment, as, for instance, the artful experiment whereby we tested the
fidelity of my favorite maid to whom, and to whom alone, we meditated
entrusting the secret of this hidden passion, considering that, should
another share it, our uneasiness, lest it should not be kept, would be
most grievous. Furthermore, it would weary you if I mentioned all the
plans we adopted, in order to meet divers situations, plans that I do
not believe were ever imagined by any before us; and albeit I am now
well aware that they all worked for my ultimate destruction, yet the
remembrance of them does not displease me.
Unless, O ladies, my judgment be greatly at fault, the strength of our
minds was by no means small, if it be but taken in account how hard a
thing it is for youthful persons in love to resist long the rush of
impetuous ardor without crossing the bounds set by reason: nay, it was
so great and of such quality that the most valiant of men, by acting in
such wise, would win high and worthy laud as a result thereof. But my
pen is now about to depict the final ending to which love was guided,
and, before I do so, I would appeal to your pity and to those soft
sentiments which make their dwelling in your tender breasts, and incline
your thoughts to a like termination.
Day succeeded day, and our wishes dragged along with them, kept alive by
torturing anxiety, the full bitterness whereof each of us experienced;
although the one manifested this to the other in disguised language, and
the other showed herself over-discreet to an excessive degree; all of
which you who know how ladies who are beloved behave in such
circumstances will easily understand. Well, then, he, putting full trust
in the veiled meaning of my words, and choosing the proper time and
place, came to an experience of that which I desired as much as he,
although I feigned the contrary. Certainly, if I were to say that this
was the cause of the love I felt for him, I should also have to confess
that every time it came back to my memory, it was the occasion to me of
a sorrow like unto none other. But, I call God to witness, nothing that
has happened between us had the slightest influence upon the love I bore
him, nor has it now. Still, I will not deny that our close intimacy was
then, and is now, most dear to me. And where is the woman so unwise as
not to wish to have the object of her affection within reach rather than
at a distance? How much more intensely does love enthrall us when it is
brought so near us that we and it are made almost inseparable! I say,
then, that after such an adventure, never afore willed or even thought
of by me, not once, but many times did fortune and our adroit stratagems
bring us good cheer and consolation, not indeed screened entirely from
danger, for which I cared less than for the passing of the fleeing wind.
But while the time was being spent in such joyous fashion--and that it
was joyous, Love, who alone may bear witness thereof, can truly say--yet
sometimes his coming inspired me with not a little natural apprehension,
inasmuch as he was beginning to be indiscreet in the manner of his
coming. But how dear to him was my own apartment, and with what gladness
did it see him enter! Yet was he filled with more reverence for it than
he ever had been for a sacred temple, and this I could at all times
easily discern. Woe is me! what burning kisses, what tender embraces,
what delicious moments we had there!
Why do I take such pleasure in the mere words which I am now setting
down? It is, I say, because I am forced to express the gratitude I then
felt to the holy goddess who was the promiser and bestower of Love's
delights. Ah, how often did I visit her altars and offer incense,
crowned with a garland of her favorite foliage! How often did I think
scornfully of the counsels of my aged nurse! Nay, furthermore, being
elated far more than all my other companions, how often did I disparage
their loves, saying within myself: "No one is loved as I am loved, no
one loves a youth as matchless as the youth I love, no one realizes such
delights from love as I!" In short, I counted the world as nothing in
comparison with my love. It seemed to me that my head touched the skies,
and that nothing was lacking to the culmination of my ecstatic bliss.
Betimes the idea flashed on my mind that I must disclose to others the
occasion of my transports, for surely, I would reflect, it would be a
delight to others to hear of that which has brought such delight to me!
But thou, O Shame, on the one side, and thou, O Fear, on the other, did
hold me back: the one threatening me with eternal infamy; the other with
loss of that which hostile Fortune was soon afterward to tear from me.
In such wise then, did I live for some time, for it was then pleasing to
Love that I should live in this manner; and, in good sooth, so blithely
and joyously were these days spent that I had little cause to envy any
lady in the whole world, never imagining that the delight wherewith my
heart was filled to overflowing, was to nourish the root and plant of my
future misery, as I now know to my fruitless and never-ending sorrow.
End of the Project Gutenberg EBook of La Fiammetta, by Giovanni Boccaccio
***
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QuoteWizard Insurance News
The Insurance Implications of the Supreme Court's Ruling on Obamacare Subsidies
Posted on: August 17, 2015 | by Bryan Ochalla
In late June, the Supreme Court ruled that the Affordable Care Act (ACA) can continue to provide tax subsidies to Americans who need help when they go to buy health insurance regardless of whether or not the state they call home operates its own exchange.
Specifically, the 6-to-3 ruling addressed concerns about a phrase in the law, often referred to as Obamacare, that said these subsidies could only be made available to those below a certain income threshold who are looking to obtain insurance via "an exchange established by the state."
"For people buying insurance on the federal exchange—and that is millions of people—[this decision means they will continue to be] eligible for tax credits," explains Margaret Riley, a professor of law at the University of Virginia. It also means "the difference between insurance costs being affordable and insurance costs that are a real burden."
"Even for those not directly affected," she adds, "the ruling gives stability, and stability in the insurance markets are good for both insurers and the insured."
A decision favoring the plaintiffs—the four Virginia residents who felt the wording above should prohibit the federal government from providing subsidies to consumers who live in states that lack their own exchanges—could have put the Obamacare law as a whole into jeopardy. The reason: nearly three dozen states currently piggyback off of the federal marketplace or exchange--as opposed to maintaining their own—and about 6.4 million people in those 34 or so states rely on subsidies worth approximately $1.7 billion a month to make their health coverage affordable.
As such, Democrats and other Obamacare supporters welcomed the ruling with a huge sigh of relief—especially since this is the second time in three years that this particular piece of legislation has been challenged at this level. (The earlier case, resolved back in 2012, focused on the "individual mandate" portion of the law, with the Supreme Court's justices declaring, by a 5-4 margin, that it was constitutional.)
Should you feel similarly relieved about this decision even if you're a regular citizen as opposed to an elected official of some sort or other? You should be able to answer that question for yourself after you read through the following information related to what the latest ACA ruling does or could or mean for American consumers in terms of their ability to obtain or maintain health coverage:
If you have health insurance through your employer, nothing's likely to change
That may not have been the case if the Supreme Court had decided in favor of the plaintiffs and as a result stifled the portion of Obamacare that allowed the federal government to supply subsidies to insurance customers in need.
How so? Well, had Obamacare been gutted in such a way, that action could have undermined and even unraveled the law in a few serious ways—one of which likely would have resulted in businesses in a number of states dropping the health insurance coverage they provide to hundreds of thousands of employees, according to a recent Time report.
That's due to one specific section of the ACA that requires employers to provide health insurance to their staffers or pay a penalty. They're only hit with such a fine if they fail to meet two criteria, though—one of which is that they provide quality health coverage to 95 percent of their full-time employees, and the second of which is that at least one of those employees buys a plan on the marketplace and receives a subsidy. In other words, if a state doesn't offer its citizens subsidies, its employers can't be fined if they don't provide adequate health insurance to their employees.
If you obtained health insurance through the federal marketplace or a state exchange, you're also unlikely to see any changes to the cost or amount of your coverage
This is true even if you currently receive a subsidy that makes your insurance policy more affordable.
Which makes sense, as the Supreme Court's ruling means that the particulars of the Affordable Care Act will remain as they've been since it was signed into law more than five years ago.
So, as has been the case since 2010, if you make more than a certain amount of money per year, and if you're not experiencing any of a number of hardships or if you're not someone who fits into any number of categories that are exempted from this so-called individual mandate, you have to buy a certain amount of health insurance or pay a fine.
In both cases, you could say that the recent Supreme Court decision has helped keep health insurance premiums from skyrocketing
Don't take our word for it. Consider the following:
Had the Supreme Court declared Obamacare's nationwide tax subsidies to be unconstitutional, a lot of people probably would have dropped their health insurance policies and paid the resulting penalty—which for 2015 comes to $325 per person, or 2 percent of someone's annual household income, depending on which is greater.
After all, a lot of people are only able to afford the plans they get through the marketplace or exchanges due to the subsidies that at the moment average $263 per person per month, or $3,155 per person per year, according to the Department of Health and Human Services.
If that form of financial aid fell by the wayside, though, the $128 that federal enrollment data says your average American currently pays for health coverage would jump to about $336, if various projections are correct.
That would have a pretty disastrous impact on the insurance industry as a whole, as with a good portion of the healthiest people removed from the system, only the sickest—and most expensive—to insure would remain. With the pool of insured individuals both smaller and less healthy than had been the case before, insurers probably would have hiked premiums to make up for the lost revenue and increased losses.
"Had the ruling gone the other way, there would have been [a lot of] uncertainty that likely would have lasted at least for months," Riley suggests. "That would have affected everyone by raising costs throughout the market—insurance companies would have had to prepare for many contingencies each with their own transaction costs—and would have left many people uncovered."
Don't take any of this to mean that this is all you're going to get from Obamacare from here on out
In fact, many experts suggest a lot still can be done when it comes to implementing and improving the Affordable Care Act.
For instance, Ira Wilson, professor of health services, policy, and practice at Brown University, recently told commonhealth.wbur.org that "there's a lot to do to continue to implement, and implement effectively, the ACA. This is a marathon, not a sprint. It's a long process. And what the ruling means is that people can keep focused on doing that implementation work rather than rethinking it altogether."
Michael Doonan, assistant professor at the Heller School for Social Policy and Management at Brandeis and executive director of the Massachusetts Health Policy Forum, on the other hand, told the same site that he expects "some of the states who are not participating [to] participate by expanding their Medicaid, [and] by looking to the federal government to see if they can cover more people in creative ways."
Riley isn't so sure about that last point, adding, "the big open question is Medicaid expansion—and whether those states that have refused to expand Medicaid will gradually do so. That affects more people than most people realize because it affects costs and what kind of services hospitals can afford to offer. Some hospitals are even closing."
That said, you probably can rest assured that the current version of Obamacare, if not an improved one, is here to stay
"Repeal of Obamacare is almost off the table," Doonan said in his recent interview with commonhealth.wbur.org, while Wilson suggested that "repeal still could happen" but accomplishing that feat just became "much, much harder" as a result of the Supreme Court ruling on June 25, mainly because "you have two major Supreme Court decisions in support of the constitutionality of proceeding in this direction."
"Although the court's ruling does mean that the ACA is likely here to stay," Riley adds, "it is still a political lightning rod. That means that it may be possible for some limited improvements at the agency levels, but any major changes that would require legislation would quickly get caught up in politics."
Still, she says, "it's certainly better than what would have happened if the court's ruling had gone the other way. If it had, every single decision would have become a political bargaining chip."
Tags: Obamacare, Affordable Care Act
9 Numbers That Explain the Supreme Court's Latest Obamacare Case
Breaking Down The Supreme Court Ruling On Obamacare Subsidies
Supreme Court Saves Obamacare
What Does The Supreme Court's Obamacare Ruling Mean For UnitedHealth?
What to Take Away from the Supreme Court Decision on Health Care
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Q: Korganizer adding events I am new to Linux and to Korganizer 4.13.3. To learn the latter, I am following the tutorial from their web site:
https://docs.kde.org/trunk5/en/kdepim/korganizer/five-minute-course.html#course-entering-events
When I click in the agenda view, and enter information about an event, it doesn't save any information, but I get the following message:
You created an incidence in a calender that is currently filtered out.
On the left sidebar, enable it in the calender manager to see the
incidence.
I am not seeing a calendar manager in my left sidebar, and have no idea how to proceed (I am using Ubuntu 14.04 LTS, and I get this same problem whether I am in Unity or KDE (I am using Kubuntu)).
Related question
I cannot enter data in KOrganizer
A: With the a window labeled Calendar - Korganizer open click View and then Filter, the last line on the drop-down menu. I have it working with (x) Nofilter selected.
Just in case it's relevant to the 'cure', before I discovered this, I also thought there being no filters present was the problem, and so, added a dummy with no rules. In the drop-down Menu, I selected my dummy entry first, and the entries popped into view on the Calendar. Then I experimented by selecting Nofilter, thinking the entries might disappear, but they remained. Probably the dummy filter was unnecessary, but I have no idea.
As far as I can see, in fact, there is no 'Calendar Manager' in the left-hand panel, unless that refers to a check-box to the left of the calendar icon?? The tool-tip's wording seems somewhat misleading, in the absence of further information being supplied -- from somewhere.
For more information, refer to Chapter 4. Views and Filters.
A: To the left of your calender is an area that says "search". Try checking "personal calendar", and then you should be able to add "incidents".
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# The Finest in Fantasy from Jennifer Roberson
BLOOD AND BONE
Life and Limb
THE SWORD-DANCER SAGA
Sword-Dancer
Sword-Singer
Sword-Maker
Sword-Breaker
Sword-Born
Sword-Sworn
Sword-Bound
_(The Sword-Dancer Saga is also available in the_
_Novels of Tiger and Del omnibus editions)_
CHRONICLES OF THE CHEYSULI
Shapechanger's Song
_(Shapechangers & The Song of Homana)_
Legacy of the Wolf
_(Legacy of the Sword & Track of the White Wolf)_
Children of the Lion
_(A Pride of Princes & Daughter of the Lion)_
The Lion Throne
_(Flight of the Raven & A Tapestry of Lions)_
THE KARAVANS UNIVERSE
Karavans
Deepwood
The Wild Road
THE GOLDEN KEY
_(with Melanie Rawn and Kate Elliott)_
Copyright © 2019 by Jennifer Roberson.
All Rights Reserved.
Jacket design and photo illustration by Adam Auerbach.
Photo elements courtesy of Shutterstock.
Interior design by Alissa Rose Theodor.
Edited by Betsy Wollheim.
DAW Book Collectors No. 1838.
Published by DAW Books, Inc.
1745 Broadway, New York, NY 10019.
All characters and events in this book are fictitious.
Any resemblance to persons living or dead is strictly coincidental.
The scanning, uploading, and distribution of this book via the Internet or via any other means without the permission of the publisher is illegal, and punishable by law. Please purchase only authorized electronic editions, and do not participate in or encourage the electronic piracy of copyrighted materials. Your support of the author's rights is appreciated.
Nearly all the designs and trade names in this book are registered trademarks. All that are still in commercial use are protected by United States and international trademark law.
Ebook ISBN: 9780756415402
DAW TRADEMARK REGISTERED
U.S. PAT. AND TM. OFF. AND FOREIGN COUNTRIES
—MARCA REGISTRADA
HECHO EN U.S.A.
PRINTED IN THE U.S.A.
Version_1
_Dedicated to three inspiring women no longer with us_
_Shera Roberson, my mother_
_Molly Hardy, my aunt_
_Pat Hyre, my friend_
# CONTENTS
_Also by Jennifer Roberson_
_Title Page_
_Copyright_
_Dedication_
Prologue
Chapter One
Chapter Two
Chapter Three
Chapter Four
Chapter Five
Chapter Six
Chapter Seven
Chapter Eight
Chapter Nine
Chapter Ten
Chapter Eleven
Chapter Twelve
Chapter Thirteen
Chapter Fourteen
Chapter Fifteen
Chapter Sixteen
Chapter Seventeen
Chapter Eighteen
Chapter Nineteen
Chapter Twenty
Chapter Twenty-one
Chapter Twenty-two
Chapter Twenty-three
Chapter Twenty-four
Chapter Twenty-five
Chapter Twenty-six
Chapter Twenty-seven
Chapter Twenty-eight
Chapter Twenty-nine
Chapter Thirty
Chapter Thirty-one
Chapter Thirty-two
Chapter Thirty-three
Chapter Thirty-four
_Author's Note_
_About the Author_
# PROLOGUE
His voice was rich, a much loved, clear baritone, as he handed his seven-year-old grandson a gun.
"All right, Gabriel. First time for everything. Don't expect to be perfect. _I_ wasn't. Let's just see what you've got, shall we? See what we've got to work with." A warm hand came down on Gabriel's shoulder, squeezed lightly. "Don't fret, Gabriel."
It was a crisp Oregon day with an afternoon rain abated but lingering in puddles and dampness as clouds receded. Tall, gray-haired Grandaddy, with neatly trimmed beard and vigorous river of thick silvering hair flowing nearly to his shoulders, had driven them out to the country in his white '63 Thunderbird with a box of empty beer bottles in the back seat, and Gabriel remembered the weight of the weapon. How it filled both hands, how it made him tense his arms, how his fingers clung.
How his grandfather slipped a hand under his, eased the gun slightly upward. Grandaddy didn't hold the gun _for_ him; he just provided a smidgen of support so that he could stop clenching his entire body with the attempt to hold the gun still. Grandaddy _steadied_ him, just a little.
"All right, Gabriel." The same calm voice. "You've got this."
He was nearly breathless with want, with the _need_ to please. To excel. "All six bottles, Grandaddy?"
His grandfather's laugh gusted from his chest in a quiet blurt of amusement, blue eyes dancing as the weathered skin next to his eyes creased into a spray of fretwork lines. "If you think you can, of course. But _one_ would be a fine beginning, Gabriel. And if you don't get it right the first time, next time will do. Or the next after that. _Baby_ steps. Baby steps, son."
But at seven he had gained an inch over what he'd been at six, and he felt strong. His body sang with it. This was, he felt within his bones, a test of some kind, even if his grandfather suggested no such thing. "I'm not a _baby_ , Grandaddy. I can do this."
"Perhaps you are not." Grandaddy's hand, once more, briefly touched his shoulder. "Then show me what you've got."
He felt his grandfather remove the support from beneath his hands, the gun. It was a measly little outdated 6-chamber .22 revolver, nothing like the 9-mil Smith & Wesson his grandfather wore in an honest-to-God holster. But it felt big. It felt heavy.
And then it didn't. It felt _right._
He _focused_. He'd heard that word before, that prompting from Grandaddy. The world narrowed. The world lost definition around the edges, refined itself in the middle. Sound attenuated, died away until all he heard was his own shallow, choppy breathing.
He stopped it, stilled it. Sucked breath again, blew it out on a huffing stream between pursed lips. It steadied his breathing; smoothed into something akin to a comfortable anticipation. It wasn't excitement. It was the need to do this, to please his grandfather, to prove he could _do_ this. And an almighty certainty that he could.
He saw six beer bottles balanced along the pitted, weather-grayed wooden fence rail, six dead little brown-glass soldiers emptied by his father, who liked to kick back after work with two or four beers; bottles appropriated by his grandfather from out of the trash. That was all that mattered. Six bottles. Six targets. Six bullets. Six _opportunities_.
What he wanted to do was what he'd seen in the movies featuring badass shooters: _one-two-three-four-five-six,_ one after another, a seamless string of reports as the gun fired, the popping sound of exploding bottles and the cascade of shattered amber glass. But what his grandfather wanted was efficiency.
Precision. Results.
One at a time. _Aim. Squeeze. Take the recoil._ Steady hands. Loosen elbows, shoulders, then do it all again: _one-two-three-four-five-six_.
Efficiency. Precision. Results.
He fired six times. The gun was empty.
He let his arm drop to his side, felt the weight dragging. He was suddenly seven years old again, not remotely a badass, and even a measly .22 felt heavy in the wake of his first experience _shooting_ the thing.
"Well," Grandaddy murmured, clearly startled. Then his hand yet again came down on Gabe's shoulder, squeezed. Firmly, man-to-man. "Guess I shouldn't be surprised, with your genetics . . . I think you're what's called 'a natural.'"
Gabe looked up at his grandfather. The words just spilled from his mouth, unwilled; and it was truth. "If you don't get it right the first time, next time may be too late."
Grandaddy smiled slowly, broadly, teeth showing in the thicket of his tidy beard. "Hardwired, boy, is it? All of it? Yes. You'll do. When the time comes." He paused, blue eyes lost in the distances. "And come it will."
Gabe blinked up at him. Grandaddy often went off on stuff he didn't truly grasp, talking about "special Mendelian genetics" and "coefficients of inbreeding," "prepotency" and "outcrossing" and any number of terms in which he staked no interest. He listened, though, because that's what one did with Grandaddy.
_"You'll understand, some day,"_ Grandaddy told him once. _"For now, place your trust in me, son, and I'll see you safe. But there is learning to do."_
Well, Gabe liked school, so learning was okay. Loved books. Spent hours lost in worlds created by others. But he also liked to move, to _do_ , to not be trapped at a desk in the classroom, or watching educational videos, or viewing documentaries on DVDs at home, or any number of things his grandfather asked of him when he came around. Which wasn't often enough, to Gabe's way of thinking. He loved his parents, loved his younger brother, but he worshipped his grandfather. Life was more interesting, was somehow brighter and louder and sharper and more _real_ when Grandaddy came. And it didn't matter that Grandaddy wasn't truly his grandfather, not in blood, but an old family friend, or some weird grownup attachment he didn't quite grasp. Grandaddy just _was_. Gabe knew in the deepest part of his soul that they were linked somehow.
He was eight when Grandaddy came again, and at eight Gabe felt like he was bursting out of his skin, that he might explode were he not given freedom—and yet he knew, too, in some weird instinctive way, that he was utterly clueless. He didn't know what he wanted. He didn't know what he needed. He just knew that he was lacking. That he was lesser. Wasn't badass. That he wasn't truly whole.
He woke up in the midst of a noisy nighttime thunderstorm and knew without a doubt that his grandfather had come. And just as he thought it, just as he _sensed_ it, in some incomprehensible way, his grandfather opened the bedroom door and came in smiling, knelt beside the bed. Took into his broad right hand the small right hand of a boy, and stroked back the wayward locks of dark hair with his free left hand.
"God's bowling," Grandaddy said, as all outdoors was loud with angry rumbling.
Gabe stared up at him from the tangle of bedclothes. "God?"
"He's bowling," Grandaddy said. "That's what thunder is, the taking down of the pins."
He frowned. "No, it's not."
Grandaddy's brows rose up beneath a cascade of graying hair. "It's not?"
"No," Gabe said with supreme confidence, because he knew the truth. "That's _Thor_."
For the first time in Gabe's brief life, his grandfather appeared to be at a loss. "Thor?"
"God of Thunder," Gabe replied. "Thor is _cool_."
Grandaddy smiled. "Is he, now?" And then he lost his smile, grew earnest and serious. "It's time we had a talk, you and I. You won't remember it, but you need to _know_ it." He tapped Gabe's chest. "Bone-deep, _soul_ -deep, you'll know it. We'll just let it sit in the back of your head for a while, buried behind everything else—you've a very busy brain, son—and one day, when it's time, I'll call it up in you, and you'll remember all of it. You'll know who you are, what you're meant to be, and what you're intended to do."
That, Gabe found intriguing. " _What_ will I do?"
Grandaddy said, as the thunder rolled behind him, "You'll be a soldier, boy. Sealed to it. Life and limb, blood and bone, heart and soul. Not a soldier like others are, for it's not the kind of war most people fight on earth. But because we're not 'most people,' you and I, it will be far more important. The fate of the world will hinge upon it."
Gabe stared at him. "The whole entire world?"
Grandaddy's voice, though soft, carried the weight of thunder. "The whole entire world—and everyone in it."
* * *
—
Remi, awkward boy's limbs asprawl in abandonment on the plank wood of the covered front porch—his father, thank the good Lord, had hung a ceiling fan out here—scooched closer to his companion, rested the back of his skull atop the soft-furred bulk of a blue merle Australian Shepherd atwitch in his sleep. The day was hot, sticky, heavy, promising no relief until nighttime, and even that was chancy. At best the house was cooled by fans of all ilk; evaporative cooling didn't run right in humid weather, and they couldn't afford 'central air,' as it was known, those big units that blew chilled air through all the rooms and gave a man a chance to breathe, to fight the moisture and dry a sweat-damp body.
Remi was used to sweating.
He leaned against the dog, who yipped and woofed in dog dreams, and wondered what the stars thought. That they did think, he knew; they must. They hung up there in the sky, glowering down upon the world, knowing what all men thought. Knowing what _he_ thought: Remiel Isaiah McCue.
_Son_ , his Nana had said, not long before she died, _You're meant for larger things. I don't rightly know what those things are, but you're meant for 'em. And one day, they'll find you_.
But when? Remi wondered. And would these things, whatever they might be, manage to find a boy in the treeless, dusty expanse that was West Texas?
He trusted Nana. She said she saw farther than others, deeper; saw beneath the skin to the heart, and swore his was larger than most. Said there was enough soul in him for two people, if he let his light shine. That maybe he'd even share it.
Times he felt like his bones downright _itched_. He couldn't put a name to it. His skin just felt too tight for his bones, like they were growing faster than the rest of him. And it was hard to sit still, at times, like his body just needed to _move_ , right along with his mind. That need ran like a river in full spate.
But he got caught up in school, and Friday night football under the lights, and learning to rope a plastic steer head stuck on a straw bale, driving a tractor, crawling out of bed with his big brother in the wee smalls of the morning to open the irrigation ditches so the water might flow to the alfalfa field. He hung out behind the chutes at the local rodeos, and Nana was long dead before he recalled what she'd said about his heart and his soul, and sharing his light, so he never got to ask her what she meant.
And then one day Grandaddy showed up and said it was time for Remi McCue to learn to throw a knife, being as he already knew how to shoot a gun.
Well, yeah. He was a Texas country boy. Shooting a gun came early.
Throwing a knife? Well, not so much. But by the time his old "grandfather" was done with him, Grandaddy said, Remi would understand how horn-and-steel felt in a hand, what balance was all about, and the pure seduction of letting a blade fly to its target.
Hell, yeah. Yeehaw. Though he wasn't sure, then, what _seduction_ meant. But that was Grandaddy's way; he didn't treat Remi like a child, or like an adult, either, when he thought about it, but just as Remi.
He was eight when Grandaddy came again after being gone for a couple of years, and that man drove up to the house out of the maw of a big old Texas thunderstorm in his aged white T-bird with red leather seats. Gray hair remained a vigorous crop flowing nearly to his shoulders.
Remi stood on the porch as Grandaddy came loping through the rain, head ducked against the worst. When he reached the porch, he gifted Remi with a display of bright teeth in the thicket of a trimmed beard darker than his hair.
"God's bowling," he said.
Remi shook his head. "No, sir."
"No?"
"It's Thor. God of Thunder."
His grandfather froze upon the step, face blank, and then he smiled. It was a broad, secret smile, with pleased laughter behind it; but not meant to be understood by any, perhaps, other than himself. "Is it?" he asked. "Thor?"
"Bring me Mjolnir," Remi said, envisioning the massive hammer as he raised his hand to grip an imaginary shaft, "and I'll conquer the world."
Grandaddy tilted his head, as if acknowledging something. "It's time we had a talk, you and I. You won't remember it, but you need to _know_ it, bone-deep, soul-deep. We'll just let it sit in the back of your head for a while, buried behind everything else, and one day, when it's time, I'll call it up in you, and you'll remember all of it. You'll know who you are, what you're meant to be, and what you're intended to do."
" _What_ will I do?" Remi asked, intrigued.
Grandaddy said, "You'll be a soldier, boy. Sealed to it. Life and limb, blood and bone, heart and soul. Not a soldier like others are, for it's not the kind of war most people fight. But because we're not most people, you and I, it will be far more important. The fate of the world will hinge upon it."
Remi stared at him. "The whole entire world?"
Grandaddy's voice, though soft, reminded Remi of a preacher at the pulpit. "The whole entire world—and everyone in it."
# CHAPTER ONE
From out of the heat of the day and into looming twilight, I pulled onto gravel and threaded my Harley through a parking lot jammed with pickup trucks. Killed the growl of the engine as I rolled up next to a handicapped spot, stayed straddled as I pulled off my helmet and gloves and let the cool pine-scented air wash over me. Pure tactile, almost atavistic relief after hours on a hot interstate.
I yanked the tie from my hair and unstuck compressed strands from my skull with a couple of quick scrubs so it fell loose to my shoulders again. Unzipped the jacket. Left my ass parked on leather and crossed arms as I surveyed the building before me.
I had to smile. Not exactly my thing.
Now roadhouses, yeah. Definitely. But in the Patrick Swayze/Sam Elliott school. This? Nuh-uh. Pickup trucks, gun racks; a lighted sign boasting live country music. Probably spittoons on the floor, for all I knew. Maybe even a mechanical bull.
The building was a bulwark of massive, stripped pines chinked together rising two stories tall, topped by a rust-patinaed tin roof. Its slab of a front door looked thick enough to bounce cannon balls off of, and the entry steps were framed by a massive split-crotch tree. Behind it loomed the shoulder of a fire-ravaged mountain, and the dying of the day.
I heaved in a breath, blew it out on a sigh as I swung a leg across the seat. "Grandaddy, why the _hell_ did you summon me to a cowboy bar in Flagstaff, Arizona?"
I clomped up the low steps in my biker boots and stepped aside as a laughing couple, nearly joined at the hip, exited. I caught the door's edge from the guy, pulled it wide, and the strains of that live country music erupted into the twilight.
I winced, thought uncharitable things about a music genre I cannot abide—all that whine and twang and mud and blood and beer—and prepared myself for an even noisier unwelcome assault upon my ears.
As always in strange places, particularly roadhouses and dive bars, which I tend to frequent, I entered carefully. Eased through the door, let it thump closed, then stepped aside and waited, marking the details of the place. Particularly the exits.
Live band, already established; parquet dance floor; booths against the wall; couple of pool tables in the back. Tables and chairs; long, polished slab of a bar; rough-hewn beams, tree trunk pillars; and so many mounted animals, trophy heads, skins, and antlers affixed to the walls that it looked more like a . . . well, yeah, the place _was_ called the Zoo Club. Though it more closely resembled a taxidermist's. In fact, just beyond my right shoulder, crammed into the corner, loomed a ginormous huge-humped grizzly bear with mouth agape to display fearsome teeth.
I did not fit here, not in this place, where I was pretty much an alien. Cowboy hats, boots, plate-sized silver belt buckles, pressed jeans, yoked shirts. Me, I wore a plain black t-shirt, motorcycle leathers, and thick-soled boots meant for the road, not stirrups. I like my bars with chrome and steel and twinned wheels parked outside, where the only hint of horses resides within engines.
A flash of movement at the end of the bar. Seems I'd caught the notice of a young woman. And boy, did I notice back. Long wheat-blond hair was slicked away from her face and tied into a high ponytail hanging down her back. I couldn't see details in bar lighting, but the assemblage of her features collaborated quite nicely, well above the norm. Red lipstick. Her brows, darker than the gold of her hair, arched as her eyes brightened, and she smiled slow and easy, the invitation obvious. She did not appear to care that I was not in the cowboy uniform, or that my hair hit just past my shoulders.
Well, then. I smiled back, raised brows, lifted one shoulder in a half-shrug that told her _Not just now_ , saw faint disappointment in the tilt of her head, the regretful twist of her mouth. Maybe later, if she were still around when business was concluded.
"Gabriel."
Even in the midst of roadhouse noise, I heard and knew that tone. With regret I shifted attention from the young lady to the man coming toward me.
Jubal Horatio Tanner, aka Grandaddy, the only one who called me by my full name. Tall, blue-eyed, clear-skinned, with a cascade of springy silver-white hair tucked behind his ears. Imposing man. To me as a kid, he'd seemed old; now, not so much, even with the near-white hair. Ageless, if anything. Rock of Gibraltar type. His brows remained dark, as did his neatly trimmed beard, though it bore a peppering of silver.
He wore, as he always did—for some unaccountable reason I kept forgetting to ask him about—an old-style frock coat, as if he'd stepped out of a Western. Which, inside a cowboy bar in Arizona, struck me as ironically appropriate. He fit. I didn't. Beneath the coat, unless he'd changed his ways, he wore a sheathed Bowie knife and a waistband holster, home to his 9-mil S&W.
We clasped hands, grinned at one another, then stepped close for a quick hug, slap of hands against backs before stepping out again.
"Too long, Grandaddy!" Couple of years, in fact. I raised my voice over the live band. "I thought you'd visit, at least." I didn't say more; he'd know what else I meant.
"Business," Grandaddy said crisply; no apology was included. "You know how that goes. I kept tabs on you." He touched a fingertip to his left eyebrow. "Your dad told me you got jumped."
I almost put my own hand up to that spot in my brow, but dropped it back down. I knew it was there: a thin, pale diagonal line, stitch-free now, but it looked like the hair wouldn't grow back.
I kept my tone light. "Too far from my heart to kill me."
Grandaddy's eyes were unrelenting. "You handle the man who jumped you?"
Handle him? Oh, yeah.
I lifted the scarred eyebrow. "We had us a 'discussion' right there in general population. Nobody bothered me after that."
What I didn't add was that it's tough to bother a man in solitary.
Grandaddy didn't respond, just gestured with a sweep of a broad-palmed hand. "I've got a table in an alcove in the back where we can talk privately. Remi's not here yet; he called to say he was running a tad late. Sorry to say that boy's _always_ a tad late; his internal clock runs about as slow as his Texas drawl."
I started to ask who he meant, but Grandaddy'd headed off through the crowd. I followed to the alcove, discovered a pitcher of beer and a half-filled mug, two empty tumblers, a bottle of Patron tequila, and another of Talisker single malt sitting atop the table.
"Unless you've changed your brand of whiskey." Grandaddy flipped aside the tails of his frock coat as he sat down and took up his beer.
I couldn't suppress my grin of delight. "Hell, no. I still drink that whenever I can get it. But it's not usually on offer in biker bars." And anyway, I'd pretty much ridden nonstop to this watering hole with time only for coffee, prepackaged convenience store sandwiches. And, well, licorice. The black stuff. The _real_ stuff.
"They don't carry it, so I snuck the bottle in under my coat," Grandaddy admitted, eyes bright with amusement.
Warm affection filled my chest. Damn, it was good to see the man again. I hooked out a chair, swung it around, pushed the back against the wall so I could keep an eye on the bar crowd, then sat my ass down and poured two fingers' worth of fine Scottish whiskey. Lifted the tumbler, let it linger at my lips as the pungent tang of spirits rose to my eyes. Took a sip.
Yeah, there it was, that complex peaty power. I just appreciated it in my mouth a long moment, then swallowed with a grateful smile and a nod of the head. I'd missed this while in prison. "So, this is all your doing, right? Early release, and now I report to you? Maybe the first time an ex-con has been assigned to his own grandfather."
Blue eyes were bright across the beer mug. As always, he _watched_ me even as his posture suggested relaxation. "Mitigating circumstances, Gabriel."
I poured more whiskey, enjoyed another swallow. "Now, who's this Remi, and why are we meeting _here_? Why not Oregon, like usual?"
"Remi's coming in from Texas. Arizona splits the difference." Grandaddy drank beer, thumbed away liquid from his moustache, then fixed me with a steady gaze I remembered very well, even if I hadn't seen it for a couple of years. "Pay attention, Gabriel."
Okay, so it's like that. I'd heard those words, that tone, so many times over the years. It always prefaced information Grandaddy considered vital, even if it made absolutely no sense. I huffed air through my nose in amusement, grinned crookedly, nodded.
And he said, by way of pronouncement, "Remi is someone you're going to come to know very, very well, Gabriel. Someone with whom you will form a bond unlike any other. Someone upon whose actions your life will depend, and whose life will depend upon _your_ actions."
For a long, arrested moment, drink suspended in midair on its journey to my mouth, I stared blankly at him. Found no illumination in his face. "My life?" I waited a beat; no answer was forthcoming. "As in, life and death?"
"Precisely life and death."
"Uhhh, okay." I set down the tumbler with a muted clunk, scratched at my bisected eyebrow. It itched now and then. "Can you kinda elaborate on that? Just—" I waved a hand in an indistinct gesture encompassing worlds of nothing much "—you know, for the sake of me knowing what the hell you're talking about?"
The eyes were penetrating. "He will have your back, and you will have his, pretty much twenty-four, seven, three-sixty-five."
I contemplated that announcement, knocked back more whiskey, then opted for candor. "That still doesn't tell me shit, Grandaddy."
Now he was quietly amused. "Not yet, no. We'll wait till Remi arrives, and then I will, as you say, elaborate."
I opened my mouth to question further, but gave up, knowing it was pointless. Grandaddy was often cryptic, and he could not be rushed. I'd learned not to push or things got more obscure. I could tie my brains into knots trying to sort out the man's intentions. "This Remi got a last name?"
"McCue. And— _ah_ , speak of the devil." Grandaddy laughed softly. "Or not." He shoved his chair back, rose, extended his hand. "Remi, good to see you, boy."
I raised my brows. Unlike me, Remi McCue fit right in with the crowd. Dark denim western shirt, tucked in; neatly pressed jeans, leather belt with big silver buckle, cowboy boots, even an honest-to-God _hat_.
This was the man Grandaddy thought I'd _bond_ with, whatever the hell that meant. Upon whom my life was to depend.
A _cowboy?_
The booze warmed my belly. I gusted a laugh and sat back in my chair, grinning. "No offense, but . . . you gotta be shitting me!"
The stranger gazed down at me a long moment, registered that he was himself the target of the irony, and raised one eloquent dark brow beneath the brim of his cream-colored hat as he made his assessment of me. In a clear tone he drawled, "Well, boy, looks to me like you're wearin' one of my steers in all that biker leather, so I wouldn't go sayin' much, was I you."
Ah. Okay. Like that, then. "You weren't me the last time I looked."
Grandaddy laughed. "Oh, in a way he is, Gabriel. While you're not related in a normal sense, there is a common genetic background. Take a harder look."
I did. Okay, yeah, the cowboy was around six-feet, one-eighty, so we were pretty much within an inch and five pounds of one another, and he had dark hair, too, but his eyes were a clear blue, not my brown. He was tanned, I wasn't; prison leaches melanin. Still, I had to concede we were of a similar physical type.
McCue smiled as he was given his second inspection. "Well, if Grandaddy says we resemble one another, then I'll have to say you _are_ a handsome devil." He paused, lips pursing. "Might could do with a haircut, though."
Beneath the hat, McCue's hair was neatly trimmed and did not remotely approach the vicinity of his shirt collar, let alone his shoulder blades. I smiled back, not meaning it; you learned to do that in prison. "And you're a poor man's Matthew McConaughey."
That, too, you learned there, to challenge before he did.
But the cowboy, patently unoffended and offering no return challenge, grinned slow, then drawled in deep tones, "All right, all right, all right."
"Remi, sit down and have something to drink," Grandaddy told him, before it went further, "and Gabriel, have another. You'll need the alcohol. I'm about to embark upon a foray into the expositional—and I guarantee you won't believe a word of it. All I ask is that you suspend your disbelief and hear me out."
I employed a booted foot to shove the empty chair toward the cowboy. He caught it, settled it, took his seat. We eyed one another in brief male-to-male consideration and evaluation, smiled blandly, poured drinks. My second went down easily. McCue drank Patron.
Grandaddy meanwhile assessed us like he was weighing our worth, marking things about the two of us I couldn't grasp. This was a man who _knew things_ , who always struck me as a secret-keeper, but not out of ill-intent. Out of privacy and a wish to control what he said when he said it. Of what he viewed was _safe_ to be said.
And just now, Grandaddy appeared to arrive at a conclusion. His smile was a brief, sardonic twitch. "Forgive me the melodrama, but I do promise that at some point, some day, all will come clear. I ask merely that you keep your minds open." His smile broadened. "I did train you for that."
Much as I wanted to, I didn't swear in frustration. Yeah, you don't push him, but Grandaddy could be more than a little frustrating at times. And a sideways glance at McCue suggested he felt the same as he smiled crookedly at me and twitched a shoulder, tilted his head in shared resignation.
But we waited. It's what you do with Jubal Tanner: you wait for pronouncements to be declared from _on high_.
He spoke quietly beneath the whine and twang of the music, but we heard him easily. In fact, everything else seemed eerily distant, muted. "You boys will go to bed tonight wiser than you are at this moment, yet ignorant of much more. That, you will learn as you go, if you survive—and yes, I mean that in the literal sense." The back of my neck prickled; Grandaddy's eyes were insanely compelling. "You know me as a close family friend, someone who _approximates_ a grandfather. The relationship is complicated, but in truth we're all related."
The cowboy and I exchanged baffled glances. Related? What the hell?
Grandaddy smiled a little. "Not by blood or birth, as humans count it, but by birth _place_ , and by certain bonds of that birthplace." He turned his beer mug in idle circles upon the table. "Where we come from, where _our kind_ comes from, we are all of one body, but not biologically linked. Not as humans know it."
I stared at my—whatever he was. What the hell?
Grandaddy—it was the only name I'd known him by, and hard to shake—stared back. There was no tension in the man's body, no indication of anticipatory concern for the reception of his words; his attitude was that he'd merely stated the obvious.
And I heard him in my head again, saying, _All I ask is that you suspend your disbelief._
# CHAPTER TWO
I finally managed the question in a carefully calibrated tone. _"Our kind?_ "
McCue's body language shouted that he was equally taken aback, but he'd latched onto something else. "Not as _humans_ know it?"
Grandaddy nodded. "The long and the short of it is—neither of you boys were born entirely of mortal man and woman."
And then the music got real loud in my head, and I couldn't say a word. Couldn't _think_ a word. All I could do was double-down my focus and stare fixedly at the man while my brain froze solid.
Remi McCue, after an equally paralytic moment, poured more tequila into his glass and sucked it right down. Then he cleared his throat and said, in a tone surprisingly casual under the circumstances, "Lick that calf again?"
Distracted, I shot him a look. "What the hell does that mean?"
McCue accommodated with a cheerful translation. "It's Texan for 'Excuse the fuck out of me.'"
Grandaddy just rolled on. "You are not entirely human. You were born of _heavenly matter_."
The bar got louder yet. I blinked hard and slow, feeling empty of everything other than a turgid disbelief, an odd unsettled tautness in my belly.
I leaned forward, stuck my face into hands as I propped elbows on the table, and rubbed my forehead roughly. Replayed the last two sentences.
Uh-huh. Yeah. Right.
Of all the people in the world I'd have bet money would _never_ lose his shit, Grandaddy headed the list. He, meanwhile, just sat there very quietly with gentle humor in his eyes and the faintest of smiles evident within a silver-peppered beard. He was, despite everything he'd stated, unaccountably and utterly relaxed.
The sky is blue. The sun sets and rises. Oh, and by the way—you're not human.
Okay, I'd inhaled legends, folklore, and mythology, grew up on science fiction and fantasy novels, movies, and TV shows. It was therefore a completely natural extrapolation from that background to wonder if maybe I'd slipped through a wormhole into another dimension, or crossed over into the _Twilight Zone_ , or hopped aboard the TARDIS. For entertainment purposes, I'd learned long ago to suspend disbelief, just as Grandaddy asked.
Well, hell. Things had gone so far sideways as to be off the planet entirely. I might as well allow myself a Mulder Moment.
I lifted my head from my hands. "Are you an alien?"
"I am not," Grandaddy answered gravely.
After a pause, I nodded and poured another drink. I hadn't eaten for hours; I wanted _effect_ , not flavor, and knew I could get it. I knocked it back fast, which was an abuse of fine single malt, but seemed called for under the circumstances.
I shot a hard glance at McCue, who gazed back in a sort of frozen consideration. I didn't know the guy, couldn't assess his thoughts, but it seemed that the cowboy was asking my opinion with a single arched brow.
I could do it. He could do it—the Spock brow lift. I wondered if we shared other habits, being sort of related, in a celestial kind of way. Which sounded—uncomfortably kinky.
It was easier to focus on a stranger than to contemplate what my— _our_ —Grandaddy had said, even if he was sitting right there. "This is all kinds of batshit crazy," I told McCue, watching him closely for reaction, "and you don't even seem to be thinking _twice_ about what bullshit we've just been told."
The cowboy shrugged, shoulders lifting the seams of his pressed denim shirt. He flicked a glance at Grandaddy, settled his focus back on me. "I'm listening to the man. That's how you learn. Open mind, he said." He hooked a thumb in Grandaddy's direction. "Besides, this man's been nothing but good to me. I owe him the courtesy of hearing him out. And if he taught you anything—and I'm betting he did _—_ so do you."
I noted how the slow drawl softened, was less pronounced. And a spark in blue eyes suggested maybe McCue was taking this more seriously than I'd thought.
Grandaddy had never, in my life, lied to me. I knew it bone-deep. I _did_ owe the man the courtesy of hearing him out.
But . . . _not born entirely of mortal man and woman?_
Open mind was one thing. Doable. Suspension of disbelief was another. Doable. But _this_ was just utter, complete, absolute, one-hundred percent bullshit.
And I, who tolerates no bullshit for very good reasons, had learned that life does not always allow for open minds and suspension of disbelief no matter how much you'd like it to. I'd been out of prison a matter of days. I'd just ridden over a thousand miles. Hadn't eaten in hours. Plus, I'd had just enough to drink, _felt_ just enough of the alcohol to allow frustrated impulse to take over. From somewhere inside, from a crack in the wall I'd built a couple of years before, anger seeped in.
I shoved my chair back with a scrape of wooden legs on wooden planks, offered him a brief acknowledging tilt of the head, and set my palms against the table and started to push to my feet.
But then Grandaddy just _looked_ at me, and I found myself somewhat vigorously reapplying ass to chair.
It hadn't been by choice.
And no one had touched me.
I sat because _physically I could do nothing else_.
I stared at Grandaddy in shock. Nah—well, maybe. "Did you just . . . did you just _whammy_ me?"
"Have yourself another drink," Grandaddy suggested lightly, freshening his own beer.
I seriously considered flatly refusing. Came close. Did not. Under the circumstances—he had somehow _forced_ me to sit my ass back down—it seemed the safest option to acquiesce. After a moment's subtle testing of whether my limbs obeyed again, and discovering they did, I had that drink, fast and hard. I craved the burn, the buffer bought by liquor that could be dropped between what I couldn't grasp and my own inner denials.
Or is it inner demons?
Just as I opened my mouth to ask a question, I became sharply aware of a new arrival at the table. At first I thought it was a cocktail waitress, then realized that no, it was someone else entirely: the ponytailed blonde I'd briefly communed with as I first entered the roadhouse.
I sat up and took notice. _All_ of me noticed.
The alcove boasted one modest, muted light glowing down from the wall. It highlighted the exotic angles and planes of her face, the slant of her brown eyes. She glanced briefly at Grandaddy, at McCue, offered them a red-lipped smile, but focused her attention on me.
"Sorry to intrude . . . well, I lie: I'm not sorry at all. But since there are no ladies with you, I thought I'd take the bull by the horns, so to speak—" She slid a sidelong, amused glance at Remi's hat, then returned her attention to me, "—and see if you would care to dance."
I took a closer look than I had upon entry into the roadhouse. Tall, slim, simple red tank top, wheat-colored jeans, red boots that in no way could be considered cowboy, a doubled loop of gold chain around her throat to match hoops in her ears. In all circumstances other than those at present, I would have been happy to depart the table for the dance floor. She was most definitely my style.
But I'd just been told _I wasn't even human._
I was not so drunk as to be dismissive of a woman, particularly an attractive one. I'd been in prison awhile. Instead, I smiled up at her from my chair, reached out a hand, took hers into it, leaned forward and pressed my lips against the back of it. "What are you drinking?"
Something flickered in her eyes. But she simply cracked her glass down against the table and said, "Whatever _you're_ having."
I was markedly aware of how both Grandaddy and the cowboy watched us. McCue's crooked smile was slight as he drank tequila, but unquestionably present. Grandaddy merely waited, expression bland. I knew that look.
I ignored them both and poured two fingers' worth of scotch into her glass. "Let's look forward to a refill," I promised. " _Later_."
Without breaking eye contact she scooped up the tumbler, knocked back the scotch, then turned and walked away from the alcove. The ponytail swung like silk against her red-sheathed spine.
"Huh," McCue said, with a world of complexities in that syllable. "Here we are in a cowboy bar, and it's the _biker_ who draws the attention of a woman like that."
I offered a slow, broadening smile; the earth felt firmer beneath my feet. This dance, I knew. "Shit happens."
Remi grinned back, saluted me with a quick tilt of his head, then focused again on our grandfather. "So, Grandaddy . . . you were sayin' something about us bein' _—heavenly matter_."
My amusement fled with that, as did lingering thoughts about the woman. Remi McCue seemed much too open to the bullshit, even if it was a man we'd known all of our lives saying it. Maybe the cowboy was batshit crazy, too.
" _Of_ heavenly matter," Grandaddy clarified. " _Of_ is not _it_."
And sometimes a cigar is just a cigar.
Loose and lazy from the liquor, I sat back in the chair, slumping with one leg stuck out. I briefly considered trying to leave again, dismissed it even as I rubbed my brow; the last time hadn't worked out so well. And Grandaddy had that _look_ on his face.
Infernal, that look. Dammit.
So. Okay. I'd play along. "But I'm still me?" I hooked a thumb in McCue's direction. "And he's still—him?"
"Of course." Grandaddy's tone suggested possibly I was an idiot. "You're flesh, blood, bone, brain, like everyone in this room. The fundamental difference in you both lies at a much deeper level. It's not physical. It's _essence_."
"Essence," McCue echoed blankly.
"When our kind is born," Grandaddy explained, "we are made of matter _,_ of essence. Celestial energy, if you will. We do not have bodies. We're not human. But many of us are intended to exist on earth, and to do so we need hosts. In certain instances when a human newborn fails to thrive, or is too ill to live and the parents pray for intercession . . . well, heaven intercedes. And the newborns survive because the dying soul is replaced with the living spark born of heaven." He smiled. "Yes, you are indeed Gabriel Jeremiah Harlan, older brother of Matthew, who was born in the perfectly ordinary human way to Will and Elizabeth Harlan; and _you_ are Remiel Isaiah McCue, younger brother of Lucas, born to Jack and Clare McCue in the perfectly ordinary human way. You didn't arrive in a clap of thunder, or spring fully formed from Zeus's brow. Your _flesh_ is human. Your bones. It's just that your souls are not."
Struck again into silence, I noted the live band was, once more, really very loud, as if increasing whenever Grandaddy laid his weird bullshit on us. So was the crack of cue ball at the break, a scattering of stripes and solids. All sound seemed filtered directly into my brain, filled it up. There was no room for anything more. Certainly not the ability to suspend my disbelief and simply _accept_ what we'd been told.
"Well, hell . . ." McCue said after a moment, seemingly at a loss.
"Not exactly," Grandaddy said with nuanced clarity. "Do you remember—and now, finally, I _mean_ you to remember—when I said to you both, on separate occasions, that we would have a talk, but you wouldn't remember it until one day when I'd call the memory up in you? That you'd know who you are, what you're meant to be, and what you're intended to do?"
The memory, now bidden, came sharp: Yes, I did recall Grandaddy had said that, and apparently so did McCue, because he nodded as well. But I didn't remember what the topic of the talk was, just that we'd had one.
I shared a brief glance with the cowboy, then looked back at the man who claimed that he was after all our grandfather of sorts—but somehow wasn't, you know, human. Minor detail.
"Now is the time," Grandaddy said, "and I'll call it up in you—but it will be done gradually. The learning curve is steep."
I asked, wanting to be very certain that the alcohol, for all I desired its buffer under the circumstances, was not completely altering comprehension, " _Heaven_ heaven? _Heaven_?"
Grandaddy silently lifted his hand in the air and pointed an eloquent forefinger upward.
Christ on a cracker. Or, so to speak. I scrubbed a hand over my face, trying to massage comprehension into a brain that preferred denial. Learning curve? How about utter craptastic bullpucky. "But—why?" I asked, trying for a neutral tone. "You're saying people just pray because their babies are dying, and all this little newborn heavenly matter gets stuffed down their gullets?"
Remi McCue laughed. "Now ain't _that_ poetic?"
"We need human hosts," Grandaddy said, "but we don't _take_ them. We're of heaven, not hell. We don't possess people. We answer prayers."
I cast him a skeptical glance, brows raised. "Every single prayer?"
Grandaddy's regret was sincere. "I wish it were otherwise, but that's not possible. There is a plan, you see . . . there's a grand design, but it's chaotic, not straight-line. Rather like evolution, it hops around, divides, ties itself into Celtic knotwork. To effect that design, to untie those knots, certain things are done. Certain things are allowed to be _un_ done. Some prayers are answered, some are not." His eyes softened. "It's not always fair, what is undertaken, what gets set aside. We make difficult choices that are often incomprehensible to humans."
McCue's tone was overtly casual, yet I heard an edge. "But you need _babies_."
Grandaddy nodded. "Yes, we need human hosts. Adults generally don't ask or desire to be— _inhabited_. Naturally they don't want to give up their souls, to have them be replaced with another. But desperate parents with dying infants _do_ pray for heaven to save their children, and newborn souls are merest threads that all too easily break." He spread his hands on the table, palm-down. "So, as they fray to the edge of breaking—and they do break all on their own; we force nothing—those threads are replaced with new. It's a matter of reciprocity. The babies survive, grow to healthy adulthood, and most bring great joy to their parents—when otherwise they die within an hour, a day, a week, and cause much grief and desolation." His eyes were brilliant even in soft light. "That is heaven's mercy, to give those parents a child. Then, at need, when those children are grown and on their own, we call on them."
I tapped the fingers of my left hand against the tabletop and glanced at the cowboy. "What was that you said earlier? About licking a calf?"
Rabbit hole time. Through the looking glass.
Grandaddy had said we wouldn't believe him. Damn straight.
I squinted into the bottom of my empty tumbler, noting a thin amber glaze of whiskey dregs. Maybe it would be easier to digest all of this if I was drunk off my ass.
McCue looked thoughtful as he drank more tequila, then absently tipped the tumbler on edge and rolled it in slow circles against the tabletop. He ventured a question. "So, we two had our so-called heavenly matter stored in dying babies, which saved them; and then we grew up like normal kids doing perfectly normal things . . . and now you _need_ us?"
"Heaven does, yes," Grandaddy agreed. "But there are other reasons. You turned twenty-eight human years old a second past midnight, the both of you. It is of significance."
McCue frowned, but I shook my head decisively, certain of _this_ much. "My birthday isn't for another week."
Grandaddy's brows twitched in dry amusement. "That's what you were told, yes. It is untrue. You were 'born,' as much as we ever are, in different states, to different families, but on the same day at the same hour and within seconds of one another. That's rare, especially for us—no one can gauge the instant heavenly matter coalesces into a soul—and it binds you." He shifted in his seat, leaning forward to compel our attention. "You are twenty-eight. Two, and eight. The numeral 2 governs certain attributes: harmony and rivalry, but also partnership and communication, shared ideals. As for the 8—" he closed one hand around the beer mug, raised it, " _—_ it represents other elements: you are armed to lead, to direct. These things are what heaven needs of you both."
I caught back a blurt of laughter, but could not control skepticism. "Biblical numerology." I shook my head. "But aren't there other bits of heavenly matter out there? Other heaven-made babies now grown to adulthood? Not just us, right?"
"Of course. And we're calling—we've been calling—upon them all, and will continue to," Grandaddy replied. "Other souls, other birthdays, other numbers. Today it was you. Tomorrow, others."
McCue's laugh rode a choppy gust of breath. "In other words, you're saying we're not fish, and this isn't a mass spawning."
Grandaddy's brows rose briefly. "Colorful but accurate. No, in fact we skew the other way. Fewer, rather than greater numbers. That's why it was so unusual that the two of you were born almost simultaneously, and why we believe you may be of significance. But we don't promise you a rose garden because of it; in fact, there is danger in it for you, because now you'll be tracked. Targeted. You need to know this."
Grandaddy said it all very casually, without the weight of portentousness, of pretension. Which meant it was important, because I had learned to read him years before, much as I could: simple statements often meant more than others.
I cleared my throat. "Tracked how, and why?"
"By hell. _For_ hell." Grandaddy downed more beer. "To simplify, the best way might be to say you have beacons inside you—"
Despite the booze, I felt sluggish alarm kindle. "Now we have _beacons_ as well as essence?"
"—or internal GPS units, if you'd prefer, and at 12:01 last night—or, more accurately, this morning—they began the process of coming online."
I leaned down and smacked my forehead into a palm in disbelief; Grandaddy could be inexorable, once on a conversational path, and I had learned to stay out of his way.
"And to keep the metaphor consistent," he continued, "that means anyone with the right kind of receiver will be able to track you. Including hell."
I didn't even bother pouring myself a drink this time. I simply picked up the bottle and began to chug.
# CHAPTER THREE
Under the noise of the bar, McCue's tone was intent. "In this case, a beacon, this GPS unit, is a light? Or maybe a shining soul?"
"You could put it that way," Grandaddy agreed.
"Uh- _huh_." The cowboy smiled. "I guess Nana really could see under my skin."
"She had the Sight," Grandaddy confirmed. "And I believe she had an inkling of who I might be. Or _what_ I might be." His brows twitched together briefly, as if he recalled something slightly less than pleasant. "I told you, we don't _possess_. But there are times we have to massage memories. I can't very well be everyone's actual grandfather . . . old family friend works nicely, but the adults must be _guided_ to that understanding." Blue eyes flicked to McCue. "Your Nana saw more deeply than most."
McCue smiled. "Well, she said my soul shone brightly enough for two people. So maybe she sensed something about me, too." He looked at me, assessed briefly, raised his brows. "Of course if he keeps drinkin' like that, he may just drown all _his_ light." He leaned forward, grasped the bottle in my hand, yanked it away and thumped it down upon the table out of my reach. " _Uisge beatha_ like that is meant to be savored, not guzzled like it's horse piss. That's a dishonor to fine whiskey."
I squinted at him, startled the Texas cowboy pronounced the Gaelic correctly: ooishkay-bah. The booze was running in me, and the question just slipped out as I flicked a glance at his hat. " _You_ know about _uisge beatha_?"
"Whiskey," McCue said. "Water of life. _Aqua vitae_. Yeah, I know all kinds o' shit. I was a Rhodes Scholar. Attended Oxford University and everything." He resettled his hat, smiled slow. Something in his tone suggested he'd found offense in my blurted question. "That would be Oxford, England, by the way. I may be just a redneck country boy to you, but I'm a damn _smart_ redneck country boy."
Apparently I'd gotten under his skin. Equally apparently there was steel reinforcement underlying his good ol' boyness. "I do know what a Rhodes Scholar is," I pointed out. "I've got a Master's in folklore."
"Doctorate trumps Master's." The cowboy resettled in his chair and said cheerily, "Comparative Religion. And are you _always_ this much of an a-hole to people you've just met?"
Well, after prison, yeah, probably I was. Especially being halfway to drunk. But _why_ I was an a-hole wasn't any of his business.
I opened my mouth to answer, but Grandaddy cut me off. "You boys done pissing? Or do you want to keep whipping it out and measuring? I know I said rivalry goes with the birthday territory, but right now there's no time for this _mano a mano_ malarkey."
I sighed, blinking one eye, then the other to clear my booze-blurred vision. Realized I should have eaten when I had the chance. "Can we cut to the fuc—" But I broke that off, because I caught the hard look in a hard blue eye; Grandaddy did not approve of vulgar language. "—chase?"
Grandaddy reached inside his frock coat, drew out two small leather boxes. "Happy Birthday. Put them on: right hand, middle finger. Then shake on it."
"And what happens then?" I asked with suspicion, rubbing my brow. Most of my face felt numb. "We go to Oz? Hoth?"
Grandaddy laughed. "You'll power up your beacons out of sleep mode, punch them to 100 percent. As I said, consider them heavenly GPS units."
McCue looked thoughtful. "I thought you said hell can track us through these beacons."
"Yes, though not at every instant. But heaven can as well. That will come in handy now and then."
I, too, wanted clarification, even inside my booze-addled head. "Are we supposed to _do_ something with them?"
Grandaddy smiled widely and bared white teeth. "Use them to save the world, boys. That's what you'll do with them. And it's time you two got down to business, because when you watch the news and read the papers you may think things seem bad now, but they're going to get a whole lot worse. Because this place has just become the devil's playground." He paused. "And I do mean that, boys. _Literally_ , Lucifer's playground. Because he's coming. The war has already begun."
McCue snorted. "You don't look much like him, but you're sounding an awful lot like the preacher back home."
"Jacob Tarnover," Grandaddy said promptly. "I know him well. A devout man."
The cowboy was startled. "You know him personally?"
"I know many people. I travel a lot." Grandaddy nudged the leather boxes closer. "Open your presents. Do as I said: right hand, middle finger. Shake on it."
I took up the small box closest tome, flipped back the hinged lid, saw the heavy signet ring: silver pentagram embedded in a matte black stone, silver bezel and band. A glance at McCue's showed the same.
We stared at one another. My mouth twitched even as his brows rose. It was just too damn obvious. Low-hanging fruit. I saw a glint in his eye.
Simultaneously, we intoned, _"One ring to rule them all."_
Grandaddy grimaced. "You are the children of pop culture, I see. Well, I can play, too: _'One ring to bring them all and in the darkness bind them.'_ And that's _exactly_ what you're going to have to do to win this war: in the darkness bind _him_. For good. Throw Lucifer back into the pit."
"This is all kinds of bullshit," I insisted, but knew, somewhere along the line, I'd lost the battle to make any kind of impact on the proceedings. Probably when Grandaddy sat my ass back down without moving a finger.
McCue grinned, held his ring up in the light. "Well, boy, you ready for this rodeo?" He had a way of slurring "boy" into two syllables, forsaking the hard "y."
What the hell. Grandaddy clearly expected something to happen. I figured it would, or it wouldn't, and I felt the whiskey enough to be willing to give it a try.
I shoved the ring on, held out my hand. McCue took a little more time, seating the ring carefully on his finger, then extended his hand. We met one another's eyes. No doubt he felt as foolish as I did. But we closed, clasped. Silver clicked faintly as the rings met.
My heart thumped hard. Ears buzzed. I was blind, then not. The world spun. Stilled.
"Shit!" I cried, even as McCue did.
I felt like I was in the middle of a marathon and out of breath. Was I glowing? Were we glowing? Were our damn beacons shining? Were we transmitting to hell?
Hands joined, silver rings touching . . . I definitely felt something. And McCue felt it, too; I could see it in his eyes. Deep inside, I sensed a certainty. A comfortable acknowledgement of . . . something. _Some_ thing. Some _thing_. Hell, I was out of words.
Maybe I'd never had words for a moment, an awareness, like this.
And I disliked it intensely. You don't just go around binding yourself to a total stranger to fight with the host of heaven to stop Armageddon. You just _don't_.
Grandaddy seemed to think you do. "This is how it shall be," he said quietly. "Always. You're bound now. Life and limb. Blood and bone. You are not immortal. You are not superheroes. Your bodies are wholly human. It's the beacon—the _soul—_ that's the heavenly matter. So look after one another. Keep yourselves alive. It will not always be easy. But you are needed. The Adversary has loosed his vanguard. Now it's our turn to do the same before the End of Days."
I wanted to accuse him of invoking Charlton Heston at the Red Sea. I did not. Images filled my head.
Satan. Lightbringer. Son of Morning. Son of Dawn.
Fallen archangel, thrown down from heaven itself.
I released McCue's hand, stared at the ring hugging my middle finger. Even drunk, I remained aware of—whatever it was. Indefinable, but, again, _something._
McCue inspected his own adornment. "Beelzebub. Belial. Diabolus. For real?"
Grandaddy nodded. "None other. This rose, by any name, doesn't smell sweet at all. Because it's corrupted, and it is _foul._ "
The cowboy thought that over, dark eyelashes hiding most of his eyes. Then he nodded, sighed. "Well, _that's_ about as welcome as an outhouse breeze."
I rested my head in one hand, elbow propped against the table, and contemplated the half-empty bottle of single malt. A vague thought drifted through that in the morning I might be convinced none of this had happened, so long as my brain did not bleed out of my head on a river of _uisge beatha_. I was so far gone now I figured I had nothing of worth left for the hot blonde who'd drunk my scotch.
I considered mourning that—and it was worth mourning, I was certain—but my unmoored attention drifted onto another topic. "So, it's Good versus Evil, is it?" I side-eyed McCue, then shifted back to my grandfather, who had said of the cowboy: _He will have your back, and you will have his, pretty much twenty four, seven, three six five._ "You want us to work together? Him and me?"
"Because of the synchronicity of your birth, it was always the intent. And now you're sealed to it." Grandaddy's eyes glinted. "Rivalry, as I said, but also harmony, partnership, shared ideals."
Again I shot a glance at McCue, noted his closed expression, returned my attention to Grandaddy. I knew what I was about to say would be considered rude, but I had a point to make. "Look—and yes, I know I sound like an a-hole—but I don't even _know_ him, Grandaddy. We're strangers to one another. Now you say we're responsible for one another's lives?"
"Everyone begins as strangers," Grandaddy countered, "except in heaven. We are of one host, boys. We're made of the same matter."
"Resistance is futile," I quoted in derision. "But, you know, I'm not really into the whole assimilation thing."
McCue smiled. "Hell, boy, you might could find me a downright sociable man."
"Oh, God," I muttered, too drunk to mind my tongue. "He really _is_ Matthew McConaughey. Or Sam Elliott. Or—" I waved a hand in a circular but nonspecific direction "—anyone else who talks that Hollywood cowboy talk."
"West Texas," Remi McCue corrected. "Never been to Hollywood."
I fixed Grandaddy with a stare bordering on stink-eye. "So if my new bestie and I are made of heaven's matter, what the hell are you? Some kind of, I don't know . . . _angel?"_
Grandaddy smiled. "Let's just say I'm an agent of heaven."
I pressed the heel of my hand against my brow and swore, but beneath my breath so Grandaddy wouldn't hear it. Then muttered, " _Please_ tell me you're not going to grip me tight and raise me from perdition."
Remi shot me an amused glance. Grandaddy just rolled on. "You're the weapons I've prepared, the grenades we'll hurl, the landmines we'll place. But in the meantime . . . well, let's tug on Superman's cape just a tad, shall we?"
McCue grinned and actually _sang_ the next line about spitting into the wind, which was a downright unhealthy thing to do as well as being messy.
I stared at him, refused to sing the next line of the Jim Croce song, because I didn't want to think about the Lone Ranger and his damn mask, and I never sang in bars. Especially if they were playing godawful country music.
I shifted against the chair back, propping a boot more heavily against the floor to keep from sliding off my ass. "Just _how_ do we go about tugging on that cape?"
"By killing things."
Holy hell. The back of my neck prickled.
The expression in Grandaddy's eyes told me he knew exactly what kind of impact that simple sentence had on me, and why.
The cowboy stirred uneasily. "Grandaddy—"
"I said _things_ ," Grandaddy said clearly. "Not humans—well, unless certain humans insist; there _are_ evil people out there. No." He looked at us both for a long moment. "I told you. Lucifer has loosed his vanguard. Well, he was never stupid, that archangel. Just too proud, too arrogant, too ambitious. He had, shall we say, an overdeveloped sense of entitlement."
McCue's smile hooked sideways again.
"And it cost him heaven," Grandaddy went on. "So, if _you_ were a smart, devious archangel stuck in hell and wanted to begin softening up the world for your long-prophesied return, how would you go about it?" He leaned forward, tapped a forefinger on the tabletop. "You would exploit _disbelief_. You would exploit _skepticism_. You would _use_ all those things people believe don't exist. Legends, stories, myths." He looked at me. "You're the folklorist. Don't you believe?"
Oh, this was undoubtedly a trap. I recognized it from childhood. With care, I allowed, "Much folklore has roots in some _portion_ of reality. Something happened, or existed, that inspired the stories."
"Same with religion," McCue agreed.
"But that doesn't mean every detail is true," I continued, launching into a lecture mode I hadn't used in years. "We're here in the Southwest, so let's take the urban legend of the chupacabra, the Mexican goat-sucker. Is there really such a beast? I don't think so. What I believe is that someone saw a big-ass feral dog, or an oversized coyote, probably mange-ridden and half hairless, killing a goat for a meal." I hitched one shoulder into a shrug. "Well, hell, _that's_ boring. And maybe the guy was drunk, or high on weed or peyote, so he made a tall tale out of it."
McCue was nodding. "We have that legend in Texas, too. Reports of sightings crop up from time to time, and animal deaths that are believed to be caused by a chupacabra."
"Sure," I agreed. "That's because these kinds of stories are like contagious infections. And there are always gullible people, or hoaxers; or total whackjobs, too, who buy anything." I paused. "And conspirorists."
McCue stared at me. "Conspirorists? That's a thing?"
"Conspiracy theorists." I shrugged. "Saves a word."
Grandaddy said quietly, "And so, too, are there skeptics who dismiss everything." Despite the pallor of bar lighting, his eyes burned brightly. "There was a man, a French poet, who understood. It's a famous quote, and you'll undoubtedly recognize it, or some form of it. Charles Baudelaire said: _The greatest trick the devil ever pulled was convincing the world he didn't exist_."
After a long moment, I expelled a chuffing sigh. This was going nowhere at hyperspace speed.
"I'm an accommodating man," McCue said, in a more circumspect tone, "and I like to think of myself as open-minded. But. You know there has to be a 'but.' Convince me."
"There were several earthquakes in the last month," Grandaddy said quietly. "Japan, Turkey, South America, Russia, New Zealand, Northern California, other places as well. The ground opened up in each of those places, and things came out."
I had been somewhat aware of the news in prison, if not intimately acquainted with it, but became more familiar on the ride down the coast via overhearing snatches of conversation in bars and restaurants. " _Things?_ " I quoted, wary.
"The devil's surrogates. And they have assumed many guises." His look on me was intended to be quelling. "Yes, Gabriel, it is indeed just as you said: everything in folklore, everything in the tallest of tales, in all the myths—even religion, as Remi noted—has roots in reality."
I wanted to clarify. "I said _portions_ of reality _._ "
Grandaddy nodded. "Which assumes at least a spark of it. Even a minute speck of reality is still reality."
I was too drunk to parse semantics with a man as clever as this one, so I just squinted at him.
"When hell's vents opened," Grandaddy said, "our reality was altered. It's been expanded upon. Those minute specks are now much larger. Therefore all those ghosties, ghoulies, things that go bump in the night, are now very real, and have been for some time. So are werewolves, vampires, black dogs, poltergeists, even your goat-sucking chupacabras, plus countless other members of the dark hierarchy. But worse than that—"
"Worse?" I yelped.
"—many now host surrogates, or will. And if a surrogate climbs inside any of those so-called mythical hosts, the threat is trebled. They are all of them _Lucifer's_ creatures, made of _hell's_ matter, not heaven's. Far worse than the stories."
" _Surrogates_?" I repeated. "I take it you're not talking about political mouthpieces and spin doctors."
Grandaddy said, "Another name is _demon_. But we avoid that in public."
McCue said after a moment, as if in passing, "Well, that's handy as a latch on an outhouse door."
I grunted. "I think an argument could be made that political surrogates might actually _be_ demons."
Grandaddy ignored me. "Faith is certainty without evidence," he said. "Faith is safety. Disbelief? Well, these days, that's a doorway to surrogates. It allows them to enter. To wreak havoc among unbelievers."
I stared at him. If Baudelaire had nailed it, Lucifer was indeed one damn smart fallen archangel. But . . . "That's a narrow mythos, Grandaddy. There are more religions in the world than Christianity, and more holy books than the Bible. Folklore is of all cultures."
"A rose by any name, Gabriel."
"And every religion has a villain," McCue said. "Don't matter what he's called. Or she. Or it. Evil's evil."
"There are questions to come, and answers," Grandaddy said, "but regardless of the name of the belief system and its own rituals, whatever they may be, the war has begun. The harbingers have risen. They've come to pave the way, to destroy. So we must destroy them. As of tonight, you two are being employed. Being _de_ ployed." He swallowed beer, brushed moisture from his mustache, then gazed benignly upon us both. "Childhood's over, boys. Happy Birthday. Happy _Re-_ Birthday. You're now in the vanguard of the heavenly host."
After a moment, McCue grinned slow, though something in his eyes spoke of an emotion other than amusement. "Well, ain't that just sweeter than stolen honey?"
I blinked hard, tried to clear my vision. He was wearing two hats. Had two faces. But I had to ask. "Is that for real? That accent?"
"I am Texas born and raised . . ." And then he broke off, appeared to reconsider that this claim, in view of what our grandfather had divulged, perhaps needed amending. "Texas _raised_ , at least. It sticks to you. And ya'll better get used to it."
"But it—" I flipped a hand back and forth, "—waxes and wanes."
McCue smiled. "Depends on when it's good ol' boy affectation or just me talkin'."
I, with a mind not as sharp as it had been before so much whiskey, decided against continuing that topic and turned to Grandaddy and cut to the chase. "Since we're angels, do we at least get wings?"
Grandaddy shook his head decisively. "You're not angels. Not yet. For now, you're almost entirely human. Baby steps, boys. It will take you time." His eyes were on me. "But as of tonight you will begin to recall certain things, and those memories will guide you."
"Almost entirely human," I quoted dryly. "Almost entirely? Isn't that an oxymoron?" When Grandaddy didn't answer, I heaved a sigh. "So, I'm an angel-wannabe. Except I don't wanna be."
And McCue, for no reason I could remotely begin to decipher, began quietly to sing. Again. But it wasn't what the band was playing.
My head was throbbing. " _Now_ what are you singing?"
"It's a country song. Charley Pride. 'Kiss an Angel Good Morning.'"
I gazed at him in frank disbelief. "A guy we believed was our old family friend is _an agent of heaven_ —and _I'd_ call that an angel, myself—we've got heavenly GPS units inside us, we're supposed to save life, the universe, and everything—and you're singing _Charley Pride_?"
McCue shrugged. "I can sing Willie Nelson, if that floats your boat into calmer waters. Maybe 'Angel Flying Too Close to the Ground.' Which might could be really appropriate, if things are fixin' to go sideways."
I grabbed up the whiskey and began chugging straight from the bottle again.
# CHAPTER FOUR
I awakened hard and fast with the thick, acrid taste of sour bile in my mouth.
Whiskey. Too much. I'd drowned in it, bathed in it, soaked my brain and gut in it. I had a hard head for booze, but not like this; not so much _gulped_ in so short a time, especially being off booze for so long thanks to an unexpected vacation behind bars.
What was it Grandaddy had said—?
Oh. Yeah. Angels and babies and heavenly matter and _beacons_.
And hell.
And Lucifer.
Shit.
I had no clue where I was. It was dark, but a glint of light pierced the room along the junction of blackout curtains. Beyond, I heard traffic. I rolled out of bed—literally—and a quick glance proved it was a motel room. But I didn't recall _getting_ a motel room.
Nonetheless they came with bathrooms, did motels, and bathrooms came with toilets, and I resolved then and there to pray to the porcelain god. I hastened to make heaving obeisances, rinsed out my mouth when my belly stilled, then lay sprawled upon the tile floor and welcomed the coolness that soothed my face.
And then, despite the muddlement in my head, _memories_ swarmed in.
It had happened so long ago—I'd been twelve—but I remembered it clearly because it hurt like holy hell.
It hurt my little brother, that is. At first. Then it hurt _me_.
It was an accident, but also an example of manifest stupidity. And it nearly got Matty killed.
After the out of control bike with its ten-year-old blindfolded passenger crashed into the neighbor's car parked at the bottom of the hill, I went tearing to the house screaming for Grandaddy, who was babysitting while our parents were away, who came out the front door with a frown upon his face asking what on earth I was yelling about and did I want to wake the dead? Not a wise thing, waking the dead. They were not pleasant companions.
Whereupon I poured out the whole story, ending with the frenzied claim that it was an accident, a terrible, horrible accident, but I appeared to have killed my brother.
Matty, as it turned out, wasn't dead, just knocked half-silly and had a broken arm. Grandaddy scooped up his disoriented charge, carried him into the house, set him carefully upon the couch, examined his injuries.
I, panic-stricken, could barely stand still. And when Grandaddy at last said that my baby brother would survive and turned his clear blue eyes upon me, I blurted, "I'll do anything to take the pain away. Give it to _me_! It's my fault!"
Grandaddy asked, in a very soft voice, "Why on earth did you tell him to wear a blindfold while riding his bicycle down a hilly street?"
I hugged myself, shoulders thrust nearly to my ears. "It was just something to do. I don't know. I don't know. It just was a dare. It was stupid, I was stupid!" I looked fearfully at my brother. "Matty, I'm sorry!" I thought the pain and the dull bewilderment in my brother's eyes might kill me. It hurt me to my soul. "Make it stop," I pleaded. "Make him better. Give it to me. Please."
Grandaddy stared hard at me, weighing me against something; I just wanted him to hurry up. "There is a way," he said after a moment, "that serves two purposes. One will relieve him of his pain."
"Do it, Grandaddy!"
"The other will teach a lesson to a very foolish older brother. One who should know better, who should always protect his younger brother. It's a stewardship, Gabriel."
I knew I'd broken a trust. While my father didn't spare the rod, Grandaddy had never laid a hand on me and never would. But his disappointment mattered terribly.
And Matty?—Matty didn't deserve what had come of his ill-advised journey down the hill with a blindfold on his eyes as his older brother laughed like hell, a journey only undertaken because he worshipped me and didn't want to let me down.
Matty was crying and his eyes looked weird, like he couldn't see straight.
I nodded frantically. "Anything, Grandaddy. Take the pain away."
His voice remained quiet, but compelling. "Will you bear it for your brother?"
I looked at Matty sobbing on the couch. "Always, Grandaddy. I will." I nodded hard. "I'll always take his pain. I don't want Matty to hurt. I don't want him to be hurt. Let it be me. Let me bear it."
Grandaddy's hand was gentle on Matty's dark hair, but he looked at me. "You consent to this?"
I nodded decisively again, and felt the weight of his gaze. I wanted to look away, but Grandaddy did not tolerate any display of submission. Obedience was required, yes; never submission.
"Do you understand?" he asked. "If you do something stupid, if you do something wrong, there are always consequences. You might have killed your brother."
I looked at Matty again, then wiped at my own welling eyes. I thought my chest might burst apart. "I know," I whispered. "I know, Grandaddy."
"Do you understand, Gabriel?"
I nodded fast and hard.
"And do you understand the concept of primogeniture?"
"All passes to the firstborn son," I answered promptly; I'd learned that from Grandaddy years before. "The eldest inherits all."
"Including the rights and responsibilities of his siblings," Grandaddy said. He looked down at Matty, ran a gentle hand through his hair. "It's far more than land, than wealth, than material goods. It's the stewardship of the younger. Matthew is yours, Gabriel, as much as he is your parents,' and mine. Do you understand?"
I nodded.
"You're old enough, now," Grandaddy noted, meeting my eyes once more. "Perhaps you're ready."
I realized then he was thinking about something deeper, something more than just healing his youngest grandson and teaching his eldest. "I'm ready."
Grandaddy wasn't quite Grandaddy anymore. He was other. He was _more_. He wore the face of a man, but reminded me of a story where a man bore the soul of something ancient, and far more powerful.
"Do you accept this responsibility willingly?" Grandaddy asked. "Even all the pain that accompanies it? It's a binding, son. A very powerful one."
I knew it soul-deep, in the parts of me that bore my name, my selfhood. "I accept it, Grandaddy. Willingly."
Grandaddy smiled, teeth glinting briefly in the beard. "Don't gird your loins quite so hard, Gabriel. It's not like that. No trumpets, no white light. You'll just know. It's duty, and honor, and love, and loyalty, and utter devotion. It's what no man can take from you. But you can surrender it should you choose."
I was certain. "I never would."
Grandaddy marked his sincerity. Nodded. "Hold out your arm, Gabriel."
I did so, and Grandaddy touched it even as he touched Matty's, and the pain passed from my brother to me.
It hurt. It hurt bad. Against my expectations, against my will, I gasped noisily and tears slipped free.
I'd done this to Matty. I'd done this to my brother.
Grandaddy didn't heal my arm once I assumed the pain. Grandaddy made me bear the consequences of my own actions: a broken arm, and a concussion.
But neither mattered. What mattered was my brother, and even as I inherited Matty's pain, stunned by its force, my brother fell asleep right there on the couch.
"Primogeniture. Or, as we say, _primogenitura_ ," Grandaddy told me. "It's now coded into your soul."
* * *
—
As I sat upon the bathroom tile and blinked myself out of the _then_ and back into the _now_ , I heard the door open. Despite the protests of stomach and head, I heaved myself upright, made it to knees, reached for the knife sheathed once again at my spine—except it wasn't there.
"Hey," called a voice. "Is it alive?"
I climbed to sock-clad feet—where the hell were my boots?—clung to the door jamb. Steadied myself, took one step into the room. Saw a man silhouetted briefly against the daylight beyond the open door. A man wearing a cowboy hat.
Okay, probably didn't need a knife with him.
"You are a walkin', talkin' example of 'rode hard and put away wet,'" announced the hat. "Though maybe talkin' is a bit beyond you just this minute." He closed the door behind him, shook the paper bags in his hand. "I picked up some grub. You want anything to eat? Grandaddy'll be here soon."
I frowned, rubbed at my brow. Picked grit out of my eyes. Heard Grandaddy again, inside my head.
_"Do you accept this responsibility willingly? Even all the pain that accompanies it? It's a binding, son. A very powerful one."_
I knew it soul-deep, in the parts of me that bore my name, my selfhood. I'd said _, "I accept it, Grandaddy. Willingly."_
I now stood upon the cusp between bathroom and bedroom and stared at the stranger. Grandaddy had said I would remember things. And I had. I recalled the day I caused my brother to crash. But this guy wasn't Matty.
"Food?" he asked. "Or maybe hair of the dog? We found a flask of whiskey in your saddlebags."
_"Primogenitura_ ," Grandaddy had told me. _"It's now coded into your soul."_
I blinked hard, thinking about Matty, an older Matty, then pushed those memories away with a hard inner thrust of denial. Not now.
I ran a hand through tangled hair, snagged some in the ring I'd donned at Grandaddy's behest. Stared blankly at it a moment, then looked back at the cowboy.
McCue? McCue.
"My bike," I rasped, throat sore from vomiting. "Where is it?"
"Outside," he answered. "Grandaddy and I hauled your ass here, dumped you on the bed, then went back and fetched your bike."
I was appalled. "You rode _my bike_?"
Remi McCue set bags and cups upon the table, moved into something approaching daylight, though dim. Then he ran open the blackout curtains, and I winced away from the vicious assault of daylight.
"Son, I been ridin' roughstock for years. Bulls and broncs. A Harley's downright polite compared to them."
I stared at him. A stranger who wasn't, quite, according to Grandaddy. And there was definitely a strong resemblance between the two of us.
" _Everyone begins as strangers—except in heaven. We are of one host, boys. You're made of the same matter."_
I felt a chill touch the base of my spine. "I didn't dream it, did I? Any of it?" Well, except for Matty's childhood incident, which I hadn't thought about in years.
"Nope, not a dream," the cowboy said. "Well, unless we had us the _same_ dream. He gave us a plateful, Grandaddy did. While you were snoring off the whiskey, I did me some thinking. The man has _never_ lied to me. Not in my life. And this—" he raised a hand into the air, displayed the ring, "—well, I felt something. And so did you."
_"You're bound now. Life and limb. Blood and bone."_
I gazed again at the ring on my right hand. The middle finger, not the ring finger. Symbolically, the longest finger represented Saturn, the Balance Wheel, a sense of right and wrong, the law, the search for truth and propriety, self-analysis, secretiveness.
Yeah, I figured being born of _heavenly matter_ maybe led one to self-analysis and secretiveness.
Silver pentagram set in hard black spinel. Five-pointed star contained within a doubled circle. Co-opted of late by some as representative of black magic, but in early Christian symbolism the basic upright pentagram represented the five wounds of Christ and was considered _protection_. Alpha and Omega. A spinel, in lore, supposedly boosted energy to power up a spiritual quest. Silver was a precious metal representing purity, purpose, vision, and strength.
Well, couldn't be a whole lot more symbolic than all of _that_.
Something flew across the room. I caught it out of instinct. "Hair of the dog," McCue said. "Settle that head. Grandaddy wants to go hiking. Has some more to explain, he says."
It was my own flask, silver, embossed with a Celtic cross. I unscrewed the cap, raised it briefly in thanks, sucked down whiskey, then gazed at McCue. I had most certainly been drunk the night before, drunker than I'd been in a long time, but an alcoholic blackout was not numbered among my experiences, not even last night. And I remembered this guy.
Then I focused on what McCue had said. "Hiking? Does the man not know what a hangover is?"
Remi laughed. "Well, that's what he announced. _We'll go up the mountain, gaze upon the world, and listen._ That's what he told me last night, when we dragged your ass in here." He smiled. "Pretty much what I've always done when Grandaddy came calling, though we don't really have mountains to speak of in Texas. So mostly we walked along the arroyo. You?"
There were mountains in Oregon, and I had climbed some with my grandfather. But I didn't say so. Just stared at him. He stared back, expression bland. _I don't know you_ , I thought. _And I don't want to know you. I've lost too much. I can't invest again._
And Grandaddy was in my head once more. ' _Remi is someone you're going to come to know very, very well. Someone with whom you will form a bond unlike any other. Someone upon whose actions your life will depend, and whose life will depend upon your actions.'_
"You know what this reminds me of?" McCue asked.
I shook my head.
"Movies where aliens show up, or elves, or witches, and no one seems ever to have heard about 'em. Like it's all _new_." He shrugged. "I mean, yeah, I'm a little skeptical about Area 51, but at least I'm _aware_ of it."
"The UFO crash at Roswell," I added.
"We _know_ about angels," McCue said. "We know about Lucifer and Armageddon. The concept isn't new. It's just—disbelieved. As reality, I mean. As possibility."
I blew out a noisy breath. "Yeah."
" _Maybe_ Grandaddy's lost his mind," McCue observed. "Maybe he's senile—unless of course he is what he says he is . . . I don't rightly think heavenly creatures possess the capacity to _go_ senile, do they?" He shrugged. "Hell if I know. But I'm going to listen to the man."
I leaned hip and shoulder against the door jamb. I hated admitting any kind of weakness, and muddled memory, in my view, is indeed a weakness. But I got no vibe from Remi McCue that suggested the man was looking for an edge, for leverage. What I sensed was calmness and possibly an inability to take offense. Which might come in handy if we really _were_ meant to work together.
I'd been alone a long time. And this guy is not what I'd pick for a shiny new bestie. Nothing against him, but—a biker and a cowboy?
I looked at him, thinking back to something Grandaddy had said. "What's your name again? Your full name?"
"Remiel Isaiah McCue," he replied. "Remi for short. And you're Gabriel—what?"
"Gabriel Jeremiah. Gabe."
The cowboy grunted. "Makes more sense, now, though, don't it? Biblical names. Seems we were named after some big-hat celestial beings."
I cast my line carefully, with single-word bait. "Grandaddy's an _angel_."
"Agent of heaven," McCue corrected. "Or so he said."
"Come on, man, seriously? Like a sports agent? I don't think so. Angel in disguise, maybe."
Remi shrugged. "He can call himself whatever he likes, I guess. I got on my phone last night, did some browsing. I'm somewhat acquainted with the whole angelic hierarchy thing, thanks to schooling, but never thought of it in terms of _family."_ His eyes were steady. "You buyin' what he said?"
I frowned. "Sure sounded like _you_ were last night."
"I got no reason to disbelieve the man," McCue said flatly. "If you look at various cultures and religions, there are commonalities of context that suggest such things as celestial beings—or whatever you want to call them—exist."
I drank more whiskey out of the flask. "Maybe."
"You're a folklorist. Master's, you said?"
I nodded.
"But I take it from what you said last night that you don't believe everything you read."
I gusted a cut-off laugh. "Hell, no."
A corner of his mouth twisted. "But I'll bet you believe that much of folklore has arisen out of 'commonalities of context.'"
I smiled, tipped the flask, then screwed the cap back on and deflected again. "I'm guessing you do."
McCue picked up one big cup, offered it. "Coffee. I don't know how you like it; I brought back creamer and sugar, too."
"Black. Thanks." I unwound myself from the door jamb, tossed the capped flask to my bed, took the steps necessary to place myself within reach of nirvana. Even fast food coffee was better than none at all, though the room likely offered a whopping two whole servings via sealed foil bag and cheapo coffeemaker. A glance at the other bed showed me a somewhat rumpled bedspread. "You sleep here?"
McCue smiled. "Grandaddy and I weren't sure you wouldn't choke on your own puke last night, so yeah. We did you the courtesy of removing your boots, but that was it. So your virtue's still intact."
I grunted, peeled back the hatch on the plastic cup lid, sucked down coffee. It had often crossed my mind that this brand was more like 3,000-mile motor oil than actual coffee. Now, in the Pacific Northwest? _That_ was coffee. And it needn't be pussied up with various flavorings, either.
_"'Commonalities of context,'"_ I said after a few swallows, quoting Remi.
McCue sat down in the chair beside the small table, opened a bag. "Kind of like saying a vast multitude buys into all the bullshit."
I smiled; I'd been in conversations like this before, in class and out of it. "But it's hard to deny when so many cultures share similar backstories. Bits and pieces, I mean. Look at all the cultures that've included stories of a Great Flood in their oral and written traditions."
McCue dug out a cardboard box holding something approximating a burger, only it was ham and sausage and egg instead. "Your Master's is in folklore . . . my PhD is in Comparative Religion. You see any parallels there?"
I captured the other bag, sought my bed. "One can always make an argument that religion _is_ folklore."
"I'm talkin' about you and me." Remi drank coffee, smiled his slow, crooked smile. "You any good with knives?"
I frowned. "Handy enough. Where's this going?"
"Grandaddy ever show up and teach you how to throw? I mean, how to use actual throwing knives?"
I shook my head. "We spent most of our time shooting."
McCue nodded. "I'm a good shot. Rifle, handgun; I'm a Texas boy, and I grew up on a ranch. But Grandaddy always wanted me to work special on knife skills."
I remembered my grandfather suggesting I learn how to handle a knife. But then, my father—my _human_ father—was a former soldier, current cop, and it was natural the man would want his sons to be skilled. I had indeed learned. Wore a knife at my belt, carried another in my Harley's saddle bags.
But Grandaddy had always taken me shooting. "You're saying he specifically targeted our training."
"I'm saying it's an _idea_ ," McCue clarified. "And something worth asking about when we're up on the mountain."
My brows rose. "You don't trust him."
"No, no . . ." He lifted a belaying hand. "I trust that man with my life. I just want to be sure I understand what he intends for me to do with it."
I lifted one shoulder in a half-shrug. "The battle of Good versus Evil. Isn't that what he said?"
Remi tilted his head, and the hat went with it. " _Is_ there such a thing?"
I'd thought on it throughout my life. I'd thought on it long and hard. _Before_ I went to school, _at_ school, when I wrote papers and my thesis, before walking away from my future. In talks with fellow students, fellow teaching assistants, even with professors caught up in the esoterica of academia.
My intellect claimed it was an exercise in freedom of thought not necessarily moored to reality.
My gut said otherwise.
I looked at McCue. Made no comment, but I had the feeling he'd follow.
Remi's blue eyes were bright. "If you accept that . . . then maybe there's truth in everything Grandaddy's ever said—and ever will."
Grandaddy an _angel_ —or, well, an agent of heaven.
Lucifer, _real_.
The run-up on earth to the End of Days.
Surrogates—demons—walked the earth, and put on the guise of legends, and myths, and campfire stories.
"So," I said, "we're ghostbusters?"
Remi McCue's smile broadened. "I'm thinkin' more like Michaels."
I squinted. "The craft store?"
"No, asshat, as in the _archangel_ Michael. Otherwise known as the Sword of Heaven."
"He's only one archangel," I said, "of the four we know best. Maybe we're Raphaels, or Uriels, or—" And I stopped dead.
Remi's tone was dryly amused. "Gabriels?"
Shit. "I _do not_ want to be an angel. Or even a proto-angel."
"Archangel might be nice, though." McCue sounded downright philosophical. "Remiel, well, he's not so high and mighty as the Big Four, but he's a good guy."
I frowned, perplexed.
"Angel of hope," he said, "and I'm sure _hopin'_ we can win this rodeo."
I squinted at him. "I'm too hung over for this."
# CHAPTER FIVE
The trail was called Fatman's Loop, for no reason I could determine, and it wound up the stony, fire-ravaged flanks of what was, Grandaddy told us, Mount Elden in the San Francisco Peaks on the northern outskirts of Flagstaff, once an active volcano. It was a rough skein of a trail attended by sentinel Ponderosa pines, scatterings of oaks and aspens, and other scrubby vegetation. An elevation of 7,000 feet sucked a fair amount of oxygen from the air, and a man working through the effects of a hangover knew it. I felt aged beyond my years, thirsty, fatigued, and sweat trailed down neck and chest and stuck the thin cotton weave of my black t-shirt to my torso.
Motorcycle leathers had a purpose—for motorcycles. They provided a tough layer against the onslaught of asphalt or gravel, should a bike go down. Not so good, though, for a man hiking up a mountain in thin air, in summer, no matter how attractive the surroundings, though at least I'd left my jacket at the motel. Here it was very dry, unlike the moist, rain-laden environment of the Pacific Northwest. The trails were no more challenging, but when hiking with Grandaddy in Oregon I hadn't been wearing motorcycle boots, or sweating whiskey out of my pores, or wrestling with the announcement made the night before that I was a part of heaven.
That I was _born_ of heaven.
That I now had some kind of celestial partner and was _destined_ —and how the hell had my life become weighted with words like that?—to be a soldier in the front lines of an angelic army intended to fight evil. To keep Lucifer from returning to take over the whole damn world.
My so-called grandfather, who _knew things_ , as I had always viewed it, offered no intel at the moment. Grandaddy climbed the mountain with a walking staff in one hand, lacking Gandalf's robes, perhaps, but none of the gravitas, the secrecy, the seeming awareness of _portents_ , of _knowledge_ , of _elements of information_ held apart from his own grandsons—or, well, whatever the hell we were to him these days. Amoebas, maybe, in the celestial pecking order.
Grandaddy went up first, with McCue just behind. I toiled at a slight remove, thinking I'd really rather be back in a motel room; or, better yet, in a sports bar, watching men meet on the fields of _athletic battle_ , rather than being aimed as weapons in a heavenly war.
It was a popular trail, and we were not alone on the mountain. But brief meetings with others consisted of water breaks, or viewings of where they'd come from and where they were bound, and the occasional curious dog.
Mountain lions, one older gentleman warned, he of the mashed hat worn against the sun; the crumpled khaki clothing; the browned, aging flesh and a sparse gray ponytail. Don't let your dogs run loose, he said pointedly, gray eyes fierce, because lions might take them down from the heights.
Since no dogs accompanied us I was not certain what this kind of suggestion actually meant, but McCue thanked the man with immense charm and courtesy, exchanging pleasantries, said farewell with a polite tip of his hat, and did not bother to point out to the gentleman that we had no canines with us that might be at risk of being taken by lions.
Not even by fleas, I reflected, though possibly flies.
Up we went again, until Grandaddy called halt at a cluster of granite boulders just beyond a crowded cluster of immature oaks in the midst of massive pines, and bade us enjoy the view.
McCue perched his jean-clad ass atop a pile of boulders, set his booted feet, removed his hat to scrub a forearm through neatly trimmed hair, hooked the hat over a knee and smiled into the day as he stared across the distances. I, on the other hand, simply stopped walking, pulled a plastic bottle from a back pocket, sucked down water. I could smell the whiskey exiting my flesh on perspiration. So much for the shower.
"Well?" Grandaddy said. "What did you notice?"
"That man," McCue observed, "was not entirely a _man_ , now, was he?"
I frowned at him. "Which man? We passed several."
"The old man. The one who warned us about mountain lions." Remi's brows rose. "You didn't feel it? He wasn't warning us about _dogs_ , you know. That was meant for us. Be careful where we go, because danger awaits."
I found it baffling. "You got that from an old man you talked to for, like, maybe five minutes?"
After a moment, McCue turned to gaze up at Grandaddy, standing amidst the sunlight in the bright backdrop of the day. "I would like to say his mama didn't raise a fool, but I'm beginnin' to wonder."
Grandaddy smiled. "Don't underestimate Gabriel. His strengths lie elsewhere."
"Who was he?" Remi asked, ignoring my side-eye.
"An angel; and yes, that was a message. You'll find the world is full of messages—and full of angels—now that you're awakened. Some will be clear; some won't be; some will be wrong. And, if I may be permitted to wax Biblical a moment, false witness shall be borne." A trace of breeze stirred Grandaddy's hair. "You've been around them all your lives, but no one knew you, either of you. Hide in plain sight, as I said. But those beacons I mentioned? Your heavenly souls? They are the seat of your strength, the homeplace of your spirits, but they are also the lodestones that others will follow to find you. There is much benevolence in the world, fragments of heaven made whole as living beings, if not wholly human; but just as he warned you, there _are_ lions upon the heights who will try to take down the dogs."
I gusted air through my nose. "Can I at least be a big tough Rottweiler? Not a wussy little Chihuahua?"
"Shows you know nothin' about dogs," McCue observed. "Those are tough little suckers. They'll chew your toes clean off your feet and then grin at you."
Grandaddy's blue eyes were bright. "You, Gabriel? Pit bull. Most certainly. But carefully bred, socialized, well-trained . . . and your purpose will come clear."
I pounced on it. "Just what _is_ my purpose, Grandaddy? _Exactly_ my purpose. Spell it out. You laid a lot on us last night."
His gesture guided our view beyond the mountain flank. "Take a look out there, boys. Tell me what you see."
It was a sun-rich day among the pines, with skies so blue it nearly hurt my whiskey-buffered head because I'd stupidly left my sunglasses in my saddlebags. Below lay a ribbon of road, and a massive lone hill of red volcanic cinders that had once been mined. Half the hill was eaten away. Beyond lay vast expanses of flat earth and occasional convolutions of cinder hills. The Peaks themselves were a huddled cluster of an extinct volcano's cones and craters, overtaken by Ponderosa, aspen, oak, various scrub vegetation, grasses.
"I see forever," McCue said, voice smooth and contemplative. "The unending earth."
"He appears to be a poet," I said with dry amusement, without rancor or sting. I didn't know the guy, but we were stuck in the same whackoid shit going down. "Yeah, it's a big world. But why is that hill so torn up? Looks like a big ol' shark chomped on it."
"They don't use salt on the roads in winter, here," Grandaddy answered. "They lay down cinders . . . which, lacking the protective nature of salt, consecrated or not, makes this area ripe for activity of the unnatural kind. But they don't work that hill anymore. Cinders for the roads are hauled in from more distant cones." He paused. "Gabriel, I want you to think about where you are. Tell me what you feel. What you _sense_."
I was hung over, tired _,_ short on oxygen. But this was nothing new from Grandaddy; he suggested this whenever we'd gone exploring on his visits. So it was with nostalgia and a certain ease of spirit, of familiarity, that I loosed my awareness of the here and now and let the sense of the immediate environs come to the fore.
I closed my eyes, made my heart slow, allowed my awareness of _self_ to be at peace.
A bright day, warm but not hot; clean, fresh air; the sound of insects, of birds, of air in the grasses; the rush of wind among the trees. My inner self caught a flash of yellow color, a reflection off something opalescent, the watery blurring of pastel colors layered one upon the other.
It rose like a tide within me, the sense of calm, of quietude, of a deep, earth-anchored spirituality.
I smiled, opened my eyes. "It's sacred. To the Navajo, Sacred Mountain of the West. Abalone Shell Mountain. _Doko'oosliid_. And to the Hopi, a home to spirits. _Katsinam._ Kachinas. A place to find a way through _koyaanisqatsi,_ life out of balance. Up here on this mountain, we're safe."
"Huh," Remi said after a moment. "That one of those strengths you were talkin' about, Grandaddy? He's a sensitive?"
"To places, but not to people. He's not an empath. People, for Gabriel,"—Grandaddy smiled a little—"remain something of a challenge. You, on the other hand, do understand people. You are an empath. You balance one another, Remi; and yes, _koyaanisqatsi_ is indeed life _out_ of balance, and that is exactly what is happening now. Your job is to help restore the balance." He looked at me. "You dreamed, did you not? Last night?"
I was unsurprised he knew. I nodded.
"That stunt you pulled putting Matthew's life at risk."
I winced; stated like _that_ it sounded so very much worse . . . but, well, yeah. I had indeed put my kid brother's life at risk.
"It was a test I knew would come in some fashion," Grandaddy said. "Matthew was your brother in flesh, so there was a bond, but not in soul. Nonetheless, you had the instinct. Some recognize it, embrace it. Others duck it, or just never awaken to it. I didn't doubt that you'd awaken to it and consent."
I frowned. "Awaken to what?"
"The stewardship of the younger. It's what you did with Matthew all of his life. _That_ day, though—that was for Remi."
I was aware of McCue's hatless head coming up sharply as he, too, stared intently at our grandfather.
" _That_ day?" I echoed. "The day I nearly got Matty killed? How the hell was that day for _this_ guy?"
"Matthew was Remi's proxy. He represented the younger brother you needed to have, to trigger the drive to protect." Grandaddy smiled. "There's always an alpha, Gabriel. That's the role of the eldest, because he or she is first. The first precedes the others, prepares the way, makes smooth the path for the younger to follow. Your instinct was always to protect Matthew no matter what, regardless of any danger . . . because that's what you'll do for Remi. That's what _primogenitura_ is."
I shot a quick glance at McCue, switched back to Grandaddy and waited for more. More always came, with Grandaddy.
"Find two soldiers who fought together," he advised. "Ask them about the bond battle creates. They are far more than 'besties.' As for you and Remi, that instinct you had for Matthew is the instinct you have, or will have, for Remi. You just don't know it yet, don't _feel_ it. That day when you offered to carry your brother's pain, begged me to lift it from him, to give it to you, I knew you were ready. And you consented. But you weren't _sealed_ to Matthew that day, Gabriel, because he wasn't heaven-born. That day prepared the way. Last night, when you put on the rings and clasped hands, you and Remi were sealed to one another."
I looked sharply to McCue, saw the cowboy staring back, equally startled. We neither of us knew what to do with the information. I could only see Matty in my head, in my memories; I had no clue what McCue was thinking.
_Sealed_ to one another?
I lifted my right hand and examined the back of it, studied the ring upon my middle finger. Rubbed the ball of my left thumb over silver pentagram, black stone. When I raised my eyes, I found McCue staring back. While his face appeared relaxed, the expression in his eyes was a myriad of complex thoughts I couldn't define.
But then the cowboy smiled crookedly, looked back at Grandaddy, an angel from on high. Regardless of semantics. The surface of Remi's tone was casual, but there were undertones within it. "So, I'm the beta, am I?"
"There's a reason you grew up with an older brother, Remi," Grandaddy said. "Birth order affects development. You're not _lesser._ Beta merely means second. Younger. Gabriel's spark simply quickened before yours. That doesn't make him _better_. Remember: even with twins, one is born first."
McCue considered it in silence, then nodded slightly. "Primogeniture is known mostly as a medieval concept. But it extends farther back than that. All the way to Biblical times." He looked at me. "Predisposition. Predestination. Guess we'll find out if they apply, won't we? Bein' together _in the trenches_ , and all."
So, Remi McCue was not entirely an amiable beta soul. I looked at Grandaddy, found him smiling faintly. I asked a silent question.
"It will come," Grandaddy promised. "It's a _process_."
I shook my head. "We have lives. Hell, I just got mine back. You can't expect us to walk away from everything."
Grandaddy's voice took on an edge unlike anything I'd heard from him before. My skin itched, and I stared at him in shock. He was _doing_ something again.
"That's exactly what I expect, Gabriel. This is the _End of Days_ I'm talking about, with the fate of the world at stake. _Everyone_ born of heaven must answer this call, if we're to succeed. Is it a sacrifice?—of course it is. But there is nothing in your lives that is of greater importance than this." His eyes were steady. "You have never disappointed me. Don't do so now."
I looked for compassion. Found none. "What about our families?"
Grandaddy didn't even attempt to hedge. "I said we could massage things. Well, I have massaged the minds of your parents and brother. They believe you are in prison finishing your sentence."
"But that's only six more months."
"And your father's reaction once you're out? Would you be welcome in his house?"
After a long moment, I said no. Because I remembered what my father had said—even if _he_ didn't because of Grandaddy's brain massage. That night on the porch, as I rolled my bike out of the garage, felt like a death-knell. My mother stayed inside, and Matty was probably out getting high.
"And what would you do, Gabriel?"
"Get on my bike and head out. Maybe for good."
Grandaddy nodded. "Well, we will free you of that. They will remember no hostilities, only that you, once your sentence is served, are on the road. And so you are free to do your duty without interference for however long it takes."
"Just cut them off like that, huh?" Though I wouldn't much miss my father. I glanced at the cowboy, looked back at Grandaddy. "What about him?"
"Remi is traveling the world undertaking research for the book he plans on writing. And he may, from time to time, call home to reassure his parents. But the calls will show overseas locations, nothing in this country. You, on the other hand, may drop postcards to your mother once your sentence is completed. Your father's a son of a bitch, but she's a worthy woman."
And there it was, all tied up in a neat little bow. The present. Our futures. An explanation for it all.
McCue sat rigidly on his rock. "But what about—?" Yet he broke it off abruptly, as if he'd seen the answer in Grandaddy's eyes.
After a moment freighted with tension, he reached down, pulled tough prairie grass from the ground, began to plait long strands. Whether he was thinking alpha vs. beta, or blue vs. red, or cat vs. dog, I didn't know.
He asked, "We have any start-date for Lucifer's return?"
"No." Grandaddy was decisive and yet regretful. "There are signs, of course. Indications. But no timeline. The world doesn't run like clockwork, despite humans swearing it does." He nodded slightly, staring into space. "I've dumped a great deal on your plates. But trust in your hearts, follow your souls, and know that I would never set my grandchildren upon a false path." He turned, reached into the pockets of his frock coat, drew forth items. One he tossed to Remi, another to me.
I caught what was thrown, frowned. Looked at my grandfather in bafflement. "A shotgun shell?"
"Powdered iron load," Grandaddy said. "The best means of dispersing or destroying a ghost. Birdshot works, so does buckshot, if it's silver, or iron. Same with bullets. Remi, show him what you've got."
McCue, smiling crookedly, held it up into the light. Displayed between thumb and forefinger, it gleamed pure and clean and silver: a .45 caliber bullet.
"You can take down a werewolf with that," Grandaddy said, "and a passel of other things. Now Gabriel, I do know you're aware of the lore, of the legends. I taught you, and so have the halls of higher learning." His smile was wry. "Even TV, movies, novels, and comic books. But what you have to remember now is that these are no longer tall tales. These beings are surrogates. They _wear the guises_ of fiction and folklore."
"Grandaddy—" But I broke off as he held up a hand.
"There's a woman in town," Grandaddy said. "Her name is Lily Morrigan. She's expecting you. She'll provide you with details of your first job, as well as with some items you'll find helpful in the coming days. She's unconventional, but completely trustworthy."
"'First job?'" I echoed.
"Why can't _you_ tell us what we need to know?" Remi asked; and I heard the first hint of tension in the cowboy's tone. "You've been doing it all our lives."
Grandaddy smiled. His eyes were ageless. "Because you're not my only grandchildren. There are others we're preparing. Go to Lily. She'll keep you updated, and I'll contact you when it's necessary."
I longed for scotch. But I longed more for the uncomplicated ignorance of two days earlier, before stepping foot inside a roadhouse in Northern Arizona and out of a life that was in no way perfect, or even comfortable, but was nonetheless free of such encumbrances as saving the world.
"How do we find this Lily?" McCue asked.
"Have drinks at the Zoo tonight," Grandaddy said. "She'll find you." He turned away from the vista then, stood with his back against the sky and a staff in one hand. He put me in mind of Moses on the Mount.
Or, yeah, Gandalf.
"You're trained," he told us. "You're ready. But it won't be easy. Trust your instincts. Trust what I've taught you. Believe in what you're doing. Commit, and never waver." He smiled, and a glint was in his eyes. "Fight the good fight."
I realized then what was in the offing. "You're _leaving,_ " I accused. "You drop all this apocalyptic bullshit on us, then walk away?"
Grandaddy marked me. Weighed me. Flicked a glance at Remi, who waited in the silence of tense expectancy. Then he nodded his head with its wealth of flowing silver hair. "Fair enough," he observed. "You're not boys anymore, to idolize every word I say without questioning it. I've never steered you wrong, but you've got minds, and were trained in school to accept nothing on faith." He bared teeth in a broad smile. "Even if that is the key to all. So. Do this job. Then decide if it's bullshit."
I was all out of patience, but schooled my tone into matter of factness. "Look, all this is normal to you. I get that. But I barely know this man you say I'm now _sealed_ to and you expect me to trust a woman I've never met to explain to us a job we're to do in order to serve heaven. Heaven! Cut me a little slack about being slow to buy into all this without questioning it, okay?" Without stirring a hair, he suddenly loomed extremely large against the sky. His shoulders stretched the seams of his frock coat. Behind him the world, the air, blurred into something like pixels. Pixels under a sheen of oily rainbowed water.
Holy crap. Those were wings. I couldn't see them, not clearly, not beyond impression, but those were _wings_.
McCue's tone was dry. "You want to ask that again, do you?"
I did not.
"It's a broken road you're on," Grandaddy said, as the pixilation died, "and you'll stumble, and you'll fall, and you'll likely be hurt, even badly hurt. Your bodies are human. You're mortal. You can die, because soldiers do. But just spit out the blood, pick yourselves back up— _pick one another up_ —and do the best you can."
I let that settle a moment. "God," I muttered finally, "we're verging on an Army commercial."
McCue quoted, "' _Sancte Michael Archangele, defende nos in proelio; contra nequitiam et insidias diabolus esto praesidium_.'"
I shot him a glance. "Seriously? Latin?"
"Timing seems appropriate. ' _Saint Michael the Archangel, defend us in battle, be our safeguard and protection against the wickedness and snares of the devil.'"_
I squinted. "When Latin evolved as a language all those four thousand years ago, I don't think it was _ever_ intended to be spoken with a Texas accent."
Grandaddy threw back his head, and his rich laughter rolled out into the day like the tolling of a bell. Maybe the voice of heaven? I shivered.
Our grandfather grinned at us both when the sound died. "Yes, Michael would approve of you both. Most assuredly." He drew in a breath, released it, and there was memory in his eyes. "I said this to you both many years ago. You needed to know it, to nudge you in the right direction, but it wasn't necessary that you recall it then. Now, it is." He looked at us both, one at a time. "There was a thunderstorm."
I smiled as memory snapped into place. "You said God was bowling."
"But he wasn't," McCue noted. "It was Thor."
I nodded firm endorsement. "God of Thunder. Damn straight."
Grandaddy looked large against the sky. "I say again what I said that night: 'You're sealed to it, now. Life and limb, blood and bone. The fate of the world hinges upon it.'"
I remembered it so clearly. And I heard Remi's Texas-accented voice overlay my own, as we said together, "'The whole entire world—and everyone in it.'"
Grandaddy nodded. "The information you'll need to survive is everywhere. Much of it you've studied, much you'll need to learn. Look to all cultures, all source material, judge for yourself what is evidence when you've experienced it—or when you're certain you can trust whoever provides you with the intel. Sort out what is empirical, what is allegory, what is metaphor. Because even the writing on the wall may be false, or intended to mislead. Remember that dead languages are _dead_ , and that a book written by many men, in many tongues, may be misunderstood, may in fact be mistranslated. And most certainly may be misquoted to serve agendas."
And then he turned away, set his staff, and began to walk down the mountain.
I squinted, closed my eyes, rubbed my brow. But I looked at the cowboy, the only man left on the mountain who understood this confusion; hell, _shared_ this confusion. And probably shared my inclination to disbelieve, which was, apparently—if one were to believe a man he'd trusted all his life—a doorway to the devil.
"You know," I reflected aloud, "he really should just _do_ it. Grandaddy."
McCue raised brows. "Do what?"
"Plant his staff and announce to Lucifer: _'You. Shall. Not. Pass.'"_
The cowboy laughed, then held up an object that glinted in the sun. "A silver bullet . . . _for werewolves_."
I inspected the shotgun cartridge I held. "A powdered iron load for _killing ghosts._ "
Remi shook his head. "Makes me wonder if we're supposed to cut our own stakes for, you know, _vampires_."
I sighed, tucked the cartridge into a back pocket. "Life just got a whole lot more interesting."
McCue rose, stretched, stared thoughtfully into the sky as he put his hat back on and, unexpectedly, quoted Shakespeare: "Cry 'havoc' . . . and let slip the dogs of war."
Well, if the Rhodes Scholar was going _that_ way . . . I rolled my shoulders, cracked my neck. "And I'm no damn Chihuahua."
Remi smiled, and I smiled back. For that brief flash of a moment, even as strangers, we were in complete accord.
McCue nodded, pocketed the silver bullet. "Come on, pit bull. Let's go see a veterinarian, make sure you're up to date on your shots."
I followed, heard McCue start singing— _again_ —as he led the way down the mountain. After a moment of disbelief, I raised a question over the sound. "That more Charley Pride?"
Remi broke off long enough to call back, "Rascal Flatts. 'Bless the Broken Road.' Grandaddy says we're on one now—figured we might need every blessing we can get."
I couldn't help the plaintive tone. "But does it have to be _country_?"
"God's music, boy. God's music!"
I wondered if maybe I could get a WiFi signal through my phone's mobile hotspot. Because then I'd download a glorious oldie, all seventeen minutes of Iron Butterfly's 'In-A-Gadda-Da-Vida,' and play it as loud as I could, even if it drained the entire battery. That song, too, was quasi-Biblical, since the song was _intended_ to be 'In the Garden of Eden' when it was written, before it got totally booze-mangled.
"Or I can sing a hymn, if you'd rather," McCue called.
I scowled at the cowboy's back. "How about we just go with the sound of silence?"
Which prompted McCue to shift to Simon and Garfunkel's "Bridge Over Troubled Water."
Well. At least it wasn't country.
# CHAPTER SIX
Saturday night McCue and I grabbed quick showers in our motel rooms, steak dinner at a place called Horsemen Lodge—it, too, featured mounted dead animals, leaving me to wonder if this was required decor for Arizona; very different from the coffee bars of Portland—then on to the Zoo Club.
It was crowded and noisy as I preceded Remi McCue inside. Live country music once again reverberated within the massive roadhouse: drums, guitars, and the broken nasal croon of a tenor voice sliding all around the notes.
Crying in his beer, I reflected sourly, forcibly resolving not to insert fingers into ears. They were always crying in their beer about a woman. Or maybe about their dogs. Or their trucks.
I worked my way through the crowd but shot a brief glance over my shoulder at Remi. I raised my voice to be heard above the music and crowd noise. "You ever cry over your truck? Or a dog?"
McCue's brows knit as he called back, "Am I supposed to?"
"You're a cowboy, aren't you? Into country music?"
"I'm more likely to cry over my _horse_ ," McCue said. "And do you really think a cowboy bar is the best place to debate the merits of what country people do or do not do? Especially when it's obvious you're _a biker_?"
Well. Possibly not.
At the back near the pool tables, I found us an empty two-seater table shoved up against one of the tree trunks serving as load-bearing support for the beamwork overhead. String lights dangled, illuminating the glass eyes of a snarling stuffed bobcat as it crouched overhead. It put me in mind of the lions the old man on the mountain had mentioned.
"I'll get drinks," I said as McCue dropped into a chair. "What do you want?"
Remi was evaluating the surroundings; maybe checking for surrogates? "House draft'll do."
I made my way through to the long slab of bar, ordered and fetched two brimming mugs, worked my way back. Handed one down to McCue, but paused as my attention was caught by the pool tables beyond, near the back exit, as someone indisputably female bent over the table.
I do like a woman with a stick in her hand.
Smiling, I wandered closer with mug in my fist.
It was the young woman from the night before, the one who'd asked me to dance, then knocked back a share of my Talisker. Still with the blonde hair pulled back in a waterfall of ponytail, but tonight she wore black tank top, black jeans, silver at both wrists and in her ears, a doubled string of chunky turquoise on silver wire around her throat. And she wielded a mean pool cue.
She played a man, and she beat him. The game had attracted onlookers; there was laughter, cash exchanged, elbows planted into ribs, muted jokes at the expense of the cowboy she'd defeated. Was she a hustler? Or just good at pool?
As she looked up and met my eyes, I smiled. Inclined my head just a tad to acknowledge her win. Sent the invitation. Hell, Grandaddy'd said nothing about ignoring the attractions while we waited to meet whoever this Lily woman was. Hell, for all I knew, _she_ was Lily.
The light over the pool table was better than in the rest of the bar. I saw now that in addition to brown eyes, she had the faintest dusting of freckles across her nose. Other than bright lipstick, makeup was minimal; but then, she didn't need it.
It was hard to look at anything other than those red lips. Probably, she knew it.
I saw the spread of her pupils as she stared at me across the table, the loosening of her mouth. She looked nowhere else as, with a slow smile, a lifting of brows, she handed off the cue stick to the man nearest her, who happened to be the cowboy-hatted man she'd just defeated. She hesitated an instant to keep my eyes on her, then turned and quietly slipped out the back exit.
I was no fool. I damn well followed.
As I stepped outside into the darkness beyond, as the door swung shut behind me, I smelled a hint of perfume. And then she surged close and her hands were on me, one reaching between my legs to cup, to tease, the other sliding up beneath my t-shirt to spread against my chest. She moved in close, stepped me back until shoulder blades and ass were against the logs of the roadhouse, and let me discover what those lips were all about.
Well, alrighty then. If this was Lily, I'd be sending Grandaddy a Candygram.
"What's your name?" she asked against my mouth.
At that particular instant I didn't know. I didn't care. My mind was on other things, such as all the blood heading south.
"What's your _name_?" she insisted.
I smiled against her mouth, then cleared my throat. Was mostly breathless. "Hell, woman—whatever you want it to be."
This was—really fast. Like, holy shit fast.
Motel. Motel would be good. Or a car, if she had one. Hell, even McCue's _truck_ , if I could get her around to the front of the roadhouse.
Or maybe right there was good. Yeah, this could work. Even against the rough logs of a wall.
Then her breath was unaccountably cold against my mouth. "You _lit it up_ last night. I felt it. And it shines so bright, it does. It's so very pretty. But now I'm going to extinguish it. I'm going to extinguish _you_."
What the hell—?
"Call me Iñigo Montoya, for all I care," she said, "I go by them all. Hell, call me Legion."
Well. Not Lily. And the woman had a _knife_.
I caught her wrist as she thrust hard with the blade toward my belly. It was all instinct, pure reaction, to catch and snap that wrist, to thump the heel of one hand against her breastbone in a short, sharp pop of impact that knocked her backward. I'd learned the hard way never to underestimate, never to hesitate, never to give anyone with a weapon in their hands the opportunity to use it—and I decided on the spot that gender damn well didn't matter when outright murder was a real possibility.
She'd have gutted me. I saw it in her eyes as she staggered, then backed away. She'd have opened me up, reached inside, yanked out my intestines.
"I'd have _tap-danced_ on them," she said through bared teeth, clearly following my thoughts. "Or maybe fried them like pork rinds and chowed right down on them."
I'd hurt her. She backed off farther with her wrist clasped against her abdomen, breathing hard. Between us on the ground, glinting in fitful light, lay the knife.
She glanced at it, looked at me. Her eyes remained brown, but the pupils stretched into an alien verticality, like weird-ass slitted cat eyes. The accompanying smile was feral. "Welcome to the war." And she stripped off the necklace and flung it at me before she turned and ran.
I ducked aside, but not before one of the silver and turquoise loops coiled around my neck and closed. I felt the throttling pressure, felt the stones bite into my flesh, felt the wire on which the chunks were strung close down hard on my throat, squeezing flesh.
I caught at it with both hands, tried to dig fingers beneath the wire. All breath was banished, and black spots crowded in from the edges of my vision.
Then my ring—Grandaddy's ring—made contact with the wire. Silver on silver.
Even as I heard the back door creak open, heard a Texas-accented voice questioning what the hell was going on, the necklace fell away. It spilled to the ground and lay beside the knife, silver wire all melted, charred turquoise chunks unstrung and scattered across the dirt. I sucked air in, gasped it back out again.
McCue was beside me. "You okay?"
I massaged my neck, cleared my throat. Swallowed painfully, sucked in and blew out a breath. "Shit," I expelled on a gust of air, "I guess—" I hacked a little, sucked air again, but the voice still came out broken, "—guess Grandaddy was right."
"About which?"
"All of it! Including the thing that old man said up there today—" I coughed, cleared my throat, flipped a hand toward the ravaged mountain looming behind us, "—about lions taking dogs from the heights. She damn near took _me_ right off. Shit." I rubbed my neck again. No blood, but very sore.
Remi bent, collected the knife, examined it in bar lighting. "Huh. SOG Seal Strike." He turned it over in his hands, admiring the weapon. "Damn near ten inches over all, and a blade almost five." He looked at me, brows lifted. "Yep, she was goin' for you, all right." He bent down in muted light, scuffed at the shattered necklace with the toe of one boot as he studied it. "What'd you do to piss her off so fast?"
I intended to answer, but the door opened again. I twisted away from it even as Remi stepped back, flipping the knife in his fingers with ease, clearly prepared to use it.
Another woman, but vastly different. Elegant inked spirals and angled geometry ran in a rainbow of colors up her bared forearms wrist to elbow in sleeve tattoos, twisted silver bands clasped her biceps, and she wore her two-toned hair in a modified Mohawk. It was dark and cropped close to her head on the sides, though not shaved; choppy and red on the top. The end of her right eyebrow sported the twin glitters of silver hoops, and a torc—an actual Celtic torc, worn in the correct manner—hugged the back of her neck as the coiled, knotted ends rested on her collar bones. Even in poor light, her eyes, set at a slight upward tilt, were a very clear green.
"So," she said, and a lilt colored her tone, "you boys outside to see a man about a horse, or got something more on your mind?"
I rubbed my throat again. "Do what?"
McCue was laughing. "That's country speech, son. She's asking if we're drainin' the lizard."
I reflected that the ponytailed blonde very nearly _had_ drained the lizard, if of something other than piss. And it sent an uncomfortable quiver through my gut. God, those _eyes_.
"You Jubal's boys?" she asked. "You look like it. You look like a lot of them." She pursed her lips, nodded. "Jubal and his grandkids. I've seen many of you. Come on, then. Let's go next door. My rig's parked over there."
McCue actually tipped his hat to her. "Would you happen to be Lily Morgan, miss?"
She seemed amused by his Southern courtesy. "Morr- _i_ -gan," she emphasized. "Not Morgan."
Morrigan. Hah.
Remi caught it, too. I found it clever and amusing as he ran with it. "Your sisters around? You come as a trio, don't you?"
Her eyes sharpened on him, accepted his challenge. "They're in the rig," she said. "Come on along and meet them."
Remi was deadly serious. "Yes ma'am."
I watched the woman continue striding ahead. "You're not suggesting she really _is_ , are you?" I asked, sotto voce. " _And_ her sisters?"
"Well, I don't think it's the Kardashians, no," he said dryly, "but you just met someone who claims to be Lily _Morrigan_. You tell me."
"Irish mythology in all its beauty," I answered, grinning. "The Morrigan is the Goddess of Battles, heroes, the dead. She's either herself and independent, or one of three sisters. The stories differ."
The woman swung around, walked backward with tattooed arms hanging loose. Her torc and arm bands glinted in starlight. "So, which do you figure you boys will be? Heroes? Or the dead?"
I cleared my throat, neck still sore. "Very much alive and victorious," I replied. "Every damn time."
"Well," she said, smiling, "that is what I'm here for. To make sure you survive to _be_ victorious."
"Where exactly are we going?" I asked, looking ahead of her to the building some distance away. "Or are you living in the bowling alley?"
"Right here." She paused, stepped aside, swept an illustrated arm out to indicate a massive black-and-red bus-like vehicle that had a front end on it like an 18-wheeler cab.
"Whoa." McCue was patently impressed. "That is one honkin' big rig. Garage unit?"
She nodded approval. "Sixteen feet of work space at the back end, the rest is all for living. Come on in. I've got whiskey." She tossed a bright glance at me. "Though it's Irish, not the Scottish Jubal says you prefer. Might help that throat."
I was astonished. " _You_ live in a motorhome?"
It was, I felt, complete cognitive dissonance to link her with the giant vehicle. Lily Morrigan, with tattoos and Mohawk, was most decidedly not what one might think of behind the wheel of an RV, based on what I'd seen at rest stops. She tilted her head a little, assessing me. "The battlefields don't generally come to me, I go to them. Now, come on inside. I'll show you a few things, find out what you know, ply you with whiskey until you're babbling like babies, and then you can crash here after you've done the job." She pulled open the door, climbed steep steps, disappeared inside.
Oh. Yeah. The job Grandaddy had mentioned.
Remi went up first, took off his hat as he entered. I waited a beat, ducked in behind him. Stepped into the impression of warm lighting, rich wood, leather, brushed metal fittings. Music was playing, something robustly Celtic.
And then the crow on its perch croaked, and the huge Irish Wolfhound rose up from the floor.
Lily Morrigan said sweetly, "Meet Nemain, the crow. The bitch is Macha. My sisters." Her smile was very wide and it set her eyes alight. " _Céad míle fáilte_ from the Goddess of Battles. A hundred thousand welcomes."
# CHAPTER SEVEN
I couldn't help myself. I was pretty sure I knew the answer, but something in me needed to ask. A couple of days before, no. But after what Grandaddy laid upon us and my encounter with the murderous Iñigo Montoya/Legion (or whatever her real name was), I was no longer so certain how the world was ordered.
"You're not . . . really," I said, and knew it sounded weak. Better I should scoff, or deny. Or simply be polite about it and say nothing. But . . . "From _mythology_?"
Then Remi McCue stepped in. "I am a man prone to belief in many things, but this, well . . ." He let that trail off, seemingly at a loss for more. Then he tacked on, "Who are you _really?_ "
I thought _what_ was she might be a better inquiry, but at least McCue had asked a sensible question. More than I had.
Her bright eyes amused, Lily Morrigan said, "This from a man whose soul is born of heaven." Then she shrugged. "Need to know basis. And what you _need_ to know, right now, is that I can set you boyos up with what's required to undertake the task. Follow me."
We followed. McCue seemed unfazed by the surroundings, by the fact we were inside what approximated, to me, a decidedly up-style hotel suite. I noted good wood, granite countertops, rich leather upholstery, flooring a split between tile and carpet. Everything was sized with infinite care to optimize limited space. Couch, recliner, big U-shaped dinette. Stove, oven, sinks, refrigerator, and freezer. Entertainment center over the front cab with speakers, flat-screen TV, CD and DVD slots.
I'd passed plenty of motorhomes on the road—always, always wallowing in an interstate lane where they did not belong. I'd seen them at campgrounds with awnings put out along with yappy dogs, but I'd never been inside one before; had never anticipated this kind of quality, of detail.
And certainly I had never expected to walk through a kitchen and into an actual _garage_ that made up the entire back end of the huge RV. It housed banks of utilitarian white-faced cabinetry, a metal ladder running up to a hatch in the roof, diamond-plated chrome flooring, plus a ceiling fan and skylight, and a roll-up door.
A rectangular platform of sorts was folded inward and fastened against one big wall, beside banks of drawers; I realized it was a bed that could be lowered and leveled. I saw, too, an oxygen tank, monitors for blood pressure and heart, a tangle of tubing that suggested IVs could be hung. On the other side, where Lily Morrigan led us now, was a stack of drawers topped by cabinet doors, and a counter.
"Jesus," I muttered, rubber-necking. "Livin' large." Certainly larger than my prison cell.
She flipped latches, pulled open foam-padded, compartmentalized cabinets and drawers, began taking items out of storage. Two handguns, an assortment of sheathed knives, even two short-barreled shotguns. She also stacked plain cardboard boxes of ammunition, various tubes and flasks, plus small dyed-leather bags with mouths snugged closed by thongs and ribbon. A motorhome that was, I realized, a combination of living quarters, field hospital, armory.
Remi voiced what I was thinking, though the vocabulary differed: "Might could come in handy, rig like this, at the end of the world."
Lily picked up two revolvers, one in each hand, held them out. I took one, McCue the other. I was conversant with handguns in general, knew these after a fashion, but had never fired either style. Mine had a short, fat, 3-inch barrel, long cylinder, a striated grip that felt like rubber. Probably 7.0 to 7.5 length overall.
I broke it open: five chambers. The one McCue held had a longer barrel; 6.5, I judged, and an overall length maybe 9.5 inches.
Lily watched me. "Well?"
I shrugged. "Taurus Judge. Two different models. Both hold .410 shotgun shells—bird- or buckshot—or .45 caliber rounds simultaneously."
She nodded. "In our line of work it's silver rounds, of course, just like the bird- and buckshot. Or powdered iron cartridges. Even consecrated salt will do, though that's only a temporary hold that gives you time to reload, or get the hell out of there."
"Only a five-round chamber," I pointed out.
She shot us both a hard look. "If you can't put down something in five, maybe you deserve to be dead. It's a revolver, not a semi- or auto, so reloading in the midst of battle is a challenge, but maybe that'll convince you to get the job done within a couple of shots."
I swapped out with McCue so he could inspect the short-barreled model. Other than the reloading in the midst of battling demons thing, I was familiar with the weapons and their features.
Lily picked up a shotgun, tossed it to me; I caught it, tucked it automatically under one arm. "Well?" she asked.
I set down the revolver, put the rifle through its paces. "It's a Remington 870. Started with an 18-inch barrel, modified with a police tactical barrel swapped in. Plug is pulled, so you can load it with five or six shells." I grinned. "It's a James Bond movie, right? You're our Q setting us up with all the cool weaponry? Where are the gadgets? The Aston Martin?"
She smiled briefly, shoved unmarked ammo boxes toward both of us. "Silver-dipped birdshot, buckshot, powdered iron, silver rounds. Keep in mind that you can't just walk into any old sporting goods store and buy this. You'll be with me a fair amount of the time, and others can resupply, but you'll need to learn to _make_ what you need. You can buy many supplies online, but you can also go to pawn shops and estate sales for silver. You can obtain silver shot online from APMEX—they're the top gold and silver retailer—but it's very pricey. Melt down your own, make the shot. Or dip what you buy."
McCue studied the gun in his hands, then looked up at her. His blue eyes held respect, but also concern. "You're serious."
"It's a _war_ ," she said evenly. " _I'm_ here, am I not—the Goddess of Battles? Wake up, boys. You're part of it, now." She picked up a sheath, tossed it at Remi. "Tell me about this."
He caught it one-handed, set down the revolver. He slid three knives out from the sheath, spread them like a fan. They were slim, elegant, oh-so-sharp, with slightly curved handles. "Hibben throwing knives." His tone was worshipful. "Gen 2. Blades four and a-half inches, give or take; overall is almost nine. Just under six ounces apiece." His grin lit up his tanned face. "Works of art, these are. Three to a sheath."
"Dipped in silver and holy oil. Get it done in three," she said, "and you don't need more. But—a little backup. For you both. Here."
McCue set aside the throwing knives as she slid a big leather sheath across the counter to me. I put down the revolver, shotgun, pulled the knife from the leather. Steel blade glinted light and dark in whorls and tangles like oil atop water.
"Bowie," Remi said, standing close. "Twelve-inch blade, almost eighteen inches overall. Serious business. Full-tang Damascus steel; look at that patterning. My God, she's a gorgeous thing."
Light glinted off Lily's piercings as she continued. "Buffalo horn handle, brass fittings; both are protective materials when it comes to certain unnatural beasties," she said. "Always, always use consecrated holy oil when cleaning. And do it every day."
McCue and I exchanged a glance, brows raised. _Holy oil?_
Now she touched the leather bags, the silver flasks. "Herbs. Oils. Some for protection, some for stopping, dispersal, or killing. Holy water; but you'll learn to make your own there, too. You can mix in a little colloidal silver for an extra kick. Consecrate the salt, bless the water, you're good to go. Holy oil: it's abramelin, made of equal parts myrrh, calamus, cassia, plus a little essential oil of cinnamon, and seven parts pure olive oil. Add a blessing, it's ready. You can burn bones with it, take out a surrogate, depending on what's needed."
"Stop." I raised one palm in a belaying gesture. "Just—slow down." I looked at the supplies laid out on the counter. Herbs, flasks, ammo, knives, guns . . . "You keep talking about 'holy,' 'consecrated,' and 'blessings.' Are we supposed to haul all this to a priest, ask him to pray over it?"
Lily's eyes were bright as her brows rose. "Doing so would probably result in your being arrested."
"No shit. And I can't really afford that. So, we have a priest on our team?"
Lily looked at McCue, then back at me. Her expression suggested she believed we were perhaps blithering idiots. "That's right," she said in tones of dry discovery, "you're newbies. My bad. Okay, so there are rituals and rites. I'll give you both cute little booklets containing those you'll use most often. Learn Latin."
"Got that down," Remi said quietly.
She looked at me. I shook my head. "A few lines here and there."
"Then you learn it, too," she advised. "Don't depend on Remi to be right there when you need it. Memorize the rites, because in the midst of demonic confrontation you're not going to have time to draw out a booklet and find the right page. In the meantime, remember something very, very important."
We waited.
Her expression was quite grave, her eyes serious. She spoke with explicit precision. "No, you do not need a priest to perform anything. Because you are both _born of heaven_ , you imbeciles. Your _souls_ are heavenly matter. Follow the recipes, recite the rituals—in Latin. Call upon that spark inside of you and imbue the hardware—breath, saliva, or blood will do—and you're good to go."
Remi smiled. " _Exorcizo te, omnis spiritus immunde, in nomine Dei Patris omnipotentis, et in nomine Jesu Christi Filii ejusmodi, Domini et Judicis nostri, et in virtute Spiritus Sancti,_ yadda yadda yadda _."_
"It's a beginning," Lily said, "but only a beginning. Not all surrogates are the same, and not all rites work on each. Sometimes it's hunt and peck."
I shot Remi a glance. "How do you know all that?"
"Studied Roman Catholicism," McCue replied, "along with a bunch of other religions. Doctorate, remember?"
"And you studied _exorcism_?"
He shrugged. "Was always kind of interested in that kind of thing. Rituals, rites. Maybe I was meant to, bein' as how we're heavenly matter supposed to battle evil." His smile crooked off to the side. "Hell if I know. But it might could come in handy, I guess, if we go runnin' into any demons."
"The Zoo," Lily said.
We looked at her blankly.
"The roadhouse," she clarified. "You know, that big building standing maybe thirty yards away?" She hitched a thumb in that direction. "It's haunted."
I stared at her. " _Haunted_?"
"Married couple, once owned the place. Benign spirits, mostly—supposedly scared some people by doing the typical ghosty things: moving stuff around, screwing with the power, changing the temperature, weird noises, popping up here and there. At least, those were the stories; who knows if they were true. But it _became_ true when the hell vents cracked open. Suddenly there were reports of paranormal activity echoing exactly what the stories claimed."
"Wait," McCue said. "You're saying you don't know if these ghosts ever _actually_ existed, but they exist _now_? How can ghosts do that?"
Lily looked at me expectantly, and I sighed deeply. "What Grandaddy described . . . if it's true that all the legends and stories are now real and among us—well, it could be rooted in thought-forms. Tulpas. What's imagined becomes real. Corporeal. And I can't believe I'm even saying this."
"Baudelaire," the cowboy murmured. "And what was it Grandaddy said about disbelief opening the door to demons?"
Lily nodded. "You must accept that these ghosts are now real, thanks to Lucifer. But now it's much worse, because demons are riding them. One person's been killed already, two weeks ago. Autopsy said a heart attack, so no one thought any different, but there are certain signs if you know what to look for."
"What signs?" I asked.
She flicked her gaze to Remi. "You've heard of the 'odor of sanctity,' have you not?"
"Sure. It's a pungent, flowery scent associated with the bodies of saints, particularly stigmata. Though I heard stories that some sick people, on the verge of death, reported smelling something strong and sweet."
Lily nodded. "Surrogates have a similar scent. It's not sulfur—that's too obvious, and they're not stupid, these demons—but a mix of resins and oils. The Egyptians called it _kapet_ , the Greeks _kyphi_. Temple incense, mostly."
My brows shot up. "Demons smell like ancient Egyptian temples?"
"Why not?" she asked. "Why is the 'odor of sanctity' sweet, like orange blossoms or other flowers? And it isn't like demons naturally _reek_ of it, though the scent intensifies significantly with death."
I huffed a dry laugh. "So, we go around sniffing people to find out who's a demon?"
"Pretty pervy," McCue observed. "Might could get in trouble for that."
Lily smiled briefly. "The woman earlier, the surrogate who attacked you, Gabriel. Did you smell anything?"
I shrugged. "Perfume." Two sets of eyes stared at me. "Oh. Yeah. Okay. Well, _shit_. Am I supposed to worry about every woman I get up close and personal with, now?"
"Can you remember her scent?" McCue asked. "Would you know it again?"
I thought back. My mind had not exactly been on _perfume_ with her hands on portions of my anatomy. But yeah, I'd been aware of a scent. "I couldn't tell you what was in it."
Lily's torc glinted as she shifted. "Do the job right tonight, you'll know what it smells like. Go over there, flush out the demons, kill 'em. Don't worry about exorcising them; the bodies aren't possessed humans, aren't actual living human hosts, but _ghosts_. The humans are long dead. You only use exorcism if a living person has been possessed, because there's a host to save."
"Flush 'em out . . ." Remi echoed dryly. "Any suggestions how we do that?"
She sent us both a glance that was clearly amused. "Oh, I think that will take care of itself."
I was reminded of how she'd described us earlier: _newbies_. "Wait," I said. "Grandaddy said hell could track us now, because our _beacons_ are turned on. Our souls? So we go after these things, and they know we're coming?"
And I recalled what the other woman had said to me. _"You lit it up last night. I felt it. And it shines so bright, it does. It's so very pretty. But now I'm going to extinguish it. I'm going to extinguish you."_
Lily said, "You can be proactive, or _re_ active. It's your choice. But yes, you are targets now. Anywhere you go where there are surrogates, they'll know you. Not all are strong enough to track you—there's a hierarchy among them as there is among the heavenly host—but all can sense you. You walk into a place where even a lesser surrogate has set up shop, made it a domicile, just as this roadhouse is, and that being will know you. But you'll learn to do the same, to sense a demon's presence."
I shook my head. "How the hell are we going to be able to do that?"
She folded her arms, leaned against the cabinets. "The way Jubal explained it to me is that you, Remi, can read people. At some point, with experience, you'll be able to recognize a host who's been possessed no matter how human it appears. _You_ , Gabriel, are sensitive to places. Kind of a living Electromagnetic Field meter. In certain places you'll sense demonic presence, whether they're in human hosts or something else. But it takes time to learn. No one is _born_ knowing everything, even with heaven's essence in you. You'll have to survive long enough to learn."
Yeah, a good goal, survival. I was up for that.
"So—tonight you'll go in after your first surrogates. Whether you live or die is up to you." Her tone sharpened from matter-of-factness to a hard precision. "But you need to realize this, to always remember this: if you die by normal means—and that's possible, because you wear human flesh—you go to heaven. But a heavenly soul that is extinguished by a demon goes to hell. Literally, _to hell_. Forever. No angelic rescue." Her green eyes were piercing. "And you really don't want to go there, because Dante got it right about all those circles and suffering. Trust me, it's no comedy, and it's certainly not divine."
"Dante emerged from hell," Remi pointed out, before I could. "Found paradise. Found God."
"And if we're made of heavenly matter—" I began, but a look in her eye made me stop. "No go, huh?"
"Jubal could tell you better than I."
"He's not here, and you're hedging," I accused her.
Lily shrugged. "I have my own agenda. I'm not an angel, remember. I'm not even a Christian." Her smile was faint, and oddly feral. "Remember the mythology, Gabe. All the various branches. Parse between them. You'll find your answers there."
McCue's tone was skeptical. " _If_ you're the Morrigan."
"Do the job," she told him, unfazed by his observation. "Survive, come back to me, we'll talk further. If you die, well . . ." Again, that feral smile, ". . . then it doesn't matter, does it?"
I eyed her thoughtfully, realized we'd get no more out of her with this particular line of questioning, so I moved on. "We were in the Zoo last night, and I didn't sense anything."
"Did you try? The way Jubal always asked you to?"
I rolled my shoulders, sensing criticism beneath her tone. "It's not a habit, and there was no reason to."
"And you wouldn't have recognized it if you had; you're still just a baby." She twitched her brows dismissively, mouth flat. "So, now you _try_. Become that EMF meter. Your grandfather started you, but it'll take time."
I remembered the day before upon the mountain, when I'd sensed the peace of the place, the slow beat of the earth's pulse.
I looked at McCue. "Since you read people—you get any kind of vibe off that woman? The demon?"
Remi cocked an eyebrow. "Other than she had the hots for you and was ready to jump your bones? Yeah, I noticed you two on the way out the door. 'Course that was before she decided to gut and strangle you, that is. Nope. Nada."
I considered that. "It might have been helpful if you had."
"What, you want to drag me around to bars where you go to pick up women so I can tell you if they intend to kill you? Not my idea of a good time, boy. Not for me, leastways. That's on you."
I frowned, looked at Lily. "Did _you_ know what she was?"
Her eyes sparked. "Who, me? When I'm just an imaginary creature out of mythology?" But she relented. "Macha might have. Dogs are very good at sensing things, far better than humans. But me, no. I didn't know what she was until she tried to extinguish you. You might do better to ask if Jubal knew last night. He's a seraph. They know far more than the rest of us."
Just like that, Grandaddy was outed as something a little more important than an agent. But her words stung. "He'd warn me. He wouldn't throw me to the wolves like that."
She laughed at me. "I saw you, Gabriel. All that woman did tonight was give you a stare, a glint, a smile. After that it was all testosterone, boyo. No throwing was done at all, to wolves or otherwise, that you didn't do yourself." She gestured at the items spread out across the counter. "Take it. Guns, knives, ammo, herbs, water, oil—call it a starter kit. You'll add your own things to it along the way. Like I said, I'll resupply you when we're in the same vicinity—and there are others of us out there, too—but you'll need to find your own sources, learn how to do this for yourself. For one another."
I cast a glance at McCue. _For one another_. When we were as yet strangers. But the cowboy looked back steadily, briefly quirked brows and tilted his head slightly as if to say he was willing to wait and see.
Lily noted the exchange. "You are not the only soldiers in this fight," she declared. "There's a _world_ out there in trouble, and neither I nor Jubal nor anyone else in this war can be in all places. Yes, your bright little heavenly beacons may— _may_ —summon help in times of great strife, should you call on them to do so, but most times you'll have to pick up that shovel and dig yourselves out of whatever hole you're in. But then, you've been taught self-reliance. Whether you knew it or not, Jubal was training you. But so was life. Now—" She waggled beckoning fingers. "—come on into the living quarters, have some whiskey, kick back for a bit. In about four hours, couple of hours after closing time, you can head over and kill a pair of surrogates." She shrugged. "Or, well . . . die."
I shook my head. "We could just walk away. Call bullshit on everything, go back to our lives. Be normal."
"There is no more 'normal' for either of you." With a gesture, she led us out of the garage into elegant surroundings. "And have you ever walked away from a fight in your life, Gabriel Harlan?"
"Well, no. But I sure as hell don't go looking for them, either."
"Doesn't matter." Lily waved us to a seat upon the couch, the recliner. She poured whiskey into glasses, presented each of us with one. "Things are different, now. This rig is protected, but the minute you step outside it, that ends. They know you're here, probably knew it last night when you put on the rings and clasped hands, but I'm betting they held back because of that woman. She must outrank them."
McCue shook his head. "Quite a picture, ain't it? Demonic chain of command."
I, in the recliner and somewhat uneasy about the nearness of perch and crow—the bird was _huge_ , with a mean eye and a nasty-looking beak—met Remi's eyes. The cowboy's expression was thoughtful.
Lily shrugged. "One hierarchy in heaven, another in hell. Balance. The two surrogates would have gone after you last night, unless someone of higher rank staked a claim. First dibs on Gabe because of _primogenitura._ You'd have been next, Remi."
I swore, ran a hand through loose hair. "So—you're saying _for sure_ she's a demon?"
"I'd think so," Lily replied matter-of-factly. "Probably possesses a human host."
I thought about that. It made me exceptionally uncomfortable. Nothing about her had suggested she was anything but all human, entirely a woman, and a highly hot one at that. But . . . now major squick factor, yeah.
I shifted in the chair and looked at McCue with an undisguised appeal. "Listen, if you're good with seeing demons in people—you gotta let me know if one's hitting on me."
The cowboy's grin was slow. "Or if you're hitting on her?"
" _Yes_ ," I said fervently. "I mean, that is, if you feel it. Smell it. Whatever it is you do."
"I don't do anything yet," McCue said, eyes bright with humor.
"Because I don't want to fu—" I broke off, slewed a glance at Lily.
"I am familiar with the word," she said gravely. "And yes, it's entirely possible a surrogate might seduce in order to extinguish you. They've been known to do that kind of thing."
"Sooo, a succubus, in essence." I heaved a sigh. "Damn, _everything's_ real, now? Succubi, incubi, sirens, kelpies, selkies . . ." I trailed off, because she was nodding. "Godzilla?"
"Probably not Godzilla," she conceded, "or King Kong. Not sure about a Transformer. But sure, Remi can get into his truck, drive away. You can get on your bike, _ride_ away. But they'll follow you. They know you, now. You walked into their domicile, and you ignited your souls."
I remembered that moment. Remi McCue and I had clasped hands, clicked rings together at Grandaddy's behest—and now everything in my life, in _our_ lives, was changed.
Lily Morrigan— _the_ Morrigan?—drank whiskey, sat down upon the carpeted floor and crossed her legs, completely at ease. "It's not an easy life, but if you listen and learn—and rely on what you've been taught—you'll make out all right." The wolfhound folded down and settled, putting a massive head into Lily's lap. She stroked the wiry hair, gazed on us both out of bright green eyes set in fair, unblemished skin. "I know this is new—but you need to understand the risks. There's no going back. You can't do control-alt-delete. You are what you are. But what you need to remember is . . . _hell knows you're here."_
# CHAPTER EIGHT
Four hours later, under Lily Morrigan's sharp, bright eyes, I wanted to laugh out loud. But McCue was being very serious, and she equally so; I doubted they'd understand my humor.
We were gearing up like some kind of serious macho testosterone movie. Or, yeah, like the elegance of James Bond discovering what new tech Q had for him.
I grinned. Biker leather and cowboy denim. Elegance, my ass _._
When it came to armaments, I opted for the long-barreled version of the Taurus Judge, preparing to load with the silver-jacketed .45 caliber rounds Lily explained had been washed in holy water, then dipped in oil of abramelin.
"Which recipe?" I asked dryly. "Samuel Mathers'? Aleister Crowley's?"
"The original," she answered. "Abraham the Jew's, as described in the Book of Abramelin. But there will be an extra ingredient in the mix just to give it a little kick: the breath of two souls born of heaven. So go on, boyos. Blow on the bullets."
McCue's head shot up. "Do what?"
I stared at her. "You're kidding, right?"
Lily didn't smile. "I don't joke about weapons of war."
After a moment's thoughtful hesitation, Remi took five bullets into the palm of his hand and blew gently upon them, looking to Lily for confirmation. I, desiring more of a flourish to make a point, picked up the bullets one at a time, huffed a brief breath at each, then loaded it into the chamber.
"Yes," Lily said, "you looked as silly as you thought you would. But it's cover your ass time, is it not?" She paused, took from a drawer a shoulder holster and a belt-mount. "Arizona is open carry. No one will give you a hard time. Gabe, you can go with the shoulder holster under your jacket. Remi, you're a Texas cowboy. Your belt will work."
After a moment's hesitation, I stripped out of my jacket, strapped myself into the shoulder holster, put my jacket back on.
I owned a Bowie, but felt most comfortable with the KA-BAR I habitually sheathed at my spine for a fast reach-around. The KA-BAR was a plain old steel knife, no silver involved, but it was comfortable and familiar, and tonight I wanted it within reach. Just having it on me offered a little ease in the midst of confusion, a mental anchor of sorts. At Lily's suggestion, I wiped the blade down quickly with holy oil.
Remi had opted for the Damascus Bowie, the Hibben trio of throwing knives, and the short-barreled Taurus. But when it came to the Remington shotguns, I recommended powdered-iron cartridges.
When McCue looked a question, I explained that folklore ghosts were most susceptible to iron. Which sounded stupid when I said it, but that's what legend and folklore claim. "Since we're doing the Cover Your Ass thing. Though I sure as shit never thought about using shotgun shells on ghosts. In stories, it's blades."
McCue seemed unfazed. "If it's folklore, it's your wheelhouse," he said. "I'll recite the Latin, handle exorcisms, throw knives as necessary, shoot the hell out of whatever—hell, I'll _pray_ over 'em, if that'll work . . . but I've never contemplated killing _ghosts_ before, so if you've got a leg up, go for it."
"Well, iron and salt are used for dispersal of, or protection against, ghosts," I explained. "Supposedly to get rid of a ghost for good you have to destroy its source material, such as bones. Now, you _do_ understand I've never actually done this before. Just studied it." I'd taught it, too, but didn't say that. "But I don't know jack about demons. Unless you're a priest, who does? It's not exactly normal."
Remi McCue grinned. "I've studied exorcisms recited in a dead language. You've read how to destroy _ghosts_. Oh, son, we are so far beyond the vicinity of normal that it ain't even funny."
I reflected that this was undoubtedly true. "So, Lily . . . we go in there, flush 'em out, then shoot 'em to back 'em off? How do we actually _kill_ them?"
"These ghosts were humans, once," she replied. "You can kill a human by shooting it with anything, if you hit the right spot. What's true in life is true in death. So hit the same right spot. But regular ammunition won't work on a ghost or a surrogate."
I nodded. "Got it. Powdered iron to knock them off-stride, then silver bullets dipped in holy oil—"
"—and don't forget we breathed on 'em." McCue was grinning. "Breath of death, instead of breath of life. Kinda like it." He paused. "Bein' as they're demons, and all."
I shrugged. "So long as we don't have to kiss 'em, I'm game."
McCue nodded, looked at Lily. "Just to clarify—if we wing a ghost, we don't kill it. But we put a shot wherever it'll kill a human, we're good?"
"If it's a _human_ ghost," Lily clarified. "And yes, you can disperse a ghost temporarily if you wing it, or even with less contact than that. One BB might do. But recall that these are _demons_ in ghost form. It won't take long for either demon to reconstitute. The first time, they may respond as a real ghost would and dematerialize, because a ghost has sense memory of itself as a human, but that won't last. Demons can't be effective if they don't establish their own powerbase within the host as soon as possible. You can't just wound a demon in the form of ghost or spirit and keep driving it away. Once, maybe. After that, bet on them being very corporeal, very strong—and seriously pissed off. Call it paranormal adrenaline. That's when iron won't work."
McCue looked at me. "Silver bullet to heart or head."
I grinned. "Works for me." But amusement died. I met Lily's eyes. "Do they know we're coming?"
Lily stared right back. "I don't know."
* * *
—
After dark, the cowboy and I crossed the lot between Lily's RV and the roadhouse. The moon was nearly full, and the swath of stars above the city was astonishing in its clarity. The only artificial illumination came from the bug light over the back door.
* * *
—
No alarms on this place," I noted quietly after we inspected the entire perimeter of the big log building and met out back once more. "Well, far as I can tell."
"So, we break a window and just two-step in there?" McCue asked in a low voice. "This is for shit, you realize. This is breaking and entering. We get arrested—how the hell do we explain anything?"
"Wanted to shoot some pool, maybe? Who can tell with drunk guys. Which is about the only cover story that might work." I attempted to peer in a window, but it was blocked by drapes. "You can tell 'em anything you want to, even the truth, but it won't get you very far. So the key is: _Don't get caught_." I knelt at the door, set down the shotgun. Gestured Remi to bend down and kept my voice very low, almost a whisper. "You got that little flashlight Lily gave you? Shine it here, on the lock."
McCue, squatting, turned on the flashlight, bent down to aim it. Like me, he spoke barely above a whisper. You'd have to be kissing close to hear us. "What the hell are you doing—wait . . . are those—?" He didn't finish, just rose abruptly and grabbed my right arm, literally yanking me away from the building. "You've got _lockpicks?_ "
I shrugged. "Well, if you don't have a key—these'll do."
It's difficult to sound scandalized in a strangled whisper, but he managed it. "You ride around on a motorcycle carrying lockpicks? Just how often do you do this sort of thing?"
"Breaking in, or attempting to kill ghosts hosting demons?" I grinned at his expression. "I know how to use them—" I'd actually learned in prison, though I didn't tell him that, "—but I don't _carry_ lockpicks, no. Or didn't until tonight. Lily slipped 'em to me. Now, come on. Let's get this done."
He followed me back the few steps to the door. "This is not what I signed on for."
I squatted again, motioning for him to aim the flashlight at the lock. "You've been a good boy all your life?"
"Damn right I have," McCue whispered back. "Well, except for that little fracas me and Luke got mixed up in at my high school graduation. But that didn't amount to much. Busted nose, a few teeth knocked out—not mine, you understand. I had some moves by then." He paused. " _Better_ than Jagger."
I felt the tumblers shift and click. I removed the picks, set my hand on the doorknob and turned. Yup. Pushed open the door but didn't enter, just knelt there and zipped the pick kit closed, tucked it into a pocket. Then I picked up the shotgun and slowly rose.
McCue stepped close against the chinked log wall. "How often you used those suckers?" he whispered. "You do this regular—this breaking in? Does Grandaddy know?"
"If I tell you it makes you an accessory."
"I'm already an accessory on this break-in, asshat!"
I ignored the conversation. "You do know how to shoot, right?"
McCue's quiet tone was dry. "Guess we'll find out. Meanwhile, you sensing anything _surrogate-like_ in there? Since you're a living EMF meter and all."
"I'm sensing a certain amount of cowboy bullshit. And anyway, she said it would take time. No, I don't get any kind of vibe, other than feeling pretty damn stupid about this whole thing." I pushed the door open with infinite care. "I think we should have asked Lily a lot more questions."
Remi leaned close again as he spoke. "You're the folklorist. Seem to know this shit."
"That doesn't make me a _ghost hunter_. Buster. What the hell ever."
"And I didn't get the feeling she was going to say much more other than telling us to come back _with_ our shields—or on 'em. You know, all Spartan-like."
"You don't look too Spartan in that hat."
"Protective coloration," came the whispered reply. "Now, of the two of us, who's more likely to be taken for a drunk cowboy wandering around a cowboy bar after hours in the dark, should the cops come by—a guy who _is_ a cowboy, or an ex-con biker dude? With lockpicks in his pocket. And guns, which he shouldn't rightly be carrying since he's an ex-con. And should we even be talking?"
"Why not?"
"Well, if ghosts—or demons—have working ears, we might could give ourselves away if we keep having long-winded conversations."
"No one could hear us from a foot away," I pointed out, well within that foot, "and if what Lily told us is true, they already know we're here and we could probably shout at the top of our lungs." I eased my way inside the door. Same door I'd exited a matter of hours before, in pursuit of the woman—who wasn't, as it turned out, actually a woman. And there came the squick factor again. I sought another topic immediately. "You know, some people believe Einstein proved ghosts exist."
McCue, trying to combine utter disbelief with keeping his voice down, sounded strangled. "Einstein? Why are we talking about Einstein?"
"But I think it's bullshit."
" _Einstein_ is bullshit, or these 'some people'?"
I shrugged. "We're made up of electrical energy. People, I mean. And Einstein said that when people die, that energy goes somewhere. Into the environment. That it can't be destroyed, only changed from one form to another."
"So, that's where the EMF meters come in. Or you. Sniffin' out that released electrical energy."
I took a few more steps inside. "That's the theory." Muted light at the back of the bar didn't illuminate much, but it kept the place from being pitch black. I advanced a few more steps.
McCue followed closely. "Mighty thin theory."
"That never stopped conspiracy theorists, or UFO whackjobs."
"You know, according to the Bible, there's no such thing as actual real ghosts. You die, you get judged, you go to heaven or hell. End of story. Which jibes with what Grandaddy said."
I shook my head. "He said a lot. What specifically do you mean?"
"I'll quote chapter and verse later, but the Bible says if there _are_ ghosts hanging around, they're demons. That any paranormal activity, if evidence of it is found, is because demons are controlling it. But there are positive spirit beings. Angels." He paused. "I wonder . . ."
I put a hand on a pool table to spot myself, moved carefully around it. Cradled the shotgun in two hands again. "Wonder what?"
McCue was very close. "Do you suppose we give off any kind of special electrical energy? You and me, I mean. Because of our beacons."
"What, you mean like Iron Man's power plant?"
"I wonder if we'd set off EMF meters. Or airport scanners. Or if an MRI or CT machine would pick up anything."
"Maybe we should go to Vegas," I said dryly, "see if we can take control of all the slot machines. Make all those little old ladies happy with their nickels."
"They might could give us a percentage," McCue agreed. "We'd be rich in no time, five cents a deposit. Now, what was it Lily said about these ghosts? Husband and wife?"
"She tripped at the top of the steps, fell down them and broke her neck," I explained. "Despondent husband killed himself a year or two later. If the impulse for these demons is to initially behave as the ghosts would, they'll hang around where they died."
"So we look for her on the stairs?"
"And for him in front of the fireplace, where he offed himself."
We moved through the pool tables, stepped out onto the main floor. In silence we waited, examining surroundings in low-level illumination.
I reflected that yes, one might term it "spooky." All kinds of dead animals with colored marble eyes and bared yellow teeth hung off the walls, crouched along the beams, inhabited corners. Antlers on mounted trophy heads stabbed the air.
I heard the clink of glass from the bar and jerked my head around.
No one was there.
Flames roared up in the fireplace where none had blazed before.
The chair before the fireplace rocked, with no one in it.
"Okay," McCue murmured, "this officially qualifies as weird."
"Or haunted."
"'Haunted' is weird." He paused. "You getting any kind of a _feeling_ about this?"
I shook my head. "You're supposed to be the one who can sense demons. You getting anything?"
"Smells a little funny."
I opened my mouth to make a comment, but suddenly a woman stood before us, appearing from out of the dark. I stared in disbelief. Her head, the neck obviously broken, lay loosely on one shoulder, which grossed me out. But it didn't seem to bother her.
Then she grabbed Remi McCue and flung him across the room.
* * *
—
I heard the cut-off yelp from McCue, the crashing of body through tables and chairs. Then she was right on top of him, and in his hands was a shotgun, not good for close-in fighting. Even as he tried to raise it, the woman— _ghost? demon?_ —chopped down hard at the barrel. It clattered out of his hands.
Definitely corporeal. Definitely _pissed_.
I yanked my revolver from the holster and fired a powdered iron shell. The woman disappeared—dematerialized?—but, dammit, in the middle of the melee I had not gotten off a chest or head shot. Winged her, maybe. Damn—I'd been right there, _right there_ , and hadn't made the shot. Not the kill shot.
Or whatever it was when you destroyed a ghost. Or a demon. Or whatever. Ghosticide? Demonicide?
I moved rapidly to the wall, putting my back against it. Heart rate was high, and I released a long breath. "Lily is _so_ going to give me much shit about all of this," I muttered, revolver cradled in both hands. "Hey, McCue?" I saw no reason to keep my voice down now; the female ghost clearly knew I was present. "Remi? Hey—cowboy!"
No answer.
Shit.
I waited a moment, listening hard. No more clinking of glasses at the bar, no more creak of the rocking chair, no more crackle of flames in the fireplace.
When no further attack seemed imminent, I opted to see if I could work my way across the roadhouse, discover if McCue had survived being flung across the room like a rag doll. Because I was certain Lily would give me more shit if the cowboy was actually dead.
Hell, Grandaddy might _murder_ me if I'd got Remi killed.
But that was the last coherent thought I had, because every dead, stuffed, marble-eyed animal in the place _came alive_.
And all of them apparently had me on their menu.
# CHAPTER NINE
Well, hell," I muttered.
I did a quick mental inventory of my personal armory. Five shots left in the shotgun, five in the Taurus; plus the Bowie and KA-BAR.
But—what the hell works on reanimated stuffed animals? That wasn't anything I'd ever read about. And could they actually see out of those fake-ass marble eyes?
Cougar. Bobcat. Lynx. Wolf. Some bristly pig-looking thing with nasty tusks. All prowled toward me from out of an almost lightless space, clearly able to see very well indeed out of their fake-ass eyes. They even _sounded_ real.
They also sounded hungry.
"Hey, McCue! You alive over there?" Backup would be good. "Would an exorcism work on these things? Maybe knock the stuffing out of 'em?"
No answer. I began to wonder if maybe the cowboy truly was dead. The idea sent a curl of nausea through my belly.
Lily Morrigan had said a heavenly soul extinguished by a demon would not go to heaven.
And for some inexplicable reason—considering I didn't truly know the man—that really, seriously, _significantly_ pissed me off. Because if heaven was real, so was hell. It blind-sided me, the anger. And it fueled me.
I took out the cougar first, then the lynx, followed up with the bobcat, the wolf. The pig-thing took my last round.
I heard Lily's voice again, ' _You've only got five shots, but if you can't put down something within five, maybe you deserve to be dead.'_
Yeah, well, that might be true if the target were just one thing, not _five_ possessed animals—that were deceased, but somehow not.
"Bear!" shouted the cowboy, who was, hosanna and hallelujah, not dead after all.
I swung around, reaching for the Bowie. I'd forgotten about the damn stuffed grizzly.
I thought about diving for the shotgun lying on the floor, but that meant I'd be going _to_ the bear. The very large bear who could indeed see me out of its fake-ass eyes. Because it stared right at me.
Shit. I wasn't good at throwing knives. But I sure as hell wasn't going to _close_ with the sucker. So I threw it.
And missed.
I was out of ammo, lacked the rifle, figured I'd miss with the KA-BAR, too. And time was just flat gone. So I grabbed up billiard balls already racked on the nearest table and began hurling them at the bear. I'd been a pitcher in high school; I nailed the sucker a few pretty good ones right in the face. Even broke out one of the eyes. But the bear kept coming.
"Duck!" McCue yelled.
I flung myself down as far from the bear as I could get in one frantic leap aside before dropping, automatically shielding my head. And then I heard a series of reports as Remi fired all five chambers from the revolver into the bear.
As the animal toppled, I shoved myself out of the path of destruction. The massive grizzly fell hard and heavy, face down. For one startled moment I was eye-to-eye with the dead—re-dead?—bear, and then I scrambled to my knees and scooped up the other shotgun. I tucked it under my arm between ribs and elbow, then dug into a jacket pocket for additional ammunition to reload the revolver.
"Mighty fine shootin,' there, Tex," I noted as McCue came cautiously forward to inspect the bear.
The cowboy looked around the field of battle. "Man, looks like we took out half a zoo."
I slid bullets into the revolver. "What's that pig-thing?"
"Javelina. With a 'j,' but pronounced as an 'h.' You know—just so you sound smart when you tell stories about this."
I was pretty damn sure I wasn't going to be telling any stories about the night stuffed animals came to life in a cowboy bar in Arizona. "Hava-what?"
"Javelina. Peccary. Not actually a pig. They inhabit Mexico, the Southwestern U.S." He gazed down at the pile of bristle and tusks, lips pursed as he nodded appreciation. "Males can mass over eighty pounds. Nasty suckers, they are. We got 'em in Texas, too."
I noticed he seemed a little shaky. "You okay?"
McCue nodded, ran a tentative hand through short dark hair at the back of his head and winced. "Caught me a good lick on the edge of a table, but it's too far from my heart to kill me." He glanced around. "Where's my hat? I feel naked."
I caught a glimpse of something behind the cowboy. _"Down!"_ I yelled, and as McCue dropped I fired the shotgun at the husband half of the ghost couple.
Who nonetheless managed to grab McCue around his throat as it went down, because I had pulled my shot upward, too worried about nailing McCue with the buckshot spread. I saw the spray of—goo?—burst from the male ghost's head as he— _it_ —collapsed, but there was nothing about him that suggested he was dead. Well, dead again. And despite the fact the man—or ghost, or demon—was missing half his head on one side from the original shot that killed him, and a divot in the other side of his skull where my spread had struck him, the ghost-demon had both hands gripped around McCue's throat from behind, digging fingers in.
Corporeal, all right. Because even as he lay tangled on the floor with the ghost-demon, Remi was clearly being strangled.
"Are you really that stupid?" I walked up to place the muzzle of the Taurus against the oozing head. "It's a silver bullet dipped in holy oil, asshole, and from this range I don't miss."
And I fired.
The ghost-demon, which resembled nothing so much as a perfectly presentable, infinitely normal man missing part of his head, made an odd discordant noise like a deflating bagpipe, rippled head to foot, broke up as if pixelated, then collapsed into a pile of dust and grit.
McCue, who was clutching his throat as he lay on the floor, croaked, _"What the hell are you doing?"_
"Killing me a demon." I reached down a hand. "Come on."
"Jesus, man, I was right in front of it—you could have hit me! And you splattered its brains all over me!"
" _Ghost_ brains," I reminded him. "Not real."
"They sure as hell _felt_ real! A little _warning_ next time. Holy Christ."
I was aware of an odor. Sickly sweet. Powerful. Jesus, but it stank. "It was a bullet, not buckshot. Went right through his head. Dude, you whine like this all the time?" I gestured with my outstretched hand. "Come on. Up from there."
McCue clamped onto my hand, allowed himself to be pulled up from the floor. He opened his mouth to say something more, and then I felt a flash of heat in my hand, startling enough that I jerked my hand away from McCue's and stared at it in alarm.
What the—?
McCue echoed me, and I glanced up to find a pair of very startled blue eyes fixed on my own.
I heard Grandaddy's voice again. _'That day when you offered to carry your brother's pain, begged me to lift it from him, to give it to you, I knew you were ready. And you consented. But you weren't sealed to Matthew that day, Gabriel. Because he isn't heaven-born. You were sealed to Remi.'_
It rose up like a tide, a slow but relentless drive of water to the shore, to break upon the land. And I knew in my bones.
This is what we were meant to do. It's literally what we were _bred_ for.
The certainty was profound.
Remi said, "Right now I'm about as confused as a goat on Astro Turf. But—this is right, what we're doing. This is necessary."
In an accord bordering on elation, I grinned at him. "Let's go kill us another demon."
And then Remi's expression changed. I saw the revolver come up, heard the click on an empty chamber; McCue had not reloaded in the aftermath of being jumped by the male ghost. Breathing had come first.
I twisted, wrenched myself aside as I brought my own gun up; saw the glint of something in reflected light: one flash, two, a third. A glitter of silver, the silence of steel as it cleaved the air.
One. Two. Three. A perfect, deadly progression of superb physical awareness, of perfect control. McCue's movements had been smooth and effortless.
The female ghost dropped. Three hilts stood out from her— _its?_ —heart, like a tight bullet grouping.
And then the body, if it could be called a body, did the wheezing bagpipe rippling pixilation thing, collapsed into dust and grit, and the three throwing knives, falling, rattled against the floorboards, chimed steel-on-steel as they landed one atop the other.
That sweet smell again, almost overpowering. And ash. Charcoal. The astringency of _heat_. A trace of burned flesh. The stench of corruption, of putrefaction.
I held a revolver in one hand, cradled a shotgun in the crook of an elbow, felt like I'd run a four-minute mile in three. But the tide rose up in me again, bringing with it an awareness of accomplishment. Of an abiding satisfaction. A trace of something edging toward euphoria.
I grinned at Remi McCue. " _Oo_ rah! We just kicked us some evil ass!"
McCue bent, picked up his knives, safed them home in the tri-part sheath. Then he went back into the dimness, returned with his hat. His grin matched mine. "How about we light a shuck and clear out before the cops come? 'Cuz I'm not too sure how they'll take to a tale about shooting the stuffings—the _literal_ stuffings, I mean—out of already-dead animals."
I tucked my revolver into the shoulder holster. Yup, bear, cougar, lynx, bobcat, a hava-pig-thing all down from their mounts and shot full of powdered iron and silver, sprawled across the floor amidst a scattering of pool balls, might cause some consternation among the authorities. I already had a rap sheet, but this? This was just plain damn weird.
I heard it in my head: _'Why, yes, Officer, we shot up all the dead, stuffed animals because they came back to life, and then we killed a pair of ghosts that were actually demons. Because Lucifer is coming and we're supposed to save the world.'_
Hell, I wouldn't go to jail for this. I'd go to a mental ward. "We gotta come up with a cover story," I murmured. "Maybe a whole boatload of them." I pushed the door open, stepped into the coolness of the night.
McCue came out behind me. "A great philosopher once said, _You may all go to hell, but I will go to Texas._ Seems an appropriate quote for demons, don't you think?"
I shook my head as I walked across the lot toward Lily Morrigan's big rig, shotgun cradled across my elbow. "No philosopher ever said that."
"Davy Crockett said that."
"Davy Crockett? Davy Crockett wasn't a philosopher!"
"He most certainly was. He also said, _Be always sure you are right—then go ahead._ Words to live by, son. Words to live by."
"I suppose you've got a country song for all of this, too?"
"Well, I might could sing the theme song to the Davy Crockett TV show—I've seen the whole thing on DVD—but there's also 'Angels Among Us,'" Remi replied promptly.
I thrust up a silencing hand. "Don't start. Do not start. And we're not angels. Grandaddy said so."
"Close enough. We're kissin' cousins."
I cast him a glance. "Actually, I'm more a fan of Camus. _Nobody realizes that some people expend tremendous energy merely to be normal_."
McCue considered that for a moment. "Then I reckon the two of us are gonna be expending nine whole boatloads of energy from here on out just to _fake_ normal."
I reached the steps of the rig first, glanced back over my shoulder. _"Only the guy who isn't rowing has time to rock the boat."_
"That Camus again?"
I pulled open the door. "Sartre."
"Well, if Lucifer plans to row the boat, we'll just have to rock the hell out of it." Remi paused, followed me up the steps. "Rock _him_ the hell out of it and right back into the pit."
Lily Morrigan, as we entered, waited in the driver's seat. "Go put those shotguns in the back. We're leaving." She started the rig, gestured for us to hurry. "Too much noise," she said. "Scanner says the cops are on their way. We're moving down the road a bit."
"My truck!" Remi protested.
I was horrified. "My _bike!_ "
"Tomorrow." She put the rig in gear, started to roll. "And don't worry about them tracing you through the plates; they'll have no idea you're with me in an RV campground, nor will they have any idea those vehicles are connected to what just happened. They'll think you guys got drunk and took Uber or Lyft home."
"Shit," I muttered; I did not want to leave my bike. I put out my hand to McCue. "Give me your gun. I'll put it up." I accepted the second shotgun from McCue, walked hastily back to the garage, found the right cabinet and hooks, safed the shotguns, latched the doors, went back to the front. Remi had hung his hat over the window valance, found a spot on the couch. I dropped into the armchair beside the crow's perch.
"How many shots?" Lily asked.
"For the demons?" I shared a puzzled glance with Remi. "Missed the first shot, sideswiped a head with a second, then took the guy-ghost out with a bullet straight through the skull."
Her gaze was level. "I heard more shots than that."
I was astonished. "You were counting? Seriously?"
She turned the rig north onto Route 66, the Mother Road. "Yes."
"Why?" Remi asked.
At this hour, the street was nearly devoid of vehicles. "Your grandfather asked me to."
I eyed the big crow, seeming unperturbed upon its perch as the RV jostled down the uneven paving. "Why would he do that?"
She was very clear. "He said Remi might at some point opt for knives, which are silent. He said you would go with the revolver, and do what needed doing in two shots. Well, I heard more than two shots."
"That's because every stuffed animal in the place came back to life," Remi protested, "and those damn ghosts, or demons, don't exactly hold still." He moved aside hastily as the big Wolfhound bitch expressed a desire to join him on the couch. She didn't ask. She just—arrived.
I was annoyed. Yeah, he had missed the first shot. The second had hit the target, but failed to put the ghost down. "Most of those shots were aimed at the animals. The stuffed animals. The stuffed _dead_ animals. That you didn't warn us about." I paused. "Did you expect that to happen? Was that some kind of test?"
Lily laughed. "What you should expect is to expect anything. And if you want to talk philosophy, here's a saying for you. About the Irish. You might find it apropos." She lifted her voice, shaped the words with a lilt more pronounced than she'd had before.
> _"Be they kings, or poets, or farmers,_
>
> _They're a people of great worth,_
>
> _They keep company with the angels,_
>
> _And bring a bit of heaven here to earth."_
She grinned at us. "That's you, boyos. That's the pair o' you. A bit of heaven here on earth."
With great seriousness, Remi drawled, "I'm a Texas boy. Of course I'm a bit of heaven." Then he smiled real slow. "Or so the ladies tell me."
Lily laughed. "I don't doubt it!"
"Oh, Jesus." I rose, walked back toward the kitchen. "I need a drink."
The cowboy quirked his brows. "It's four o'clock."
I began opening cabinets. "That's close enough to five."
"In the morning!"
"That's close enough to five."
# CHAPTER TEN
I did indeed pour myself whiskey, once I dug a bottle out of the kitchen cabinetry, even as Lily pulled into the RV campground site assigned to the massive rig. I was more than a little impressed as she wheeled the big motorhome through the narrow road hedged by huge pines, seemingly unfazed by the challenge.
I reflected then that I should probably say nothing whatsoever about being impressed, since she would, in all likelihood, consider it some kind of sexist comment, but I rode a motorcycle, after all, which was infinitely easier to park, so what did I know? I was pretty damn certain _I_ couldn't handle the motorhome in close confines.
And I also reflected, with a weird twitch of disbelief, that the most impressive thing about the whole situation was that the woman driving the giant RV claimed to be _an ancient Irish goddess._
Christ, and I'd thought it was culture shock when I went to prison.
The big RV swayed, and I decided it was easier to ride out the parking job from where I stood in the kitchen, rather than falling over on my way back to the armchair, as she maneuvered the rig into the narrow site. So I clung to a cabinet, braced myself with spread legs and loose knees; saw Remi's amused eyes on me. But since the cowboy was currently pinned down by the weight of half a huge dog, I wasn't certain McCue was looking any more badass than I was.
So, we take out demons and undead dead animals, and we fall down in motorhomes or get engulfed by snoring dogs _._
Yeah. Badass, all right.
Lily parked, turned off the engine, opened the driver's door and climbed down, out of the rig. I turned loose of the cabinet and made my way back to the armchair.
"You want help hooking up?" Remi called.
"I do not!" Lily called back.
"Yes, ma'am," McCue said. "Not sure this gal would let me stand anyhow." And he placed a hand on the wolfhound's head, threaded fingers through wiry hair.
I heard the sounds of compartments being opened from outside, rapping and banging, and shot Remi a glance. "What's she doing?"
"Hooking up water, sewer, power," the cowboy answered matter-of-factly. "You ever been in an RV before?"
"No."
McCue grinned. "It's not exactly rough camping. 'Specially not a rig like this."
Lily came back in after a bit, cast us a quick smile, then triggered toggle switches above the door. "Lap of luxury," she said.
And the motorhome, on the whine of hydraulics, _grew_.
"Damn," I muttered. When the sound shut off I realized the RV was now twice as wide as it had been. "It's like a pregnant house. On wheels."
"Slideouts," Remi explained, which told me exactly nothing.
"Couch, dinette," Lily said. "Both convert to beds. You two can crash on those. Bedroom's mine, of course." She plucked the glass from my hand, downed what was left of my whiskey, gave me back the empty glass. "It's five o'clock," she said simply to my raised brows. Then she moved on through to the kitchen. "Coffee coming up. You two go clean those guns, oil up the knives. And from here on out, any time anything calls for oil, make it holy oil. Water?—holy water. Always. Leave nothing to chance."
I shot a glance at McCue. "Holy water go for showers, too?"
She was taking a coffeemaker out of a cabinet, pulling grounds out of the refrigerator. "Screws up the plumbing, pretty much, because of the salt in it. But, you know, you can always buy _bath_ salts, exfoliate your heavenly skin so you'll be smooth and pretty for the demons. 'Course you'll need to bless it first."
McCue intoned _,_ " _We humbly ask you, almighty God: be pleased in your faithful love to bless this salt you have created, for it was you who commanded the prophet Elisha to cast salt into water, that impure water might be purified._ " He paused. "Well, leastways, that's the beginning."
I clapped a spread hand over my face in disbelief. "Stop quoting!"
Lily laughed. "Havin' a hard time of it, are you, Gabe? Not quite reconciled to what you are? Here, let me help." She walked over with a bottle, poured more whiskey into my glass. To Remi's glance of disbelief, she said simply, "I'm Irish. What do you want?"
"Coffee," he answered pointedly.
She grinned. "Irish? Or straight?"
"I think I'll just go clean my weapons," Remi observed, but didn't immediately move. He pointed to the dog sprawled half across his lap. "Is she going to let me up?"
" _Faigh suas ó ann_ ," Lily said. _"Lig cinn an buachaill bocht."_ The wolfhound lifted her head, and from gold-brown eyes stared at Lily, who then said, "Yes, I mean it." And the dog rose up slowly, slithered off the couch, collapsed upon the floor.
McCue stared at her. "That first part wasn't Texan."
"I told her to let the poor boy up. More or less. So, now you're unencumbered by the terrible great dog." She waved a hand. "You're free to move about the cabin."
Remi got up, hesitated a moment as he touched the back of his head, proceeded on through the motorhome and into the garage.
I watched him go by. Blood. There was blood on the collar of his shirt, spotting down the back. I set down my glass with a thunk, followed McCue into the garage. "You're bleeding."
Remi unholstered his gun, placed it on the cabinet counter. "Like I said, I got up close and personal with a table."
"You didn't say you cut yourself."
McCue shrugged. Lily came through from the living quarters, carrying mugs of coffee. "Cream and sugar are up front," she said, then frowned at Remi. "What'd you do to yourself?"
He looked a little like a deer caught in the headlights. "I'm fine!"
"You're bleeding." She set the mugs down, tapped her hairline above her forehead. "Right there. Just now."
"Back of his head, too," I said. "Whacked his head on a table. I don't know what that is in front."
Lily stepped in close and caught a handful of McCue's shirt, holding him in place before he could back away. She pushed back his hair, uncovered a small wound. "That," she said, "looks like buckshot caught you." She turned her head to look at me.
And just like that, I was flung back years to the day I stupidly tricked my brother Matty into riding his bicycle down a hilly street while blindfolded.
When Grandaddy had said, _"Do you understand? If you do something stupid, if you do something wrong, there are always consequences. You might have killed your brother."_
It rose up in me abruptly, the same wave of fear and remorse and terrible sense of guilt that had overwhelmed me when my brother had been injured. Here. Now. Again. It wasn't Matty—and yet, oddly, it was.
I'd pulled my shot intentionally when the ghost had grabbed McCue, trying to keep the buckshot spread away from human flesh. But I'd gone into the fight with an unfamiliar gun, unfamiliar ammunition; had walked into the middle of a wholly unfamiliar kind of battlefield—and got bitten in the ass.
McCue, who'd been bitten a little worse, smiled crookedly. "Guess you might could use a little target practice." And then his smile faded. "Hey. Hey, it's just a joke. This ain't nothin,' trust me. What—are you squeamish? One of those badass studs who passes out at the sight of blood? Because you sure look like you're goin' to. You okay?"
No, I was not okay. I remembered with visceral clarity how sick and frantic I felt when Matty had been injured, and how grateful I was when Grandaddy had lifted the pain from my brother and transferred it to me. Because pain didn't matter when it was my own. So long as it wasn't my brother's.
I could have killed Matty.
I could have killed Remi.
Lily grabbed a mug from the counter, pressed it into my hands. "There's whiskey in it," she said. "You look like you need it."
Grandaddy had said, _"Matthew was Remi's proxy. He represented the younger brother you needed to have, to trigger the drive to protect."_
Well, if I was supposed to _protect_ McCue, I'd done a piss-poor job of it.
"Can I take this from him?" I asked. "Grandaddy did it for me years ago—lifted it from my brother, gave it to me. Can we do that?"
"Jubal can," Lily answered. "I can't. Neither can you."
I stared hard at the cowboy a moment, cast a fleeting glance at Lily, then turned abruptly and headed toward the front of the motorhome.
I heard Remi ask a question. Lily said simply, " _Primogenitura_. But it's a process, a growing into it, and not a perfect one."
I brushed the scar in my eyebrow. Hell. It had been a process with _Matty,_ but at least I'd figured that one out. I'd learned how to bring order to the chaos that was my brother. All I had to do was make it my _own_ chaos.
Which was how I'd ended up in jail, by saving my brother and killing a man in the doing of it.
* * *
—
Not long after sunrise, I hoped to hitch my way to the Zoo, maybe a mile, mile-and-a-half down the road from the campground, but few vehicles were out so early Sunday morning. So I hoofed it. And was grateful beyond measure to discover, when I got to the big roadhouse, that my Harley was exactly where I'd left it. The only other vehicle in the parking lot was a bright silver extended cab Ford F-250 pickup bearing Texas license plates, parked over on the side between the roadhouse and a large pine. Remi's, I figured.
I picked up my helmet, donned it, pulled on gloves, threw a leg over the saddle, lifted the bike, started to tap up the kickstand, then paused.
Something. _Something_.
After a moment I let the bike settle, checked out the surroundings. Still felt twitchy. Finally I swung my leg back over, put down the helmet, and walked around to the rear of the building.
If the cops had come to check out reports of gunfire, they hadn't been concerned enough to mark the place off with crime scene tape. Maybe they'd chalked up everything to vandals. After all, the only things killed were dead animals, and ghosts.
Well. If dead things could be rekilled. Still wasn't clear on that.
I paused outside the back door, glanced around, saw no one. A hand on the knob found the door locked again, but that was no impediment. I wielded the lockpicks again, then stepped inside the bar and shut the door behind me.
I lacked the shotgun, but had collected the short-nosed Taurus revolver on my way out—silver bullets—the Bowie, and my KA-BAR.
The stuffed trophy animals were still dead, and they still lay where they'd fallen after being filled with consecrated silver and powdered iron. It was, my watch told me, around 6 a.m. Maybe the cops didn't care about bodies other than human; and obviously the owners either hadn't come at all, or had come and gone.
I walked quietly through the alcove hosting two pool tables. The balls I'd hurled at the bear remained scattered on the plank-wood floor. I moved past them, beyond the dead animals, onto the dance floor in the center of the roadhouse.
Up on the mountain, I'd sensed the deep peace of the environs, had recognized that the area was sacred to both the Navajo and the Hopi. But this place? A roadhouse? It was where people went for a vast variety of reasons, including simple enjoyment, but also to fill the otherwise empty hours.
How often had I done the same? Days on the bike eating up miles of interstate from sunrise to sunset, hours wasted in bars, a cuisine of crappy food. A come-down, my father told me, after all my promise. My mother had said no such thing, of course, but I saw the look in her eyes, the worry, the trace of disappointment. I'd missed none of what they felt, in their eyes and postures. They'd expected better of me. Matty was the one who'd worried them. And then . . . well, then their eldest had seemingly "gone bad," too, only worse.
I'd never told them the whole truth. And my brother? No, Matty never would. Matty just enjoyed the hell out of the fact that his older brother had rushed to his rescue yet again, had taken the fall; and then Matty Harlan just kept on _keeping on_ the way he always had, knowing his big brother would always pull his fat out of the fire no matter what.
And now? Now that brother had been told he wasn't even human. Wasn't actually _related_ to Matthew Harlan. Was related, instead, to Remi McCue. Who maybe, it occurred to me, was worth more than Matty.
Guilt flooded me. Where the hell had that come from? Matty was _blood_. Matty was my baby brother.
Wasn't he still, despite the whole heavenly matter bullshit?
Huh. Maybe not.
I stood in the center of the dance floor and let myself go quiet, the way I always did when Grandaddy asked me to. Remi had referred to me as a _sensitive_. Yeah, if you wanted to hang a name on it, even if it did sound a little wussy. I'd always gotten vibes from certain places, and then Grandaddy asking me what I felt, what I sensed . . . I'm more open to surroundings than most, yeah.
And here, now, early on a Sunday morning in a place where the night before McCue and I had killed two ghosts who'd become demons, and shot the hell out of reanimated dead animals—well, what did I sense _here_?
I closed my eyes. Let myself go very quiet, go inside myself. Invited the roadhouse to speak its own language.
Peace, upon the mountain. Safety. Here, there was joy, and noise: the sound of live country music; the crack of cue ball against solids and stripes; the shuffle, thump, and slide of boots against parquet; the clink and chime of glasses and bottles; laughter, the rare but not unheard-of bar fight. I smelled beer and hard liquor—this was most definitely not a wine bar—a trace of ancient tobacco predating no-smoking laws, raw wood, wood stain, lacquer and leather. Perfume and aftershave. Felt warmth upon my skin from the crush of bodies, the breath of cold and snow blowing in an opening door in the midst of winter; felt the stomp of snow-caked boots against matting and wood to free them of cold-caulked encumbrances. And heat when summer was upon the place.
And I sensed evil.
Not the place. Not the roadhouse. But something that had inhabited it. Not for long. But evil had entered. Benign ghosts, Lily had described the former owners. Doing no harm. Things moved around, a woman on the stairs or in a booth, a man in the rocking chair before the fire. No reports of trouble, merely some kind of _presence_. But what I felt now was something malignant.
The bulk of it was gone. It was just a smear, a stain, the taste of, well, of _afterness_ , for all that wasn't a word. Maybe it should be.
The afterness of evil.
A woman's voice said, "I need to talk to you."
# CHAPTER ELEVEN
I spun, snaked my hand up beneath my jacket to pull the revolver, cradle it high in both hands. Beyond the raised gun I caught the quick impression of shape, of a woman: black hair, dark Asian eyes, hands lifted palm-up before me, as if to ward off bullets. But other than raised hands, nothing in her posture spoke of fear, aggression, or hesitance. It struck me that she merely wanted me to stay put.
"You can shoot if you like," she said, "but it won't harm me. Bullets can't. Nor can the knives you carry."
I sniffed surreptitiously. No discernable scent. "Well," I began warily, figuring the answer might be vital to my survival, "what _can_ kill you?"
She blinked at me, then smiled. And laughed, showing good teeth. "Like I'll tell _you_ that?"
I shrugged. "Had to ask."
"You can put the gun away."
Yeah, right. "I kinda like it right where it is."
"You can't hurt me with it."
"Call it my security blanket."
She smiled faintly. "Yeah, well, a blanket can't do anything to me, either."
Her black hair was chin-length and she wore bangs cut straight across her brow. Striking, if not pretty. And though she was clearly Asian, the epicanthic fold was not pronounced. Her tip-tilted eyes were so dark as to appear pupil-less, but the _impression_ of pupils was there, even in poor light.
I stared at her, wondering if the gun maybe _would_ hurt her—or if she might be telling the truth. Because I felt pretty damn vulnerable at the moment even with a gun and two knives on me.
The thing about shooting stances is, at some point you get tired of holding a revolver at the end of outstretched arms. I'd trained enough to be good at holding the stance a fair amount of time, but I was out of practice after eighteen months away from weapons.
But I wasn't about to lower my arms. "Who are you?"
"I'm Grigori."
I raised my brows. "Not a woman's name I've heard before."
"That's not my name," she said impatiently, "it's what I am. Grigori. A watcher. And I shouldn't even be here, because we're supposed to _watch_. But—much of that ended eons ago, when some of us fell." A gesture seemed to indicate she wanted to brush away anything nonessential. "You need to know, you and your soul-brother—you're not being given the whole story. It's not just good versus evil, heaven versus hell. There's far more to it than that. You need to know this. Because they'll try to use you, the fallen Grigori. The _other_ Grigori. Some of them escaped from hell, too, when the demons climbed out amidst the earthquakes. Some are here on earth."
I stared at her, then finally lowered the gun. If she wanted to harm me, she'd have done it already. "What the hell are you talking about?"
"I'm talking about _hell_!" she snapped. "And heaven. It's not straightforward, any of it. The man you call Grandaddy's only given you half the story."
That lit a coal in my belly. "You're suggesting he's hiding information?"
"Every angel in heaven has an agenda," she said. "Every angel who is here on earth, and in hell. Lucifer is an archangel. He was God's favorite, which is part of the reason God was so upset by the betrayal. Do you believe it can't happen again? He fell, was cast out—so were others. Many of the Grigori—"
I interrupted. "Which you claim _you_ are; so how do I know you're not playing me?"
"At this moment you and Remiel are innocents being set up for manipulation," she went on tensely, as if running out of time. "I'm not telling you which side to take. I'm just warning you not to accept everything at face value. Yes, the war has begun—but it's not just black and white. Everything is—"
I cut her off sharply. "—shades of gray? Okay, I get that. It's what life is. But given a choice between heaven and hell, how many people—other than the evil whackjobs—are going to opt for bad over good? Dark over light?"
She shook her head. "When most can make a choice, yes, they choose good—or at least the lesser evil. Happens all the time in politics. But I'm just saying there's a much larger picture than what's been painted for you."
I blinked at her, brows raised. Dryly, I said, " _Larger_ than good versus evil? Larger than the End of Days and man's continuation upon the planet as we know it?"
" _Quality_ of life," she said sharply. "It's not a question of euthanasia because life's gotten really crappy for a pet, or even a person. It's not a choice of whether survival is a desirable goal, but whether that survival, in the worst-case scenario, maintains even the smallest trace of hope. Because without hope, it's all despair. And that's deadly. That's what kills, in the long run. So _think_ , Gabriel. What are you truly fighting for?"
So, she knew who I was. And as absurd as the words sounded, it's what Grandaddy had impressed upon me. "To save the world."
"And just which world is that?" she asked. "The one you know now, the one Lucifer desires, or the world that is merely collateral damage after the immoveable object hits the irresistible force? Rock and a hard place, Gabriel. Who _cares_ whether the rock hits the hard place if nothing lies between to be harmed? Then it doesn't matter. So, the rock loses a chip or two, the hard place cracks a little . . . _it doesn't matter_. It only matters when there's _flesh_ between the two, flesh that can bleed, can cry out, can die. Humans, Gabriel. Never lose sight of humanity. This world was made for them."
"You're saying heaven doesn't care about humanity?"
"I'm saying that it may become a case of the needs of the many outweigh the needs of the few. And humans are vastly fewer in number than are the inhabitants of heaven or hell. So, yeah, humans may wind up getting hammered."
In sheer incredulity I'd gotten stuck on her first sentence and missed everything else. "You're quoting _Mr_. _Spock?"_
_"_ Or Jeremy Bentham," she said impatiently. "Philosopher, jurist, social reformer. ' _It is the greatest good to the greatest number of people which is the measure of right and wrong.'_ "
God, we were having Philosopher Wars. McCue would love it. "'The greatest number of _people_ ,' is what you said. That suggests it is about humanity."
"It was a human who said 'people.' Heaven may view it differently."
I stared at her. "Whose side are you on?"
"I'm _Grigori_ ," she emphasized. "Are you totally ignorant? I'm an angel! Of course I'm on the side of heaven."
I grunted in derision. "I'm supposed to buy that, after you've just announced that some of you joined Lucifer in hell, and that a fair number of heaven's inhabitants may not give a damn about humanity."
"They all give a damn," she snapped, "but a portion of humanity may simply get in the way. What part of 'collateral damage' do you not understand? Are you a total dipshitiot?"
"'Dip'—what?"
"Dipshit. Idiot. Dipshitiot."
The absurdity completely swamped me. I could find no words. None at all. In any language. In fact, the only thing I was physically, mentally, and emotionally capable of undertaking at that moment was to go prop myself up against the bar clutching a useless gun. Where I broke into long, slightly hysterical laughter.
Like a dipshitiot.
When I wiped the tears from my eyes, she was gone.
* * *
—
Okay, yeah, I could have reentered Lily Morrigan's rig with a little less drama, but the door really did get away from me as I yanked it open. It crashed against the side of the RV, and by then I was through it and standing in the entry way.
Remi, who apparently had dozed off on the couch, sat up swiftly even as Lily pulled a gun from somewhere, the crow flapped its wings and loosed a horrific sound lodged somewhere between a gargle and a horror film shriek, and the wolfhound leaped to her feet growling, peeling back her lips to display a fine set of highly intimidating teeth.
I sliced the air with the edge of a flattened hand. "Okay. My life over the last forty-eight hours has become something of a cosmic joke. I'm not even sure it's real anymore. I'm not even sure _you're_ real. For all I know I got slammed by an 18-wheeler, and I'm lying in a hospital somewhere in a coma dreaming up this whole entire screwed-to-hell scenario—a scenario _about_ hell!—or maybe I'm actually dead; or maybe somebody slipped me a roofie; or, for that matter, maybe Rod Serling is standing in the wings while they play the spooky damn _Twilight Zone_ music; or maybe this is even the holodeck on the fucking _Enterprise_ and I'm a token redshirt, but I've about had it. I got mysterious women coming out the wazoo: a she-demon trying to gut me or strangle me; one sitting _right here in front of me_ who's playing Need To Know with Irish mythology as she arms me with magic guns, bullets, and _holy oil_ , for Christ's sake; and some chick named Gregory informing me it's not just good versus evil anymore but humanity caught in the middle _between_ the forces of good and evil and, as it turns out, heaven has a few angels with _agendas_ and the needs of the many outweigh the needs of the few, and humans are outnumbered and are therefore _screwed all to hell_. Literally!"
I stopped then, because I was out of breath.
Remi, who'd been staring at me in alarm as I began, relaxed back into the couch and eyed me in something akin to compassion. "Could you have possibly fit any _more_ movie and TV references into that?"
" _Yes._ "
Lily put the gun away, said something soothing to both the crow and the wolfhound, then fixed me with wide green eyes. "A woman named Gregory?"
I reached around, caught the door, pulled it closed. "Well, not exactly. What she said was Gri- _gor_ -i. That she's an angel."
"A _Grigori_?" Remi asked in shock.
I eased between bird and dog to reach the armchair, dropped my ass into it. "You know what that means?"
"Watcher." Lily lowered herself to the floor to sit beside the wolfhound. "And also fallen angels."
"She said _some_ are fallen. She said she wasn't. That not all are." I shrugged. "Though we don't know that she's telling the truth."
Remi rubbed a hand through short hair. "The ones who fell started out okay, here on earth. But they kind of screwed the pooch when they got a little up close and personal with womankind, and got cast out for good. Some believe that was one of the deciding factors for the Great Flood."
"Nephilim," Lily put in. She noted my blank expression. "You're not all that up on your Bible, are you?"
"Basics," I answered. "I was into folklore, not religion."
"The Grigori were sent here to be actual guardian angels," Remi said, "when humanity was still wearin' diapers. All kinds of conflicting stories exist, but in Judeo-Christian texts it says they started sharing knowledge not yet intended for humankind: astrology, divination, herb lore, even magic. I can see where magic might could be a bit chancy, but I don't know why the first three were considered risks."
"Too much too soon," Lily put in. "You don't generally ask a toddler just learning to stand to go out and run a marathon."
"And?" I prompted.
"Nephilim," Remi said, repeating the word Lily had used. "They're the offspring of human women and angels. Grigori specifically." He paused. "Bad juju, according to some of the ancient texts, though others were kinder. But the upshot is, all those watcher-angels who dared to sleep with human women got cast down from their places, and the kids, all male, were considered genetic no-nos."
"So God, being pissed off with a lot of things by then," Lily went on, "including his rebellious son, Lucifer, from whom He took back the car keys and tossed out of the house, decided it was time for a do-over. But he didn't want to wipe out all of mankind, so he sent Uriel to Noah."
"You should know the next part," Remi noted. "About Noah. It's kind of famous."
I shot him stink-eye. "What concerns me is that she said Grandaddy hadn't told us the whole story." Now I looked at Lily. "Why would she say such a thing? Why would she suggest humanity might get caught in the crossfire and end up collateral damage?"
"Because humanity might," Lily replied. "He didn't lie to you, Jubal didn't. It _is_ about Good versus Evil, Heaven versus Hell. But if you want a metaphor, look at traffic lights." She shrugged when we both stared at her. "What falls between red and green?"
We exchanged blank glances, then chorused dutifully, "Yellow."
"And what happens when a light turns yellow at a very busy intersection when everyone's in a hurry?"
"Drivers step on the gas," Remi answered promptly.
I, who had experienced several near-misses on the bike, added, "And cars often collide."
Lily nodded. "Humanity is the yellow light."
"Then what are we?" I demanded. "In the metaphor, I mean. Since we're not demons _or_ angels _or_ humans."
Lily laughed. "You're the _traffic cops_ , boyos! For when all the lights fail."
After contemplating that a moment, Remi said, "Well, that just tickles me plumb to death."
And I, whose annoyance and adrenaline had run out at pretty much the same time on a wave of weariness—we'd been up for something like thirty-six hours, plus hiked a mountain and killed shit—slid down deep in the chair and stretched out my legs. Planted an elbow on the chair arm and propped up my head against a hand. Around a yawn, I said, "At least they're hot."
Lily was not following. "What's hot?"
I waved my hand in an indistinct gesture. "You three. Legion, Lily, and Greg." I allowed my eyes to drift closed.
"Well, I've only seen two of the three ladies," Remi said, "but I'll agree with you on those counts. Yup. Pretty as twelve acres of pregnant red hogs."
For a moment I assumed I'd heard that wrong because I was just on the verge of a coma. But no. I hadn't.
My eyes snapped open. "Did you seriously just compare women to _hogs_? Do you even _know_ how many women would have your 'nads for that?"
Remi quirked a brow. "It's just an old Texas saying. I didn't make it up."
I looked at Lily to gauge her reaction.
She smiled, lifted one shoulder in a casual shrug, said, "If all the boyo wants are hogs in his bed, that's one way to make sure of it."
A slow but distinct parade of considerations crossed Remi's face. Finally he said, "I might could retire that one."
"Uh-huh," I agreed pointedly; preferring, in the interests of male solidarity, that Remi survive.
He said, rather mournfully, "But to a man who loves bacon—"
Lily advised, "Shut up, Remi."
"Yes, ma'am."
Smart man.
# CHAPTER TWELVE
With the dinette transformed to a bed—McCue took the pull-out sofa bed—I slept hard for several hours, rousing only when the scent of coffee permeated my senses. I discovered Remi up and pouring mugs full; Lily, the cowboy said, had taken the dog for a walk.
I crawled out of the sleeping bag Lily'd thrown at me, staggered my way to the bathroom, staggered back out and accepted the mug Remi offered. After drinking half of it, the staggers went away, and I felt alert again. It was verging on afternoon, but it felt like morning.
"You drink this much booze all the time?" McCue asked in an idleness underscored with—something. "I mean, God knows I party now and then, and I can hold my beer, but you been hitting the hard stuff."
"Three days," I murmured, rubbing my forehead.
It baffled him. "Three days what?"
"Three days of drinking after eighteen months with no booze," I explained. "I think I may be forgiven."
His eyes went blank, and I knew what he was thinking.
"I'm not on the wagon," I said. "Not working the program. Best explanation: the wagon got stolen and locked in someone's barn." I stretched, cracked my spine.
"If you say so." Remi swallowed coffee. "Can you run me down to my truck in a bit? I feel lost without my wheels."
That's right, his truck was still at the Zoo Club. "Sure." I wandered into the garage, squatted at the cabinetry, began pulling open drawers.
Guns, ammunition, knives, flasks, bottles of a clear liquid and something oily, myriad tins and packets of herbs, bags of silver shot, boxes of bullets and cartridges, a loose substance that smelled like iron, an array of equipment used for making bullets and filling shotgun cartridges, knife sharpening stones, polishing cloths, chamois. In the deepest drawer was a stack of multi-sized bowls made of wood, brass, silver, steel, iron, and heavy glass, some carved with glyphs, some markedly plain. Small tripods, bowl rings with legs, clear glass containers of various liquid substances including something the dark, rich red of aged blood.
I found, too, a drawer containing sharpened stakes and crucifixes.
"You gotta be shitting me . . ." I took out one of the stakes, examined it. It was perhaps eighteen inches long, of a gnarled, striped wood, smooth, polished to a sheen akin to glass. It was twisted, with silver set throughout. The wood felt silky in my hands, and warm. Almost alive. Carved into the butt end was a Christian cross.
Remi had followed me. Now he leaned a shoulder against the wall, threw most of his weight to one leg, crossed a booted ankle over the other casually, cradled a mug in one palm. "A week ago, I was at a rodeo. Rode me a saddle bronc, bareback bronc, even a bull. Made it to the buzzer on the broncs, but that big old bull threw me off like I was a fly perched atop his back. He swatted me good. There I was in the dirt, and that old boy ducked his head and tried to hook me. I saw that horn comin' down from above, comin' _right at me_ , and I thought: I'll die, or I won't, but I'll know one way or another in under two seconds."
I looked up at him. McCue's face was pensive, but his eyes strangely calm, considering he was talking about nearly being _gored by a bull_. "Yeah?"
"Clown distracted him, another got me on my feet, and I jumped up on the fence as that old bull headed through the alleyway. It wasn't but ten minutes later, just as I picked up my gear bag, that Grandaddy called. Said I was _needed_ , summoned me here. But he didn't say why, and I didn't ask. He trained me just to answer whatever he suggested."
I nodded. "Didn't tell me anything, either, just that he wanted to meet me here. I was up in Oregon. I've always done what the man asked of me, never even asked 'how high' when he suggested I jump. Just did it. But had I known about _this_ . . . I don't know. Not sure I'd have come." I ran a hand through long hair, pushed it off my face. "Hard to wrap my head around, you know? But I remember things—things he said when I was growing up. They made no sense then, but . . ." I let it trail off, because I still wasn't sure what I thought of anything, of anything at all to do with angels, demons, Lucifer, the End of Days, and any other crazy thing Grandaddy had alluded to.
"And now they do," Remi finished. "Make sense, I mean. It's the craziest damn thing I've ever heard, what he told us the other night. They oughta lock me up for even thinkin' it might be real."
I knew what was coming. "But."
"But," Remi agreed. "Then I stand right here and look at what's in those drawers and cabinets, and I remember what we did last night. What we killed. Or whatever it was we did to those—things. And right now that piece of wood you're holding might could be something meant to kill vampires, if all the folktales are true." He smiled, drank down some of his coffee. "That's your wheelhouse. So— _are_ they true, those stories? We going all Buffy the Vampire Slayer on their sparkly little undead asses?"
I dropped the stake back into the drawer, closed it and rose. "Hell if I know. But something tells me we're going to find out. Because beacon or no beacon, _something's_ happening. Grandaddy sure as hell seemed to sprout wings out there on that mountain. Then there's our own personal Irish goddess, a demon in a hot chick's body, and a third chick warning me that you and I are being set up by celestial beings with ulterior motives. Not exactly Charlie and his angels." Skewed humor bubbled up. "Hell, with Grandaddy in the mix, in this case Charlie _is_ an angel."
After a moment, Remi's tone changed. "I heard what Grandaddy said about your parents. But you've got a brother, right?"
I thought about my parents, who I hadn't seen for eighteen months—my cop father not being inclined to visit a son in prison, and backed up by my mother—and my brother, home for Sunday dinner, whom I'd floored with a single punch the day before I left to answer Grandaddy's summons.
And while in prison I thought about having a normal life to go back to, but I didn't, really. Just a bike and the open road. Matty would know why, because it was his fault my normal life was over. It always was his fault. But now he wouldn't have me to pull his ass out of the fire.
I hitched one shoulder in a half-shrug. "Yeah, I've got a brother. Let's leave it at that."
McCue was very quiet for a long moment. "I've got a daughter. She's two. But she lives with her mother, who's got a new beau, and I don't hardly get to see her. Maybe twice a year, if I'm lucky." He grimaced. "I guess now I won't even have that."
He straightened, pushed off the wall, headed swiftly back toward the living quarters while I stared after him.
* * *
—
Remi left his hat behind when I ran him down to the Zoo Club on the motorcycle. When the bike rumbled its way into the gravel-and-cinder lot, we discovered a police SUV parked next to Remi's silver pickup. I felt a familiar pinch of apprehension in my gut, even though it was wholly illogical; I was out of prison by some arcane maneuvering of a celestial being, so I didn't think that was the issue. Probably someone suspected the truck had been abandoned, since it had been parked between the tree and building for well over twelve hours. I goosed the throttle enough to roll the bike slowly toward the truck to drop off Remi.
An officer stood at the driver's door, hands cupped around his eyes to peer into the interior. As I halted the bike nearby, I felt Remi unwind himself from behind me, the shift of weight off the bike; felt the brief clap upon my shoulder that spoke of thanks. I intended to pull out again, but something made me pause. Something made me turn off the engine, remove my helmet, and wait.
"Officer!" Remi called, raising a hand as he walked beyond the patrol SUV toward the driver's side of his truck. "Sorry about that. I got a little liquored up last night, and this feller—" he hooked a thumbed over his shoulder in my direction, "—kept my drunk ass off the streets. I do apologize. I'll get that outta here right quick."
The officer straightened, turned to McCue with a smile, took a few steps away from the driver's door. He was young and fit in a dark navy uniform, with a peak-crowned, round-brimmed trooper's hat, and a spray of freckles across his face. "Sure thing. But can I see license and registration, just to make sure everything's in order?"
"'Course." Remi unlocked the truck and pulled registration info out of it, then handed it and license to the cop, who accepted them with thanks but didn't even glance at them.
In the same friendly tone, as he looked across the truck bed at me, he said, "You lit 'em up nice and bright two days ago inside this building. I think all of us within a five hundred square mile radius felt it. Now you're broadcasting loud and clear."
I felt a wash of ice down my spine as I realized the magnitude of our stupidity. Over two days we had been warned multiple times by multiple people, and yet here we were, both exposed with no weapons that might work against this kind of enemy. We'd left everything at the motorhome. Hell, Lily didn't even know where we were.
Newbies, indeed. Dipshitiots.
Swearing prodigiously, I dropped my helmet even as I threw a leg over the bike and pushed upright onto both feet, reaching back for the KA-BAR at my spine as I took three long running steps toward the truck. Remi vaulted up into the cab and dove for the open glovebox.
The surrogate in a cop's body, meanwhile, just smiled at us benignly, even as I paused at the back of the truck with the knife gripped in my hand while Remi dug a revolver from the glovebox.
I exchanged a glance with McCue, and saw the same question in the cowboy's eyes. _What the hell do we do now?_
"Broad daylight, here, where anyone can see you," observed the demon, "and just what are you boys planning to do? Kill a cop? Because you're not currently packing what can kill _me_. It's the host who'll die. Besides—" he spread his hands in a non-threatening gesture, "—right now I'm just saying hello to the new kids on the block."
The cop stood several long paces away from the open truck door, where Remi had placed himself behind the wheel with his torso turned outward and two hands steadying the revolver. He flicked another quick, questioning glance at me, and I felt a sense of helplessness well up along with anger.
"That was quite an auspicious beginning last night, taking out two of us," the demon continued, "particularly for rookies. You are to be commended. But don't get too cocky; those were just rank-and-filers. Lesser demons. Cannon fodder, as it turns out. Now, my paygrade's higher, and if you boys want to take a few shots at me, cut me with that KA-BAR, you go right ahead. A report on a certain silver Texas truck has already been called in. My host ends up dead or wounded? It's not me who'll suffer."
"Then what the hell do you want?" I asked. "We just going to stand out here all morning monologuing like in a bad TV show?"
The young cop still looked friendly and cheerful, lacking the alpha aggressiveness of some law enforcement officers even if he did rest the heel of his hand upon the holstered gun. "Well, actually I was here as cleanup detail. While those two ghosts didn't leave any bodies, per se, you did cut the connection between the hosts and their cohabitants. That means there are the remains of two of my brethren still inside that building. They're dead, as humans reckon such things, but there are _remains_ , just as with human bodies. We look after our own." He shrugged. "Everyone comes home."
Remi, still tucked up in the open truck cab with the gun in his hands, sounded scandalized. "Home to _hell_?"
Which did strike me as a rather odd thing, now that I thought about it. Did hell view its demons as soldiers? Were there rituals and rules?
"Or Hades, bottomless pit, underworld, the _habitation of fallen angels_ . . ." the cop went on, then waved a dismissive hand. "It doesn't really matter what name you hang on it. It's still just home to us, and the sooner I get those two back there, the sooner they'll be reconstituted. So no, I'm not here for you; I don't have the time. There'll be others along directly who feel a little differently. In the meantime, have a nice day. Oh, and—happy birthday. Welcome to the war."
I saw a pixilation in the air, the displacement of delineated image, now all jagged and broken up. A sudden blast of grit-laden air at my face made me duck aside and thrust a warding hand into the air, but too late to prevent the wash of irritation in my eyes. I smelled the rich, cloying stench, tasted it as I swore and wiped involuntary tears away from my cheeks with the back of my hand.
When my eyes cleared, I saw the cop had disappeared. "Where the hell did he go?"
Remi slid out of the truck and dropped to the ground, holding the gun down along one thigh. "He said he was here for the _remains_. I'm betting he's inside the building collecting what's left of those demons."
"Yeah, well . . . now what?"
"We take him out," McCue replied, heading for the roadhouse. "Ain't that what we're for? Besides, he's got my license and registration. Come on, son, let's go in after him before he grabs up those remains, takes 'em to hell, and they get . . . what'd he say—reconstituted? Yeah. That."
In two jogging strides I caught up. "Take him out with what, Tex? You can't use your six-shooter on a cop, now, can you?"
"I could," McCue offered, moving swiftly. "Or you could, since you're the one so handy with a gun. You could shoot him in the leg."
I was astonished. "Shoot him in the leg? What the hell for?"
"I reckon a demon would rather have a host that wasn't ventilated," Remi answered. "We might could maybe drive out the demon that way."
"Oh yeah—and then when the cop realizes I _shot_ him, he'll thank me? I don't think so. Because how do we know if the guy will realize his body was borrowed by a demon? I'm not shooting him!"
"Well, it was just an idea," McCue said. "But I got a better one anyway. Let's go throw Latin at that sucker instead."
# CHAPTER THIRTEEN
It was a half-assed plan, but I couldn't think of a better one. We really couldn't shoot or stab the host. I mean, did the cop even know he was being, what—ridden? Possessed? Would he be sane once we exorcized the demon? Hell, would he even be _alive?_
Grandaddy had said the learning curve was steep. Lily'd said we'd pick up details as we went along. Of course, that was supposing we _survived_ long enough to learn anything at all.
"You go around back," McCue suggested. "Pick that lock if it needs doing. I'll try the front door."
We moved stride for stride toward the building, then split up. I headed around back, wondering if there was any way I might distract the surrogate if I beat Remi inside. I didn't know the Latin for the rite. Hell, I didn't even know if exorcism actually _worked_.
The surrogate could have killed us, I suspected. He hadn't. I wanted to know why. I mean, as a cop he could have shot us both, I suppose, then crafted a story after the fact, but that would draw a lot of attention from police department to reporters and prevent him from collecting the remains of the surrogates Remi and I had killed. For some reason that seemed to matter a great deal. _Everyone comes home,_ he'd said.
Home to hell for reconstitution? Then sent back into the field to find more hosts?
Around back, I pulled open the screen door quietly, then closed a hand around the knob. Tried it. It turned, the tongue-latch gave, and I eased the door open.
Remi was already inside. Remi was _chanting_.
I moved swiftly through the pool tables and found the cowboy standing in the middle of the dance floor as he let the Latin of the _Rituale Romanum_ roll from his tongue. I heard nothing of Texas in it, just smooth vowels and consonants, an easy familiarity with the cadences and inflections.
Before him, on hands and knees, the cop was vomiting. But it wasn't food or liquid he was bringing up. Wasn't yellow-frothed bile. Neither was it smoke, nor light. It was gout after gout of something black and shiny, glinting in the light.
No, not something. Actual _things._
Things that were _alive_.
I stopped dead, appalled. From the cop's wide-open mouth, from his nose, his ears, poured streams of black, glittering carapaces. Like an army of ants on the march, the beetle-like things clustered upon the floor, then moved en masse in my direction.
McCue cut off his ritual long enough to yell at me. "Get some water! Throw salt in it! Spit in it! Then pour it all over those suckers!" And went right back to chanting.
Okay. Water and salt I could do. Spit, too; half-assed holy water, since I didn't know the blessing, but Lily had pointed out we were ourselves a kind of blessing—hence, the spit, as with breathing on the bullets—so maybe it would work.
I rounded the bar, grabbed a pitcher, filled it at the sink, dumped salt into it and basically hocked a loogie of saliva into the pitcher.
I swung back around the bar, saw the river of—holy shit, were those _cockroaches?_ —flowing toward the back door. I sucked in a breath, stepped right into the middle of that river, and began sloshing salt- and saliva-laden water over the roaches. Beneath my boots, they crunched. And once the makeshift holy water hit them, they burned.
Carapaces cracked open like popcorn. Each gave up a spark. And they _stank_.
I swore, took a couple of long staggering steps away, then stood staring at the field of roaches turning into crispy critters. The stench was acrid and eye-watering. I thrust an elbow across my mouth and coughed into it, knuckled involuntary tears away.
McCue was no longer chanting. He coughed too, spat, then knelt beside the sprawled body of the young cop and placed fingers against his neck. After a moment he nodded, then carefully rolled the guy over onto his back. "Pulse is steady, so's his breathing. Far as I can tell, he'll do."
Yeah, well, maybe physically. But otherwise?
I surveyed the field of the fallen. Little curled-up, crispified bodies. The stink was fading, but I still tasted it. Acrid, metallic.
I leaned forward, spat. Wiped my mouth against the back of my hand. "Maybe we ought to add Roach Motels to the armory."
Remi, now standing, was shoving a booted foot this way and that, as if gathering dirt together. "I don't get it. There aren't any _other_ roaches, or roach remains, from those two we killed here last night, right in this room. What the hell was this demon coming back for?"
I went over, saw he'd boot-scraped together a pile of something. I had a vague memory of the two ghosts, once they'd ceased their weird deflating bagpipe noises and shimmering, basically collapsing into exactly what McCue had scraped together.
I cast a glance over my shoulder at the carpet of burned little roach corpses, then squatted to take a closer look at Remi's pile. Ash. Dust. Some sparkling grit, like obsidian. "I don't get it, either."
The woman's voice said, "That's because they were ghosts, not possessed humans."
She startled me enough that as I lurched up I overbalanced and landed right on my ass. Lily Morrigan, standing just inside the main door, gave me a long look, then shook her head in resignation.
"That matters?" McCue asked.
Her green eyes were bright. "It does."
I pressed myself up off the floor with a one-armed thrust. "Why? I mean, ghosts aren't _real_. I mean, not _real_ real."
"Are they not?" She lifted her chin. "I'm an ancient Celtic goddess, and _I_ am real. Now, come along. The young boyo will be awakening soon. He'll be confused, but he's still a cop. Best we be elsewhere to avoid awkward questions."
McCue indicated the swath of burned roaches with an outflung hand. "Can't another demon come along and gather up the remains? Take 'em back to hell and bring 'em back to life?"
Lily shook her head. The neck torc gleamed. " _Rituale Romanum_ , holy water imbued with heavenly essence, and salt. No demon of this paygrade's coming back from that. Same with the ghost remains. They were lesser demons." She tilted her head. "Now, come. I hitched here; I'll catch a ride with Remi, and we'll go back to the rig. Time for debriefing."
* * *
—
Debriefing consisted of Lily interrogating us. What had we said; what had the surrogate in the host's body said; what was our original plan of action; how had we carried it out, and how closely did our actions align with the plan for the evening, this whacking of two ghost-demons, and so on and so forth. I was bored within five minutes, and let Remi do most of the talking.
The crow, upon the perch, rotated its head and fixed me with a bright and beady eye. I noted the powerful beak—throughout history, they pecked eyes out of dead and dying bodies, did crows and ravens—and shifted slightly backward in the armchair.
I rubbed the pad of one thumb against the leather upholstery. "Look. We did what you wanted. We took out those surrogates in ghost form, killed off the reanimated stuffed animals, and exorcised a demon out of a human who'd been possessed. We _saved_ him, right? So what more do you want? Didn't we pass your tests?"
"Barely," Lily said, from the driver's seat turned backward. McCue was behind the dinette table. The wolfhound lay close by. "You were sloppy last night, then walked into a situation this morning without a single weapon to aid you."
"We got it _done_ ," I declared.
She flicked a glance at McCue. "Did you?"
He'd hooked his hat over the window valance, scrubbed a hand through short dark hair. "End result, yeah."
"And what might you have done differently?"
He shrugged. "Thrown Latin at him sooner? Before he went into the building?"
Lily leaned forward. Her legs were planted, thighs spread, and she rested her forearms atop them, hands dangling. "Do again what you did today," she said, "and you'll end up dead."
The wolfhound rose. She shook herself off from head to toe, then turned toward Remi and glowered at him across the dinette table.
So, okay. We had the Morrigan mad at us, and her sisters equally pissed.
I stroked loose hair behind one ear. "Just exactly how long are we to do all this surrogate-killing shit? What's our tour of duty?"
Lily said, "Until the job is done."
"And how long is that supposed to take?"
Her tone was bland. "How long do you suppose ending the run-up to Armageddon _should_ take?"
"Overnight would be preferable," I told her pointedly.
"Well, those odds aren't exactly good," Remi observed. He looked at Lily. "You got a Bible on hand?"
She rose. "Which flavor?"
"King James, or New King James."
"Why does it matter?" I asked, as Lily walked past us to the cabinetry.
Remi shrugged. "Scholars say both the King James versions are closest to the original texts. Ninety-eight percent. Other versions are a little more casual about the vocabulary." McCue took the volume Lily handed him as she returned to the big driver's seat. "Okay, let me look something up. I think it's Mark . . ." Remi paged through rapidly, murmuring to himself about days and hours.
I frowned. "What are you looking for?"
"Here." He planted his index finger against the page. "It's Mark 13:32. _'But about that day or hour no one knows, not even the angels in heaven, nor the Son, but only the Father.'_ So it looks to me like the angels have no idea how long this might last. Just God."
"Huh." I scratched an eyebrow. "What is it you say, 'might could'? Well, it might could be helpful if God let us in on that intel."
"Signs," Lily put in. "War, for one."
Remi quoted, _"'Nation shall rise against nation, and kingdom against kingdom.'"_
Her eyes were bright. "I'm not the only one here. I have sisters other than Nemain and Macha. Co-workers, you might say, in the art of war. Athena. Kali. Bellona. Enyo—and no, not En _ya_ , the Irish singer; I've heard that joke already. Ifri. Neith." She shrugged. "Many of us have come. Male gods, too."
"So . . ." I thought about it a moment. "If you're saying all the gods and goddesses of war are here . . ." But I broke off and clapped hands to my skull, squeezed my eyes shut. "No. No. I'm reverting back to the theory that I'm in a coma somewhere and this is all a hallucination."
"False prophets," McCue added. "Fake news, maybe?"
"But not all of it's fake," I said between gritted teeth. "The news."
"Politicians," he went on. "Religious leaders. Misleading the elect."
I opened my eyes. "Okay, so now you're saying some religious leaders predicting _the Apocalypse_ are false prophets? I mean, how does that even make sense? If we're on the cusp of it, according to Grandaddy and other angels, then aren't the false prophets telling the truth?"
Remi shrugged. "I reckon it depends on what they're sayin,' and how they're sayin' it."
"Moral decay," Lily said. "Another of the signs."
I couldn't help myself. "Cats and dogs living together."
"Yes, my kind are here," Lily said. "We are here, but not all of us are working with the angels. Many are on the other side."
"Why?" I asked. "I mean, what brought you here from—wherever it was you were?" I caught Remi's eye, shook my head in disbelief before turning back to Lily. "What would induce gods and goddesses to pop out of some kind of alternate universe, the whole space/time continuum thing, to join this battle between good and evil?"
She shrugged. "We're being paid."
It took half a minute to process that and then I was astounded. "So, what, you're saying _gods and goddesses_ are mercenaries? Seriously?"
"In a manner of speaking."
"Then what the hell are they paying you with?"
Lily smiled. "War."
I wanted to growl in frustration. "But the goal is to _stop_ the war."
"The goal is to stop the end of the world," she pointed out. "We're using war to do it."
"But why are some of you on one side, while others are backing the devil?"
"War," she repeated. "Are you totally ignorant? There must be _opposing sides._ "
I let that insult slide. "Besides war," I said, "what do you get out of it?"
Her pupils spread. They were still round, unlike those of the cat-eyed demon chick, but it was unsettling nonetheless.
"To be _known_ again," she said; and her tone suddenly deepened. "To be in the world again. We are far more than words on a page or on a computer screen. _People worshipped us._ "
The crow mantled, shifted on the perch. I eyed it uneasily. It eyed me back. Hackles rose on the wolfhound's spine.
Holy shit.
"But this isn't heaven," Remi said.
I blinked, the moment broken, and looked at him blankly. "What?"
"The battle is to take place in heaven. Listen up—' _And there was war in heaven: Michael and his angels fought against the dragon; and the dragon fought and his angels.'_ "
Lily had a quote of her own: " _'On earth as it is in heaven.'_ "
Well, hell.
I looked at the woman who called herself the Morrigan. "Goddess of Battles, is it?" I asked. " _Prove it_."
The sleeve tattoos on her bare arms writhed. "All right," she said; apparently she'd been waiting for it. " _Agus mar sin beidh mé. Téigh abhaile liomsa."_
And the world whited out.
# CHAPTER FOURTEEN
I woke up to the whistle of the wind, face-down, conspicuously asprawl, in grass wet with heavy dew. For a moment I was horribly disoriented, as if rousing up from a four-day bender: pounding head, mangled brain, dicey belly, shivers inside and out—and then the fragmented pieces began to coalesce.
I opened my eyes. Peered at what I could see of the world.
Grass. Tall. Brown. Dead.
I lay there a long moment, putting together the concussion protocol: name, birthday, date, mother's maiden name, the whole password recovery bullshit. Except, well, no capital letter, number, or an oddball grammatical symbol, apparently not understood by computer coders. Were they just _unaware_ of the ampersand, caret, and tilde? Okay. Yeah. Me. Among the living.
I dragged my elbows to my ribcage, took a queasy breath, pushed partially upright. Ducked my head and remained bowed down, elbows planted, skull clutched in my hands and ass stuck up in the air.
After a long moment of even longer dire mutterings, I pulled my knees under my belly, rocked backward, and pushed myself into a seated position, more or less, butt planted, and boot heels, knees bent upright. I was dew-soaked head to toe, hair loose, tangled, soggy. I wiped vaguely at my face to clear it.
My hand came away red.
_Not_ dew. Blood.
Wet head to foot with _blood_.
The breath gusted out of me, followed immediately by the F-bomb. And three more, raspy and slurred save for the hard _k_ at the end, as I shoved myself upward, stood unsteadily.
My t-shirt, beneath the open leather jacket, was wet and chill against my torso. I pulled it away from skin, felt at flesh, did a mental inventory, but other than a headache nothing hurt. Nothing suggested any part of me was bleeding at all, let alone enough to wet hair and clothing. I let the tee slop back.
The wind yet blew, whispering now. Otherwise the quiet was almost uncanny. No birds at all, no insects. Just—silence. Save for the breeze and the rustle of tall grass.
I smelled blood. I reeked of it, as did everything around me. And it's a sharp, metallic taste, like copper, or iron, with a thick, throat-cloying fug. I pulled the loose flaps of my jacket aside, rattling metal buckles and studs. Noted that my black leather pants were smeared from belt to ankle.
I scrubbed the back of a hand against a cheek, though probably all I did was rearrange blood. My gag reflex engaged, then eased before I hurled, though bile burned partway up my esophagus. With effort I swallowed it back down, then finally took a good look around.
I wasn't alone. Thousands of men were present. Before me, behind me, beside me. All lay sprawled upon the battlefield in attitudes of death: prone, supine, some limbs reaching skyward, others twisted, or hacked away. Eyes closed. Eyes open. Mouths agape in frozen grimaces, or features utterly slack. Long hair, braided hair, subtle plaid trousers and tunics of earth-born colors. Torcs glinted, as did brooches and wristlets. The wind caught at cloaks, fluttered cloth.
I drew in a very deep breath, then blew it out hard. The sky above was a hazy blue, the sun muted. Still no sound beyond the wind and the sibilance of waving grass.
No man moved, or moaned. No man cried out. No man gave any evidence of being alive. Empty of blood, what had been warriors were nothing now but sacks of flesh shaped like men, fallen helter-skelter across the plain.
I began walking, stumbling, because there was nothing else to do. And as I walked, a slow, turgid urgency rose up.
Remi McCue. The cowboy. I needed to find him; and that need set a band around my chest, squeezed my ribs and organs. Was he dead? Alive? Had the Morrigan also brought him here to this field of death? That they were Celts, I knew; she was a Celtic goddess, and this was her home. Was McCue, like me, surrounded by bodies? Or was _he_ a body?
I walked the plain, squelched through soil turned to mud because of all the blood. Felt the whip of grass against motorcycle leathers. No loom-woven, woolen trousers and tunic for me, no cloak, no torcs or brooches. I was as I had been in Lily's motorhome, in leathers, a t-shirt, and motorcycle boots.
Except for, well, the blood all down my front.
Shit. Shitshitshit. Where the hell was McCue?
I could search for days, pull bodies aside, tip sprawled forms over, and not find him, not amid so many. So I stopped where I was, cupped hands around my mouth and shouted for the cowboy.
No answer. Nothing. Only the rush of wind across the plain.
The quiet was just _wrong_. This was a battlefield. Dead lay everywhere. There should be carrion birds plucking out eyes, flapping wings, shrieking warning to others to stay away from claimed human prizes. And insects buzzing, chirping, rattling, humming on noisy wings.
Then came sound. A voice, shouting my name.
I spun quickly, saw a man in the distance. He was mostly indistinct, but I could see the unmistakable outline of a cowboy hat.
Relief was palpable. I flung an arm upright, swung it back and forth, then shouted his name and broke into a jog even as he came on.
When he was close enough, I saw that he hadn't escaped the carnage. His clothing was soaked and richly red, and his hands and face were smeared. Twins in survival, we two: men who did not belong here.
Remi halted. I saw a bloody handprint against the cream of his hat. He eyed me up and down, registering my condition.
"Not mine," I said. "You?"
The tension in his body eased. "Mine, neither." He took off his hat, rubbed a forearm through matted hair, and left another hand print upon the pale felt, though this one was lighter, smudged. "Well, hell. I reckon we got us our answer. She's the real deal, all right. Let's just hope she sends us back when she's done making her point." His eyes fixed on mine as he put his hat back on. "Guess she took it personal when you asked her to prove it."
I stretched out arms from my sides, let them slap back down. "Do you blame me? I mean, come on! This whole thing is unfuckingbelieveable."
He smiled a little. "That's right, you believe you're in a coma somewhere and hallucinating. So my question to you would be: Why the hell am _I_ in it?"
I swore, raked a spread-fingered hand through damp hair. Grimaced because it was blood, nothing so benign as rain or dew. "Hell if I know. But what I _do_ want to know is where Lily is. I'd like to take a shower, change clothes, you know? Get rid of all the blood, knock back a couple of drinks."
His brows ran up. "You always drink so much?"
I scowled at him. "Give me a break. Making up for lost time. It'll pass."
He nodded thoughtfully. "Might could do with a drink myself, seein' as how we got, what, translocated?"
I felt helpless, which made me surly. "Let's just say Scotty didn't beam us here. But I'd like to figure out how to get back. Because unless Grandaddy means for us to hunt demons in _Roman-occupied Britain_ , we've got no angelic mission here."
Remi looked across the field of the dead. The look on his tanned face was pensive. The wind rippled his denim shirt, snatched at the points of his collar. "Which battle do you suppose this was?"
I stared at him in surprise. We'd been taken through time, and he wanted to know what battlefield we were smack in the middle of? "Why does it have to be a _specific_ battle?"
His gaze came around to mine. Something bright sparked briefly in his eyes, and then he smiled faintly. "Just trying to figure out where—and _when_ —we are. So we know what comes next. I'm thinkin' it might could be important."
Well, there was that. Hands on hips, I turned in a full circle. Still no carrion birds, no insects. Still just the wind hissing through the grass.
Finally I shook my head and met his eyes. "Could be one of many battles. The Celtic tribes were always killing one another."
He shrugged. "Seems like a lot of men for mere tribal squabbles."
Okay, yeah. McCue was right. I looked around again, saw the narrow plain was bounded by trees on two sides, and the distant uncertain darkness that suggested trees behind us. I turned in a circle, then gave up. "Yeah, no, I've got no idea where _or_ when—"
But I wasn't allowed to finish, because suddenly Lily was standing six feet away. The wolfhound was at her left side, and the crow flapped and soared idly in the air. Piercings and torc glinted in the sunlight, and the red shock of her hair was bright under the sky. The remarkably fine bones of her face stood up beneath pale skin.
"Look again," she said. "Are all of the dead, men? Are all of the dead, Celts?"
Remi stared at her a moment, then moved hastily to several bodies. Two he turned over, then straightened, turned, and looked once again at Lily. His face was taut. "Okay," he said; and there was no lazy drawl in it, only tension. "Women, too. Is that what you want us to see?"
Lily's smile was luminous. Otherworldly. She looked beyond—or _through_ —me.
I swung around abruptly, following her line of sight, saw the field shift before me. I blinked my eyes wide. Men, yes, but now also women, as McCue had noted. The body count abruptly increased.
"Holy shit," Remi said.
"Look again," Lily repeated. " _Feuch dè bha, faic dè tha._ See what there was, see what there is."
McCue and I glanced at one another, then did her bidding.
Now _chariots_ , where there had been none. Unhitched, upended, broken, wheels missing, and the scattered bulks of slain horses.
Ten, maybe fifteen paces away from me, a Roman soldier, complete with armor of leather and metal strips, the iconic helmet with cheek-guards and upright plume. But only _one_ Roman. All the other bodies I could see, male and female, were Celts.
I closed my eyes, burrowed inside myself to find the part that _sensed_ , as Grandaddy had called it. Beneath the sun of a younger earth I let it come to me, the knowledge. Heard the clatter of shields, of javelins; the wet slop of Roman gladii against wool-wrapped flesh. And the war cries, the screams of horses, the shouts of men, the ululations of women who wanted nothing more than to throw off the Roman yoke.
I opened my eyes. I knew.
"Boudicca," I said grimly. "Her final battle against Gaius Suetonius Paulinus, 60 or 61 Common Era." I gestured, indicating three directions. "Narrow plain hemmed by trees, and a forested gorge behind, cradling the Roman formations. No Britons could sneak up, or lay ambushes. It could only be a frontal attack. And it was disastrous."
"Why disastrous?" Lily asked, watching me closely.
I gazed across the plain. " _Eighty thousand_ Britons died, with maybe, and _only_ , four hundred Roman soldiers lost."
"Jesus," McCue murmured.
"I was there," Lily said simply, "as I am here. At war, I am always present. And this _is_ a war, boyos, never doubt it. Apocalypse, Armageddon, End of Days, Ragnarok . . . call it what you wish. This war is the end of all, unless it is _won_."
The Morrigan, Goddess of Battles, stood before us clad in knee-torn jeans, a tank top, bare feet. The multi-colored spirals and angles of her sleeve tattoos writhed upon her forearms, as if alive. Her eyes were vastly bright.
She smiled faintly at McCue. "Those cowboy boots of yours aren't precisely ruby slippers, but I'll wager if you tap the heels together three times, we'll go home." She paused; the wolfhound yawned prodigiously. "Well, to what _you_ boys know as home."
Remi blinked at her, looked at me with brows raised.
I shrugged; who the hell knew? _I_ sure didn't.
So he gave it a try. Three taps of his boot heels, and the world once again whited out.
# CHAPTER FIFTEEN
I came to my senses much as I had in Roman Britain, save there was no blood in my hair, on my face, or a blood-glued t-shirt stuck to my torso. Nor was I lying face-down upon the ground. I was however, sprawled partly upright and wholly inelegantly in the motorhome recliner, arms hanging over the sides, legs outstretched loosely, and my ass nearly off the edge of the seat.
I was so close to landing on the floor that I flailed without anything approaching coordination, grabbed hold of the chair arms and shoved myself upright, which wasn't necessarily the best idea in the world for my innards, though it took precedence over doing an ass-plant on the floor. No, what this abrupt movement did was set my head to aching and my belly to churning. Bile—or something more—once again climbed my esophagus; and once again I swallowed it down. Then closed my eyes tightly, pressed a hand across my eyes, and cradled either side of my skull with thumb and fingers.
All I could manage was a tight-throated rasp. "Can we _not_ do that again? Jesus Christ. That's not how they do it on TV and in movies, the whole—" I waved a hand in an indistinct circle, "—scrambling the atoms thing. It never appears to be _painful_ , you know? Nobody ever hurls."
"You challenged Lily to prove something," a deep voice said. "That's a very foolish thing."
My eyes snapped open. I registered McCue across from me, clean again, hat bearing no bloody hand prints, forearms resting on the dinette table, but looking tight around the mouth. He squinted at me, then poked a finger in the air to direct my gaze elsewhere.
Oh. "Hey, Grandaddy."
He stood tall and solid in the aisle, shoulders thrown back in his frock coat, silver-white hair flowing to shoulders. He was patently displeased.
"Well, what the hell did you expect us to do?" I disliked my defensive tone at once. Better to be a little belligerent, under the circumstances. "Just _believe_ some tatted-out chick in a red Mohawk and piercings who claims she's a Celtic goddess?"
"I _am_ a Celtic goddess," Lily stated. "Do you require another trip to a battlefield? Perhaps the American Civil War?"
It was Remi's turn to speak, and he sounded no more pleased than I as he looked at Grandaddy. He seemed relaxed as he leaned back against the dinette cushions, but the flesh at the corners of his eyes was lined. "You piled our plates high. Might could explain a _little_ more, don't you think?"
Grandaddy's blue eyes weren't unkind, but certainly serious. "No man—even one born of heavenly matter—who is raised as a human can take everything in at once. It's not like sticking a flash drive in your USB port and downloading, since you don't _have_ USB ports."
I muttered, "The number 42. And Johnny Mnemonic."
Grandaddy looked perplexed, and McCue just gazed at me blankly.
I waved my hand again. "You know—Life, the Universe, and Everything. Could be cool if it _was_ just downloaded, you know? Into our heads."
Remi looked at Lily, who was seated on the floor with a wolfhound head in her lap. The rest of the dog lay stretched out behind her. "Did he order scrambled eggs for brains during that trip back?"
"It can be disorienting," Lily replied, "but I think this is just Gabe."
"Just me," I agreed, then clasped hands behind my head and cracked my neck. "So, Grandaddy, we don't know everything there is to know about angels and demons and gods and goddesses and monsters and all kinds of shit that goes bump in the night, and even the kitchen sink being thrown at us, for that matter. Oh, wait—can a sink be haunted? Hey, maybe you've got a textbook for all this bullcrap. That would help. I'm a pretty fast reader."
Grandaddy didn't crack a smile. "You'll be given information as circumstances require it, as time goes on. At this point, it's for you both to find out what you can do, how to do it, and how to back one another up."
"Hey, _he's_ the one got me translocated," Remi protested. "I was just fine with believing she's a fictional character out of fictional Irish folklore, no questions asked." He flicked a wary glance at Lily, then looked at me with accusation in his eyes. "I don't call that 'zactly backing me up."
"How the hell did I know she could actually _do_ it?" I threw back. "But now we know, and we can proceed accordingly." I looked up at Grandaddy looming in the aisle. "Is there anything in particular that brought you back here? Like, it's time to provide more of this angelic intel we'll need?"
"Not intel," he said. "An assignment. Time to clean out the rat's nest. Clear the domicile."
Remi shot a frown at Lily. "You used that word before. 'Domicile.' Beyond someone's house, what's it mean?"
"Demons _move in_ ," Grandaddy said, before Lily could speak. "It may be into a body—or, now, into creatures from legend and folklore—or into a _place._ If it's the latter, they set up shop, make it a home. That's what they did at the cowboy bar. Two took on host bodies, made them corporeal, and damn near killed you both."
"'Host ghosts'?" I suggested. "Naw—'ghost hosts' sounds better. So, what about them?"
"The ghosts were owners of the roadhouse. Or rather, the _ghosts_ of the owners, right?" McCue seemed to be picking his way through a minefield, apparently unwilling to just blurt things out the way I did. So much for having _my_ back. "So the surrogates, in ghost-form, legally—and spiritually?—now own the bar. And they made it a domicile?"
"We killed them," I pointed out. "Or disappeared them. Or whatever it was we did to their demonic little asses." I shot a glance at McCue, was inspired. "All they were was _dust in the wind_ when we were done with them."
"Don't start up with Kansas," Remi suggested, "or I'll be singin' country music again."
"Oh, God. No. I'll even beg." I started to cross forefingers in a warding sign, but Grandaddy's big hand came down and closed upon mine, damn near crushing my fingers.
"No," he said; it was enough to stop me cold. "Do no such thing!"
I sat frozen in place. So did Remi, and Lily. Grandaddy was bent low to capture my hands with their linked fingers, now folded down. Behind his shoulders I saw the shimmering pixilation, the prism, the pressure, the impression of wings.
"This is _real_ ," Grandaddy said; there was neither kindness nor patience in it. "Do you not understand? _Everything_ is real! Would you place a _warding sign_ on him? To consign him elsewhere, to damn him, when he is your greatest support?"
Holy shit. I swallowed tautly, coughed a little. "Uh—no?"
His hand did not release my own. My fingers protested. " _Think_ , Gabriel. You studied folklore. Legends. History. Remi knows how cultures believe, how they worship. You are neither of you stupid, but _foolish_ , oh yes. Now, grasp this. As I told you explicitly, but let me say again: Everything you've read of legends, fairy tales, folklore, mythology, even Bible stories, _is now made real_."
In the sudden absence of speech, I heard the wolfhound's heavy breathing. She slept hard with her head in Lily's lap, twitching and damn near snoring. Not exactly goddess material.
Okay. Okay. Lily Morrigan, _the_ Morrigan, had proved herself real.
I drew in a deep breath, nodded, and Grandaddy released my hands. I shook them out, rubbed knuckles, plowed on. "So, we also took out that other demon. The one in the cop. He said the others were lesser demons, and his paygrade was higher." I debated saying the next thing, said it anyway because it was true. "We still took him out."
Grandaddy straightened and gazed down upon us. I decided then and there we should never sit in the man's presence again, unless _he_ was sitting. The man, as seraph, was downright intimidating.
He could be dryly, and subtly amused, could Grandaddy. I'd seen both. But he was neither at this moment. "Your mission tonight is to go back to the Zoo Club. Eat, drink, dance, shoot pool—it's immaterial. Merely be there. And wait."
I was wary. "Wait for what?"
"For the surrogates, of course."
Remi went wide-eyed. " _More_ demons?"
"Oh, shit," I muttered, remembering. "That demon chick knew we were there. She said she felt it when we clasped rings and lit our heavenly asses right up."
Remi sighed deeply, rubbed a hand through his unhatted hair, then absently attempted to finger-comb it back into order. "That cop demon said something, too. That we'd be hunted."
"You are now prey," Grandaddy said, "but also predators. Tonight you'll be bait. And any who come to you, any and all, who attempt to snatch the lions down from the heights, are to be destroyed. The bait shall bite, and clear the domicile." He paused. "You have an hour before you go. Make use of it."
I chewed my bottom lip for a long moment, turning things over in my mind. Then looked at Lily. "You got a whisk broom and dust pan? You know, for demon remains?"
"Stop," Grandaddy said sharply. "Gabriel, _stop_ this. You must understand—"
It exploded from me as I shoved myself to my feet and stood but two feet away from him. "I _do_ understand! How could I not? I was _raised_ to handle guns, _raised_ to understand folklore and legends, _raised_ to _—_ or imbued _with—_ this primogeniture bullshit, and a useless ability to _sense_ places _._ Holy shit, Grandaddy, I was just at Boudicca's final battlefield! I _tasted_ death there. Heard it. Smelled it. And it reeked. It _stank._ Blood, bowel, urine: men, women, and horses . . . yes, I get it! I _get_ that a stranger is now bonded to me; I _get_ that I have a holy GPS unit in me; I _get_ that you want me to destroy surrogates . . . but I was raised a human and I'm scared shitless. What makes you think I can do this?"
The syllables were exquisitely clear. "You killed a man."
My knees literally went weak. I locked them rigid and held my place. I felt hollow, a little sick, and hyperaware of McCue's sharp, startled attention. Limbs prickled, like cold sand running out. "And I went to prison for it."
Grandaddy's face was terrible. " _Who do you think sent you t_ _here?_ _"_
# CHAPTER SIXTEEN
I stared at the man I loved more than my own father. Felt tears gather in my eyes. Literally felt sick to my stomach.
Thought: How could—? Why did—? But couldn't finish either question, aloud or in my head.
"Yes," Grandaddy said, knowing exactly what I couldn't bear to ask. "Yes, I arranged it. I didn't force it, but I _manipulated_ certain things, I nudged, made sure various details were in place so the goal would be reached as it was intended. Yes, the man is dead; yes, you killed him; yes, you went to prison. Because all that was necessary."
I stared at him, still in disbelief. "It was _necessary_ for me to kill a man? _Necessary_ to go to prison?"
_"_ It was."
"Why?" I was now able to ask it. "Why on earth would you do such a thing? Why was it ' _necessary'_?"
His tone was unrelenting. "Because you must be strong. Because you must understand. There will be hardships. Challenges. Things you wish not to do. Times when you are helpless, perhaps even a prisoner. You will be required to kill, Gabriel, and you must not hesitate."
Okay, creepy cult leader talk. "But—"
He cut me off. "When a man comes at you with a gun in his hand, it's possible he's not a man at all, but hosting a surrogate. Until Remi can tell demons from humans at distance, you must be prepared. No, we do not advocate killing possessed humans—always do your best to save them—but there will be occasions when killing surrogate _and_ host is best for the greater good."
My breath ran choppy from my lungs. I stopped seeing Grandaddy, saw instead the Asian woman, the Grigori, warning me that not all angels were on the same page. That some had an agenda, and humans could very well be collateral damage.
I swallowed hard, refocused. It was Granddaddy before me again. I cleared my throat, looked at McCue. "You ever kill anyone?"
His face was tense, eyes unblinking. I couldn't read his expression. "No."
Grandaddy said, "It was not required of him. You are the alpha, Gabriel."
I did not look away from McCue. His opinion mattered. " _Could_ you kill someone? Someone who looks like a human?"
I saw a shift in his blue eyes, some unnamable emotion. "I don't know."
I weighed him, as best I could; he was a stranger to me, whatever he was intended to become at some point. _Sealed_ to me.
I looked at our mutual whatever-he-was, who had never been a stranger but now was both less and more of a man—a man?—than I'd known. "You said he'd be my backup."
Grandaddy didn't hesitate. "Yes."
I shot a glance at Remi, then just plowed on through. "Well, in this war you've thrown us into, what good is he if he can't kill a demon wearing a host body?"
Remi sat abruptly upright. He was not pleased. "I saved your ass the other night when the ghost—the _demon_ —had ahold of you. And the bear. Remember the bear? And I exorcized the surrogate from the cop's body so the cop, the host, is still alive. Hell, boy, I'm not sure you'd have survived had I not pinned the host to one place and spouted Latin at him. And it was _me_ who had to tell you what to do: water, salt, spit. You want backup? Start there. Seems to me you've had plenty." He flicked a glance at Grandaddy. "But he's the alpha, is he?"
"Give him time, Remi. You _might could_ be surprised."
"Okay." I was exhausted all of a sudden, dropped down into the armchair. "Yeah, okay; you're right. But I'm just not understanding why Grandaddy—or whatever he is to us, being as we're all cut from the same heavenly cloth—felt it was necessary to _set me up_ so I'd kill a guy." I remembered the moment I'd pulled the trigger, the shock on the man's face as the bullet struck; and from my brother the whoop of exultation, of relief, the cheers of victory, as if he were an athlete who'd scored.
Scored—because a man had died.
Grandaddy's tone lost its edge. "You killed that man to save your brother. It's what you've done all of your life, made Matthew's problems your own. And he knows it; he knew it then, that night." I stared hard at the floor beneath my boots, thumbed the back of one hand over and over again in a nervous gesture. "He has abused your trust countless times, Gabriel. But Remi will not. Remi is not your brother. He is _more_ than brother. And when the time comes, he will always save you."
I leaned forward, slid elbows down my thighs. Kept on rubbing the back of one hand. "Hell, Grandaddy—I don't even know if I _can_ shoot a demon wearing a human host. I didn't exactly enjoy killing that guy, regardless of what he was. And prison?" I straightened up again. "Prison cost me time, a job, a girlfriend, my self-respect, and it lost me my father, basically. He didn't mince any words when he told me what he thought of me. He'd already raised one son he couldn't respect, in Matty. And then . . . then it was my turn. You'd think a cop could understand that the shooting was justified. But he threw me under the bus. Hell, he told the judge he should throw the book at me!" I couldn't help the bitterness. "Some support _that_ was. Some backup, huh?"
"And now you are out," Grandaddy said. "You are free, according to the law, and accountable only to me. You made sacrifices, yes. But now you are needed, and your particular skillset is required. Yours and Remi's."
I looked at the cowboy, caught a level stare unlike any I'd seen from him before. I needed confirmation he felt as lost as I did. "This is all kinds of fucked up."
Before Remi could say anything to indicate agreement or repudiation, Granddaddy stepped in. "That's what the End of Days _means,_ Gabriel. And now it's up to us—you two, me, Lily, all the other angels and our allies—to _un_ fuck it."
* * *
—
Remi and I drove to the roadhouse in his truck, fully armed, revolvers loaded with what I dubbed magic bullets, plus knives bathed in breath and holy oil. Remi said not a word on the way, probably, like me, contemplating what the hell was about to happen and what we were supposed to do about it. He didn't say anything even when we pulled into the parking lot. McCue shut off the engine, but made no move to get out. I stayed put, too.
Early evening, but the sun was down. Monday was not a party night such as Friday and Saturday, but going by the parking lot a fair crowd was there nonetheless. Even as we sat in the truck, staring through the windshield, another pickup pulled in. Two Native American men climbed out, headed toward the door. A Toyota Forerunner turned off the highway; when parked, it disgorged a couple of college-age boys dressed cowboy style, and one thin, blonde girl probably not old enough to drink. Then a minivan arrived, and two middle-aged white women got out, settling purse straps over shoulders and moving close together for confidence as they walked toward the entrance. Too old for soccer moms; and anyway, soccer moms would be home with their families at this time of night, instead of at a roadhouse on a Monday night. I'd never played soccer, myself.
Finally I broke the silence. "What do you suppose the staff thought when they found those animals lying all over the floor? Especially the bear."
"Vandals."
"What about a field of crispified roaches and the remains of two ghosts who were possessed by demons?"
"I reckon that likely did not cross their minds," Remi replied. "Whose mind _would_ it cross? Three days ago, I was just a cowboy on a West Texas ranch who likes— _liked_ —to rodeo on weekends."
I stared out the windshield and said nothing. McCue now knew what I'd been and where.
He drew in a deep breath, blew it out on a gust. "How's this going to go?"
I didn't avoid it or beat around the bush. " _Should_ it go? Seriously. What's to keep us from splitting up and leaving? You've got a daughter—go back home to her. Me, I'll just . . . I don't know, have you drop me off at a biker bar. I can hustle pool for some bucks, hook up with someone who'll help me collect my Harley from the campground without alerting Grandaddy or Lily." Well, I hoped.
McCue didn't look at me, just stared out the windshield with both hands hooked loosely over the top of the steering wheel. The big neon Zoo Club sign illuminated him in profile. Cheekbone, the shine of an eye. Gaze fixed on the building. A glint of silver from his ring.
It made me look down at my own hand, see an identical ring on my finger. What the hell were we doing?
"I remember," he said, "when you tried to walk out the other night, and Grandaddy sat you back down."
I kept my tone level. "Grandaddy's not here."
"Yeah, well, maybe his _angelic powers_ can still reach this far."
I chopped a sharp laugh. "If they can, why doesn't he take care of this domicile clearing operation on his own?"
"Because _we're_ supposed to." McCue turned his head, looked at me. "Get it?—this is a test. And I reckon it's as much about taking hard advice as it is about taking out a demon."
I rubbed at my forehead to stall, then met his eyes. "And if it's in a human host?" Remi said nothing, and I knew what he was thinking. " _Yes_ , I killed a guy. I had no choice, not if I was going to save my brother's life. But I don't go around shooting people."
McCue's tone was on the dry side. "Not what I was going to suggest. Was gonna say, what's to stop us from escorting our surrogate outside into the dark and exorcising him? Just like we did with the cop."
Relief washed through me, that he didn't after all consider me eager to shoot humans hosting demons. Exorcism seemed like a very pro-active, healthy, positive approach. "So, do we steal a pitcher of water, throw salt and spit into it, sneak it outside when we do our escorting?"
Remi said, dryly, "I figure a glass might be a little more inconspicuous. There's salt and pepper on the tables, we'll ask for water, and then, well, one of us can spit into it."
It was automatic, as if I were with Matty. " _I_ did it the last time."
McCue's expression was deadpan. "Guess that means I can't do me a little snuff. Who knows if tobacco might mess up the holy water."
I stared at him. "You chew tobacco?"
"No, no, you don't _chew_ what I use," Remi replied. "That'd get you sick right quick. It's powdered, just sits in the mouth between the lower lip and gum . . ." Then he grinned at me. "Greenhorns, I'm tellin' ya. Fall for the stereotype every time. Come on, then. Let's go visit us a bar and see what turns up."
I grabbed the door handle. "And if nothing does?"
"Grandaddy seemed pretty certain-sure about it. I'm betting he knows something."
I elbowed open the door, climbed out, shut it. "So, how's this going to go?"
Remi exited the truck, shut and locked it, then glanced at me across the hood. "I already asked _you_ that. Figure it's up to you to call the shots. You're the alpha, right?" And he strode off.
After a hesitation, I stretched my legs and caught up, heard him humming. Heard _what_ he was humming. "Is that what I think it is?"
He ignored the question and sang a couple of lines.
Yes, it _was_ what I thought it was. "You're not old enough for that song, and neither am I."
He broke off to comment. "But you know what it is, don't you?"
Well. Yes. "Oldies station. Don't judge: sometimes it's all I could get."
He didn't alter his relaxed, loose-limbed stroll across the parking lot. Maybe it's the boots that make cowboys amble so slow and easy, or what _looks_ like slow and easy. "I figure this demon-baiting is something of a gamble, isn't it?"
Okay, I knew what was coming. Or something like it. "Here we go."
And he said, in his slow Texas drawl, "I just found myself in the mood for a little Kenny Rogers. This being a gamble, and all."
I squinted up at the stars as we climbed the steps to the entrance. "Let's just hope we don't do any _folding_ tonight. Of ourselves, I mean."
# CHAPTER SEVENTEEN
It felt odd being back inside a roadhouse that over the past three days had hosted a hot demon chick, two demons doubling as ghosts, a Grigori—oh man, gotta ask Grandaddy about that—a Celtic goddess, stuffed dead animals that had come to life, a possessed cop, and a seraph.
And, well, _us._
The animals were all back in their original spots, including the bear, though I noted if you looked hard enough you could see a broken-out eye. And judging by the customers wandering around the place, there'd been some vacuuming up of demon remains. Which amused me; how ignominious was that? Sucked into a vacuum bag, then thrown out with the trash.
Actually, come to think of it, fitting.
Remi led the way to the same alcove we'd inhabited when Grandaddy laid down the law about the end of the world and what we were and what it was we were intended to do. Both of us automatically spun the chairs so the backs were angled against the wall and we could watch the customers as well as one another.
We'd managed quick showers and fresh clothing before heading out. Once again, I wore biker leathers and boots, and McCue his cowboy duds from head to toe. And again I stuck out, though the clientele did not appear to be as full-bore cowboy as Friday and Saturday nights. Naturally Remi fit right in. A pretty waitress came by, gave us menus, took a drink order—water included—as McCue picked up the salt shaker and gave it an appraising once-over. Then he studied the ring on his forefinger.
"Looks a little weird, huh?" I asked. "Two guys wearing matching rings."
He got it after a moment of blankness. His smile was crooked. "I suspect no one'll think we're married. We're too similar. Brothers, maybe, and a family crest."
I didn't think we were _that_ similar, though yes, one might assume we were relatives of some kind. Same dark hair, though mine was past my shoulders and his was cut short; bone structure and overall build that echoed one another despite slight height and weight differences; but he was tanned while I still bore the jailhouse pallor. Plus our eyes did not match, with Remi's a clear blue, mine mid-brown.
"Genetically speaking," I said, "I'm not sure it can happen."
He looked at me, frowning faintly. "What can't happen?"
"Blue eyes. Brown eyes. Siblings."
McCue smiled. "Missed out on Mendelian genetics, did you? Yeah, it can happen. Has to do with dominant and recessive genes. But anyway, I don't imagine it applies to us, bein' as how we're not, you know, _human_."
It kinda creeped me out, thinking about that again. I mean, I'd heard and seen much over the last couple of days that persuaded me the world was not, after all, completely as I had viewed it, but _born of heavenly essence_?
The waitress stopped by and set down a pitcher of beer and two mugs along with a glass of water for each of us. Remi actually tipped his hat to her in thanks. She smiled, asked if we'd had a chance to look at the menus.
No, we had not, but pretty much every restaurant served burgers and fries, and both of us asked for that. McCue very clearly specified mustard and waited for her to confirm she would bring it.
As she collected the menus and walked away, I asked, "You got a thing for mustard?"
Remi drank down a third of his beer, then nodded at me. "You ever notice how everyone brings ketchup, or it's sitting on the table already, but you have to _ask_ for mustard?"
I blinked at him. "That would be a 'no.' I have not spent any time at all noticing things about mustard."
"Well, pay attention," McCue directed. "Happens all the time. And then usually the server forgets to bring the mustard, and your burger goes cold while you wait for it. Or you have to ask for it again."
I mulled that over a moment. "I never really put much thought into it."
Remi said, with deep gravity, "Mustard is the forgotten condiment."
I lifted my mug, contemplated him over the rim a moment. "We have to carry salt around with us these days. Should we add packets of mustard to our supplies?"
He smiled his slow smile—I suspected many ladies fell for it—then picked up the salt shaker and tipped it over the surface of his water glass, tapped the butt end with the other hand. White granules poured out and clouded the water.
I grinned. "Don't forget to spit."
"You know," he said, "you'd think blood would be better. I mean, there's iron in blood. More protection for us, more threat to surrogates."
"Well, you can't really go around cutting into your hand or arm to spill blood into your water glass without alarming people, despite what they do on TV," I explained. "And anyway, the hand has more nerves and pain receptors that many other areas on the human body."
"So spit it is," Remi murmured, and lifted the salted water glass to his mouth to attempt the business at hand without grossing anyone out.
I drank beer as he set down the prepared glass and pushed it up against the wall for protection. It wasn't much to look at, but the pitcher of salt and saliva I'd poured over all those cockroaches exiting the cop's body had sure done a number on them. I wasn't sure I'd ever forget the sound of carapaces popping, and the stench.
"So Lucifer has a thing for cockroaches," I mused.
"Makes sense," McCue said. "I think they're something like three hundred million years old."
"And I'll bet they were disgusting even then." I paused. "Do we even know how old Lucifer is?"
"Depends on who you ask."
"If he's really old, maybe he'll tire easily. I mean, maybe he has a bad prostate by now, has to piss sixteen times a night. That would interfere with ending the world."
McCue eyed me over the rim of his mug. "You're talking about an archangel. _Do_ they piss?"
"I'll ask Grandaddy next time we see him."
The waitress arrived then bearing plates of burgers, fries, and a bottle of ketchup. She set everything down with a flourish, then asked if she could get us anything else.
Remi said very politely, "Mustard, please." And shot me a look as she hurried off.
"The forgotten condiment," I noted. "It's very sad. Adding it to the armory might not be such a bad idea. Along with Roach Motels." I blopped ketchup onto my burger along with salt, rearranged ingredients and stuck the bun on top, then poured ketchup over the fries. "Do you think Lily will be open to that?"
Remi, still waiting for his mustard, watched me bite into the burger. "I'm not sure we could convince her or Grandaddy that mustard is a ward against evil, but we can give it a try." He waited a moment, then asked, "You sensing anything? And by that, I mean _sensing_."
I swallowed burger, washed it down with beer. "I know what you mean. And no, but then I haven't really tried. And back at you: you sensing any demons around?"
After a moment he said, "Maybe we should both do our respective things and see what turns up."
I paused before taking another bite of burger. "Can we finish eating first? I mean, Grandaddy didn't prohibit us from eating. In fact, he said we could even shoot pool. So let's eat, finish our beers, then do our respective things."
"I can't," Remi said. "I don't have any mustard."
I stared at him over the burger clenched in my hands, then carefully set it down. I pushed my chair back, rose, glanced around, spotted the bottle, took two long steps to another table and liberated the bottle of mustard from the inhabitants. "Sorry," I said. "Popeye needs his spinach."
Of course by the time I turned back to our table the server had brought the bottle Remi requested and now we had two.
McCue was squirting mustard onto his hamburger. "You should give that back to those folk. It's the polite thing to do."
"I'm keeping it," I told him. "I'll put it in one of the cabinets with the guns, holy oil, holy water, knives, ammo, even the stakes meant to kill vampires."
McCue bestowed a lazy smile. "That'll do."
I went back to my fries, burger, and beer.
* * *
—
Once done eating, with plates collected and mugs refilled, we stared at one another. McCue said, "Do you sense anything?"
"I haven't started yet. You?"
"Me neither."
I thought it over. "Maybe we ought to walk through the crowd, see what we feel. Meet back here."
Remi pushed away from the table, grabbed the homemade holy water, headed around the busy dance floor. No live music tonight, thank God, just a juke box. I could hear myself think.
I wandered back toward the pool tables, remembering the blonde demon chick. It was not an entirely welcome recollection, since she'd come close to doing both a hand-job _and_ trying to kill me almost simultaneously. Talk about mixed messages.
I hung out by the tables, trying to sense something. Anything. And for the first time since being released from prison, all of ten days ago, I felt vastly uncomfortable having people behind me. The hairs twitched on the back of my neck and my shoulders itched.
I reached under my jacket to the butt of the gun, then realized I couldn't exactly pull it out if the demon showed. Or a knife, either. Arizona was open carry, but I assumed drawing a gun in a dancehall full of people would not be well-received. It might even get me shot, if there were any wannabe heroes.
Remi was right: we needed to grab the host body and take him—or her—outside into the dark. Hell, if you had to say _surrogate_ instead of _demon_ in a room full of people who might overhear, you couldn't exactly shoot or knife the host in the middle of a bar.
But I had an idea a demon might not feel the same reluctance. I didn't know what one could do to us. The woman had tried to gut, then strangle me. So maybe demons had to use human means to kill us.
I went to a wall and placed my back against it to ease the itch, then unfocused from the surroundings and tried to feel out the heart of the place. But I heard the clack of balls, the jukebox, laughter and loud voices, which made it difficult to concentrate. When Grandaddy had asked me to sense stuff, it had been outdoors in quiet places, such as on the mountain looming over the roadhouse, and when I walked through this place the night the Grigori showed up.
I wondered if there would be bleed-over, if I did get a sense of demonic presence. Two demon-inhabited ghosts, and a cop. Could I cancel those out and track down the new threat?
I went inside myself. Eased my breathing, tried to relax my whole body. Slowly let the surrounding environment fade out.
"Hey dude, you okay?"
Vision focused again. Standing before me was one of the college-aged boys I'd seen in the parking lot. Red hair, blue eyes, freckles, boots, and a big buckle. He looked honestly concerned.
"What?" I asked.
"You okay? I mean, you look like you're about to blow chunks."
Well, that was an illuminating image. And then it crossed my mind to wonder if this kid might be whoever—or _what_ ever—we were supposed to 'clear' tonight.
And could Remi sense him, if he was? That would be awesome. But a quick darting glance told me McCue wasn't handy, nor the glass of holy water. Maybe I could use the Vulcan nerve pinch or something, get him outside, and do the exorcism.
I decided to go for it, hoping I might see something in his face, his eyes. "You a surrogate?"
He was baffled; but then a demon would know how to use the host's expressions and movements. "A—what?"
"You here because it's a domicile?"
He blinked. "I'm here because it's a bar."
I scowled at him. This was going nowhere fast. "Don't you have school tomorrow?"
His brows knit. "It's summer."
Well, there's that. "Okay, then don't you have to work tomorrow?"
His eyes were wide and concerned, but not necessarily with me. He took one step backward. "Dude, you're not my father, you know? And I just meant you looked like you were going to puke, so I was going to get you a chair. 'Scuse me for caring."
A demon could be polite. The cop had been. They wouldn't necessarily _look_ like they served Lucifer.
I remembered how the eyes of the blonde demon had shifted, with pupils elongating into something resembling a cat. I was tempted to ask the kid if he could do that, but it would undoubtedly sound completely off the wall and thus draw even more attention. And what we didn't need was attention once we found the surrogate.
So, to put the kid's mind at ease, I told him I was not about to blow chunks so his shoes were safe, and thanked him for thinking to get me a chair. He nodded a little, then wandered off. I wandered off, too, looking for Remi.
He found me before I found him. He was still carrying around the full glass of cloudy holy water. "Any luck?"
He shook his head. "I'm smelling cologne and perfume, booze, and I know what everyone at the bar is drinking, and that apparently the steak here is mighty fine, but nothing more than that. You?"
"That I look like I'm going to blow chunks when I'm trying to sense something."
His eyebrows shot up. "You what?"
An idea occurred. "Come with me."
"Where?"
"Just follow me."
He obliged, and I led him to the pool tables. I glanced around, then told him to lean his back against the wall nearby and close his eyes.
"What the hell—?"
"Just do it. I want to try something."
Remi rolled his eyes after squinting wariness at me, then leaned against the wall.
"Take your hat off," I directed. "And mess up your hair."
"Do _what_?"
"Take your hat off, mess up your hair, then lean against the wall and look like you're going to hurl. Or pass out."
"Look—"
"Just _do_ it," I said urgently. "I want to try something."
McCue drew in a very long breath, let it go slowly, took off his hat and set it on the nearest table.
I looked at it, then back at him. "It's upside down."
He ran a hair through his hair, ruffling it. "Yup."
"Why is your hat upside down? Is it a cowboy thing?"
"If you set your hat down flat, you can warp the brim. You know these hats are steamed into a specific shape. So if there's nothing to hang 'em on, you rest them on the table upside down."
"I did not know that," I replied. "Are you telling me cowboys in the Old West set their hats upside down?"
"No. Hell, them boys used to tote water in 'em sometimes. That's where the term _ten-gallon hat_ came from, for a hat with a tall crown."
"So it _is_ a cowboy thing."
"It's a cowboy thing." He paused. "My hair good enough for you?"
I gave it a look-see. He had a cowlick that stuck up in back, now that he'd mussed his hair. "You'll do. Now lean against the wall and look like you're sick."
"You gonna tell me what this is about?"
"Look, there's a guy—probably a college kid—who walked up to me, said he was checking on me because, according to him, I looked like I was about to 'blow chunks,' and he wondered if I wanted a chair."
McCue considered that a moment. "Well, that was downright neighborly," he observed, no more taken with this explanation than others, "but it could just have been a nice kid looking after a stranger. We do that in Texas."
"It could," I conceded, "in which case you'll be able to tell. But I've still got a hunch. Since you're supposed to _sense_ things in people, I thought I'd send him over here."
"Why don't I just walk over by him?"
"Because he noticed me."
"You're a long-haired, leather-clad biker in a cowboy dancehall," he pointed out. "Of course he noticed you."
"What if it was more than that?"
"You mean, that he's the surrogate?"
"As you would say, he _'might could'_ be."
Remi shifted against the wall. "Okay. Let's get her done."
The kid was shooting pool against his friend, whom I looked over, too. Watching them was the thin blonde girl who I believed was too young to drink, but she had a beer in her hands, probably bought by one of the boys. None of them was looking at me. None of them was looking at Remi.
I dug a ten dollar bill out of a pocket, walked slowly up to the table and laid it out on the top rail. "I'll play winner."
Both boys paused and looked at me in surprise. The red-head recognized me; the other, a sandy-haired boy, did not, but now he paid attention. "We're not playing for money."
"Why not?"
They exchanged a glance. "It's just for fun," the red-head told me. "We play for who buys the pitcher of beer."
"I lose, I'll buy you two pitchers."
They were clearly uneasy. The blonde girl came up, gave me a big smile. "I'll buy you a pitcher."
"I don't think you can buy _you_ a pitcher," I said, and saw her face fall. "So you boys go ahead and finish your game, and I'll play winner."
The sandy-haired kid was so nervous he scratched on his next turn. Well, that worked for me. "You rack," I told the red-head.
He nodded, began pulling billiard balls from the pockets and collecting them at the foot rail to move them into a rough triangle with his forearms. Then the rack went down.
I walked over to him, like I was inspecting his rack as he rolled the balls inside it. "Hey," I said. "That guy look sick to you?"
He brought his head up. "Who—?" And then he followed my line of sight.
I'll admit it, Remi was doing a good job. He mostly leaned against his left shoulder, hunched a little, the water glass dangling from the rim in his right hand.
"Bad food?" I asked. "Or too much booze." I clapped the kid on the shoulder. "Look, you were thoughtful enough to check on me . . . how 'bout you see if he's okay. I'll just wait here."
The kid nodded, went unerringly to McCue. I couldn't hear the conversation over the sound of balls clacking and voices, plus now someone was speaking into a microphone in the main room, but I did see Remi nod at the kid, wipe a hand across his face as he straightened up, grabbed his hat, and wandered off just a shade wobbly on his boots.
The kid came back. "He said he had two shots of tequila too fast. He's gonna go outside, get some air."
McCue had headed toward the main room and the entrance, which wasn't the plan. It was the _back_ door we wanted to use if we came across the surrogate. And I needed to know what he thought.
I looked beyond the kid, gave a low whistle. "Shit," I said, "I just saw one hot chick . . . dude, gotta check that out." I scooped up the ten dollar bill. "You mind?"
He'd barely said no before I headed off to work my way through the crowd. No more dancing. A mic was in the center of the parquet dance floor, and a couple of guitars were on stands close by. I crossed in front of the guy talking into the mic, but I wasn't paying any attention to what he said.
I found Remi waiting back at our table. I dropped into the chair, raised brows in a question.
"I got nothin'," Remi said. "He could be Lucifer himself, for all I know, but I didn't get any vibe off him. Offered to get me a chair."
"I wonder if that's code."
McCue stared at me. "Code for what? That he's a good kid? Some just are, you know. _I_ was a good kid."
I sighed, turned back to gaze across the main room. The bar stools were full, and everyone had swiveled to face the dance floor. The man behind the mic announced they were ready, and called a woman forward.
It became abundantly clear, as a woman walked out of the crowd and stood behind the mic—I was pretty sure it was one of the middle-aged non-soccer moms—what was going on.
Remi read my stricken expression. "You didn't see the sign? Open Mic Night."
"In a _cowboy_ bar!"
"Yeah. So?"
"They'll be singing country music!"
He gave me a long, assessing examination. "Probably," he agreed. Clearly, he thought I was pretty stupid.
"And butchering it." I paused. "Not that country music isn't already pretty butchered."
He set down the glass and drank beer, which now was room temperature. "So, what do we do now?"
"I don't know. Leave?"
"You want to risk Grandaddy's heavenly wrath?"
I turned my mug around and around on the table, watching it glumly.
The woman did indeed sing a country song, something about trashing a cheating boyfriend's pretty little souped-up 4-wheel-drive, which I thought was a weird way to describe a pickup. Pretty? (Of _course_ it was a song about a truck.) The crowd rather raucously joined in on the chorus. I debated stuffing fingers into my ears, but decided I was badly outnumbered by country fans, and I'd do better to keep my mouth shut and mind my manners.
Remi abruptly sat bolt upright in his chair. "Listen, all we want to do inside is ID the surrogate, right? Then get him—or her— _out_ side where we can do the deed in private."
"Well, yeah. That was the plan."
"So you get up—take the water with you—and wander through the crowd. Look for someone starting to choke, like maybe cockroaches are going to start spilling out of him."
"I thought they had to be dead for that to happen. Kind of obvious, don't you think? And people do cough just to cough."
He ignored me. "Make your way to him, threaten to spill the holy water over him, and escort him toward the back door. Can you whistle? I mean, like hailing a cab?"
"Yeah."
"So start checking people out."
Still struck me as lame. "What are _you_ going to do?"
"I got an idea. Just look hard for someone acting funny. Don't pay attention to me."
And McCue walked away, making a beeline toward the man who had called the woman to the mic.
Oh, shit. I was pretty sure I knew what was coming.
Sure enough, Remi stepped up to the mic. He nodded his head around at the crowd as if thanking them in advance. The applause was tepid at best; likely the singers were regulars, and McCue definitely wasn't.
As he introduced himself, his voice was diffident, though the drawl was a little stronger. "I am a stranger in town, out of West Texas and headin' down to Phoenix. I hope ya'll don't mind if I sing somethin' for you. It was my nana's favorite song. Probably some of you know it, too, if you're church-going folk. Christmas services, at least. I know it's not what you usually hear performed in a roadhouse, but today would have been her ninety-fourth birthday. I hope ya'll will give me a listen."
Church-going folk?
"Schubert's 'Ave Maria,'" Remi announced earnestly, as he removed his hat. "Give glory to God." Then he drew in a deep breath, smiled shakily, nodded nervously—all an act—and opened his mouth.
Yeah, I'd heard him sing before. Just snatches, but I knew he had a pretty good voice. This was a little more than pretty good. But why now? Why that?
_Oh._
Tune the same, still in Latin, but different lyrics. The guy was _singing_ the exorcism rite.
Clever bastard.
But I didn't listen for more, because I was on the move with a glass of makeshift holy water in my hand.
# CHAPTER EIGHTEEN
It's not easy looking for someone beginning to vomit cockroaches in the middle of a roadhouse. I mean, it would be pretty damn obvious, but I assumed there was a step before the roaches actually put in an appearance. Remi'd specified someone just beginning to choke. And that's even tougher, because any number of people might start coughing or choking for all kinds of reasons. People just do that sometimes, without being demons.
So I threaded my way through tables, walked the length of the bar, paused now and then to take a closer look around.
Pretty much everyone's attention was on Remi, except for those shooting pool. Because patrons were fixed on the performance, I could look into faces more easily. "Ave Maria" is sung in Latin, which is probably why Remi picked it, but I did notice a few people exchanging puzzled glances.
Well, it's true that the exorcism rite sounds nothing at all like the song celebrating the Lord's mother.
I saw nothing in the main room, heard nothing suspicious. Then I headed back toward the pool tables, where I found that someone _was_ coughing. It was one of the college kids, the sandy-haired guy, not the red-head who'd come to check first on me, then Remi.
He paused, thumped himself on the chest and said something about beer going down the wrong way, began coughing again. The red-head looked suitably concerned, and the blonde girl was not present.
Okay, Remi had not covered what to do if the surrogate had _friends_. Maybe all of them were demons. Grandaddy had not specified number. I'd ask about that next time, if we survived _this_ time.
I made my way to the table, put down the same ten dollar bill, called the next game—then with feigned concern looked at the kid who was coughing. "Hey, dude, you gonna be okay?"
It came out hoarse, and oddly suppressed. "Yeah. Fine." Then he clamped a hand over his mouth like he was about to puke.
I rounded the table, stepped close. "You need some help?"
Now he shook his head.
I closed a hand around his upper arm, as if I meant to help steady him. Then I leaned close and said, very quietly, "There's holy water in this glass. I'm suggesting we step outside so no one gets grossed out by the sudden appearance of a river of roaches spilling from various orifices, and have a conversation."
Then I whistled, piercing and clear.
In the kid's eyes was speculation, as if he were evaluating me and what I might do, or maybe daring me. Then, clearing his throat, he looked at the red-head, hovering to make sure his friend was okay. "Hey Rick, I'm going to go outside for a minute, get some fresh air. You go ahead and rerack 'em, okay?"
After a moment of doubt, Rick apparently saw something that put him at ease. He shrugged, nodded, bent over the table with his back to us. Remi arrived—I heard someone else singing in the background. He paused briefly after giving the surrogate a hard look, then walked right on by and out the back door so it wouldn't look suspicious with _two_ of us escorting the kid out. I clamped down a little harder on his arm and guided him toward the door. If he tried anything at all, I'd dump the holy water over his head, get him outside, and let Remi turn him into really ugly bugs.
The kid had stopped coughing, maybe because Remi wasn't singing the rite anymore. He'd have to start it over once we were outside, though probably without the 'Ave Maria' performance.
As we stepped out the back door, I pressed the glass firmly against the surrogate's back. A little water slopped out as we navigated the big step down, splashed the back of his shirt and soaked through. He hissed in pain, and the fabric began smoking.
Yup, we had the right guy.
Remi was standing beneath the big overhead bug light, which painted him slightly jaundiced. His face was tense, eyes very serious. "You took this host without permission. Now it's time to leave."
And he opened his mouth and began the Latin.
Nothing happened. The kid didn't cough, didn't choke. He just stood there in the spill of illumination, and smiled at Remi. "I hate to break it to you rookies, but I'm not a surrogate. I faked it."
Didn't believe him for a minute. "Uh-huh. And when the holy water hit you, it hurt," I pointed out, "and your shirt smoked and now has a hole in it." I stuck a finger through that hole, poked the surrogate's back. "See there?"
And out of the darkness came the blonde girl. "Angels can manipulate such things when it's needful. And we judged it was."
I stared at her. Shit. "So, _not_ college kids, or a girl too young to be drinking."
Remi took an assessing look at her. "Well, aren't you half as big as a minute."
"I'm Candy," she continued; perhaps she didn't speak Texan, "and I'm older than dirt." She indicated the sandy-haired kid. "That's Dick, and that—" she gestured with her chin, "—is Rick." The red-head opened the screen door and exited.
My brows shot up. "Rick, Dick, and Candy?" I snickered. "What, you double as porn stars?"
"Well, our angelic names are difficult to pronounce." Red-headed Rick was matter-of-fact. "So we'd be explaining all the time to humans how we came by them, which is inconvenient. Easier this way."
Remi and I stared at one another, trying to figure out what the hell we were supposed to do now. I raised my brows in a question, and he shrugged back. Dick—the sandy-haired guy—removed his arm from my loosened grip. He smiled at Remi. "Singing the rite was genius. I admit, we weren't expecting that."
"This was a _test_ ," McCue said, sounding tense.
Candy replied, "We cleared the domicile earlier, but you still should have _sensed_ something. Surrogates leave an essence for a while after they die."
Now Rick. None of them sounded like kids anymore. "Remiel, you were intended to find us by sensing our presence, to recognize all of us as angels, not surrogates, the way you did up on the mountain with the old man. You didn't."
Dick said, "And you, Gabriel, should have been able to sense that the roadhouse had been cleared, was no longer a domicile. You didn't."
It pinched; in fact, it reminded me of when Grandaddy was disappointed in me, but I didn't let it show. Just looked at Remi. "Guess we flunked."
He was frowning a little, attention distant, clearly thinking something over. Finally he refocused and said, " _Prove_ you're angels. Isn't Lucifer the Great Deceiver? You could well be surrogates just playing us. Or playing _with_ us."
I remember what Grandaddy had done up on the mountain. His wings had never been corporeal, nor did they exactly look feathered. All we'd seen was an impression in the air, a pale, pixelated image rising from his shoulder blades.
"Show us your wings," I challenged. "All three of you."
"No." Candy looked straight at Remi. "What do you _sense_?"
I heard the frustration in his tone. "You already said I failed in there. I don't sense anything yet, other than occasional hunches. And none of them have ever been about surrogates _or_ angels. I don't have visions, I can't get any kind of read on people other than the normal kind of stuff, like expressions, tone, body language. Just like everyone else."
She tilted her head a little. "Try again. Try _now_."
Apparently he did, and apparently this time he truly did sense something. I saw the stunned expression in his eyes, on his face as his mouth opened. He took two steps backward, then stopped himself. "Holy crap . . ."
Rick was amused. "Holy, yes, but not even remotely connected to crap, thank you very much."
"What did you feel?" I asked McCue sharply, because he looked like he might pass out.
He tried to speak, failed, closed his mouth, then tried again. "It's—impossible to explain. There just aren't any words."
"Not in human language, no," Candy agreed. "But in ours, yes. And it will become yours, too." She paused, engaged Remi's eyes. "In which language was the Bible written?"
He shook his head. "It wasn't a single language. Plus there's a difference between Old and New Testaments. Old Testament was primarily archaic Hebrew, though Ezra and Daniel wrote parts in old Aramaic. New Testament was Koine Greek."
Yup, handy to have a Biblical scholar on the team.
"Do you speak it?" she asked.
Remi made the _maybe/maybe just a little_ gesture with one flattened hand, tipping it back and forth. "I looked up some translations for my doctorate."
_"Eno no qyomto w hayo."_
He smiled faintly. "Gave me an easy one, huh? _'I am the resurrection and the life.'_ "
Rick said, "Learn it." He looked at me. "And you need to learn Latin. Aramaic can come later. In the meantime, Remiel will be able to translate for you. You'll have dictionaries and phrase books."
I quirked a brow at him. "What's wrong with English?"
"There are texts and concepts that can't be encompassed by English," he explained. "Surrogates read and understand English, plus their own language. They don't know Aramaic."
I was skeptical. "Wouldn't Lucifer? I mean, he was an archangel."
Rick shook his head. "Archangels speak an older form. No surrogate can possibly understand it."
"No grade school for demons?" I asked.
"Their brains are not wired to understand or to learn it."
I remained skeptical. "But humans are?"
Simultaneously, all three declared with impatient vehemence, " _You're not human_."
Oh yeah. Forgot about that part. "Okay, Latin's first on my to-do list. I'll learn Aramaic from Remi."
His tone was dry. " _I_ have to learn it first."
I looked at the angels. "Can't you—I don't know—download Aramaic into his brain, or something?"
"We could," Candy replied, "but it would kill him."
"Pass," Remi said, predictably.
I sighed. "Now what?"
Dick said, "You will go back inside. Remiel will choose a person and try to pick up something of his or her feelings, while you, Gabriel, should concentrate on getting a sense of what was the domicile. You will need to know it."
I frowned. "I thought you said you cleared it out."
"You need to _see_ it," he said. "Go behind the facade. You need to feel it, bone-deep. Every time you go someplace new, Remi will try to track surrogates, while you will sort out whether it's a domicile, or simply a place."
Candy said, with deadpan humor, "There is much for you to learn, but learn it you will."
I was incredulous. "A Yoda joke from an _angel?_ "
"Of course," she replied brightly. "We get every first-run movie in heaven before anyone else does. Perks of the job."
"Huh." I thought about it. "That would be cool."
"Too," she elaborated, "it's a good way for angels intended to take a human host to learn something about humans first."
Remi and I looked at one another, brows raised. McCue said, "Ya'll do know movies aren't real, right?"
"But they provide insights nonetheless," she explained. "Now, go inside. Do your thing."
"Wait," I said sharply. "You're possessing the hosts, right? Yet Grandaddy said you didn't."
Rick said, "We borrow, from time to time. But where demons eat out the brain over time, we leave the hosts intact—and healthier than before—when we depart."
Eat out the brain? Holy Jesus.
# CHAPTER NINETEEN
As commanded, Remi and I went back inside. Our table in the meantime had been appropriated by a couple helping themselves to what was left of our pitcher of beer.
Rude, dude. But I had a mission, and it didn't include arguing over booze and a table. At least, not this time.
McCue gave me a pointed sidelong look and a slight tilt of his hat—well, head—which I took to mean he was going to park himself close by and try to get a read on either the guy or the girl.
I went to the bar and planted my ass on a stool. Ordered a beer, hunched over it slightly upon its arrival, registered that the bartender was black and big, but didn't pay more attention than that. Mirrors along the bar back provided a view of the room behind me.
Unfortunately, Open Mic Night continued. As I sat in silence nursing my beer, on the verge of trying to _go behind the façade_ , whatever the hell that meant, I heard portions of various songs, all country. Some singers butchered them, which made me wince and want to plug my ears, but I figured that was not a good idea in a place full of locals and regulars; others were passable, despite their poor choice of genre; and a couple were actually pretty good. One chick sang about a chick named Jolene begging someone not to take her man, a guy performed a raucous number about friends in low places, and two rough-voiced men sang a duet about mama not letting her babies grow up to be cowboys.
I wondered if Remi knew about that one. Probably he wouldn't like it. His mama _had_ let him grow up to be a cowboy.
Okay. Time to try my thing.
It had been easy on the mountain to reach out, to let a sense of the place come to me. As then, I saw brief flashes of color around the edges of my vision, but this time red, which symbolizes passion, blood, and war. Not happy about that, though it didn't surprise me.
The red was superseded by something that felt—well . . . green. I sensed a flow of health, renewal, vigor, which made sense for a place cleared by angels. But the feelings were fleeting, leaving me trying harder to get a sense of the bar. Before, alone in the building, I had "remembered" feelings I'd never experienced. It was the _place_ that exuded memories, but only in snatches. There was nothing I could latch onto, because the moment I tried, blankness washed over any sense of color, of place, of memory. The roadhouse felt weirdly empty despite the customers.
That is, until someone started caterwauling the high-pitched chorus of The Police's "Roxanne," which was considerably better than a country song, but the screech that came out of the guy's mouth was ear-shattering.
I winced in involuntary reaction, noticed the bartender was smiling as he came toward me. Now I paid attention. Big black guy, really big, probably six-four and maybe two hundred forty, fifty pounds. His head was clean-shaven, his voice deep. A black t-shirt was stretched over the bulk of muscular chest, shoulders, and arms. Looked like he could be a former NFL player.
"Yes, he is not one of our better singers, but he enjoys it and is a good man, so he is forgiven." The bartender had an accent, one with which I was not familiar. But he didn't sound disapproving, nor did his expression show any kind of hostility. "May I present another beer?"
Several nights before, a demon in the guise of a hot chick had tried to kill me. A Grigori showed up to warn me the war might relegate humans to collateral damage. A surrogate inhabited a cop. And tonight angels had actually chastised us. All four experiences made me wary of people in general, especially in the roadhouse. The back of my neck twitched again.
And asked if he could _present_ another beer. Not normal phraseology for a bar. I shot him a searching appraisal. Angel? Demon? Grigori? Nephilim? A god? Just a human pouring booze?
He noted my assessment. "It was only a question," he said mildly. "There is no need for you to be biker threatening."
Biker threatening? Me? To him?
But if anyone else took it that way . . . well, I was in a bar full of locals and cowboys. "No," I said hastily, "no, man, sorry. Just—thinking about something. Bad memory." I placed a ten on the bar as tip, hoping to soothe him, slipped off the stool and headed over to where Remi ought to be.
He saw me coming, walked to meet me just off the dance floor. "You get anything?"
"Green." His brows inched up. "I think it means this is a safe place, now. Cleared of threat, or surrogate presence. But it wasn't clear, wasn't like the sense of a place I usually get." I checked his expression then smiled benignly. "Too much country music."
He eyed me. "You gotta get over that prejudice, son."
"Why?"
"It's just not a healthy thing."
I ignored that. " _You_ get anything?"
"Yup." He was solemn. "The guy is thinking about bed sports later, and the girl is thinking about a favorite TV show she's missing."
"Really?"
"No."
I gave him the side-eye. "Seriously, did you sense anything?"
"Caught traces here and there, but not anything like the angels want. I think what we're doing is boot camp."
Interesting. "So none of this counts?"
"Oh, I reckon _all_ of it counts. And I also expect we could be killed if we take a major misstep; there's no urgency, no incentive otherwise. I caught a good lick when that ghost threw me into a table."
I was visited by the image of Remi being actually, actively licked, which left me grossed out, and I attempted to mentally bleach my brain immediately. "So Grandaddy basically started grooming us for this job back when we were little kids. Now we're being challenged to show what we know."
Remi nodded. "To show what we know _and_ whether we can do the job without getting ourselves killed. It's a GRE test for grad school. We get in, or we don't."
It was a startling concept. "You think they'll _kill_ us if we don't pass?"
Remi shook his head. "I reckon they won't have to. I _reckon_ a surrogate will take care of that for them. At any rate, how 'bout we go back to Lily's rig? We could—"
And he stopped dead, looked startled, a little tense; but a moment later it was like his hackles went down. He smiled, eyes going warm, as he looked beyond me. I turned, and there was Grandaddy.
I was a little uneasy about having him at my back. The last time we'd been together, he'd announced that it was his doing I'd gone to prison.
Remi said, damn near breathless, "So _that's_ what it is!" At my frown of incomprehension, he explained. "I felt him. Almost like a vibration in the air. But—it's not like it was outside. It's—" he groped for words, "—just _bigger_." He looked at Grandaddy. "I never felt it before, when you came a'visiting."
Grandaddy nodded. "We angels can shield ourselves. I dropped my shield just now so that nothing blocked you."
I stared at him. "What, like a Romulan cloaking device?"
He ignored me. "Remi, did you sense anything of the angels here earlier?"
"No, sir. At least, not until the girl did the shield thing."
Grandaddy nodded. "At some point it won't matter; you'll be able to tell when an angel is in the building."
Said I, "Well, _I_ thought Dick—or Rick?—was a surrogate. But I didn't feel anything. I was just—suspicious."
Grandaddy declared, "Suspicion, when your gifts are yet too young to function properly, is healthy, and a sound safeguard."
"When _will_ our gifts start functioning?"
"When they do."
Well, that was helpful.
Grandaddy smiled at my annoyance. "Now, you boys come with me. There is someone you should meet."
Remi and I followed him across the main room, threading our way through tables, and the someone turned out to be the big bartender. He saw us coming and I caught a glimpse of teeth as he presented Grandaddy with a smile.
"Is it time?" he asked.
"It's time."
He nodded. "Drinks?"
"Oh, I think it best," Grandaddy agreed. "Your best single malt, and best tequila; I'll have a Guinness."
I studied the bartender. "That was a joke, then, wasn't it? The crack about me being 'biker threatening.'"
He smiled again as he poured two fingers' worth of single malt, slid the tumbler across the bar toward me. "Not precisely so. I imagine others would indeed find you 'biker threatening,' on occasion. But you are no threat to me."
His accent was fascinating, and the clarity of his speech in a resonant voice. "I imagine not," I agreed. "I don't imagine _anyone_ is a threat to you."
He poured tequila for Remi, then the dark beer for Grandaddy. "That is true."
I looked at Grandaddy. "How do you drink that shi—stuff. It's like motor oil. Used."
He smiled benignly and ignored me. "Now, raise your glasses and toast this man, for he is like none you have ever met."
"Angel?" Remi asked.
"No, indeed," the bartender said. "And I thank you for your courtesy."
Grandaddy, McCue and I clinked glasses, then we each took a swallow. The Scotch went down smooth. Our grandfather—or whatever the hell he was—said, "He's an Orisha."
The big guy nodded. "I am Aganju. Our pantheon is ancient." His dark eyes slid in Remi's direction. "But even you, with your doctorate, won't know me, or the others. We are forgotten."
McCue looked right back at him. "You said pantheon. The word refers to a collection of gods."
"And so it does." The big man smiled. "Orishas are of Africa."
"Huh," I said thoughtfully. "What's an Orisha?"
"A god."
"A _god?_ "
He nodded, going solemn. "I am Aganju, warrior king, and lord of volcanoes and deserts. This place is my responsibility, you see. The volcano behind us, though it sleeps, and all the others; and the desert surrounding us."
"This is a desert?" I asked skeptically. "There are pine trees all over."
Grandaddy said, "It's what known as 'high desert,' here. But go two hours south toward Phoenix, and you will see a true desert."
Remi sounded thoughtful. "The Morrigan and an Orisha."
Aganju nodded. "Many of us are here, those made for war. We have been freed from banishment. Men long ago stopped believing in Orishas. But now they shall worship again."
I remembered Lily's impassioned speech about wanting the same. Apparently this was a regular thing for old gods and goddesses, this needing to be worshipped.
"Finish your drinks," Grandaddy said. "We're going upstairs."
Remi and I exchanged puzzled glances, then knocked back what was left in our glasses, put them on the bar, and followed Grandaddy to a dark area against the wall. There was a stairway, and while not truly hidden, neither was it obvious. Just—out of the way. I guess the one the woman fell down and broke her neck, thus becoming a ghost.
And I guess Aganju's presence explained why there'd been no questions about demon roach remains. But where the heck had he been when that was going down?
Oh yeah. A test. He could have been watching, for all I knew.
The staircase was narrow, with steep risers. As we climbed after Grandaddy, it became clear that Open Mic Night was over. The jukebox was playing again, and a thread of music followed us up.
At the top landing, Grandaddy led us down a hall, indicating various open doorways. "Common room. Kitchen. One bathroom, two bedrooms." He stepped across the threshold into one of the bedrooms, and we followed. Smallish, but with a bed, a dresser, closet, desk. The north and west walls were constructed of large-diameter chinked logs, the others paneled in flat, grainy wood.
"Okaaay." I frowned. "What are we doing here?"
Grandaddy said, "This apartment is where you'll live. You can't stay with Lily, as she has her own missions and cannot always be here. For now, your territory is Arizona, so this will be both headquarters and home. And when you are ready, you will be responsible for all of the Southwest. No rent, no mortgage; free food and drink, if you choose to eat downstairs. TV. Wi-Fi."
It was muffled, but now and then, depending on volume, the jukebox downstairs was audible.
Oh, hell no.
I shook my head. "How about not here? How about a motel, or a rental house? I can live in a motel, no sweat. It's still better than a cell."
"Here," Grandaddy said.
I said with no little urgency, "I'd really rather not."
Remi was frowning at me. "What's wrong with here? Nice bedrooms, a kitchen . . . free room and board."
I glared at him and said with great clarity, "It's a cowboy bar!"
He didn't get it. "Yeah. So?"
In a strangled tone, I said, "They play _country music!_ "
Grandaddy sounded inordinately cheerful. "Every night but Sunday, when the place is closed. Live music Friday and Saturday nights, Open Mic Mondays."
McCue, the bastard, was laughing at me. "Guess you'll be learning plenty of good ol' country songs, then. And maybe a little respect."
"I don't think so." I appealed to Grandaddy. "Please say you're joking."
He was amused but adamant. "Here."
I took two steps to the bed and dropped ass, planted elbows on thighs and leaned forward to grasp my head in disbelief and despair. "Shit. Shitshit _shit_. Just kill me now."
# CHAPTER TWENTY
Grandaddy took his leave. Remi wandered off to tour the place while I sulked on the bed. He came back and said, "There's beer in the fridge." He offered a bottle. "Drown your sorrows."
I took the bottle but didn't crack the top. I just kind of stared at it in a daze. "It's too much."
"What, the beer?"
"No. _Everything._ Angels, gods, that Grigori chick—"
Remi interrupted. "To be accurate, the Grigori are angels, too. So are Nephilim, though admittedly they're also half human."
I began again, this time poking thumb and fingers into the air one by one to enumerate. "Angels. _Other_ angels. Demons. Gods. Goddesses. What the hell else?"
"Creatures out of folklore. History. And, I guess, literature, from what Grandaddy said."
Now I opened the beer and downed half of it in two huge swallows. I looked at McCue in silence, then drank the rest of the beer. "If an African _god_ works here, how come surrogates managed to establish a—what did they call it? Domicile?—without being killed off by this Aganju? I mean, he appears to be working for our side."
Time for more pointed questions. Remi smiled slow as honey. "Well, let's go downstairs and ask him."
I rose, set down the empty bottle. "He is a very large man."
"Big as all hell and half of Texas."
Colorful, but true.
We trooped downstairs into music, laughter, raised voices, the clack of billiard balls, which reminded me of the angelic trio, Rick, Dick, and Candy, who _did_ have names like porn stars no matter what they said.
"How the hell are we supposed to live here?" I complained to Remi, who descended ahead of me. "Country music or no country music, the place is loud. And we're supposed to sleep above all this racket?"
He stepped off the final step and threw a comment over his shoulder. "I reckon you'll get used to it."
"No, I don't think so."
"You got used to prison, didn't you? A man can adjust to anything."
Well, that shut me down. I followed him to the bar in disgruntled silence.
Remi and I found two stools at the very end of the long slab of bar counter and sat down. Aganju was in the middle of a discussion with one of the patrons, an ancient, weathered cowboy who was going on and on about something. Aganju just listened and nodded, spoke a time or two, though I couldn't hear what was said.
"He an angel?" I asked.
"He's a god."
"No, not him. The old man."
"Well, if he is, he's got his cloaking device up and running."
I roll-tapped my fingertips against the bartop repeatedly, impatient. And eventually Aganju came down to us.
Before he could ask what he could get us, like any normal bartender would do who wasn't a god, I wasted no time. "Why was it you didn't clear this place of surrogates?"
"Ah." He nodded, as if he'd expected the question. "That requires angels. It is of opposites, you see. Black and white, good and bad. Angels and demons. I am not of your heaven."
"Well, then why are you here? You said you are an African warrior-king-god."
"And so I am. My day will come, and I shall wade through blood. But it is not the time."
Wading through blood, huh? Not exactly on my bucket list. "So, why are you here rather than in Africa?"
His teeth appeared again. "The war here will be of greater consequence than the petty coups of Africa. And nowhere else boasts six hundred volcanoes."
It was startling. "Six _hundred_?"
In his deep voice, he said, "Now they answer to me."
Holy shit.
Aganju continued. "Yes, I may kill demons. It is only the domiciles that are not in my purview."
I considered him. "Do you, what—smite them?"
He looked a little surprised. "Oh, no. That is for angels. I have a sword, you see. I take heads."
"Ah. Like the Highlander, movie and TV versions? Do you have Quickenings?"
The big man smiled. "I do kill them quickly."
"No, that's—" But I waved it off. "So, where is it?"
"My sword? It is here."
"You mean, beneath the bar? Like a baseball bat or a shotgun?"
Remi shifted on the stool next to me. "Are you done yet with your interrogation? You wanted to know why he didn't clear this place, and he's answered."
But I wasn't satisfied. "Could you have helped us with the ghost-demons? With the demon-cop?"
Aganju was solemn. "These things are for you to do."
That figured.
He added, "I go up upon the mountain when I am not here."
It was McCue's turn to question him, if with a little more decorum than I. "And what do you do up upon the mountain?"
His teeth shone pearlescent against dark skin. "I sing to her, and worship. But she sleeps, so she says nothing. I hope to wake her soon, and the others."
After a moment of arrested silence, Remi got there before I did. He straightened abruptly on his stool, tight as a bowstring. "Hold up. If you mean you want it to come back to life and erupt, you'll kill people, destroy the town." His sweeping gesture encompassed the bar. "These people right here. That old man you were talking to."
Aganju was unperturbed. "Such a thing will destroy demons, stop the Lucifer, and save the world. This is a good thing, is it not? To be victorious?"
McCue just stared at him, apparently struck dumb by the realization that Aganju was a bloodthirsty son of a bitch.
"Victory is good," I agreed after a moment, "but not at the cost of all these people."
"Men die. That is war."
Remi's tone was pointed. "And women and children."
Aganju said, with a weird sort of serenity, "That is the cost of war. We must all bear it."
Neither I nor McCue spoke as the big man went to another customer, but I could sense the cowboy's tension echoing my own. I was reminded yet again of the Asian woman, the Grigori, warning me that not all angels were on the same page, that humans would die.
And this guy, this Aganju, wanted to wake an extinct volcano.
Had Grandaddy brought Aganju and other gods of war to the here and now merely to serve a purpose without regard for the risk to humanity?
That made me uneasy. _Very_ uneasy.
The Grigori had told me, _At this moment you and Remiel are innocents being set up for manipulation._
"They'll do anything," I said, and felt a chill. "Everyone involved in this. Anything at all."
"It's Lucifer himself," Remi said. "It's the End of Days. Yes, I think they'll do anything, because _every_ thing's at stake."
I gave him a sidelong glance. "And us?"
He stared down the length of bar at the African Orisha. "I have a feeling we'll probably do anything the angels tell us to."
Well, hell. "I would suggest, with all gravity, that we are well and truly fucked."
Aganju came back to us, set down a tumbler of scotch for me, tequila for Remi. "On the house. As all drinks and food are, for you. Call me Ganji; it is easier. And we shall be seeing a great deal of one another."
Oh, joy.
"And yes, you are." His gaze was unblinking as he stared at us both. He was certainly the kind of man who'd intimidate others easily—big, black, shaven skull—but that wasn't what he intended at the moment despite his size. He was perfectly matter-of-fact.
I was wary. "We are what?"
"Fucked."
Remi and I waited till Ganji was serving other patrons before we looked at one another. When we did, McCue's expression was a mixture of resolution and concern.
He said, "He's dark as the devil's riding boots—and I ain't talking about the color of his skin."
# CHAPTER TWENTY-ONE
We departed the roadhouse after that, keeping our silence. I'd mentioned to Remi that if we _had_ to stay at the Zoo, we ought to collect our clothing from the motel, and my bike from Lily's rig. So we climbed into his truck and headed north.
"Six hundred volcanoes," I pondered aloud. "I had no idea Arizona was so active back in the day. Or that this area had pine trees, for that matter, instead of big-ass cactus. Still doesn't look like a desert to me, high or low."
He didn't say anything until we turned onto the motel parking lot, and then he sounded subdued. "I don't like it. Not at all. But, like I said, we may have no choice 'cept to do what they tell us to. And what happens down the road? Grandaddy said we weren't angels _yet_. Suggests that if we live long enough, we might could sprout us some wings."
That was not a happy thought. "And we've got no idea how long this war is supposed to last, or what happens if Lucifer _does_ get out of hell."
"I think that's what Lily and Ganji are for," he said, "and any others tied to this mess. Demons are the target now, but if Lucifer goes bustin' out, I reckon the angels and all these gods and goddesses will unite to destroy him."
I mulled that over a moment. "Cannon fodder. That's what we are."
"Canaries in the coal mine."
I muttered a curse, unlatched the truck door and shoved it open. McCue did the same on his side, and together we walked to our respective rooms.
I had very little to collect beyond toiletries. When saddlebags are the only option for carrying stuff, you limit what goes into them. I had a pair of jeans, tactical BDU pants, a few tees, couple of Henley shirts, underwear. My world on a bike.
I'd done eighteen months in prison. Throughout, the bike was parked at my parents' house. When I went to collect it, my father—ex-military, current cop—gave me a piece of his mind about what a big disappointment I was, and did I know how embarrassing it was for him to deal with questions at his station about his imprisoned son?
I'd been a little busy trying to survive inside, so no, I really didn't spend any time thinking about my father's disappointment and embarrassment.
My mother visited a couple of times. He did not.
* * *
—
Remi and I met back at the truck and headed up the road to the RV campground. As we arrived at Lily's rig, I was astonished all over again by how large it was. Multiple _rooms_ that slid out from the chassis, and that garage. I wondered how often anyone was banged up enough to use the cot, the medical equipment in the back end. I wondered, too, how much Lily knew about field medicine. She was in the business of war, not in saving lives.
Lily wasn't there—and then she was. She walked out of the trees with the wolfhound at her side. I heard the crow overhead.
It was dark because the campground was studded with massive pines that blocked much of the moon, but she had exterior lighting on the big motorhome. Her face was shadowed, then it came clear as she walked out of darkness into the sphere of light. I noticed again how the sleeve tattoos seemed to move on her arms.
"Have you seen the apartment?" She climbed the steps and swung open the door. She held it there, then stepped aside so the crow could fly in. The wolfhound followed, and Lily gestured for us to climb in behind her. "We'll hope it brings you better luck than the last two of you."
That grabbed my attention in a hurry. I stopped short just inside the doorway. "Last two? _What_ last two?"
"The two boys before you," Lily replied. "Did you think you were the first, then?"
Remi poked at me to suggest I move, which I did, then he stepped in, removed his hat and hung it over a window valance. "Okay, so we're not the first. What happened to our predecessors?"
Lily's eyes were bright. "They were killed by demons, so no heaven for them. Now, shall I pour you some Irish whiskey?"
At the roadhouse, we'd consumed most of a pitcher of beer and a couple of tumblers apiece of tequila and scotch, so we both refused. "I thought you said this all began with the hell vents opening," I reminded her. "The earthquakes, and so on. That's only been a matter of a few months."
"It doesn't take so long to die, now, does it?"
I wanted to raise a salient point as comeback, but a wave of weariness reminded me I was way overdue for a decent night's rest. Besides, I didn't want to think too much about those two guys who died before us. I pushed back the sleeve of my jacket to check my watch, noted it was almost two. Last call, back at the Zoo.
Remi felt as I did. "We need sleep."
I nodded. "You know, it's going to be difficult for us to rest with all that racket going on. Hard to sleep, hard to stay sharp." I tried to turn it into a joke. "Maybe the reason the other two guys died was for lack of sleep."
Lily walked into the garage, came back with a small zippered duffle that she dropped on the floor by the door, and two sawed-off shotguns. Her tone was matter-of-fact. "They died because they were stupid. They enjoyed success for a while, but got cocky. They were torn into pieces by a demon, then eaten. Raw." Her smile was odd, like the death amused her. Well, maybe it did; she was a creature of war. "This is why you both have come to be here. Your predecessors died in the line of duty. Horribly. So there's a lesson for you: Lead with humility."
Remi and I exchanged a glance. In his eyes I saw the decision made not to get cocky, and he probably saw it in mine. Getting torn to pieces was, well, kinda undesirable.
Lily poured herself whiskey but did not immediately take a seat. She stood by the cabinetry, and the interior illumination highlighted the red of her Mohawk, painting her eyes greener than ever. It glinted off her piercings, off her arm cuffs.
With nothing else immediate to say, I mentioned we had met Aganju.
Her smile went wide. "And what was your impression?"
Remi resorted to a Texas-ism again. "Big as a Brahma bull."
I had no scale for that, so I asked. "How big _is_ a Brahma bull?"
"Between eighteen and twenty-four hundred pounds."
I was stunned. "A _cow?"_
"Bull."
My disbelief was manifest. "And you _ride_ those things?"
McCue nodded. "Bulls and broncs."
I reassessed him. He was tanned, and his hands were callused. A couple of shallow squint lines at the corners of his eyes suggested a fair amount of sun and good humor. Good-looking son of a bitch; and I say that because we do resemble one another.
I'd known him four days. It felt like four months. Still much to learn about him, but I was far more comfortable than I had been. I barely knew him, but trusted him. And trust had absented itself from my feelings during my trial. If that trust was an effect of _primogenitura,_ it might come in handy _._ Grandaddy had said McCue would have my back, and I'd have his. It felt right.
So, there was biker tough, ex-con tough . . . and, apparently, cowboy tough.
I looked at Lily again. "Ganji is a very imposing man. God. King. Whatever."
"He sings to volcanoes," Remi added.
Lily nodded. "He can sing them to life."
McCue looked doubtful. "This one is extinct, not dormant."
"Doesn't matter. All wake to him."
"Okay, we don't really need the singing and waking," I said. "We'd prefer he didn't do that, actually."
She disapproved and apparently thought we were terribly naïve. "If Lucifer climbs his way out of hell, there will be worse than volcanoes and lava fields. Would you have everyone on earth die of spontaneous combustion?"
Remi cocked his head. "Dying is dying, ain't it? Means don't matter."
Lily studied him a moment, then said, "Just don't make Ganji angry."
Well, yeah. Okay. A good goal. He looked badass all on his own, but as a god who could raise volcanoes? Yeah, let's keep him friendly. I attempted a casual tone, but had the feeling she saw right through me. "So, you know one another? You and Ganji?"
Her smile and the bright snap in her eyes suggested she knew exactly what I was asking. "We all of us know one another, now that we are aware again, and here. The angels made it so."
"That's another thing," I said, "you speak with an Irish accent—"
"I'm Irish."
"—and he speaks with what I assume is an African accent—"
"Igbo."
"Ee—okay. But Gaelic is nothing like—Igbo. You're ancient. Yet you both speak English."
Lily was highly amused. "I'm speaking Gaelic."
Remi stared at her, frowning, and alert as if he were listening for subtleties. "It don't sound anything like Gaelic. Trust me."
Lily smiled. Once one looked past the tattoos, the piercings, the Mohawk to the woman beneath—she was striking in an almost eerie sort of way.
She said, "He speaks Igbo to you. We don't speak in English, we speak in our own languages. But English is how you _hear_ it, as does anyone we speak with. Angels—and others born of heaven—know all languages on Earth."
"But we _don't_ know," I told her. "I don't speak any foreign language."
"I speak Spanish," Remi said, "and, well, Latin. But we're hearing nothing like either."
She nodded; apparently this topic was expected. "You will. When you're older. For now, you will understand the language of those you work with directly. When you become angels, all languages will be yours."
Huh. Heavenly Rosetta Stone, maybe? "About that," I began, again attempting to sound casual, "Just when does this—promotion—happen?"
Lily was amused, but not blind to my play for information. "That is for Jubal to tell you—but he probably won't. Angels are secretive."
"No shit," I muttered.
She opened a drawer, retrieved a slip of paper. "Now it's time for you to leave. Take the bag—it's your starter kit—and return to the Zoo Club. Settle your things. Then go to this website."
I frowned. "What web—wait, we have a computer?"
"In the common room." She held out the paper.
The URL was foreign to me, except for the triple W. I handed it to Remi who took it, frowned over it, was clearly comprehending no more than I had.
"Wupatki," she said.
"Woo what?" My natural inclination was to think of the term _woo-woo_ , denoting crackpot New Age followers and UFOlogists.
She spelled it, letter by letter. Oh, it was _wu_ as in woo, _pat_ as in pot, and a _kee._ Which told me absolutely nothing. "What's a Wupatki?" I'd never heard of a Wupatki in folklore. Some kind of monster?
"Not _a_ Wupatki," Lily clarified. "It's a place. A National Park. Indian ruins. Where there is a barghest." She enunciated clearly. "A black shuck."
"Legends," I said tightly. "Real, now? All of them?"
Lily nodded. "Mythology, tall tales, fictional villains: all _alive_. The surrogates are riding them."
Remi looked perplexed. "What's a barghest? What's the legend?"
Lily's tone was sharp. "It's not a legend, it's a demon. Gabe will fill you in on the folklore. Now, go home. Type in the URL. The site is deep web, and you will find what you need to know there." Her smile was wide, showing small white teeth. "You must kill it, this black shuck demon."
Okay—or, well, maybe not okay. I'd go along for now, but I was curious. "What's it doing?"
"What demons do," she replied. "And in that form, it's shredding humans. Tearing out throats. Adults, but also children."
I hadn't looked at a newspaper for days, or watched TV. Remi's stunned expression suggested the same. If people were being killed by a beast, surely there would be coverage. We needed to check it out.
Lily's tone was level. "You must kill it tonight when the moon is full."
Now my brows ran up. "What, like a werewolf? That's not in any of the legends."
"The full moon has nothing to do with the demon in barghest form." She looked at me, then at Remi. "It has to do with how, in the night, you're going to _see_ well enough to kill it. Now— _go_."
So go we did.
# CHAPTER TWENTY-TWO
The Zoo's parking lot was empty; last call had come and gone, which meant it was after two in the morning. After a prison regimen with lights out at ten p.m., all the action and revelations were catching up to me.
We parked bike and truck around back. I unhooked my saddlebags; Remi climbed out of his truck with the bag Lily had given us as well as a larger duffle. He'd put the two shotguns in the gun rack across the truck's back window, safed below a handsome long gun stripped of bells and whistles. Winchester 1873, a model based on the original made _in_ 1873, known as "The Gun that Won the West."
I tried the back door and found it unlocked. Not a particularly safe thing, but then I imagined a god could strike down—or, in Ganji's case, behead—anyone who came looking to steal stuff. Or maybe wake the volcano and lava the thief to death.
As we entered and thumped across the wooden floor in motorcycle and cowboy boots, supersized Ganji, who currently was not out singing volcanoes to life but rather incongruously washing glasses at the sink, greeted us with a smile.
He pushed a manila envelope across the bar. "Something came for you."
My brows ran up. "I thought with the domicile cleared—I think that's the terminology—no one knows we're here except for Grandaddy, Lily, and you."
He put his finger on the envelope. " _Gabriel_ and _Remiel_. Are there more of you with the same names?"
Remi said in a dry tone, "You might could actually open it and find out."
I walked to the bar, dropped my saddlebags on a stool, took up the envelope. Remi put his duffels down as well and leaned slightly toward me to view the envelope as I tore it open.
A single sheet of 8.5 by 11 paper. Just plain old paper. I tilted it so Remi could read it, too. The brief message was formed of letters cut from newsprint, magazines, all in different fonts, different colors. Just like ransom notes in the movies. But this was not a ransom note.
Five words changed everything: _'I know who you are.'_
"What the hell?" Remi reached out, took the paper from me. "No signature, no name at all, not even in cut-out letters."
Ganji said, "Demons know you are here."
I shook my head. "Like I said upstream—the domicile is cleared. We killed the demons in ghost form, and Remi exorcized the one in the cop. Then I burned up the bastard's cockroaches."
Ganji's eyes were knowing. "What of the woman whose arm you broke that first night, the one who sought to strangle you? What of the Grigori?"
So, he knew about those instances. God grapevine? Or: "Lily told you."
Ganji nodded. "She told me, yes, but would not have spoken of it to anyone else other than your grandfather. And the domicile was cleared. No demons are here, nor will they be."
"Huh," I said. "So, they can't just gang up on us when we step outside? We're safe here, even if our blinking neon beacons are transmitting?"
"You are safe _here_ ," Ganji clarified, "and the clearing affords some protection immediately outside this building. But they know you are here, and they have access to you elsewhere. And since you are to seek them out, you must be vigilant."
Vigilant. Yeah. "They could just lay in wait for us. Watch, and wait for us to leave."
Ganji's tone was dry. "Vigilance." He drew two mugs of draft beer, set them on the bar, implicit invitation. Apparently a god didn't care if he was breaking the law by serving after last call. "When you, Gabriel, can discern a domicile, and you, Remiel, can discern a demon before it strikes, you will find it easier to survive."
"And when is that supposed to happen?" I asked. "Tomorrow? A week from now? A month? It probably would help us with the whole vigilance thing."
"It happens when it happens, as all things do."
"Not very helpful," I muttered, and Ganji laughed.
Remi's turn. "And what exactly is 'clearing a domicile'? Clear it how?"
"I do not know how," he answered, "only that it must be done by angels—or those born of heaven—after a demon's death, or the place might be renewed as domicile." His smile was slight, but a spark lit up his eyes. "Not very helpful."
I looked at the sheet of paper with its simple but discomfiting statement. "I'm assuming you didn't see who brought this."
"I did not. It was in the mailbox."
With no address or stamp on the envelope, it meant whoever it was had brought it personally to the Zoo. Which suggested that yes, other demons knew exactly where we were.
"Well, shit." I cracked an involuntary yawn, then sighed. "I'm going upstairs. I'm beat." I grabbed the beer with a nod of thanks to Ganji, grabbed my saddlebags, then headed toward the stairs. Remi was right behind me.
Grandaddy had shown us the bedroom on the right. I poked my head in the one across the hall, saw it was identical, said, "I'll take the other."
Remi didn't care. Just walked in and dropped his duffle on the bed, put Lily's bag on the desk. I went into the opposite room.
Now that it was mine, I took a closer look at the room's decor. All was rustic, made out of lodgepole pine. The woven bedspread and desk chair's upholstered seat seemed to be some kind of Native American design, but not like the omnipresent Pendleton patterns back home.
Dresser, and closet. I needed neither, though I guess I could throw my two pairs of pants, handful of shirts, and underwear into a drawer. I had lived so long in prison clothing with no true possessions that it felt odd to look back on my life before and remember things I'd owned. I'd discovered I didn't need any of them. But I missed my books.
Missed, too, the girl who'd broken off our engagement after I was charged with manslaughter. Around eight months after I went to prison, a brief letter arrived with interesting news: _'I got married. His name is Paul.'_
Now I stripped off my jacket, threw it over the chair, and was in the midst of downing my beer when Remi appeared in the open door, tapped the jamb with one hand. "You need to come see this."
"No, man. I just want to crash. If we're going demon-killing tomorrow night, I need some rest. It can wait until morning."
"First of all, it's demon-killing _tonight,_ since it's past midnight; and no, it can't wait."
He looked deadly serious. I heaved a sigh, nodded, followed. He led me into what Grandaddy had indicated was the common room.
Four-seater table in the middle. A computer on the desk. Floor to ceiling shelving jammed with books. A brown leather nailhead couch, and against the wall a flatscreen TV in a modest entertainment console.
"I input that URL." Remi gestured toward the computer. "Take a look."
Crime scene photos. Autopsy photos. Two adults—male and female—and two children maybe eight, nine, boy and girl.
Newspapers do not print graphic crime scene photos, and definitely not autopsy photos. Lily had said the site was deep web, not accessible to others. We, Remi and I specifically, were meant to see these.
The throats were torn out so badly that pearly neck vertebrae were visible in the mess of shredded flesh. The children were the worst because they had smaller necks. Long stripes, looking like claw marks, also scored all four bodies.
I said all I could manage. Nothing else sufficed. "Holy _fuck_."
Then I stood up, turned, took three steps away. Stopped and stood there feeling nauseated. I sucked air in, then blew noisy breaths out repeatedly.
Christ, if we could stop things like _this_ , the "deployment" was worth it.
Remi's tone was subdued. "I tried to access other sites. Facebook, Twitter, Dallas Cowboys homepage, etc. Page errors all over the place. The only website comin' up is this one."
I returned to the chair beside Remi's. Abruptly, with neither of us touching the keys, the computer ditched the photos and posted a one-word message: _Table._
"Table?" I found it utterly baffling. "What?"
Then another sentence replaced the first word: _Phones pre-loaded with pertinent numbers._
I turned around and looked at the table. Two cell phones were lined up with precision, side by side. A tag on one said ' _Gabriel,_ ' the other _'Remiel.'_
And next to them, credit cards. Black, with a silver pentagram and our names stamped in silver.
More words showed up on the monitor: _Phones will not dial other numbers. Cards not for frivolous expenditures._
I needed some levity in the midst of serious shit. "So, what constitutes a—" I made elaborate air quotes "— _frivolous expenditure_ when we're trying to save the world?"
Remi picked up his phone, studied it. I grabbed mine, turned it on, pulled up Contacts, found four names and numbers: Remi, Grandaddy, Lily, Ganji.
"Check out the back," McCue suggested.
I did. Impressed into the black case was the same design as on our rings, and on the credit cards: silver pentagram. I turned the phone back over, looked at the front of the bezel and screen. "No brand."
Remi nodded. "Maybe iAngel?"
I selected McCue's name, pressed a fingertip against the screen. Sure enough, his phone rang. Or, rather, it emitted a series of five tones.
"Oh, come on!" Now I was getting aggravated. "That's from _Close Encounters of the Third Kind_."
Remi looked thoughtful. "Maybe those aliens were actually angels."
I scoffed. "No angels we've seen look anything like those Roswell rejects aboard the Mother Ship."
Remi's face went very still for a moment, eyes distant. "What if the humans who got on the ship were intended to be angel hosts?"
"It was a _movie_! With actors. It didn't happen. It wasn't real!"
"It might could be," McCue said. "What if Steven Spielberg was hosting an angel when he made the film?"
And then he grinned.
I called him a vulgar name, turned on my heel and stomped back to my bedroom—but not before I picked up my credit card—where I closed the door firmly. Maybe he'd get the message.
Steven Spielberg—an angel? My ass.
# CHAPTER TWENTY-THREE
I woke up to someone knocking on my door. Sunlight poured through the windows. I found my phone—my normal phone, the one that _did_ dial other numbers—and saw it was 8:00 a.m.
I did not feel refreshed. I needed more sleep. I bellowed, " _What?"_
Remi's voice sounded odd. "You need to come see this."
Summoned by computer yet again. Muttering about rude awakenings, I found my leather bike pants, pulled on a fresh t-shirt, and opened the door.
Remi was not there. Barefoot, I padded down the short hall to the common room and found him sitting at the computer.
"What?" I repeated. "More cryptic statements?"
He was hatless, but dressed. Still all cowboy. "Exactly the opposite. Everything's normal. I can reach all kinds of webpages."
I massaged one eye socket, then ran a splay-fingered hand through loose hair. "And last night you couldn't."
"Last night I couldn't."
"Well, can you reach the site we saw before? The photos, stuff about the phones?"
"Page error."
I grabbed a chair from the table, dragged it across wooden floorboards, sat down just off Remi's right shoulder so I could see the monitor. "Now what?"
"Found a website for the place we're heading."
"Woo pot?"
"Wupatki. Look." He typed it into the search field, then clicked on a link. It brought up a page hosting a large photo of a substantial pile of rocks.
But a pile of sunset-colored stone, shaped kind of like squashed bricks, and organized one atop another to make walls, windows, doorways. It reminded me a little of castle ruins. "Native Americans lived there?"
"Anasazi and Sinagua. Built the pueblo in the twelfth and thirteenth centuries. And I think we ought to head out there now in the daylight, so we can scope out the place before we go lookin' for a demon wearing the shape of a—" he paused, gave me a sidelong glance, "—whatever the hell a barghest or black shuck is." Another pause. "What _is_ a barghest or black shuck?"
"Also Hairy Jack, Padfoot, Churchyard Beast, Hateful Thing. Among other names."
McCue sounded annoyed and repeated himself with excessive clarity. "What _is_ it?"
"It," I said, "is a dog."
"A dog? We're hunting a _dog?"_
His reaction didn't surprise me _._ "You know _The Hound of the Baskervilles,_ by Arthur Conan Doyle? Features Sherlock Holmes?"
Remi nodded. "Never read it, though."
"The book's about a fearsome, ferocious beast believed to be straight out of hell. Tons of legends about it from all over England, which inspired Conan Doyle. Basically it's a harbinger of death, though some legends say different." I stretched, cracked my spine. "Mostly it haunts places, doesn't actively kill people."
"It is _now_ ," Remi put in. "Adults _and_ kids."
Yeah. So it was. Subdued, I went on. "Best known name is plain old 'black dog.' Mostly it haunts places and _wasn't_ particularly known for killing people, though obviously that's changed." I thought about the photos. Thought about the kids. Felt sick all over again.
"So we're gonna hunt us a dog."
"No, we're gonna hunt us a demon." I pushed the chair back and rose, shoved it into place at the table. "I need a quick shower, a little food." I looked him over. Neatly pressed clothes and different from the day before. "You good to go?"
"Already showered. There's coffee made," he said, "and eggs, sausage, hash browns. Also cereal. I had Wheaties."
"Breakfast of champions. Or also, I guess, breakfast of demon hunters." I left McCue reading about Wupatki, headed to the kitchen still barefoot. Boots could wait on coffee.
Coffee. Stuff of the gods.
But not of Ganji, I hoped, or it might taste like lava and ash.
* * *
—
We hit a minor glitch when it came time to head out to Wupatki. Remi had assumed we'd ride together in his truck. I said no way.
I told him, "I like the freedom of a bike on the open road."
His look was quizzical. "You're wearing leathers, a full coverage helmet, gloves, and boots. It's like armor. What it's _not_ like is feeling the wind in your hair. And if you really want that, I can lower my windows."
"You have a horse, I think you told me."
Remi, frowning at the apparent non sequitur, nodded.
" _That's_ not like riding in a truck, either," I pointed out. "And when on horseback, don't you feel different?"
Apparently he'd never considered that. After the brief parade of thoughts visible on his face, he arrived at an understanding and climbed up into his truck as I threw a leg over my bike.
Wearing my armor, as Remi described it, _was_ an encumbrance. Zooming down the interstate in shirt, jeans, without gloves or helmet, was far more freeing than doing it all geared up. But bodies bounce when they come off bikes at high speed, and when head meets road it usually results in death or a persistent vegetative state. I wanted neither, thank you very much. Around towns at much reduced speeds, yeah, I might go without the whole shebang, but the interstate could be a vicious beast.
We headed north on Highway 89 about twelve miles, turned off at the sign for Wupatki and something called Sunset Crater.
Sunset Crater turned out to be a massive blackish purple-red cinder cone left over from an eruption. With my mind on Indian ruins, not on a mountain of cinders, I didn't much give it attention as I rode toward Wupatki—that is, until something reached out and grabbed me.
Not a person. Not a demon. A _feeling_.
Wrongness.
Evil.
I braked, rolled my bike onto the shoulder, stopped. The road was crowded by pines, firs, juniper, and wild grasses. Patches of cinders broke free of the soil, of the deadfall of limbs and leaves. The remains of jagged, broken lava flows had turned much of the area to black, blistered rivers of volcanic stone, intercut with trees. Trees and grasses also climbed most flanks of the crater.
Like an ice cube dropped down my back, I was abruptly chilled. I heard thin, high-pitched screaming.
Remi, too, had pulled over. He drove up beside me, dropped his windows, leaned low toward the passenger door to speak out the window. "What's up?"
I did not want words. I waved him into silence, concentrating on what I had felt. Climbed off my bike, took two long steps toward the huge cinder cone, then stopped short and blinked heavily, because it felt like my eyes weren't behaving properly.
The colors that crowded my vision were black and red. And I sensed a thrumming buried deep. Saw flashes of fire, of lava, of an undulation in the earth. Heard again the screaming.
Remi, out of his truck, came up beside me. His tone was low, and infinitely quiet. "What do you feel?"
I worked my shoulders, trying to shed the chill lodged in my spine. "Did you hear anything?"
"Sure," he said. "Rabbit. Maybe more than one. Probably a mountain lion. They hunt this whole area. You ever hear a rabbit die?"
I shook my head.
"They scream. Sounds like a human baby."
I thought again of the dead children and their dead parents. Neither were babies. But they were too young to die, and no one deserved to be torn apart by a demon in the guise of a massive dog.
"Do you sense anything?" I was tense, unsettled. "Any kind of—a human host? A surrogate?"
McCue shook his head. "But I suppose in dog form it could wander back and forth between the ruins and the crater. They're only about fifteen miles apart. I checked the papers, news sites—there've been no reports of anyone being killed _here_. Just at the ruins."
"And what do the papers and news sites say about it?"
"Mostly, they believe it's a mountain lion. Maybe a bear. They've gone looking, even on horseback, but found nothing. A few whackjobs claim it's extraterrestrials." He grinned. "That a space ship crashed and caused the crater in the cinder cone when it buried itself, and if the government would just allow people to hike up the cone to the crater on top, evidence would be found."
"So you can't hike the cone?"
"Nope. What I read said it used to be allowed, but it was breaking down the ecological integrity of the area, so it's closed off to the public now."
Anger surfaced. "I want this son of a bitch."
Remi said, "Then let's go kill it."
Back on my bike, the sense of wrongness bled away from me. I heard nothing more, _felt_ nothing more than the warmth of a bright summer day mitigated by the shade of tall trees and a breeze ruffling through.
I pulled back onto the road with a pickup truck behind me.
# CHAPTER TWENTY-FOUR
At the turn off to Wupatki, I cut the engine, rolled to a stop. The road that led to the ruins was blocked by a wide metal gate across the entrance, a three-strand wire fence, and featured a big posted notice stating the ruins were temporarily closed to visitors. Apologies for the inconvenience.
Yes, having four people killed by— _something_ —would indeed be inconvenient.
Remi got out of the cab, climbed up into the truck bed, and opened one side of the big hinged tool box, which ran from one bed rail to the other. Kind of like steel saddlebags, in a weird sort of way.
He jumped back down, brandished a pair of wire cutters, and proceeded to cut away the fence beside the gate. He bent it back so that my bike and his truck would fit through, then motioned me onward.
I took the bike off asphalt onto deadfall made up of pine needles and other detritus, then returned to the entrance road. A little way down I came upon a parking lot, rolled into a space.
As I shut down the bike, I couldn't take my eyes off the ruins. Wupatki in the flesh, so to speak, was identical to photos that captured the place but not the _feeling_. Wupatki was old bones with an older soul.
Considering the builders cut stone into flat rectangles back in the twelfth and thirteenth centuries, and mortared them together with clay into stacked walls, I was damn impressed. The big building was built around massive sandstone rock formations swelling from the earth. The color throughout—stacked stones, the formations, and the dirt all around it—was a pale, dusty sunset gone rich beneath the sun.
We had the place to ourselves, since access was closed. I pulled helmet and gloves off as Remi drove in. I felt for the Taurus in my shoulder holster, decided to strip down to t-shirt, which left the gun and holster visible. Legal open carry is all well and good, but some people get nervous around guns. My ex hated them. Had we come where others, such as tourists, gathered, I'd have kept the jacket on even though the day was warm.
Supposedly the beast attacked at sunset, so we had some time. And no guarantees it didn't continue to hang around until well after dark. Lily seemed to think so; she'd said we would need the full moon in order to see.
I felt better for having my KA-BAR and revolver loaded with powdered iron shells. Remi checked himself as well: belt-mount holster, big damascened Bowie, the tri-part sheath of throwing knives.
Ghosts, as had been established, could be killed if shot or stabbed in the vulnerable parts of human anatomy. At the Zoo, I'd taken out the male ghost with a point-blank head shot, which certainly would have killed a normal human. Remi had thrown knives into a tight grouping in the heart, almost like arrows or darts in the bulls-eye, and killed the female ghost. And he had exorcized the demon in the cop, leaving the host alive and unpossessed.
Now we faced an animal. Because black dogs had always been in the realm of folklore rather than reality as ghost-beasts haunting various areas, I wasn't quite sure what it would take to actually kill the thing.
"Well, let's get this checked out," McCue said, heading out. I hesitated a moment. Wondered, because I couldn't help it, if I would scream like a dying rabbit if the Churchyard Beast took us.
It wasn't a steep climb up into the ruins because the place was built on a modest plateau and twentieth-century contractors had put in wide, shallow, framed steps. At some point they'd also mounted hand rails throughout various parts of the trail in the ruins, and in some areas natural mortar gave way to modern cement to shore up the place.
A winding trail, converted to powdery red dust by the soles of thousands of shoes, led me inside. The ruin was open to the skies, though at some point a wood-pole ceiling had kept out the sun and rain, because the remains of those poles were scattered across the ground. And the walls were easily two stories high, no second floor. Again, I figured the builders had used wood to create a floor that now lay in ruins upon the ground.
The windows and doorways were shaped with great precision, considering the material was hand-hewn stone. I walked into a square tower-like room, saw the high second-story window. I tipped my head back to see what I could see. The sky beyond was a blistering blue.
I closed my eyes. Let my awareness of the here and now fade.
There was warmth in my head, like a banked fire, and the edges of my vision were a comforting green yet again. I caught the faint trace of fire, of ash, of water; of ground-growing plants such as squash, plus beans and tall, rustling corn. But that was not all. Piñon pines bore seeds that were roasted, or pounded into flour for baking. Wild plants offered flavor and spice, and there was meat aplenty from deer, rabbits, and prairie dogs, those weird little chipmunk-like animals who popped out of burrows and chittered to sound the alarm when children came hunting them.
It had been a vital, thriving community, a place where multiple native cultures came together to trade, until the climate changed. Water became less available, the land more arid, and eventually Wupatki could no longer support its residents. The tribes moved on, leaving behind them a massive structure surviving to this day, if depleted of roof, ladders, and flooring.
"Gabe!" Remi's voice snapped me out of it. "Gabe—come take a look!"
I followed his voice some distance, found a huge, round, deep dug-out area in the ground lined with stacked stone walls and mortar, plus a surrounding wall up top. Those top walls were probably three feet high; below the outer surface ring was a lower structure forming circular bench-like seating around a large area of flat, dusty ground.
I stood atop the wall, found Remi down in the—well, hole wasn't the right word. It was a planned and hand-built structure, not a natural opening in the earth. It struck me as incongruous to see a hatted modern cowboy standing in the middle of something built centuries before by Native Americans.
"It's a ball court," he said.
"Ball court?" That seemed unlikely. "How do you know?"
Remi stared up at me as if were too stupid to live _._ "I read the sign."
Oh. Yeah. Signs are handy.
"Anyway, come down," he called. "There's an opening about ten paces to your right."
And so there was, and so I walked through a walled entrance that struck me as similar to the entry tunnels football teams use to head out onto the field.
I strode toward Remi, until he told me to stop. "Look," he directed, and pointed at the powdery, gritty earth.
Boot prints all over the place. But overlaying them were animal tracks. Big-ass animal tracks.
"Too large for mountain lions," Remi noted, "And the shape's wrong, anyhow. Those prints weren't made by a cat."
Or a bear, I thought—though I doubted there were any out here in the middle of a waterless, very warm nowhere.
"Dog _like_ ," Remi continued, "but too large for a coyote or wolf."
I squatted, looked more closely at the paw prints. The depressions were clear. Even the holes left by toenails were well-defined. The creature had returned after the investigators departed.
I shook my head. "They're nothing more than folklore."
"Grandaddy and Lily said—"
I cut him off. I hadn't exactly meant for him to hear that. "I know what Grandaddy and Lily said."
Remi's expression was serious. "Then why do you want to deny it?"
It wasn't denial, exactly. More like an expression of hope, but I didn't know how to explain it. From my squat, I looked up at the cowboy. "Because black dogs are _not supposed to exist_."
He was unsmiling; this was no joke. "Neither are demons."
I looked at the paw prints again. "This is a domicile. I can feel it, even if I can't quite do it in Grandaddy's way. You sense a demon in any form hanging around?"
Remi's eyes went blank and unfocused, but finally he came back to the here and now and shook his head.
I said, "It would be helpful if you did—" I gestured at the tracks "—but I guess not really necessary."
"I reckon that's the truth of it."
After a moment of quietude where all we heard was the buzz and whine of insects, I said, "I'll admit it . . . this scares the shit out of me. Some big-ass _thing_ out of folklore killed four people."
"Yup."
I rose. "And it'll kill more."
"I would bet my horse on it."
That was stone-cold significance, coming from Remi. I stared across the high desert vista, saw endless blue skies. I was trying to convince myself that all this was real, when what I wanted was to forget the last few days.
It was a fucked up errand, maybe, but people had died. "We're coming back tonight and taking this thing out. One way or another."
# CHAPTER TWENTY-FIVE
In Arizona, I'd discovered, the curtain-fall of twilight came late. No wonder the state didn't do Daylight Savings Time. The light lingered much longer than in the Pacific Northwest, as did the heat of the day. Daylight here most assuredly did not need to be saved.
We were in Remi's bedroom back at the Zoo, and he was checking the slide of throwing knives in their sheath. The big Bowie was on his belt, as was his Taurus. The smooth efficiency of his movements made him dangerous. In a way I wished it otherwise—because then it made _me_ dangerous, too. Maybe that was necessary now, but it played hell with memories of a quieter life. College student, college prof. Now ex-con and killer.
He glanced up. "How do you think we should kill it? I mean, what will work? Iron? Silver? Holy oil or water?"
After a moment's consideration, I remembered Lily's duffle. I placed it on Remi's bed, unzipped it. We both looked into an array of cardboard boxes containing bullets and shells, flasks, small bags of various herbs, a few bottles of God knows what. Also two of the twisted, silver-wrapped wooden stakes.
"It's not a vampire," I noted, "so we probably don't need to bring the stakes."
Remi's tone was subdued. "I'm not lookin' forward to the day we have to rely on those stakes. That's close work. _Too_ close."
The black dog had been close enough to take down every member of the family. It could well try the same thing with us.
"We'll take a shotgun," I said, "but your rifle may be the best bet. I think it's advisable to keep our distance from this thing. You can play sniper. Or hunter. Whichever works for you."
McCue didn't buy it. " _You're_ the one with the extra special, fancy-dancy gun skills."
"You're the one with the skills for _that_ gun," I pointed out. "Later, I'll take the chance to learn its weight, its kick, check out the sights, the lever action. But not tonight."
Remi eventually agreed, then suggested we mount up. I shot him side-eye, grabbed a box of silver bullets and a second of iron buckshot shells. I opened it, considered the ammo, then blew a breath all over bullets and cartridges. It still felt ridiculous, but I wasn't going to argue.
"I imagine they were already dipped in holy oil," McCue remarked dryly.
"Let's just be on the safe side. Bless it with spit, you know?" Sexy image.
I handed him the box, and he followed my example as if he were blowing out birthday candles, then nodded. "Let's head 'em up, move 'em out."
"In case you hadn't noticed, I am not a cow."
"In all that leather you might as well be."
* * *
—
By the time we arrived at Wupatki, sunset was upon us. The witching hour, so to speak, but heralding instead the incipient arrival of a black dog, a barghest, a black shuck and all the other names.
I pulled off helmet and gloves, left them on my bike. McCue was taking down a shotgun and the Winchester from his gun rack. He handed me the shotgun. I broke it open, saw it was loaded with what we needed. I tucked more shells into my jacket pockets.
Remi's hunting rifle wasn't prepped for killing demons; he took the time to eject the old rounds from the repeater, replaced them with breath-blessed silver.
Abruptly, in my bones, in my consciousness, I sensed it again.
Evil.
Malevolence.
But tenuous. Coy. Ephemeral.
_Domicile._
I settled the shotgun in my hands and against my right shoulder, turned in a circle, twitched shoulders against a chill. "You getting anything on the demon?"
McCue paused, and his eyes went distant. After a moment he shook his head. "I got nada. I don't understand how come your heavenly engine is running while mine isn't."
" _Half_ running," I said. "It's still—intermittent. But as to why, maybe it's an alpha thing. And maybe in five minutes you'll know exactly where the demon is and what it's doing."
The moon was rising as we walked the path toward the ruins. My chest felt constricted, breath ran short. We'd killed two ghosts, exorcised a demon, but all had been wearing human form. This was a dog, and black dogs could indeed pull down humans from the heights, even those of us with sparkling little celestial beacons.
When McCue, in his drawl, said that he was fixin' to break off from me and circle the ruins from one direction, suggesting I go the other, I declared that idea horror-movie stupid. "If we do that, and I was watching this at home, I'd be yelling at the screen about how stupid we are. We stay together."
"And give the surrogate a tight-grouped target? Once we get in there and spread out, at least one of us'll get 'im."
Well, he had a point. "I really, really wish you could sense where it is."
He shot me an annoyed glance. "So do I, but I can't. Yet. So let's do this thing. If the surrogate's not inside the building, I'll find a spot to set up with the rifle. If that sucker is inside, we corner it and take it out."
We walked up the trail leading to the ruins. We did not rush. We made our steps quiet.
Remi watched the left, I the right. Both of us employed a sweeping visual assessment, checked the immediate environment, then one of us swung around and walked carefully backward to check our rear while the one in front kept moving.
Shitshitshit. I wanted not to be here.
Beside me, Remi said, "I wonder where the kids were when the parents got taken. Or if the kids were killed first."
And that eased me, in kind of a sick way, because I had a purpose. It smoothed down the hair on the back of my neck, settled my belly. I blew out a breath and let go of the tension. I knew we needed badly to save other children. And we could.
Kids. Parents. _Kids_. Boy and girl, maybe seven, maybe eight. Or both at once, possibly twins.
Adrenaline ratcheted up. I wanted this done. Over. And it was enough to fuel me. As we reached the big multi-roomed building, Remi said he was going left to circle around, and I should go right. The plan was to meet after checking out the perimeter, then go together inside the ruins to investigate individual rooms.
The moon remained on the rise, not yet at its zenith, but already it illuminated my surroundings. Lily had been right; tracking a surrogate in a black dog's clothing required good light.
I was halfway around the perimeter when the beast showed. Beneath the moon, not shunning its light, the thing paced slowly forward. Its head was lowered as it scented track, the tail up and rigid.
It was sleek, and slick, and oily beneath the moon, as if colors could not stick. Thanks to the lunar illumination I could see it clearly: angles, muscles, curves, a startling definition even against encroaching darkness.
I like dogs. Dogs like me. But this . . . oh, hell. _This_ wasn't a dog. It was the End of Days. The end of my life. The thing was Mastiff. Great Dane. It was pit bull and Rottweiler with Doberman thrown in. It was _huge_ , and the sheen of its coat, beneath the full moon, the delineation of its muscles strung tight yet rippling, promised a strangely beautiful, economical killing machine. An almost elegant death even in the midst of horror.
If, that is, one didn't mind having a throat ripped out and the vertebrae, like knuckle bones, like _gristle_ , chewed to powder.
It stilled. Lifted its head, air-scented, looked directly at me and lowered its head in a hackled, stiff-tailed threat display. Teeth glinted, but that's not what I was struck by.
White. _White_ eyes. Not the red of legend.
The pupils ceased their roundness and bled into vertical slits. Cat eyes in a dog's body.
Well, hell.
Here it was, and here _I_ was, and my throat was there for the taking.
# CHAPTER TWENTY-SIX
Only a fool stands in front of his ending and does nothing about it. I might still die, but I'd go out with a bang. Literally.
Even as I brought the shotgun up to the hollow of my shoulder, I heard a rifle report. The black dog staggered, cried out, dropped briefly, then surged back to its feet. It drew its lips back in a ferocious snarl.
Well, Remi might could be a better shot.
The black dog leaped, and I took off running.
I dumped the shotgun; it encumbered me too much as I ran, banging against my hip as I clutched it in one hand. Dogs, as far as I knew, even with a demon aboard, lacked the capacity—and the opposable thumbs—to gather up a gun and fire it.
Please God, let Remi's shot, even if not center mass or buried in the brain, slow the sucker down.
The ground was a conglomeration of grass, grit, dirt clods, stone rearing out of the soil. My knees and ankles took the brunt as I scrambled to remain upright, worked to keep my footing. The moon bathed the earth, but behind me I heard the dog.
No. The _surrogate_.
The chant gained volume in my head. _Remi Remi Remi._
Breath, all tangled amidst need and fear and effort, came up from lungs into the air. I sucked oxygen, hearing the scrape of it against my throat.
There, before me . . . the ball court wall, crouching in moon and shadow. It would hardly stop the beast, but it nonetheless offered an opportunity.
_"Remi!"_
I thrust weight upward, leaped, felt my boots hit stone, then crouched and planted a hand at the end of a stiffened arm, propelled myself outward and down. I most assuredly did _not_ want to hit the circular bench built low.
It was horror-movie stupid. The ball court featured only two narrow entrance/exits. The walls surrounding the court were not insurmountable, but definitely a challenge to someone badly winded and running for his life. But I believed it might give Remi another chance.
I hit ground, went down, tucked and rolled, came up to my feet just as the beast leaped onto the wall. The moon painted its body, turned it a tarnished, silvered gray, slick sheen against the darkness.
Just as I pulled my Taurus from its holster, Remi's rifle shot took out the surrogate.
I stood there with lungs heaving. I wanted to swear, but I lacked the breath.
The black dog, struck through the heart, had tumbled down the wall, bounced off the bench-like structure, and landed in a heap on the ground. Blood spilled out of its mouth, attenuated, stopped. The heart was no longer pumping.
I bent over, planted hands against my knees, though one still clung to the gun. I felt my hair fall forward to obscure both sides of my face. After a moment, breathing better, I sat down all of a sudden with no grace at all. I was still not acclimated to the elevation.
_"Gabe?"_ came the call.
I wanted to answer. Nothing in me could.
Remi jumped up onto the wall with its wide, flat rim and stared down at me. The rifle hung from his hands. "You okay?"
I waved at him, trying to communicate my okayness through the gesture, then kind of folded up, tipped, lay flat on my back against the ground.
Holy shit. Holy _Christ_.
I didn't feel anything like a scrap of heaven's grace, neither essence nor spark. I was just a man, no more. But Remi and I had killed a demon.
McCue came into the ball court opening, walked over to me. He didn't squat, didn't kneel, didn't even bend down. Just stood over me with a rifle in his hands.
Then, waxing poetic, he said, "You scrambled like a chicken tryin' to outrun a fox."
I presented him with an eloquent middle finger.
After a moment, as my arm flopped back to the earth, he asked if I would live. I said yes, levered myself into a sitting position, then with the help of a stiffened left arm, I thrust myself up from the ground and onto my feet.
Said I, as I recovered my breath, "Couldn't you have taken him out _before_ I scrambled like a chicken trying to outrun the fox?"
"Hell, son, where's the fun in that?" He grinned, and added, "Betas _rule._ "
I was sore from my tuck and roll. Apparently I'd come down hard on my left shoulder before doing an imitation of a hedgehog rolling itself up against a threat.
I looked at the sprawled dog again. Definitely dead. And since a black dog wasn't truly real, there was no host to save.
"You know," I began thoughtfully, "this is really, really weird."
Remi's eyes went wide. "Wait—wait. _Gabe_ —"
He was supposed to get a sense of demonic presence and seemed clearly alarmed, so when a woman appeared atop the wall I didn't even think about it. On edge, jittery, trembling from an adrenaline high, I shot her five times.
# CHAPTER TWENTY-SEVEN
She didn't go down. What she did, with markedly angry eyes, was unfold a pair of huge, black, shimmering wings, then snap them forward to meet and emit a thunderclap of sound so powerful that it sent Remi and I both to the ground from the force of it.
She put up a fist with the forefinger extended and stabbed at the air. " _Don't_ piss me off!"
Remi brought his rifle up, though he was seated on the ground.
I reached out and caught the barrel. "No. No, that's Greg."
She glared at me. "I am Grigori; my name is _not_ Greg."
"Who?" Remi asked, and then he got it. "Oh, the angel from the Zoo Club? The one who told you angels have differing agendas and humans might end up as collateral damage?"
We were still sitting in dirt. "That's the one. She says she's not one of them, but at this point who knows?" I rose, brushed dust from my pants, tucked my revolver back in its holster and extended a hand down to McCue. He grasped it, and I pulled him up.
Black eyes glittered as she stared at us. The moonlight was kind to the bone structure of her face, with its high, delicate cheekbones and wide mouth, though it was set very grimly at the moment.
"How did you not die from that?" Remi asked. "Heart high through the center of your chest."
"Apparently they don't," I murmured. "Die, that is." I reconsidered. "Then again, maybe they do; she wouldn't tell me how it might happen the last time we met." I looked up at her. "You know, those wings are awesome. Grandaddy never really did show us his, just kind of a rippling, an _impression_ , and displaced air."
The wings were outstretched. Dressed in black and standing atop the wall with those things unfurled, she resembled a . . . well . . . yeah: an avenging angel, Asian style. Then she furled the wings back almost like an accordion. They dimmed into the darkness beneath the moon, then disappeared altogether.
She gazed down on us both. "Grigori watch. Grigori record. Grigori keep track of what others are doing, most particularly when they are infants, as we reckon age." Now she glared. "My name is not Greg, or Gregory, and I'll ask you to stop using them. My name is Ambriel."
I gestured, indicated the black dog, hoped to mollify her. "We took out a surrogate."
"I saw." Her black hair was glossy in the moonlight. "I was watching; that's my job. But yours isn't finished."
Remi asked, "Are we to bury it? Burn it? What do you want us to do with it?"
"You are to clear the domicile. The demon will go with it."
"Uhh," I said. "They haven't taught us that yet."
She didn't look any happier. "Can you not sort it out on your own?"
"Remi knows Latin," I pointed out. "Will that help? If there's a ritual, I mean."
"Latin will do," she said in bad humor. "So, Remi, what do you think you should say to clear the domicile?"
He gazed up at her, ruminating on word choices. I could see it in him. Finally he offered: "Out, damned spot."
Greg— _Ambriel_ —looked no happier. "This isn't _Macbeth_ , idiot boy."
Remi looked thoughtful. "Would a rite of exorcism work?"
"It's already _dead_ ," she pointed out. "But we can't very well leave the body out here for tourists or Park Rangers to find, so we must get rid of it as well as its lingering influence. Erase it from this place. _That's_ clearing the domicile."
I looked at the very dead black dog, empty now of demon. "Why is it still here?" I asked. "Black dogs aren't real. Without the surrogate animating it, shouldn't it just disappear?"
"The demon _made_ it real," she explained. "Therefore it is."
I looked at Remi. He looked at me. We both turned to Ambriel. I said, "We haven't got a clue."
She was annoyed, sent both of us a sweeping glance of sheer disgust. "You should be able to figure it out."
"Okay." I looked again at the very dead dog, gave it a command. "Scram."
Naturally 'scram' was not the magic word, any more than Lady Macbeth and her damned spot was. Ambriel finally seemed to understand that we really, truly didn't even know where to _start_.
"Watch and learn," she commanded. Dutifully, Remi and I did just that.
She remained atop the encircling wall. Though her elbows were bent, she lifted and extended her forearms, bent her wrists upward, displayed the palms of her hands. She resembled nothing so much as a fugitive surrendering, hands in the air.
Then commenced the Latin: " _Patri bonorum omnium creatorem_ _omnium sanctorum deponentes pestilentibus locis visum vocemque bestias inferni."_
I expected bolts of light, or laser beams, or smoke, or _some_ thing, _any_ thing visible, shooting from her hands. Something dramatic.
Nada. Nothing at all came from her hands.
And then the black dog's body exploded.
It was literally blasted apart. Portions of the body were flung up into the air and across the ball court. Remi and I both ducked, hunching shoulders and protecting our eyes from flying blood, bone, and viscera.
All of the scraps, pieces, and traces of the exploded body were immolated. Fires burned here and there, illuminating the ball court rather like lighting up a baseball stadium. Each pocket of flame consumed the black dog's scattered remains.
When the body was gone, when at last it was truly gone, I caught that overwhelming smell of sickly demon deodorant. Remi and I both coughed hard enough that I thought I was going to puke. But I swallowed my belly back into its cavity.
"Oh," McCue said in a hoarse voice. "Yeah, I can recite that. I just don't know how to explode the thing without C-4."
She stood with hands on hips, looming over us. "You need to learn it. I suggest you do that very soon. Because so far there have been angels to clear the domiciles, but that won't last. You're being given a little latitude right now because you are infants, but next time—and there _will_ be a next time—you will have to do this yourself."
I opened my mouth to ask another question, such as why was it angels couldn't do all of this on their own and leave us out of it entirely, but she was just— _gone_. Between one moment and the next. "You know," I mused, "it would be helpful to be able to do that. Just translocate our heavenly asses right out of here."
Remi was staring at the ash and grit now smeared across the ground. Then he looked at me. "Well?"
"Well what?"
"Is it clear?"
So I let myself go, lost myself, sensed nothing of malevolence, nothing of evil. Just an _absence_. No colors at the edges of my vision, no blurriness, only an impression of emptiness. Then a little of the now-familiar comforting green eased itself in.
"It's okay," I said. "It's gone. Nothing at all left here." I looked at the scuff marks, boot impressions, paw prints. "So, what was it?"
"What was what?"
"What she said."
"Oh, the Latin? It means _'Our father, creator of all things good, of all things holy, rid this place of the sight and sound of this pestilential beast of hell.'"_
I laughed. "Oh, _that's_ good, even if it is a mouthful. I like that _._ Pestilential beast of hell. Very cool."
Remi lifted his rifle, tipped the barrel back against his shoulder. "Let's go, pestilential human. I'm fixin' to go horizontal for a while, then eat half a cow."
I looked again at the absence of dog, absence of surrogate that had killed four people. The entire complex of ruins was free of demon, completely cleansed. Free, too, of the cloying scent of its perfume.
I followed Remi up the low ramp leading out of the ball court.
# CHAPTER TWENTY-EIGHT
Fifteen miles up the road from Wupatki, my vision went black, then red. I blinked hard, eyes wide, then squinted, tried to clear away the colors washing across my vision. Eventually I braked, slowed carefully and veered off road onto shoulder. My eyes felt foggy, but I could see.
Swiftly I pulled my helmet, planted it on the handle bar, swung a leg over the saddle. Even as Remi in his truck pulled up I walked straight across the road, stood on the other shoulder. In the moonlight, against the stars, the huge cinder cone bulked black against the sky.
I heard the truck door open, heard Remi climb out, slam the door closed. He crossed the road to stand beside me, but said nothing. He seemed to know instinctively when not to speak.
My vision was clear again, but I knew all I had to know. "It's a domicile."
"What, the cinder cone?"
"I _think_ it's another black dog, though I can't be sure."
"I don't feel it," Remi said. "I mean, I felt the angel, but nothing now. No sense of surrogate."
I looked at the top looming over us. "It's up there somewhere. It owns this place."
Remi turned, walked back across the road, got into the truck and hauled out his rifle, a shotgun, another box of shells and bullets.
His tone was completely casual. "Well, then let's go kill us another demon."
* * *
—
The full moon helped us make our way. It bloodied the cinders, lit up swatches of yellowed grass, illuminated dark vegetation upon the flanks and sentinel pine trees. I knew that surrounding land was always more fertile after an eruption, but above the treeline the cone was all cinders, nothing more.
We maneuvered across the broken lava field, trying not to sprain or shatter ankles in the process, until we reached the base of the massive cone. On the hike from road to base, Remi had nearly killed himself several times while Googling on his phone. Now that we had reached the cone, he was full of facts.
"It's one thousand feet high," he said, "and one mile in base diameter. It erupted twice in the twelfth century. One lava flow traveled _six_ miles."
"Does your phone tell us how to kill a demon in a black dog guise?"
Remi sounded amused. "Well, I'd have to do a search on that."
"Do it," I suggested pointedly, " _after_ we've killed this thing."
McCue carried his rifle, not a shotgun. He had tucked a few spare bullets into the pockets of his jeans. I, on the other hand, _did_ have a shotgun as well as my Taurus. Blessed cartridges in the shotgun, blessed bullets in the revolver.
"Blessed" sounds a whole lot better than saying they were bathed in breath and spit.
"You're sure it's up there?" McCue asked.
"I'm sure one has been here, or _is_ here. I'm the one who handles places, remember? You're the one who's supposed to identify demonic presence. Preferably from a significant distance."
Remi tipped back his head and gazed upward, following the line of the round-shouldered pyramid from base to top.
"We're not splitting up," I declared.
"Hell, no."
"So let's climb this sucker."
Okay, so climbing up a mountain of cinders is not the easiest thing I've ever undertaken, especially at night with only the moon to see by. It's like planting boots in dry, loose, pebbled sludge, and every time you try to push upward the cinders give way and you lose ground. It was one step forward, four sliding steps back. Slamming boot toes into the cone, as if we were climbing Everest with crampons, didn't do much, either, because the cinders simply crumbled away under our boot soles.
Remi said, in a breathy tone, "The elevation here is over eight thousand feet."
I was panting now, trying not to slide and lose ground I'd gained. "Do I need to know that?"
"Well, it explains why we're both breathing like a bellows." He sucked air. "We're not acclimated yet, either of us."
_Plant—dig_ — _slip—slide_. Over and over again.
McCue was right. The altitude was sucking breath and strength from us both. "Anything?" My voice was hoarse. I wanted water badly.
"Hold . . . Gabe, _hold_ —"
I stopped climbing. Felt myself slide a few inches. Remi was a couple of feet down from me. He brought his rifle up to cradle it across his right elbow, the barrel snugged into his left palm.
He shushed me when I tried to ask him a question, his face shadowed beneath the brim of his hat where moonlight couldn't reach.
"Oh yeah," he said at last. "It's here. And it's another of those big bastards."
"Black dog?"
"Yup."
"Two? They're traveling in _packs_ now?" I lost another inch or two as a shift of my weight stirred the cinders afresh. "That's not like anything I've ever read. They are solitary beasts."
Remi said tautly, "It's coming."
He and I went down on our bellies. I hooked my right elbow back so I had access to the shotgun trigger. All I had to do was rise up onto my knees and blast the sucker.
That is, _if_ it came close, and _if_ it offered a large enough target.
Moonlight painted the cinder cone. I saw the beast upon the flanks, literally _slinking_ downward. Like water, it flowed.
I saw, too, white eyes shining, broken now and then by a blink. And then it stopped blinking and just _stared_ , even as it continued to flow like dog-shaped mercury down from the heights.
"Go left," I said tersely, "I'll go right."
Because if we stayed where we were, so close together, it would be nothing for the black dog to leap into the middle of us and shred both our throats with no effort at all.
As we split, it stopped moving, sat down on haunches and elbows, butt uphill, chest heading downward, and contemplated us. I saw it lift its nose to scent, as nostrils flared. Jaws opened.
And it leaped.
Remi stayed where he was, flat on his belly save for the rifle butt tucked against his shoulder. A Winchester '73 is not a sniper rifle, it couldn't be set up like a true-born sniper rifle, but it did fire bullets.
He got off two shots. Both missed, digging out divots and sending a spray of cinders into the air. As the black dog came down toward him, I rose up onto my knees and let go with one barrel, but as the beast fell sideways, too near Remi for comfort, I knew it wasn't enough.
"Keep your head down!" I shouted, taking huge sliding strides through cinders. And as the beast reared up, blood running black beneath the moon, I fired the other barrel and took out much of its head.
"Okay," I croaked, realizing how out of breath I was. I turned, sat down on my ass with boot heels planted downhill to arrest a slide, and tried to breathe. "By the way, _alphas_ rule."
Cinders whispered, crunched, tumbled as Remi, on his knees, made his very awkward way closer to me. He collapsed on his belly, head uphill, rifle in one hand as he lay against the massive, mountainous pile of cinders.
He was breathing hard. "Do you think Greg will come?"
"Why would Greg come?"
"To clear the domicile. I'm assuming this place remains fouled by the demon."
Oh, it was fouled. Definitely unclean. "Well, you know the Latin for it now, right, since she said it? Can't you do it?"
Remi shook his head. "I didn't hear it in Latin."
"She said it in Latin."
"I didn't _hear_ it in Latin. It was, I don't know, mumbo-jumbo."
"Aramaic?"
He contemplated that a moment. "Better learn it right quick, hadn't I?"
" _I_ heard it in Latin."
"Maybe you can do it, then."
"And risk saying something totally wrong? I don't think so. We could end up clearing our own asses right down a rabbit hole." I inhaled, tasted that awful perfume of demon death and decomposition. It coated my tongue. I spat, spat again, gagged, finally was able to speak. "Okay, we have to figure out how to do this. Because if we don't, another surrogate will show up to re-establish a claim, people will die, and we'll have to come back to kill _it,_ too. Can you sound out what she said by ear? Do you remember enough of it?"
Remi clambered to his feet. "I reckon I can try."
I summoned Yoda again. "Do or do not, there is no try."
"You're just a regular fanboy, aren't you?"
"Pot, kettle. You _get_ those references."
"Okay," Remi said, ignoring that, "she assumed a specific position, put up arms and hands just so. Then she spouted the mumbo-jumbo."
We both set down our guns and stood facing upward, where we could see the body. I felt a little stupid putting my hands up in the air, and thought McCue looked stupid, too, but we copied what we'd seen her do.
It looked better on a woman.
So there we were, hands raised in the air, palms turned outward. Remi commenced chanting. And I heard it as Latin, even if he was trying out the Aramaic, or whatever language Greg had invoked. I waited for the whole _pestilential beast_ part, but nothing happened. The body remained where it was, not exploding and not burning.
In the quiet of the night, we looked at one another. "Go again," I suggested.
Remi rolled out the chanting a second time.
Nothing happened.
"Well," I said, "apparently we need more schooling in the fine art of blowing up demons. I've got the magic phone; I'll call Grandaddy. If he can't come to clear the domicile, maybe he can at least tell me what we're doing wrong." Remi was frowning, not paying attention. "Earth to McCue. I'm calling Grandaddy."
"Wait," he said. He looked at the ring on his middle finger. "High-five me."
"What?"
"High-five me." He lifted his arm up in the air, displayed his palm.
I stared at it a long moment, looked at him, finally started the arc that would fulfill the gesture.
Remi caught my hand, flattened both, one against the other, began the chanting a third time.
Our rings met, clicked together like magnets, and the corpse blew up.
We were close enough that we could not escape the flying body parts. As portions slid down our faces, fell off our shirts, were kicked free of boots, all the bits of the exploded surrogate caught fire. _We_ did not, fortunately, though I didn't understand why we hadn't been burned at Wupatki, either. Maybe it was more of our celestial energy.
Remi and I retreated as best we could, sliding in cinders. As the stench of the perfume rose, I pulled the top of my jacket across the lower portion of my face.
"This is downright _gross_!" Remi exclaimed, unsticking his shirt from his torso. "And we can't go around in public blowin' up demons. It might could be a tad bit obvious."
"We didn't blow up the ghosts at the Zoo. We didn't blow up the possessed cop. Maybe it has to do with the Latin. What you just recited was different from the rite of exorcism. This was about clearing the domicile. Rick, Dick, and Candy did it for us at the bar—though it was cockroaches, ash, and grit—and Greg did it with the other black dog. And yes, it _is_ gross." I thought about it. "I guess if we kill a surrogate rather than exorcizing, we can just haul the body out of wherever it is, find a safe place, _then_ do the Latin and blow it up there."
"We've got to ask more questions," Remi declared. "They've got to tell us more than they have. It's like going in blind. The guys before us who got torn to pieces? Maybe it happened because they didn't know enough. In which case, the angels set them up for failure, and now they're doing the same with us."
" _Why?_ It makes no sense."
"How the hell should _I_ know! Even Lily said angels were secretive . . . and Greg tracked you down in the bar to warn us about differing agendas."
My hair was damp and tangled, my leathers slick with—monster goo. Though I guess I should be grateful that my clothes hadn't burst into flames. "Let's go," I said. "We've saved the world from two more demons, and I want to shower, have a celebratory drink, hit the sack. This whole sneaking around in the middle of the night is wearing." I held out my hand, palm down, looked at my ring. "That was pretty cool, though, the ring-thing."
Remi thought about it. "Grandaddy gave us rings that help us. That don't sound like a man who wants to get us killed."
"Maybe it's not Grandaddy. Maybe it's someone else." I turned, began the descent. McCue came along behind me, crunching cinders as he slipped and slid. Hell, maybe it was _everyone_ out to get us.
But still. Greg was an angel—and it would probably piss her off that we called her Greg instead of by her true name, but Ambriel sounded too much like Ambien for me—and had indeed warned us. The angels with the porn star names helped us. Grandaddy had been helping us in one way or another since we were kids.
Yeah, time to have a talk with the man—
I stopped short, turned around, slipped a little, then fixed Remi with a glare. " _You_ need to work on your aim."
McCue glared back. "You ever try to kill a charging black dog while lying on your belly praying not to slide down a volcano even as you do, _and_ working a lever-action rifle?"
Well. No.
"Okay," I said, "maybe we both need the practice."
But not tonight.
# CHAPTER TWENTY-NINE
We returned to the Zoo after last call, which meant an empty parking lot. Once again we pulled around back, found the door unlocked—needed to ask Ganji about this—and went in. I wanted to go straight to the stairs, but Ganji—man, the dude was _big_ —appeared to be waiting for us.
"You have had an eventful evening," he noted, viewing the state of our clothing. "And also, this came for you."
On the bar lay another manila envelope.
"Well, damn," Remi said wearily. "This shit's startin' to make me feel meaner than a skilletful of rattlesnakes."
"Did you see anyone?" I asked. "I mean, did you keep an eye on the mailbox in case you could see the person delivering the envelope?"
"It was not in the mailbox," he answered. "It was placed just outside the back door."
This time Remi grabbed the envelope and tore it open. He read it, then handed it to me. Same as the first time: letters cut out and stuck to the page by Scotch tape.
But this message was different: _'I know_ _what_ _you are.'_
I looked at Ganji. "Can you give me a drink? And then another one? Maybe even a third. Because I'd like to get some sleep, and now my brain will be making the jump to lightspeed."
Remi turned down a drink, said he was going to shower and go bed. I made it through one drink before the activity of the night caught up to me.
Ganji said, "Go to bed. I, meanwhile, will go up the mountain and sing to her."
I rose from the stool. "Uh—don't wake her up yet. At least let me catch a few hours of sleep. Then you can do a Pompeii number on Flagstaff."
As I started to turn, Ganji said, "I was there."
I blinked, turned back to him. "At _Pompeii?_ "
"I am at all eruptions. They happen because I sing them awake. I am their lord. But here, now—it is not yet time. You may sleep without fear."
I stared at him a long moment. "Give me another drink."
* * *
—
My brain eventually did slow down from light speed, and I slept. I dreamed about volcanoes erupting, black dogs exploding, and some whackjob sending us mysterious messages. And since I hadn't eaten before going to bed, just showered after Remi then collapsed, I awoke ravenous and starved for coffee.
Eggs, bacon, country potatoes, toast. I was feeling human again on my second cup of coffee when Remi showed up in the doorway. His face was grim. "The deep web's working again."
"Just like that?"
"Just like that. And it's talkin' to us again, so to speak."
I stuck the cutlery and plates in the sink, took the coffee with me as I followed him down the hallway to the common room.
This time a two-word opening message: _Consecrated ground._
Okay. We looked at one another blankly.
Then Remi seemed to get it. "A church is consecrated. And _un_ consecrated if the building is sold."
Before I could say anything, a series of photos appeared. Two churches, a mosque, and a synagogue. All were charred skeletons, but identifiable.
"Phoenix," McCue said. "Catholic, Protestant, Jewish, Muslim. All on the same night. _Last_ night. They've ruled out hate crimes, because no one religion was targeted."
"We're supposed to go to Phoenix?"
Remi shook his head. "Grandaddy seemed to think we'd be busy in Flagstaff for a while."
I thought it over. "I'm pretty sure demons can't go inside a holy place. Consecration is its protection."
More text appeared: _Chapel of the Holy Dove._
Nothing more came up. The screen went blank. "I guess that's our assignment," I said. "Find out if a surrogate is targeting it, stop him/her/it before the chapel burns down. A holy place can't be a domicile, but there's nothing to stop a demon from getting busy from the outside with gasoline and a lighted match or a Molotov cocktail."
First we had to find out what and where was the Chapel of the Holy Dove. That turned out to be easy. The regular internet was working again, so Remi typed the name into the search field. Photos along with info came right up.
There wasn't much to it. It was twenty miles from Flagstaff, neither large nor imposing. Just a small, simple A-frame building, with a shingled roof and a red-painted door hosting a cross made of mirrors.
"It burned once before," Remi said, "Got rebuilt in ninety-nine, all volunteer labor."
"Demons?"
"In ninety-nine? The hell vents didn't open until recently. Probably just a careless human."
"Soooo, we go out there to see if you can get a feel for any demonic activity?"
Abruptly the screen went dead. Remi depressed the power button a couple of times. Nothing happened.
Well, until one word appeared: _Now._
I ran a hand through my hair. "Well, at least it's daytime."
"Maybe not so good. Other people will be there."
I rolled my head back until I could see the ceiling, released a noisy breath. "Collateral damage."
* * *
—
Once again I said I'd take the bike. Remi mentioned that it was a waste of gas to use two vehicles. I just looked at him, threw a leg over the saddle, pulled on gloves and helmet. I figured that was message enough.
The ride out U.S. 180 to the chapel reminded me of back home, which I was not expecting. Back home in Oregon with its great tracts of forests, yeah, but here? Ponderosa pines, fir, aspen, juniper. Brilliant blue skies. A winding asphalt road. I felt sorry for McCue, walled inside his big pickup. Bike and an open road. What more can you ask?
Twenty miles out, I saw the sign on the right for the chapel. I pulled off, rolled slowly into a cinder parking area that was actually just a wide spot in the road. Tire tracks proved that people did park here, but with no delineated spaces. It was just dirt and cinders.
I was beginning to hate cinders. Maybe I should have a talk with Ganji about it.
Up close, you could see just how small the chapel was. And its simplicity. At the west end, the end with the red door, the entrance was quite low. Beyond, soaring upward on an angled line, the roof gave way to windows. The entire east wall was made up of glass. I climbed off the bike, waited for Remi to park his truck, and then we walked inside. The view was astounding. The windows and A-framed roof embraced the San Francisco Peaks, the volcano cluster Ganji wanted to wake. Meadow, wild flowers, scatterings of pines. And the A-frame shape was carried through with interior, sharp-angled upside down V-shaped beams marching from door to windows. No formal pews, just wooden benches with backs, the kind you'd find in parks.
Graffiti covered the interior. Every inch of wood contained commentary in markers, pens, even rough knife-blade carving, and notes pinned to the walls. People from all over the world had left their marks. At first I was shocked that people would deface a chapel so badly. But as I walked the gravel floor and read a few of the notes, the messages written on wood, I realized most were of hope, prayers for others, or requests asking for intercession. There was great respect for the little chapel built atop mortared natural rocks.
No one else was present. We stood alone within the walls of stone, wood, and glass, upon gravel in place of manufactured flooring. The chapel featured a modest wooden pulpit, also covered with graffiti, a low stone altar, and a rough-hewn wooden cross mounted high in front of the windows, as if it were floating.
Remi found handouts explaining how the chapel had come to be built, then rebuilt after the fire, information he naturally shared with me. He ended with, "Weddings are held here all the time."
I walked to the windows and stood there gazing upon the Peaks. As I had on the mountain, when Grandaddy had hiked my hungover ass up and down Fatman's Loop, I sensed the peace of the place, the sanctity. Here, there was no _koyaanisqatsi_ —no life out of balance. Life in this chapel was very much _in_ balance.
Which explained why surrogates wanted to burn it down. Out of chaos comes Lucifer. Burned churches, synagogues, mosques, even this little roadside chapel, would make people think less of worshipping God and more of Lucifer. And to think of him was to open the door.
I stared at the cross. Thought again about what McCue and I knew now of Grandaddy, of ourselves: That he was a seraph, and we were born of heavenly essence, of celestial energy.
I asked what the girl in that kid's book had: "Are you there, God?"
God did not, as far as I could tell, offer any answer.
Remi was wandering around the chapel, reading notes and carvings. "I don't feel anything," he said. "I don't sense any demons."
"But you're not running at full capacity yet."
"We're in a _chapel_ ," he clarified. "Consecrated ground. Any sense of surrogate activity around here ought to be obvious, regardless. It would be like a black stain on white linen."
I shrugged. "Maybe the demon hasn't arrived yet."
"Well, if it wants to burn down the place, let's make sure it can't."
As Remi turned and headed for the red door, I called, "How the hell are we supposed to do that? And wouldn't someone _else_ have already thought of it?" But McCue was gone, so I followed and found him up in his truck cab, pulling out Lily's duffle. He dug through it, came up with two flasks I recognized: holy oil and holy water.
"It's possible." He shut the passenger door, "Maybe after the fire the new chapel wasn't reconsecrated. And even then, if they did, it may only have been the building _itself_ , not the land surrounding it." He tossed me a flask. "You man the oil. I've got the water. Let's walk a large circle all around the chapel, say twenty feet out from the building, and dribble oil and water every three feet."
All things Biblical I left to Remi, so I didn't object, didn't ask questions. Just eyeballed the twenty-foot distance from the chapel, and began to walk in a large circle. Remi started on the other side. We met, passed, wound up back where we'd begun. Both of us stared at the chapel.
I asked, "What now? Anything more?"
Remi nodded, then spoke quietly. But not to me.
_"We present this building to be reconsecrated for the worship of God and the service of all people."_
I waited.
_"This house shall be known as the Chapel of the Holy Dove."_
It was markedly quiet all around us.
And finally, after a fair bit in Latin, he added, in English:
_"By the power of your Holy Spirit consecrate this house of your worship._
_Bless us and sanctify what we do here,_
_that this place may be holy for us and a house of prayer for all people."_
And that was that. Remi said the land around the chapel was consecrated, and the chapel itself _re_ consecrated.
"So nothing inside the circle will burn?"
"Nothing inside the circle will burn."
A minivan pulled up. I glanced at it, saw Remi concentrating, eyes distant as he went inside himself. "Anything?"
He shook his head. "There might could be a demon—but there also might not."
The minivan doors slid open and a family got out. Man, woman, older woman who was probably a grandmother, three kids. I thought of the family the black dog had killed, and I wanted none of _this_ family to be at risk—nor did I want any of them to be a demon in disguise.
They all went into the chapel, which answered _that_ question.
A moment later the mother came back out. Petite, brown-haired, wearing a University of Arizona football jersey. She looked at the truck, the bike, and us. Clearly noticed the motorcycle leathers, the cowboy attire. In her eyes was reticence, confusion, a hint of shyness, but also a need to know something.
"This is going to sound really weird," she said, "but are you by any chance Gabriel and Remiel?"
I felt a cold finger at the base of my spine. "Why?"
"Are you?"
"Yes," Remi said.
"Wait." She put up a delaying forefinger. "He gave me something for you."
Remi and I exchanged concerned glances as she went to the minivan, unlocked it, reached in and pulled out a manila envelope. She relocked the vehicle, then walked back to us.
"We stopped for gas," she explained. "Just on the outskirts of Flagstaff. A convenience store. The cashier told us about this chapel, said it was very pretty, and added that if Gabriel and Remiel were here—cowboy and biker; he was explicit—I was to give this envelope to them." She held it out, eyes doubtful.
I tamped down foreboding, managed a smile. "It's a scavenger hunt," I told her, manufacturing a false amusement. "The clues come in the envelopes." I took it from her. "Thank you so much. The cashier's in on the game, too."
That eased her concern. "The kids love scavenger hunts!"
And then one of those kids started yelling inside the chapel, and she excused herself to deal with it.
As before, our names were written on the outside. Nothing more. I glanced at Remi, who nodded solemn encouragement. I was wearing gloves, so I used my KA-BAR to slit the envelope open.
A note, prepared as all the others were. The message was simple: ' _More to come.'_
I looked inside the envelope again, discovered something else. Pulled it out, held it so both of us could see it.
Oh, holy shit.
Remi was swearing also.
The 8 x 10 color photo was of a young woman, very dead. Blood obscured her face, soaked half her clothing. Her limbs were sprawled.
I turned the photo over. In red ink, two scrawled words: _Mary Ann._
Remi wasted no time _._ "We've got to go to the cops."
"Wait." I thought back, recalling the other two notes. "The first said, _'I know who you are,'_ the second, with bolded emphasis, ' _I know_ _what_ _you are.'_ Whoever this is—I'm assuming a demon, as Ganji noted—it's playing a game _with us_. If we go to the cops, there will be all kinds of awkward questions we can't answer—and I'm pretty sure we'd become suspects. After all, I've got a record."
"Then we mail it," Remi's words were clipped, his drawl reduced. "Clean it of prints, send it to the police. Let them take it from there."
I held up one hand, wiggled my fingers. "Wearing gloves."
"So let's go back into town, get it done."
I nodded, slipped note and photo back into the envelope, placed it carefully in one of my saddlebags. I had no affinity for cops, after eighteen months inside, but McCue was right. This was police business.
# CHAPTER THIRTY
As we headed down the road with its elegant asphalt curves winding through the forest, I found my mind making the jump to lightspeed again.
It had to be a demon playing us—unless it was an angel. I hated the latter idea, but Greg had sounded absolutely certain Remi and I were in pretty deep, that we'd be used by anyone who wished to manipulate us. Then again, what if _she_ were the one playing us? Some Grigori had fallen, she claimed, and made Nephilim by sleeping with human women. For all I knew, _she_ was a fallen Grigori with an agenda all her own.
I thought again of Grandaddy. There was nothing in the man now, nor had there ever been, that made me distrust him. But he himself had alluded to Lucifer's greatest trick—convincing people he didn't exist. Because then he could work in private, observed by no one until it was too late.
What if _Grandaddy's_ greatest trick was convincing us he was working to save the world, when he wasn't?
And why on earth wouldn't he?
I thought of the magic phone tucked into my jacket. Perhaps it was time we summoned Grandaddy, instead of him summoning us.
And then a woman on the right-side shoulder darted out into my lane.
Oh, _shit._
Laying down a bike is not something anyone ever wants to do. You veer, you swerve, you do everything you can to keep it upright, especially a big twin Harley weighing over seven hundred pounds. Laying down a bike also is _nothing_ like in the movies. For one thing, it hurts like hell. And a human body can break. The late Evel Knievel could attest to this after his attempt to clear the fountains at Caesar's Palace in Vegas. He broke more than forty bones.
So I veered sharply, then again even more sharply as the woman, who apparently had a death wish, or was drunk, or high, or just downright _stupid_ , took three more steps into the road, turned toward me with her hands up. I hit the brakes, tried to keep the bike upright with sheer strength and balance, but I lost it.
I lost it.
You can't perform miracles with a big twin Harley, can't stop as short as with dirt bikes and the like. I really, really wanted not to hit her, so this meant I needed to control the machine the only way I could: laying it down in order to avoid her, and also not destroy myself in the doing of it. I hit the rear brake because now I _needed_ the bike to go down, in hopes of sliding by without clipping her, rather than plowing right into her.
In seemingly slow motion the rear tire locked up, the bike slid sideways, tipped, leaned over—gravity was winning—and the rear assembly rotated forward. Both wheels now faced the way I'd been going because the bike was turned sideways across the lane.
I saw her face, very pale, surrounded by clouds of black hair. She appeared to be in shock. Well, yeah; I would be, too, with a huge motorcycle bearing down on me.
All I could think about were two things: Getting my left leg free before it got turned into ground-up hamburger sandwiched between road and bike; and keeping my head up. Helmets do a lot of good, but they're not impervious.
So as the bike went down I yanked my left leg out of the way. Metal hit the ground, sparks flew as the foot peg dragged. I slid with the bike a few feet, then got loose from it and commenced an awkward, involuntary series of tumbles.
I flipped, flipped again, rolled, strained my neck as I tried to keep my head off the ground. Asphalt grabbed at leather, which slowed me, but also changed the arc of my body, this time me _without_ the bike, and thus considerably lighter than I was with the Harley. I flipped off the road entirely. I ended up on the shoulder of the lane across from mine.
Oh crap. Oh Jesus. Oh shit.
A woman. A _woman_ in the road.
I tried to lever myself up. Saw my bike, uncontrolled, miss a curve and slide straight ahead at 50 mph, where it disappeared into trees and underbrush.
By this time Remi had stopped and was out of his truck, running across the road toward me. I pushed myself upright with my left elbow until I realized that was not wise, since I'd gone down on my left side. I tried to suck back the pain on a series of stuttered breaths.
As Remi arrived, I blurted, " _Woman._ She okay _?"_
McCue knelt at my side, eyes frantic. "Hey—hey . . . don't move. Okay? Just don't move. I'm calling 911."
"Wait," I said urgently. "Go find the woman. See if she's okay."
" _What_ woman?"
"The one in the middle of the road. I didn't—didn't just fall over, you know. Go look. I'll call."
McCue was unconvinced and focused strictly on me. "I didn't see any woman."
I moved a little, wished I hadn't. "Well, _I_ did. Go look. Go back and look. I'll call, I'll call. If she's hurt, the ambulance can pick up both of us."
I didn't want to think about her actually being _dead_. But I didn't recall contact. Maybe she'd jumped out of the way even as I swerved and laid the bike down.
Remi stared hard at me, assessing, reluctant. He still didn't move to follow my instructions, and now I was frustrated.
"I'm okay! I don't think I broke anything. Bruises, scrapes—I'll live. Got my left leg out, kept my head off the ground. _Go look for the damn woman!"_
He didn't like it, but he rose and started jogging back the way we had come. I fumbled at a jacket pocket, pulled a phone free, hit the Contacts screen.
I didn't call 911. I called Lily.
After five minutes or so, Remi jogged back. Once he reached me he knelt and said, "There was no woman. Nothing, Gabe. No scrap of clothing, no tracks—nothing. It was the right place—I saw your skid marks."
"I saw her, dammit!"
"Well, I didn't."
I let it go. We could come back another time. If she wasn't lying in the middle of the road, or unconscious on the shoulder, I probably hadn't hit her. Or else, if I _had_ struck her, she'd been rescued by a Good Samaritan, though I didn't recall any vehicles going by.
"Okay." Damn, I hurt. "Get my helmet off. Everything moves okay; there's no back or neck injury. It's okay to take it off."
Remi freed me, set the helmet aside. "Don't get up!"
Because I was trying. I got as far as sitting, stared down the road. "My bike."
"Forget about the bike! You need a hospital."
I made an awkward attempt to climb to my feet. Remi called me all kinds of names, but in the end he helped me up.
"I'm okay," I told him again, holding myself stiffly and trying not to overbalance one direction or the other. "I'm—I'm gonna hurt like hell tomorrow, a few days after, but this is why I wear all the leather and a helmet. This isn't the first time I've wrecked."
Though none of them had been this . . . this extravagantly dramatic.
McCue said nothing, just stared at me.
I ran a gloved hand through my hair, scrubbed, unstuck it from my head. "Let's go look for my bike."
"We're _not_ looking for your bike, dammit! We're waiting for the ambulance."
I improvised. "They said an ambulance is thirty minutes out. And it'll take twenty or so back."
"Christ," Remi muttered. "Okay, let's not wait. I'll take you to the hospital myself. Can you make it across the road?"
I made it across the road. Remi hovered, but saw I was walking okay, if stiffly. I managed to haul myself up into the passenger seat, and McCue closed the door for me.
As he turned the engine over, I inspected my left-side leathers. Sleeve was badly scuffed, but the pants showed significant damage with leather scraped thin from hip to knee. My left boot was deeply scuffed. I rubbed a hand up and down my left arm, checking for pain, then pulled off my gloves.
Yup, I hurt like hell already. Bruised underneath the leathers, probably some scrapes as well, but I wasn't broken.
We were about ten yards down the road when I asked McCue to stop, and it pissed him off. I'd never seen him angry before, just irritated.
He shot me a disbelieving glance, then turned his attention back to the road. "I'll come look for your bike tomorrow _,_ okay? I'm just thinking about _you_ right now."
I explained the facts of life in Motorcycle Land. "If we leave it, even if they can't lift it—and they can't; it's a heavy son of a bitch—people will steal parts. Strip it. I need to see where it is, see if it's hidden well enough until we can get someone out to collect it." I paused. "It left the road right about here. See the broken branches?"
"Will you shut up if I stop?"
"Yes."
McCue pulled over on the shoulder and parked. "Stay here. Wait here. _I'll_ look. I don't need you staggering around in a forest and keeling over when it all catches up to you. Don't want to haul your tough-guy black-leather ass back to the truck."
I shut up. Remi scowled at me, then went off to track the path of my bike.
I leaned my head against the window. Okay, yeah, the truck seat was better than hiking around a forest. The aches were ramping up.
Remi remained gone for some time, and I began to fret. And when he showed up, his expression verged on blank.
"Nobody can strip it," he told me. "No one can _reach_ it."
"What?"
"There's a ravine."
I was horrified. "My bike's in a _ravine?"_
_"_ And there's a creek at the bottom. Lots of boulders."
I felt numb. "No. No no no. You're shitting me."
He shook his head. "I'm not. I'm sorry."
I planted my elbow on the arm rest, stuck my head in my right hand, and closed my eyes. I wanted to mourn my bike. But I hurt too much, and I was concerned about the woman.
The woman who _had_ been there.
And then it occurred to me that maybe she hadn't been there at all.
Or, at least, not a human.
# CHAPTER THIRTY-ONE
It was quiet in Remi's truck. I heard only the engine, the rumble of the road. I'd carefully shifted positions any number of times, uncomfortable. I'd become more certain that I hadn't broken anything, but skin and muscles were definitely insulted, and it was difficult to remain in one position for much longer than a few minutes.
I was biding my time before I told Remi we weren't going to a hospital. I had my ammo ready to go.
I shifted again. Needed distraction. "How about some tunes?"
Remi shot me a glance, smiling, then reached forward to turn on the radio.
Sure enough, shrill, twanging country. Should have known it. I thought about asking him to find another station, but it was his truck, after all, and I'd learned long before that, in the succinct words of a TV character: Driver picks the music, shotgun shuts his cakehole.
Then that song came on. "Okay," I said, on a stuttered breath, "tell me the truth."
He glanced at me, frowning. "Yeah?"
"Mamas and cowboys and babies. You know— _this_ song. You know it?"
"Willie Nelson, Waylon Jennings. It's a classic."
I thought about it a minute. "Doesn't it bother you? A song suggesting babies should never grow up to be cowboys? I mean, isn't that against the code, or something?"
Remi smiled for the first time since I'd wrecked my bike. "Well, that's not actually what the song is about. And a lot of us flipped it."
I started to turn my head toward him, stopped because my neck hurt. "Flipped it?"
He was looking through the windshield, but his smile grew wider, broke into a grin. "Mamas, don't let your cowboys grow up to be babies."
Hah. Very clever. And it provided the perfect opening. "So, you've been hurt a lot riding all those animals?"
"Bulls and broncs? Well, for me most injuries have been bruises, maybe a couple of broken ribs now and then, stretched ligaments in an arm. Other cowboys sometimes run into something a little more serious, but for the most part it's nothing we can't take care of ourselves after a quick look-see with paramedics at the arena."
"You all opt for the hospital when it's bumps and bruises?"
He rolled his eyes. "Okay, okay. I get it."
"Suck it up tough, huh? Cowboy tough?"
"We _are_ tough."
"Okay," I said amicably, "the same can be said about bikers. And that's why I called Lily, not 911."
He shook his head. "You're an asshole."
I grinned. "Yes. I am."
* * *
—
By the time we reached Lily's rig at the RV campground, I could barely move. Remi opened my door, took a good look, said he'd get me down.
I told him to go to hell.
He ignored me.
And so it took Remi's aid to get me out of the truck no matter what I preferred. Lily descended the motorhome steps part way through the attempt. She eyed me up and down, assessing my condition, then told Remi to take me into the garage and have me sit on the cot.
The first order of business was to learn medication allergies. I said I had none. Then it was family medical history—and I told her, rather sharply, that the history probably didn't matter because, hey, I wasn't _human_ , was I?
Then she told me to take off my jacket, which I did with great care. Followed by holster. Then, I was told, it was t-shirt and leather pants.
I stared at her. "What?"
"Take off your t-shirt and your pants. I need to check out bruises, scrapes, etc."
"I'm bruised, and I have scrapes. Okay? Nothing new."
She was adamant. "T-shirt."
Remi stood all of two feet away, watching over Lily's shoulder. He'd put his hat elsewhere, and his head looked naked, hair standing up from his efforts to undo hat hair.
I sat on the edge of the cot, let down from its position against the wall. "I just want a hot shower, then ice."
"And you shall have them," she said. "But first let me treat the contusions and scrapes. There is no weakness in it; d'ye think many brave men have scorned a women's touch after battle?"
Well, me laying a bike down wasn't exactly battle. "I really only need a hot shower and a bed. I'll just head back to—" And I stopped. Because now I lacked the freedom of my own transport.
Remi wasn't laughing or smiling. "Let her take a look," he said, "and I'll chauffeur us both back to the bar."
I'd lost eighteen months with the bike, a year after I'd bought it. And it wasn't just the _bike_ in and of itself, but what it represented: the me who had been, the me I yet might be. The me I'd never be again.
I made a valiant attempt to shed t-shirt, which was seriously uncomfortable since it required lifting my arms. The true difficulty arose as I realized that I simply couldn't _do_ for myself. I needed Lily. I needed Remi. They worked me out of the tee, and I was bare-chested.
Okay, so a few bruises and scrapes. Minor road rash, seeping lymph-diluted blood a little. My left elbow, a strip down the side of my forearm, tenderness low on the left-side ribs. Red bruises were blooming as broken blood vessels made themselves known.
Lily used alcohol wipes on the scrapes, which only added insult to injury. I sucked in a hissing breath, then glared.
She smoothed antibiotic ointment over the scrapes. "All right. Off with the pants."
"I'm not taking off my pants!"
She shook her head, scoffed. "I've slept with a thousand men. D'ye think you've got anything they hadn't?"
I looked at Remi in appeal, but he was no help at all. In fact, he was studiously looking elsewhere.
I caught Lily's eyes and glared. "You may have slept with a thousand men, but none of them was _me._ "
After a startled moment, she laughed in joyous abandon. Then she told me I could take off my pants willingly, or she'd cut them off.
Okay, taking them off was going to be a challenge. I was stiffening more and more by the minute. But I got them unzipped and shoved down past my hips, refusing help. Fortunately my boxer briefs were intact.
More contusions. Point of hip, knee, also seeping a little. I had a few dings on my right side from tumbling, but most of the damage was confined to my left.
"Aren't you the fortunate man," Lily observed, wielding wipes again, followed up by the ointment.
"Considering what could have happened, yeah." I worked the pants back over my hips, asked for my t-shirt. It wasn't easy to put on, but I managed it.
Lily looked me up and down. "You know the RICE protocol?"
"I've been banged up before, you know. Rest, Ice, Compress, Elevate. I can't very well elevate my whole body, but I'll go for the others, okay?"
"You can stay the night here," she offered.
Just what I wanted; sharing grunts of effort, the hissing expulsion of breath if I moved too quickly, with a woman who reveled in war. And I was pretty damn sure her offer wasn't an _offer_. "Thanks, but no. I'll head over to the roadhouse." I looked at Remi, hating my lack of freedom. I'd only just gotten it back.
He looked back. "If it were me, you wouldn't fret over giving me a ride. Well, I'm not frettin' over giving _you_ a ride."
He didn't understand. Unless he'd been in prison, he couldn't. Everything there was regimented. _Everything._ Now I was free of it. Riding the bike almost 1,300 miles had given me myself back.
Lily dug something out of one of the cabinets, handed me two pill bottles. "Tramadol," she said. "Painkiller. Be sure to take it with food, otherwise you'll vomit all night. The other is Flexeril, a muscle relaxant. You'll be stiff tomorrow."
In the movies, the hero would turn down medication, intending to tough it out. But I was no hero, and I knew how I felt now, how I'd feel the next day. I took the bottles.
"RICE," Lily reminded.
"I got it." Now I was getting testy and tried to soften my tone. "Thanks, Lily."
Remi pulled the truck remote from his pocket and walked back into the RV living quarters with me on his heels. I didn't wear either holster or jacket; less to take off when we got to the Zoo.
McCue had the RV door open when he stopped short on the first step. I started to ask him what the hell he was doing, since I almost walked into him, but then he descended the rest of the steps, and I saw at the bottom a manila envelope.
Shit. _Shit_.
I turned, walked back into the living room and eased myself into the recliner. Remi came up behind me, pulled the door closed.
Lily looked perplexed. I warned her this wasn't going to be pretty.
McCue tore open the envelope, removed a note, read it then handed it to me. Once again letters cut from various sources. Two words only: _And another._
Remi slid the photo from the envelope, looked briefly, closed his eyes. I took it from him.
Another woman, messily dead. I didn't linger on the image, just turned over the photo. Again in red ink, a name: _Annie._
Lily took the photo from me and studied it. I remembered then that as Goddess of Battles such images were familiar and probably would not bother her.
She said, "This is not new to you."
"The envelopes began arriving a few days ago," Remi explained. "At first just notes. Said 'I know who you are,' then _what_ we are. Now two photographs. We were going to mail the first to the police, but it was in Gabe's saddlebags and the bike's in a creek at the bottom of a ravine. Though now we have this one. You see anything in the local papers?"
"No," Lily said firmly. "Nothing at all about murdered women, and you know the news would be full of it with reporters descending on us. The time is spent on the killings at Wupatki." She handed the photo back to Remi, who slid it and the note back into the envelope. "I don't think this happened locally."
Remi nodded. "I'll get on the computer, try a few searches under various keywords. I might turn up reports."
I shook my head, anxious. "How is this guy finding us? It's almost like he's tracking us with a GPS unit, or stalking us. Two envelopes delivered to the Zoo, one to us at the chapel outside of town—" And then I remembered. "Our beacons. Grandaddy said hell could find us."
Lily's expression was solemn. "And so you are found. And followed. You are safe here, safe in the Zoo. And at Wupatki, because you cleared it."
Well, no—Greg had. But I opted not to say anything.
"Sunset Crater," Remi said, sounding tense. "It was a second black dog. We killed it, then cleared the area."
"The chapel," I added. "We installed a protective circle. Remi reconsecrated the building, then consecrated the earth surrounding it. It can't be burned."
"And so the photos." Lily nodded. "I believe this is called _gaslighting_. Psychological manipulation."
"But why us?" I asked. "I mean, we're barely born. Newbies, as you said."
"Oh, not just you," Lily replied. "I'm sure this is happening elsewhere, if not in the same fashion. But _especially_ you, yes. Newbies or not."
"Especially us?" Remi asked. "Why especially us?"
Lily looked at each of us. "You were born on the same day, at nearly the same hour, minute, second. That is of significance."
"Again, why?" I asked.
"Because it's never happened before. No one, including the angels, quite knows what you are, or who you'll be. I imagine you are at the top of the demon hit list for no reason other than that."
I felt a chill pimple my flesh. "And the angels? Are we on their hit list, too? If no one knows what or who we are, including the angels, wouldn't it be safer just to kill us?"
Lily shrugged, utterly matter-of-fact. "Oh, I'm sure some will try." She flicked a bright glance at Remi. "Might could even succeed."
# CHAPTER THIRTY-TWO
The ride back to the bar, a mile away, was short. The only thing said at all was a comment Remi made. "Guess we don't have to mail the new photo to the cops."
Remi's computer search might turn up something. I thought at first it would indicate the surrogate's whereabouts, then remembered we received the notes and photos locally. I supposed it might be two demons, with one doing the killing while the other delivered the envelopes. Hell, if we were at the top of a hit list, maybe dozens of demons were after us. It all began the night Remi and I put on the rings; the next night the hot blonde with the whackjob eyes had nearly killed me.
Or maybe it was angels.
I didn't want to think about that. I just didn't. Angels were supposed to be the good guys. But Greg had planted the idea, and it hadn't gone away. On one hand, if Grandaddy were right, it made no sense that angels would want to kill us—unless what Lily said was true, and Greg, that none of them knew what McCue and I were capable of.
Then again, _we_ didn't know what we were capable of.
The parking lot was crowded as we turned into it. At twilight, bar-goers headed out, even on weekdays. It was a social activity, and offered escape for many into beer or the hard stuff.
It was now habit for us to park in the back, though this time it was only the truck. And that made me anxious all over again. Landing in a creek might have softened the bike's impact. It would most likely need a lot of work, depending on the damage, but some motorcycle mechanics are wizards.
I opened the passenger door, looked at the step down. Oh, this was _so_ not going to be fun. Remi offered to help, but I just shook my head, reached carefully for the overhead handle, began to slowly lever myself out.
As expected, and despite the brevity of the ride, I'd stiffened. I felt a hundred years old as I worked my way down from the seat to plant feet on the ground.
With great care I stretched, rolled my neck. Muscles and bruises protested, but I needed to move. While bedrest was recommended, I knew from experience I'd feel better if I didn't just lie around.
Since I had no desire to spend the night puking, I did indeed intend to grab some dinner so I could take the tramadol. But as we walked into the bar, with billiard balls clacking, loud country music playing on the jukebox, people laughing and dancing, I resolved that dining in one of the booths downstairs was not a good idea.
Remi realized it, too. "I'll have them fix something, and I'll bring it up to you."
I nodded my thanks. "Something juicy."
"Steak, or burger?"
"Definitely steak. Medium. Baked potato, loaded. Corn on the cob if they've got it."
"It's a cowboy steakhouse/dancehall. They'll have it."
"Okay. Just don't flirt with the waitress and let my dinner go cold."
He grinned. "Only if she flirts first."
Not fun climbing the stairs because it put pressure on my body in different areas. But I made it, dropped off the holster and jacket in my room, walked down the hall to the common area. I figured while I waited for dinner I could start Remi's search for murdered women.
It booted into a black screen. I thought maybe the video card had gone bad, but then a name appeared: _'La Llorona.'_
When Remi brought dinner he had two of everything, set the plates down on the table. He saw me staring at the screen and looked himself. "La Llorona? Well, it's Spanish for _weeping woman_ , but after that I've got no clue."
I remembered he spoke Spanish, so he knew to turn the double 'L' into a 'Y' sound. "It's a legend," I said.
"I figured."
"Hispanic in origin, but has since been adopted by others. There are variations, but the basic story says a woman killed her two sons by drowning them in a river, then felt terrible remorse and now wanders around looking for them. Mourning them. Weeping."
Remi collected a beer out of the fridge for himself though nothing for me. Painkillers and booze don't mix. "So she's a ghost," he said.
"So she's a demon."
He stopped dead, saw the expression on my face. "What? What is it?"
I'd been thinking about it, knew I was right. "The woman who ran out into the road. White shirt, white pants, black hair. La Llorona is always seen in white." I looked at him. "She came out of nowhere, ran right into my lane. It's why I wrecked the bike, why she wasn't hurt, why you couldn't find her."
Remi took off his hat and set it on the table upside down, away from the plates. He deliberately avoided meeting my eyes.
"I saw her." I was absolutely certain. "It was her, Remi."
"I'll take your word for it, but I didn't see anything."
I shrugged, wished I hadn't. It hurt. "I don't know the reason for that, but she was there. White clothes, black hair."
Remi pulled out his chair, sat down to start in on his steak. When I didn't move, he glanced up, stilled. "You want to go back there tomorrow."
"Apparently we're supposed to," I pointed out. "The name was sent to us. If we weren't intended to look for her, why would the name appear? She's a ghost, a demon, and we're supposed to take her out. But not exorcism; if the story is true, the host is much too old to survive. So it's guns or knives."
He paid deep attention to cutting his steak. "Not tomorrow. You need the rest."
"I'll be fine. Painkiller tonight, ice, the muscle relaxant tomorrow. And anyway, we need to kill this demon before it can cause more havoc."
"I'll go. On my own."
Something rose up in me, something imperative. Not just a wave, but a tsunami. "No." I didn't know what it was, just that I _couldn't_ let him go alone. Even the idea of it was highly unsettling. _Primogenitura_ , maybe, that need to protect. "Besides, she's not visible to you. I'll be your seeing-eye dog."
In consternation, he stabbed his fork so hard into a piece of steak that tines scraped on the plate. He told me to type in a simple question, said what I should ask.
I stared at him, then did as he suggested. _Gabe too?_
The screen replied: _Always both._
I thought that over, typed in something else. Kept it clean. _And if we are entertaining? Ladies, that is?_ Because I sure wasn't going to have an audience.
After a delay, the screen blanked, flickered, then disappeared and left me with a normal website.
Remi's tone was philosophical. "Well, I reckon that can be taken as either a 'yes' _or_ a 'no.' I prefer the latter."
I dragged the chair back to the table, sat down with care. "And while we're out there, after we kill the demon?" Remi asked a question with raised eyebrows. "I want to find my bike."
He opened his mouth to protest, saw my face and took the better route. He didn't try to talk me out of it.
Smart man.
The tramadol took the edge off and, thankfully, did not make me sick. I didn't want to contemplate how that would feel after my contact with asphalt earlier. I left Remi doing a search on missing and dead women who fit his parameters and went to bed.
The only one sharing it was ice.
* * *
—
Remi had promised to wake me. He didn't. I woke up on my own, discovered it was ten o'clock and levered myself out of bed. Yup, very stiff, very sore. I wobbled my way into the hall, swearing I'd kill McCue if he'd gone by himself after the message on the computer: _Always both._
Then I found him in the kitchen making coffee, and revised my intention. "I thought you were waking me up."
"I let you sleep in. We've got time."
"Did you find anything about murdered women online?"
"I found _too_ much about murdered women online. But nothing identifying a Mary Ann or Annie. Well, that I saw. I didn't have much to go on."
I cracked multiple eggs into the skillet, pushed two pieces of a sliced bagel into the toaster, poured myself coffee.
"How you feelin'?"
I couldn't resist. "It's not the years, it's the mileage."
"You got pills for that."
"So I do. And I will not shirk them."
Remi opened his mouth to ask something more, then we heard the familiar five tones from _Close Encounters._
"Yours," I said. "My cell's in the bedroom."
Remi answered, went very still as he listened, then said, "Thanks, Ganji."
His expression was odd. I got a funny feeling. "What is it?"
"Another envelope. This time tucked under one of my windshield wipers. Ganji saw it when he took the trash out."
I pulled the skillet off the burner and slid it onto a cold one, popped the bagels up, said, "Let's go find out what this one says."
"You eat. I can bring it up to you."
"I've lost my appetite." And I had.
I took the muscle relaxant first, which had the effect of oiling my rusty joints, then got myself downstairs without falling. Remi was behind me.
Ganji was sitting on a stool in front of the bar. We joined him, and without speaking he slid the envelope down to me.
I dreaded it. I just knew what was coming.
I opened it, pulled the note free. Just like all the earlier ones. This time, it said: _And another._
I couldn't look at the photo. I just couldn't look. I handed off the envelope to Remi. After a moment he removed a photo, glanced at it quickly, then turned the picture over. Red ink, a woman's name: _Elizabeth._
I dropped an F-bomb, and Ganji's brows rose. Remi set the photo face down atop the envelope. "Mary Ann, Annie, Elizabeth. Not exactly unusual names," he said. "Without last names, I doubt we'll find them."
"If he isn't killing locally, how is he putting the envelopes all over the place?" I asked. "Can demons translocate, or is there an accomplice?"
"Not only that," Remi said, "but the duration between deliveries is getting shorter and shorter. He's escalating."
I paced because I had to, and because it knocked more rust off my bones. I was loosening up but not quickly. "What's the end-game? What do these photographs _mean?_ Why involve us?"
"Maybe he figures that's what he'll do to us," Remi said. "He's previewing his work."
"We're not women, so probably not targets."
"Seeing photos of women dead is worse than seeing photos of dead men."
He was right. I didn't want to look at grisly pictures of dead men, but it definitely bothered me more that the victims were women. "Maybe he thinks it'll flush us out," I ventured. "We're safe here, at Lily's, those other places. But that doesn't make sense. If we keep taking assignments, doesn't that make us vulnerable? And easy to get to us, I'd think."
Remi shrugged. "Maybe he doesn't want to get to us yet. Maybe he's just screwing with us. I mean, who knows what kind of games surrogates play, or how their minds work?"
I thought about it. "Maybe he figures it'll unnerve us, be easier to take us when the time comes, as the anxiety builds and builds. He's playing with his food."
Remi look vaguely ill at that.
"We've got to find him," I said. "Or her. Kill the mofo before we get nailed, before the death-toll hits double digits."
"How do you reckon we can do that?"
"Clues," I answered. "We can talk to people. Details on the envelopes, the paper used for the notes, the cut-out letters, even the ink and handwriting. Even the photo paper the image is printed on."
"We're not the police," he said skeptically, "and this sure as hell is not _CSI_."
"No, but there are specialists. And maybe, with a nudge from Grandaddy, they'll cooperate with two guys who aren't angels yet." I paused, stared at Remi. "Hell, why didn't we think of this? We should have asked Grandaddy first."
Remi pulled his the magic phone from a pocket, hit "Contacts." His cadences suggested no one was on the other end. "Grandaddy. Hey. Remi. Listen, Gabe and I got some questions. We need a little help. Can you come over to the Zoo tonight?" He disconnected, slid the phone back into his pocket. "Eat something, Gabe. Take a shower. We don't know how long it's going to take to kill off La Llorona _or_ get down to your bike. Which proves you've got a big hole in your screen door."
_"What?"_
"Texan for crazy," he said. "Cuz that's what I reckon you are."
# CHAPTER THIRTY-THREE
Lily's starter kit provided another box of cartridges and bullets. To be on the safe side we bathed them in holy water, then oiled them. No spit was necessary this time.
My Taurus, shoulder holster; his Taurus, belt. My Taurus, iron buckshot; his, silver bullets. Then I rubbed holy oil into the blade of my KA-BAR. It wasn't silver, wasn't iron, but now it was half-assed blessed.
"Should I make the sign of the cross over it?" I asked, thinking of a little more celestial backup.
"You Catholic?"
"No. I'm not anything."
"Well, I was raised Southern Baptist. We don't do that."
I thought about it. "I guess it doesn't matter. I mean, God is supposedly everywhere. Maybe he's even in our guns."
Remi was feeding rounds into the chambers. His tone was dry. "Yup, I don't doubt at all that God is a bullet." He closed the cylinder, tested the rotation. "Now let's get ourselves on down the road and take care of this woman you saw."
"And my bike."
He slanted me a glance from under lowered brows, then headed toward the stairs.
* * *
—
When I told Remi again to make sure he reacted as soon as I said I saw the woman, he suggested I shut up.
I protested. "Well, _you_ can't see her."
"We have established that," he said. "As you've told me _three times_ : You see her, you tell me, I hit the brakes. Hell, it would be easier just hitting her with the truck."
"The truck's not iron or silver."
"You sure you didn't clip her with your bike?"
"It's not iron or silver, either."
Remi had latched onto a topic. "I mean, can ghosts feel things? The demon inside, I mean. And then if it's exorcized, does the host feel pain?"
I shifted in the seat, hid an involuntary grunt of discomfort. The shower and muscle relaxant had helped, but I'd be creaking a while longer. "Mark that down as another question for Grandaddy."
McCue took us back to the chapel to be certain we wouldn't miss the area where the woman caused my wreck, then headed us back toward town. We passed a few cars, but not many. McCue suggested I get online with my phone and see if there were reports of car accidents along this stretch.
I checked. Was astonished. "Four in two weeks. Nobody died, but they're all pretty banged up. But the accidents didn't happen in the _same_ location. Close, but not the same."
"What's a few yards matter to a ghost?"
"Or a surrogate." I thought it over. "In the stories, she's always near or in a river. She drowned her kids, so she doesn't travel far from water."
"There's no river here."
"But there _is_ a ravine with a creek in the bottom. You said so."
" _Why_ did she drown her kids?" Remi asked. "Does anyone know? That is, if the story were true?"
"It varies. But she was reported to be so beautiful that all men lusted after her. She attracted a rich man, who married her, and she had the two kids. Then the husband admitted he never wanted kids and was going to leave her. La Llorona drowned them to hold onto her husband, but he left her anyway. So she died, and her ghost grieves to this day."
Remi shook his head. "What mother drowns her own kids?"
I thought of a couple who had. "Andrea Yates drowned five in the bathtub, including a six-month-old. Susan Smith rolled her car into the lake and killed both children. Smith did it because her lover didn't want kids and was going to leave her."
Remi was silent a moment. "Awful lot like the legend, ain't it?"
"And if it happened now— _now_ now, I mean—I'd say both were possessed. But back then, the hell vents were closed and no surrogates were loose on earth."
"So she wanted you to wreck, but not necessarily to die?"
I thought that over. True that no one had died. Then I remembered something Grandaddy had said about the master plan of heaven and all the Celtic knotwork. "Chaos. It's _chaos_."
Remi took his eyes off the road long enough to shoot me a glance. "Chaos?"
Counting off, I stuck thumb and two fingers into the air one by one. "It's mythology, it's philosophy, it's religion. Chaos is said to be the first formless matter that existed before Creation. It's cosmology. Chaos magic is a branch of occultism."
Remi got on board. "So this demon in ghost's clothing doesn't care if people die, only wants to cause chaos?"
"I think so."
"So she won't be actively trying to kill us."
"Well, she might if we actively try to kill _her_."
* * *
—
La Llorona, as hoped, eventually darted out in front of Remi's truck. I blurted that information, and he didn't try to swerve, as swerving and overcorrecting is probably what caused all the wrecks. He drove the pickup right through her.
She was not corporeal. Therefore not killed or injured.
And then, to my shock, she ran in front of the truck again. When I shouted, he didn't swerve. Or, well, he did, but it was an intentional, controlled swerve. He took the truck to the side of the road, threw the gear shift into neutral, yanked on the emergency brake, told me to start driving as soon as he got into the bed of the truck. Up and down, back and forth.
He was out by the time I shouted after him, "But you can't _see_ her!"
"Then tell me where she is! Use a clock face!"
I climbed across the console, which wasn't particularly easy or comfortable, slid in behind the wheel. Dropped the emergency brake. Threw the truck in gear, goosed it to roll it off the shoulder into a turn without laying down rubber, drove back toward the chapel, turned around again. Heading toward town, she'd so far run into the right-hand lane, never the left.
"She'll be on the right!" I shouted. Saw in the mirror Remi's body shift over.
She ran onto the road. I slowed, shouted, _"Two o'clock!"_
I saw a throwing knife zip by the truck, but she did not fall. And now we were heading right by her. _"Three o'clock!"_
He missed again, but she nonetheless paused for an instant, eyeballed us, stared hard at Remi as if evaluating him, then turned and ran back toward the trees.
"I can see her now!" McCue pounded on the truck roof. "Stop! I'm going after her!"
I stopped. Before I could say anything Remi swung himself over the bed rail, jumped down, and ran after her. Amidst an array of blistering curses I shut down the engine, got out of the truck without falling flat, and ran stiffly after _him_.
She led us a chase. It wasn't merry, that's for sure; in fact, it was painful. Remi was a fair distance ahead of me, and I had no second gear. The body just wouldn't give it to me. But I did my best to make my way over fallen trees, through screening branches, and dodged stumps.
"Remi! _Wait up!"_ But I was pretty sure he wouldn't.
And yet, he did. He was waiting at the lip of the ravine, poised to take off again at any instant. "She went down here," he said curtly. "There's a way, but it's chancy. I don't think you can make it."
"I'll make it. Don't _lose_ her!"
He took off his hat, spun it onto the ground, went over the lip.
When I got there, I saw why he called it chancy. It wouldn't even qualify as a true trail, but there were opportunities to make your way down if you were careful.
Even as I started descending, my brain worked overtime. Why would a surrogate run from us? They wanted us dead. All of them, I assumed. But this one apparently wanted nothing to do with us.
Chaos. _Chaos_. A state of utter confusion, disorder, total lack of organization, unpredictable behavior.
We had a demon on our asses who was very methodical, mostly predictable when it came to notes and photos, and certainly organized. Maybe chaos demons were only good for causing mayhem, not committing murder.
Shit, the climb down was _not_ fun. I clung precariously to rocks, brush, grass, leaned close to the wall of the ravine. I could see Remi's boot prints in patches of soil and followed them. Granite outcroppings allowed for safe handholds now and then, and dirt-filled crevices permitted very careful steps.
McCue was now down from the trail, hastily stepping toward the creek over water-polished rocks. I caught a glimpse of white above the water, as if La Llorona was floating there. And then she was. White clothes, black hair, a strikingly beautiful face with large dark eyes.
Dark eyes, not white. Not like the black dog.
According to the legend, she now and then left rivers to visit people, which explained why we'd tangled with her. Now she was back in the river, more or less, and I wasn't sure if it would provide additional power, additional chaos.
Remi splashed into the creek, took a stance.
_"Twelve o'clock!"_ I shouted. "Straight on!" Then recalled he could see her.
He let loose of the third and last throwing knife.
It went home in her heart. As had the ghosts in the bar, she pixelated, groaned like a deflating bagpipe, then fell right down into the water, where she dispersed. Remi hesitated, then waded out farther.
I slid down the last step before level ground. "What are you doing? You got her. She's dead!"
"Lookin' for my knife!"
"Do you think you can _find_ it in the water?"
"It's clear and still," Remi called back. Then cried " _Hah!_ " and bent down, pulled the knife from water. He was grinning to beat the band. "Okay. Now we go back for the other two knives."
"Well, they're on the road somewhere."
"I'll find 'em. And my hat."
I stood stiffly by the river rocks, trying to ignore overtaxed muscles. I waited until he was almost out of the water. "Don't _do_ that again!"
" _You_ would."
"That's beside the point."
"That _is_ the point." He glanced back at the creek as if searching for remains. But La Llorona was gone. "Hey, look yonder!" Remi exclaimed, pointing. "Your bike!"
Indeed, my bike, in water close to the bank. Wheels submerged, left side in shallow water, gas tank visible and part of the saddle, one handle bar sticking up.
I wanted to cry. Of course, it was possible the bike was yet salvageable; we needed to get to it so I could take a look. But still.
I waded through the water and reflected that at this moment I was behaving more than a little like La Llorona, hunting and mourning my drowned child.
# CHAPTER THIRTY-FOUR
As we stood in calf-deep water, Remi bent and spread his legs, braced, reached down to grab part of the bike.
"What are you doing?" I asked.
"Don't you want to get it out of here?"
I agreed. "I'd _love_ to get it out of here, but even working together we can't lift a bike weighing over seven hundred pounds."
Remi, who had straightened, gazed down upon my drowned child. "Then—what do you do? I mean, you're hardly the first who's totaled a bike."
"First, it may not be totaled; and second, a wrecker. Here, though . . ." I shook my head, vastly unhappy. "I don't know if they've got a winch cable long enough, and I seriously doubt anyone can drive down here. And a helicopter—"
_"Helicopter?"_
"—doesn't have enough room to maneuver," I finished. "But I can't just leave it here."
Remi was exasperated. "You just named all the things that can't be done! Is there anything that _can?_ "
I stared down at my half-submerged bike, then looked up at Remi. "Levitation?"
McCue remained exasperated. "Then let's levitate our asses out of here, and we'll figure out what to do with the bike later."
I looked back once again at my poor motorcycle, reached down and patted the part of the saddle that wasn't submerged. "We'll be back."
Remi and I turned around to begin the sloppy walk back to the bank, when two white-swathed figures abruptly burst upward from the water and floated in midair. McCue and I both took a couple of steps backward made ungainly by the water.
Children. In christening gowns, despite the fact they were much too old for the ceremony. Five? Six?
They stared at us out of dull black eyes rimmed with blue bruises, faces a sickly pallor. As one, mouths moving simultaneously and tonal pitch identical, they said, _"You killed our mother."_
It took me no time at all. "La Llorona's sons."
Remi was almost stupefied. "But she _drowned_ them! Why are they mad at us?"
"Because ghosts are pissy, and they're not really ghosts anymore." I eased a hand toward my shoulder holster. Two sets of eyes fastened on the movement. "Okay—if I distract them, can you shoot them? Or whatever?"
"They're kids, and considerably smaller than we are," Remi said. "What harm can they do?"
"They're _not kids_!" I shouted, and then one of them came at me while the other flung itself at Remi.
Oh, hell. I grabbed the gun out of the holster as I went down to keep it out of the water, managed to push myself into a seated position, snatched a quick look at the kid-ghosts to figure the angles, and fired twice.
One of the shots went off-center. The other did not.
I fought my way to my feet, bracing against submerged rocks, and saw Remi handling the big Bowie as if it were a butter knife. He sliced _his_ kid-ghost across the throat, then stabbed him in the heart.
Kind of overkill, you ask me.
Black blood flowed down the ghost's christening gown. Then, merely bodies in the water, he and his brother-ghost simply dispersed and disappeared. No sound. No smell. They just—became threads that were spun by the water into nowhere.
I straightened, water running from me, still-dry gun held in one hand. "Okay," I said, "Mom and the kids are one big happy family together in—hell. Can we go home, now?"
* * *
—
It was the most uncomfortable ride I'd ever experienced, and I was willing to say it was the same for Remi. Both of us were soaked from mid-chest down. Our boots, so waterlogged, weighed a hundred pounds apiece. And jeans, when wet, adhere to the body, even feel like they are shrinking through an area where you really don't want them to. And leather is worse. Neither of us mentioned it, but we were vastly uncomfortable.
Remi rolled the truck behind the roadhouse and parked. As the evening before, plenty of customers. And since the back door led us into and through the pool tables, we got plenty of weird looks. Both of us tromped upstairs and disappeared into our respective rooms to shed clothing, particularly the boots, and dry ourselves off. I wanted a shower, but I was too hungry to wait. In dry clothes, sock feet—I really didn't care—I left my bedroom to check out the common room, see if the man behind the curtain was sending us messages again.
I found plenty of normal websites, but the deep web didn't come up again.
It had been some time since Remi had left that message for Grandaddy. I didn't know that I'd have any better luck, but I called him. Reached a voice with a one-word command: _"Speak."_
So I spoke. Told the phone that Remi had already called him, got no answer, and now it was me. Would two calls work better than one?
But it was early yet, as time is measured in establishments serving alcohol, and it was entirely possible Grandaddy didn't respond to phone messages. I'd never thought of calling him before. He just showed up now and then, without any advance warning.
I'd never once wondered what he might be doing when absent from my life. For as long as I remember, he came and went. Never any explanation other than a nebulous _I have business._
Now I knew what that business was.
* * *
—
Remi and I ate downstairs. For some reason the crowd of patrons provided a certain amount of relief. They were _normal._ They weren't black dogs, or a ghost woman hunting drowned ghost children, or a Grigori, or a surrogate.
Grandaddy did eventually arrive, and he carried a manila envelope. Remi and I both stared at it. Neither of us moved to take it.
I was seated. Grandaddy was not. Ordinarily it was a deliberate positioning that might intimidate the ones who remained seated. In this case, whether or not that was his intention, it simply wasn't successful.
He dropped the unopened envelope onto the table, just shy of landing in our plates. "A young woman," he said, "came to me as I arrived and asked me to deliver this to you."
I looked at the seemingly inconsequential envelope. Our names, as always, were printed on it. Gabriel and Remiel, not the shortened forms.
McCue shoved his chair back and stood up abruptly. "Best do this in private upstairs."
I followed his lead. Grandaddy, with the envelope, brought up the rear. We went into the common room, where all three of us sat down at the table.
"We need to know," I said plainly. "What our roles are, what this war is, what the true agenda is—and how many angels are opposed to Remi and me specifically?"
The light was kind to his silver-white hair, kind to his face. I had no idea how old he was. I had no idea if angels aged, or if they remained as they were, stagnant. "The agenda is to stop Lucifer from returning."
Remi's tone was cool. "And just how do we do that?"
"You _are_ doing it," Grandaddy said, "by being who you are, and by your actions."
I shook my head. "Not good enough. A Grigori came to me, warned us that we may be used for other purposes than the one you just stated."
His head came up, and his blue eyes burned. "A Grigori." When we said nothing more, he looked at each of us individually, weighing us. He'd always done that with me, and I assumed also with Remi, but this time there was a sharpness to his gaze. "Who?"
"Ambriel," I told him.
He pushed away from the table, rose, took three steps away, then turned back. "Your job, the one for which you were conceived and trained for, is to kill as many surrogates as is possible. Legends, mythology, tall tales, fictional characters, historical figures, and so forth. We must gain control before they do."
"Aren't you omnipotent?" I asked.
"God is. The rest of us are not."
I flicked a glance at the computer. "And just where _is_ God in all of this? Hiding behind a dark web and feeding us a few scraps now and then? Is this _Mission Impossible_ , with orders filtered by unseen beings? _Charlie's Angels,_ where Charlie was never seen, only heard?"
Grandaddy neither confirmed nor denied. "In a handful of days, you have removed from the devil's chessboard eight pawns. You did this working with very little information and only part of your powers; and be assured, they will grow. _No one_ has accomplished what you have in so short a time. Yes, it's true you will threaten some, even angels, because we are a political hierarchy as well as a celestial one."
Remi's tone was very clear. "Are we targets? Targets of our own kind?"
"No."
I stared him down. "Are you sure?"
The lines in his face deepened. "What precisely did Ambriel tell you?"
"That humanity may be caught in the middle of this heavenly battle, and instead of being the beneficiaries of a world without Lucifer's threat, humans become collateral damage. Irrelevant."
"Irrelevant," he echoed. Then he was back at the table without us seeing him move, and he bent, slammed a fist down upon the surface. " _Who do you think we are doing this for_? Not us; _we_ hold heaven _._ But to save humanity, to keep the sacrifices of humans to a minimum. To stop Lucifer, to kill his children. _That_ is your job."
"Then what exactly is Ambriel? Has she any stake in this?"
He shook his head. "Grigori are to remain neutral. They are not to interfere. They have caused enough trouble in the past."
"When they fell."
Grandaddy was more than a little annoyed. "God doesn't kill his children. Even Lucifer was merely flung out of heaven, not killed. But the male Grigori, who were merely to watch, began providing information to humans long before they could possibly be ready for it, and then they lay with human women and got children on them."
"Nephilim," Remi said.
"They should not exist," Granddaddy said flatly. "They were never _meant_ to exist. The Grigori in effect created a new race, and did it without permission."
Absently, I rubbed fingertips against the table. "Ambriel said—"
"I don't care what Ambriel said! No, she is not one of the fallen, but she takes their side. She wishes the Nephilim to live."
Both Remi and I stared at him, stunned. After a moment McCue took his hat off, set it upside down, ran both hands through his short hair and fixed Grandaddy with hard eyes. "So much for saying God doesn't kill his children."
"He doesn't. But the Nephilim are not his. They are not the children of God, nor are they the children of man."
I shook my head in disbelief. "So they are expendable."
"They should never have been born in the first place."
"Yeah, well, they're here now," I said. "That horse is out of the barn."
He was stone-faced. "Your job is not to hunt Nephilim. Be glad of it. When one dies, a piece of heaven dies, and it damages us all."
I assessed him. "So Ambriel is the enemy?"
"Ambriel has been misled. What she doesn't realize is that Nephilim may at any time turn their backs on God and join Lucifer. They owe no allegiance to anyone."
"Mercenaries," Remi said.
I rubbed fingertips against the wood again. "If that is true, why doesn't heaven pay the Nephilim, either to fight for heaven or at the very least to stay out of the argument. Then you wouldn't have to worry about them joining Lucifer."
Grandaddy said, "That is not our way."
"But sending us off to kill for you is?" I shrugged. "Maybe it should be. Desperate times, and all that. I would think the End of Days is the most desperate time of all."
Remi indicated the computer. "If it's not God on the other side, is it you?"
"It is not. What I have to say to you can be said in person, not through a computer."
"Then who is it?" I asked.
Grandaddy shook his head.
The lazy drawl was back in McCue's voice. "Either you can't, or you won't. Look, it ain't no shame to admit you can't _when_ you can't."
Grandaddy shook his head again. "I can't, because I don't know. I suspect, but I don't know the truth."
I looked at the sealed envelope he'd dropped onto the table. "Do you know what's in there?"
"No."
"We do. Go ahead. Open it."
Grandaddy studied us both a moment, then picked up the envelope. He slit it open with a fingernail, removed the note. As he unfolded it, read it, his eyebrows rose. He dropped the note to the table where Remi and I could see it, cut-out letters once again: _And a fourth._
I felt a crawling in my gut. Grandaddy removed the photo, was clearly concerned by what he saw. "Who is this poor woman?"
Remi said, "You want to know her name, turn it over. It'll say."
Grandaddy did so. "Catherine."
"She's the fourth," I told him. "Two notes without photos, followed by four additional notes _with_ photos. All women, all butchered—and each manner of death worse than the one before."
My phone went off. Remi and I both jumped. I answered.
Ganji said, "There is a delivery for you. A package."
I looked at the note, the photo. "Do we—do we have to sign for it?"
"No. The delivery man is gone."
I looked at Grandaddy. "But you're sure the delivery guy is not a surrogate?"
He shook his head. "This building has been cleared. No demon may enter here."
I said into the phone to Ganji, "Grandaddy says no."
"Shall I bring it up?"
"Please."
I disconnected, looked at Remi. "I take it you weren't expecting a package."
He shook his head. "And I take it _you_ weren't."
I looked again at the note, the photo. "Catherine" was nearly unrecognizable as a human.
Ganji knocked at the door at the top of the stairs. Remi rose, went to open it. He came back with a small parcel and yet another manila envelope.
God, I did not want to open it.
I said to Grandaddy, "You really don't know what's going on with the notes and photos?"
He shook his head. "This is not the work of angels. It's the work of Lucifer's children."
Remi set the package on the table. He stared at it as if it might be a rattlesnake, every bit as reluctant as I to discover what was inside, what the note said.
Grandaddy unsheathed his knife, cut open the package. It was a styrofoam container much like doggie bag boxes. That, he opened as well.
Holy Christ. A recognizable—and recognizably human—body part.
I recoiled. Remi looked appalled.
"Oh, man," I muttered. "This is—this is fucking _sick_."
After a glance at Grandaddy, who seemed as ignorant of answers as we two, Remi slit open the envelope, looked inside, then pulled out a note.
This one was not made up of cut out letters from multiple sources. This one was handwritten in red ink, just like the names on the back of the photo.
McCue read it quickly, color fading. Then, in a squeezed, uneven voice, he read it aloud.
> _"From hell._
>
> _Gabriel and Remiel_
>
> _Sirs_
>
> _I send you half the kidney I took from one woman, preserved it for you the other piece I fried and ate it was very nice. I may send you the bloody knife that took it out if you only wait a while longer."_
Remi let the note drop from his fingers. He lost his tan entirely, swallowed convulsively. "Oh my God. It's the text of the original letter . . . and the names are right. The first names of the women in the photos—they're the same. Mary Ann _Nichols_. Annie _Chapman._ Elizabeth _Stride_. Catherine _Eddowes._ " He drew in a breath, still pale. "The next one—I think the next one will be a woman named Mary Jane. For Mary Jane Kelly."
" _Who?_ How do you know that?" I picked up the note, scanned it quickly—old-fashioned cursive writing and filled with spelling errors, almost indecipherable—then saw there was more writing on the other side. Four lines only, printed with careful clarity.
> _Call me Legion._
>
> _Call me Iñigo Montoya._
And the final two sentences:
> _Call me Jack the Ripper._
>
> _Catch me when you can._
_Original in the Records of Metropolitan Police Service, National Archives, MEPO 3/142_
# AUTHOR'S NOTE
It's believed, of the thousands of letters received by police that were purportedly written by Jack the Ripper, the one most likely to be genuine was indeed accompanied by a human body part identified as a kidney. And while most of us tend to misquote the letter's famous line as _Catch me if you can_ , the actual text says _when_.
Urban fantasy began for me, as it did for so many of us, with Charles de Lint. I read most of his work before the subgenre had a name. Read more urban fantasy by other authors over the years since de Lint's work first appeared.
After three decades of writing novels set in imaginary worlds of my own devising—the Cheysuli _,_ Sword-Dancer, and Karavans series—I got an itch to try my hand at fantastic elements set in our world, relying on reality as a backdrop to the _un_ reality I would serve readers. I wanted to play with the paranormal, the supernatural, as well as legends and lore, myth and magic, history both oral and written. And to use modern slang and pop culture references was great fun.
I did ask my editor if I could use the occasional F-bomb. It's not a regular part of my personal lexicon, but I felt it would definitely be part of Gabe's. She was fine with it. Said she was born in New York City and f*** was her middle name. Unfortunately I could not quote song lyrics word-for-word because of copyright and licensing issues. It's a shame, because country music lyrics are the most colorful on Planet Earth.
My minor is in British History, but cultural anthropology also fascinated me in school. I loved learning about different civilizations, religions, rituals. I very nearly switched my major from journalism to anthro, but to do so would have required me to take a statistics class, and that meant the arcane and incomprehensible rituals and recipes of mathematics would be involved. Thus, my BS in journalism.
The initial impulse was to make my two protagonists a male and female pairing. But I realized it would probably end up sounding very like Tiger and Del of my Sword-Dancer series, and while I love writing about those two characters, I didn't want to do a knock-off of my own work.
I had Gabe already in mind, reflecting my interest in folklore and mythology. As for Remi, well, I grew up in Arizona and, for many years, owned horses and attended rodeos frequently. This even led to three rodeo queen titles in the '70s: Scottsdale Parada del Sol, Phoenix Rodeo of Rodeos, and Miss Rodeo Arizona.
I spent pretty much every Friday night dancing to country music at cowboy bars, and Wednesday nights after Rodeo Club when attending Northern Arizona University in Flagstaff. The latter venue included the Museum Club on Route 66, which is the model for the Zoo Club. So when Remi talks about powdered tobacco tucked inside his lip—well, yes, I tried it a few times. And I did shots of tequila, and have eased my way through good single malt _uisge beatha_. ( _Alba gu brath!)_
As for guns, I've taken classes, hit the shooting range (revolver preferred to semi-auto), researched, repeatedly picked the brains of gun experts James Kosky and Simon Hawke. As for motorcycles, I've ridden but never driven, and certainly never laid one down. There, Simon Hawke's experience with All Things Motorcycle was invaluable to me. But all errors, always, are mine and mine alone.
Upcoming projects, in addition to the further adventures of Gabe and Remi, include a new Sword-Dancer novel and the final installment of the Karavans series. I can be reached through my website, www.jennifer-roberson.net.
> — _Jennifer Roberson_
>
> _Tucson, Arizona_
# ABOUT THE AUTHOR
**Jennifer Roberson** is the author of the Chronicles of the Cheysuli, the Sword-Dancer Saga, the Karavans series, and the new urban fantasy series Blood & Bone. She has also published three historical novels and several in other genre, and a short story collection titled _Guinevere's Truth & Other Tales_. She creates mosaic art and jewelry, and lives in Arizona with an assortment of Cardigan Welsh Corgis and cats.
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# Contents
1. Cover
2. Also by Jennifer Roberson
3. Title Page
4. Copyright
5. Dedication
6. Contents
7. Prologue
8. Chapter One
9. Chapter Two
10. Chapter Three
11. Chapter Four
12. Chapter Five
13. Chapter Six
14. Chapter Seven
15. Chapter Eight
16. Chapter Nine
17. Chapter Ten
18. Chapter Eleven
19. Chapter Twelve
20. Chapter Thirteen
21. Chapter Fourteen
22. Chapter Fifteen
23. Chapter Sixteen
24. Chapter Seventeen
25. Chapter Eighteen
26. Chapter Nineteen
27. Chapter Twenty
28. Chapter Twenty-one
29. Chapter Twenty-two
30. Chapter Twenty-three
31. Chapter Twenty-four
32. Chapter Twenty-five
33. Chapter Twenty-six
34. Chapter Twenty-seven
35. Chapter Twenty-eight
36. Chapter Twenty-nine
37. Chapter Thirty
38. Chapter Thirty-one
39. Chapter Thirty-two
40. Chapter Thirty-three
41. Chapter Thirty-four
42. Author's Note
43. About the Author
# Landmarks
1. Cover
2. Cover
3. Title Page
4. Start
5. Table of Contents
6. Copyright
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{
"redpajama_set_name": "RedPajamaBook"
}
| 6,309
|
Q: Can't set initial value of input element with type="time" in Chrome I like that Chrome gives you the masking/format automatically when you use the type="time" attribute/value. But I can't seem to set an initial value for the control.
This:
<input type="time" value="05:00 PM" />
Just renders as blank (with the masking) in Chrome. And when my form is submitted, it submits it as blank.
I'm guessing this has to do with the format of the string I am setting in value. But I don't know what the correct format is (and why the format I used wouldn't work - seems like a reasonable time format to me).
Here's a JSFiddle to play with just in case you want it.
Any suggestions?
A: Try something like:
<input type="time" value="17:00:00" />
Read this.
A: So, per the spec:
Value: A valid partial-time as defined in [RFC 3339].
Referring to RFC 3339 specifically section 5.6 in a footnote. Here you will find that it is defined as
partial-time = time-hour ":" time-minute ":" time-second
[time-secfrac]
So, practically this means that you can not use anything like PM or AM in the value, although the browser may decide to present the time in any way it wishes (depending on how the user chooses sometimes even). The value you will receive when you submit the form will be always in the above format however.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,489
|
package uk.ac.ed.ph.snuggletex.dombuilding;
import uk.ac.ed.ph.snuggletex.SnuggleLogicException;
import uk.ac.ed.ph.snuggletex.definitions.CombinerTargetMatcher;
import uk.ac.ed.ph.snuggletex.definitions.CorePackageDefinitions;
import uk.ac.ed.ph.snuggletex.internal.DOMBuilder;
import uk.ac.ed.ph.snuggletex.internal.SnuggleParseException;
import uk.ac.ed.ph.snuggletex.internal.TokenFixer;
import uk.ac.ed.ph.snuggletex.internal.util.StringUtilities;
import uk.ac.ed.ph.snuggletex.semantics.InterpretationType;
import uk.ac.ed.ph.snuggletex.semantics.MathBracketInterpretation;
import uk.ac.ed.ph.snuggletex.tokens.ArgumentContainerToken;
import uk.ac.ed.ph.snuggletex.tokens.EnvironmentToken;
import uk.ac.ed.ph.snuggletex.tokens.FlowToken;
import java.util.ArrayList;
import java.util.List;
import org.w3c.dom.Element;
/**
* Handles matched parentheses encapsulated within {@link CorePackageDefinitions#ENV_BRACKETED}
* environments. These have either been explicitly specified with <tt>\\left</tt> and
* <tt>\\right</tt> or have been inferred by the {@link TokenFixer}.
*
* @author David McKain
* @version $Revision: 742 $
*/
public final class MathFenceHandler implements EnvironmentHandler {
public static final class BracketCombinerTargetMatcher implements CombinerTargetMatcher {
public boolean isAllowed(FlowToken target) {
boolean isAllowed = false;
if (target.hasInterpretationType(InterpretationType.MATH_BRACKET)) {
isAllowed = true;
}
else if (target.getMathCharacterCodePoint()=='.') {
/* Check for special case of combiner being a '.', which signifies "no bracket" */
isAllowed = true;
}
return isAllowed;
}
}
public void handleEnvironment(DOMBuilder builder, Element parentElement, EnvironmentToken token)
throws SnuggleParseException {
/* Create <mfenced> element with correct attributes */
ArgumentContainerToken contentContainer = token.getContent();
String opener = getBracket(token.getArguments()[0]);
String closer = getBracket(token.getArguments()[1]);
if (opener!=null && closer!=null) {
/* Have explicit opener and closer so can make a proper <mfenced/> */
makeMfenced(builder, parentElement, contentContainer, opener, closer);
}
else {
/* Spec says we've not to do <mfenced/> if missing an opener or closer so we
* revert to long form here.
*/
makeBracketed(builder, parentElement, contentContainer, opener, closer);
}
}
private void makeMfenced(DOMBuilder builder, Element parentElement,
ArgumentContainerToken contentContainer, String opener, String closer)
throws SnuggleParseException {
/* Create <mfenced/> operator */
Element mfenced = builder.appendMathMLElement(parentElement, "mfenced");
mfenced.setAttribute("open", StringUtilities.emptyIfNull(opener));
mfenced.setAttribute("close", StringUtilities.emptyIfNull(closer));
/* Now add contents, grouping on comma operators */
List<FlowToken> groupBuilder = new ArrayList<FlowToken>();
for (FlowToken contentToken : contentContainer) {
if (contentToken.getMathCharacterCodePoint()==',') {
/* Found a comma, so add Node based on what's been found so far */
makeFenceGroup(builder, mfenced, groupBuilder);
groupBuilder.clear();
}
else {
/* Add to group */
groupBuilder.add(contentToken);
}
}
/* Deal with what's left in the group, if appropriate */
if (!groupBuilder.isEmpty()) {
makeFenceGroup(builder, mfenced, groupBuilder);
}
}
private void makeBracketed(DOMBuilder builder, Element parentElement,
ArgumentContainerToken contentContainer, String opener, String closer)
throws SnuggleParseException {
/* We'll be putting everything in an <mrow/> here */
Element mrow = builder.appendMathMLElement(parentElement, "mrow");
/* Maybe do an open bracket */
if (opener!=null) {
builder.appendMathMLOperatorElement(mrow, opener);
}
/* Then add contents as-is */
for (FlowToken contentToken : contentContainer) {
builder.handleToken(mrow, contentToken);
}
/* Finally, maybe do a close bracket */
if (closer!=null) {
builder.appendMathMLOperatorElement(mrow, closer);
}
}
private void makeFenceGroup(DOMBuilder builder, Element mfenced, List<FlowToken> groupContents)
throws SnuggleParseException {
builder.handleMathTokensAsSingleElement(mfenced, groupContents);
}
private String getBracket(ArgumentContainerToken argumentContainerToken) {
/* Ensure fence endpoint is either nothing or a single bracket */
List<FlowToken> contents = argumentContainerToken.getContents();
String result = null;
if (!contents.isEmpty()) {
/* (Logic here follows the combiner logic above) */
FlowToken bracketToken = contents.get(0);
if (bracketToken.hasInterpretationType(InterpretationType.MATH_BRACKET)) {
/* This is a proper bracket */
result = ((MathBracketInterpretation) bracketToken.getInterpretation(InterpretationType.MATH_BRACKET))
.getMfencedAttributeCharacter()
.getChars();
}
else if (bracketToken.getMathCharacterCodePoint()=='.') {
/* "No bracket" */
result = "";
}
if (result==null) {
throw new SnuggleLogicException("Bracket combiner was not of the expected form");
}
}
return result;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,816
|
View For TOP Online Promo Codes for Anon Google from Every Shop at Online Store KingdomOfTheSun. Buy Greatest Anon Google with Free Shipping to United States, United Kingdom, Australia, Canada, Europe, and All Over The World in Online Store KingdomOfTheSun.
KingdomOfTheSun Suggests Outlet Az, Oakley Or Ray Ban and Polarized Sunglasses Brands In India - Buy Online at Amazon, Walmart, Ebay. Enjoy your shopping for discount prices? Want to compare prices on Most Popular Sunglasses online before making a purchase? Looking for ways to save money? All of this and more is possible at KingdomOfTheSun.net - your premier source for online comparison Sunglasses shopping: compare prices on 300 000+ models from leading stores Glasses, Lookmatic, Ray-Ban, WayFair, Steven Alan , Target, Warby Parker, BestBuy, Dior, SunglassHut, SunglassesShop and other.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,485
|
Dubai, United Arab Emirates, September 19, 2016 — Following the successful rollout of its cloud-first strategy in the Middle East, Epicor Software Corporation, a global provider of industry-specific enterprise software to promote business growth, today announced it will showcase its flagship ERP (enterprise resource planning) solution at GITEX Technology Week 2016. Epicor can be found in Hall 7, Stand B7-35.
"Over the last few years, we have witnessed strong growth in the Middle East and today boast more than 650 customers and over 30 channel partners. To support this growth, we have made significant investments in the region both in terms of resources and products and our presence at GITEX is in line with that commitment. GITEX gives us an unparalleled platform to continue to build our brand, showcase our latest solutions, create mind-share with partners and our industry peers and most importantly, help customers choose an ERP solution that will serve as the backbone of their business processes and a platform for growth," – concluded El Komy.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,533
|
from django.contrib.auth.models import User
from django.db import models
# Create your models here.
class UserCargo(models.Model):
name = models.TextField(max_length=200)
def __str__(self):
return self.name
class UserProfile(models.Model):
user = models.OneToOneField(User)
date_of_birth = models.DateField(null=True)
phone = models.PositiveIntegerField(default=0, null=True)
avatar = models.ImageField(upload_to='user/')
cargo = models.ForeignKey(UserCargo)
def __str__(self):
return self.user.username
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,501
|
\section{Introduction}
Gravitational-wave standard sirens (GWSS) have been proposed as a powerful method for the determination of the Hubble constant (see e.g. \cite{schutz,Holz2005,gwss1, nissanke,holz,ref1,ref2,ref3,ref4}). The feasibility of the method has been experimentally confirmed by the recent spectacular observations of the event GW170817 \cite{GW170817} and the detection of an associated optical counterpart \cite{2017Sci...358.1556C,2017ApJ...848L..16S,2017ApJ...848L..12A}, yielding a constraint of $H_0=70_{-8}^{+12}$ km/s/Mpc (maximum a posteriori value with minimal 68.3\% credible interval) \cite{natureH0}.
While this constraint is much weaker than those currently obtained from measurements of luminosity distances of standard sirens or observations of the cosmic microwave background (CMB) anisotropies, it is expected to significantly improve in the coming years with the discovery of additional standard siren events.
Moreover, this kind of measurement is clearly of particular interest given the current discrepancy on the value of $H_0$ between standard candle luminosity distances of Cepheids and Type Ia supernovae, that report a value of $H_0=73.24\pm 1.74$ km/s/Mpc at $68 \%$ C.L. \cite{R16} (R16, hereafter),
($H_0=73.52\pm 1.62$ km/s/Mpc at $68 \%$ C.L. in the new analysis of \cite{R18}), and CMB measurements from the latest Planck satellite 2018 release that gives $H_0=67.27\pm0.60$ km/s/Mpc at $68 \%$ C.L. (\cite{Aghanim:2018eyx}, see also \cite{plancknewtau, freedman,planck2015}). Current observations of baryon acoustic oscillations (BAO) are in agreement with the Planck cosmology and a combined Planck+BAO analysis gives $H_0=67.67\pm0.45$ km/s/Mpc at $68 \%$ C.L. \cite{Aghanim:2018eyx}, also in strong discrepancy with the standard candle results of \cite{R16,R18}.
While unidentified systematics could be clearly present, this tension may indicate the need of new physics beyond the standard $\Lambda$CDM model. Indeed, since the Planck constraint is derived under the assumption of $\Lambda$CDM, simple extensions to this model could relax the CMB constraint. For example, new physics in the dark energy or neutrino sectors can significantly undermine the Planck constraints on the Hubble constant, solving the current tension on $H_0$ (see e.g. \cite{R16,papero,paper1,paper2,
Zhao:2017urm,Yang:2017amu,Prilepina:2016rlq,Santos:2016sog,Kumar:2016zpg,Karwal:2016vyq,benetti,Ko:2016uft,Archidiacono:2016kkh,Qing-Guo:2016ykt,Zhang:2017idq,zhao,Sola:2017jbl,brust,mena,boehm,wekiang,paper0}). Moreover, a time varying dark energy equation of state could also alleviate the tension between the Planck+BAO and the R16 constraint (see e.g. \cite{paper2,Zhao:2017cud,DiValentino:2017gzb}).
Clearly an independent and accurate future determination of $H_0$ from GWSS will play a key role in confirming or rejecting the possibility of new physics beyond $\Lambda$CDM.
It is to emphasized that an accurate measurement of the Hubble constant, even though it is a low-redshift quantity, can have important consequences for other higher-redshift cosmological parameters such as the dark energy equation of state~\cite{2005ASPC..339..215H}.
The possibility of constraining cosmology with GWSS has been already considered in several previous works (see e.g. \cite{gwss1,gwss2,gwss3,gwss4,gwss5,gwss5,gwss6,gwss7,gwss8,gwss9,gwss10,gwss11}). Some of these studies analyzed ``far future'' experiments such as the LISA satellite mission \cite{lisa} expected to be launched in $2034$ or third generation interferometers such as the Einstein Telescope \cite{ET} or the Cosmic Explorer \cite{CE}. However, recently, in \cite{holz} it has been estimated that, depending on the discovery rate of binary neutron stars, a sub-percent determination of the Hubble constant from GWSS could be achieved by the Hanford-Livingston-Virgo (HLV) network as early as during the second year of operation at design sensitivity ($\sim2023$~\cite{2018LRR....21....3A}). Given the rate uncertainties, a sub-percent measurement may have to wait for two years of the Hanford-Livingston-Virgo-Japan-India (HLVJI) network, which is expected to commence operations $\sim 2024+$.
On the other hand, a significant improvement in the observational data is expected from the next CMB and BAO experiments. Future satellite missions such as LiteBIRD \cite{litebird} and ground based experiments such as CMB-S4 \cite{stage4} will improve the Planck results thanks to cosmic variance limited measurements of CMB polarization. The LiteBIRD satellite is a JAXA strategic large mission candidate in Phase-A1 (concept development) and is currently scheduled for launch around 2026--2027. A complementary ground-based CMB experiment with the sensitivity of CMB-S4 is at the moment planned after $2023$. Similarly, galaxy spectroscopic surveys such as DESI (\cite{Levi:2013gra}, expected to be completed by $2023$) will observe BAO with unprecedented precision.
The level of accuracy on the Hubble constant expected from future CMB+BAO observations can reach the $0.15 \%$ level (see e.g. \cite{core1}). This could naively appear as an order of magnitude more accurate than future projections for standard sirens constraints. However the CMB+BAO constraint is obtained under the assumption of $\Lambda$CDM and, as we show below, can easily be more than one order of magnitude weaker in extended cosmological scenarios. These extended scenarios are of particular interest as they may offer a solution to the existing tension between different measurements of $H_0$.
We emphasize that standard sirens constitute a direct measurement of the luminosity distance, obviating the need for a distance ladder. The absolute calibration of the source is provided by the theory of general relativity. The possible systematics associated with standard siren measurements are expected to reside primarily with the instrument, and in particular, with the calibration of the photodetectors which lead directly to the measurement of the amplitude of the gravitational waves \cite{abbott1,abbott2}. This calibration is expected to be achieved to better than 1\% in the near future~\cite{2016RScI...87k4503K}. Gravitational wave standard siren measurements thus have the potential to provide a particularly clean and robust probe to the sub-percent level. This is to be compared with the case of Type Ia supernovae standard candle measurements, which involve astronomical calibrators such as Cepheids, and multiple rungs of the distance ladder. It remains unclear whether the supernova systematics can be reduced to the $\sim1$\% level~(see, e.g., \cite{2016ApJ...826...56R,2017ApJ...848...56U,2018RPPh...81a6901H}). However, if supernovae achieve this level of accuracy on the measurement of $H_0$, then our results apply directly to them as well. Of course, supernovae also offer the opportunity to probe to much higher redshifts than GWSS, and therefore offer additional cosmological constraints.
It is therefore timely to investigate what kind of additional constraints a direct determination of $H_0$ with $\sim 1\%$ accuracy from GWSS can bring, with the expected completion of new CMB and BAO surveys within the coming decade.
In this paper we address this question by forecasting the cosmological constraints from future CMB and BAO surveys in extended cosmological scenarios and by discussing the implications of an additional independent and direct $H_0$ measurement at the level of $ 1\%$ from upcoming GWSS sources.
Several previous works have presented forecasts on $H_0$ from a variety of potential future cosmological datasets. Most notably, Weinberg et al. 2013 \cite{weinberg} performed a thorough analysis of future CMB, BAO, weak lensing, and supernovae data, and presented future constraint on dynamical dark energy and explicitly discussed the impact of a future $H_0$ prior. In this paper we complement and, in some cases, extend these studies; in detail:
\begin{itemize}
\item While most of the previous forecasts have adopted a Fisher Matrix approach, we base our analysis on a Monte Carlo Markov Chain method. This is needed when the posterior distribution of the parameters is strongly non-Gaussian. As we discuss later in this paper this is the case for several key parameters when CMB and CMB+GWSS datasets are considered.
\item We use an extended $10$-parameter space including not only dynamical dark energy but also possible variations in neutrino masses and in the neutrino effective number. We also discuss the impact of the improvement of the $H_0$ prior on each of the parameters, as well as on the global Figure of Merit (FoM, hereafter).
\item We consider $4$ future, post-Planck, CMB experiments (one satellite and three ground-based) discussing the relative advantages and disadvantages of these missions. This is the first time that a similar comparison is presented in these extended parameter space
(extended parameter spaces considering the CORE-M5 proposal have been already studied in \cite{core1}.
\item In addition to the CMB data we conservatively adopt a single additional cosmological probe, namely a BAO dataset from the DESI experiment; we do not incorporate future supernovae or weak lensing measurements. The main goal of this paper is to demonstrate the kind of improvement a GWSS measurement could bring to a very conservative framework, therefore considering at the same time the largest number of parameters and the smallest number of datasets in order to minimize the presence of theoretical biases and experimental systematics. We choose CMB and BAO data since they should be, in principle, less affected by theoretical and experimental systematics (see e.g. Table I and the discussion in \cite{2018RPPh...81a6901H}) letting us to produce more accurate forecasts \footnote{For example, while future weak lensing measurements from large surveys as EUCLID (see e.g. \cite{euclid}) are extremely promising, systematic errors could limit an accurate determination of the galaxy shapes (see e.g. \cite{mandelbaum}) and redshifts (\cite{huterer2}). The accurate description of non-linearities and non-Gaussianities in extended scenarios could also become a relevant issue since at the moment most of the current predictions are computed from N-Body simulations that assume $\Lambda$CDM. An accurate modeling of all these systematics for a given experimental configuration is currently under study from the weak lensing community and it goes beyond the scope of this work. We therefore do not consider cosmic shear data in our study.}. We complement these measurements with the standard siren measurements, which enjoy a similar level of theoretical and experimental purity.
\end{itemize}
Our paper is structured as follows: in the next section we discuss our methods, in section III we present our results, and in section IV we present our conclusions.
\section{Method}
In this section we describe our forecasting method. We start with a description of the assumed theoretical framework, and then discuss the generation of forecasts for CMB, BAO, and standard sirens constraints.
\subsection{Extended models}
As discussed in the introduction, in this paper we consider parameter extensions to the standard $\Lambda$CDM model. These models, as we discuss below, are physically plausible, compatible with current observations, and able to solve in some cases the current observed tensions between cosmological datasets.
The standard flat $\Lambda$CDM model is based on just $6$ parameters: the baryon $\omega_b$ and cold dark matter $\omega_c$ physical energy densities, the amplitude $A_S$ and the spectral index $n_S$ of scalar primordial perturbations, the angular size of the sound horizon at decoupling $\theta_s$ and the optical depth at reionization $\tau$. Following \cite{paper0, DiValentino:2017clw} we consider variations with the addition of $4$ additional parameters:
\begin{itemize}
\item Curvature, $\Omega_k$. Most of recent analyses assume a flat universe with $\Omega_k=0$ since this is considered as one of the main predictions of inflation. However, inflationary models with non-zero curvature can be conceived (see e.g. \cite{linde}). Moreover the recent results from Planck prefer a closed model $\Omega_k>0$ at more than two standard deviations \cite{planck2015}. Including further data from BAO strongly constrains curvature with $\Omega_k=0.0002 \pm 0.0021$ at $68 \%$ C.L. and perfectly compatible with a flat universe \cite{planck2015}. However this result is obtained in the framework of $\Lambda$CDM+$\Omega_k$, i.e. in one single parameter extension while here we want to analyze a larger parameter space, varying ten parameters at the same time. In this scenario the current Planck+BAO constraints on $\Omega_k$ are weaker.
\item Neutrino mass, $\Sigma m_{\nu}$. Neutrino oscillation experiments have demonstrated that neutrinos undergo flavor oscillations and must therefore have small but non-zero masses. However the neutrino absolute mass scale and the mass hierarchy are not yet determined (see e.g. \cite{capozzi} for a recent review). Usually, as in \cite{planck2015}, the total neutrino mass scale is fixed to $\Sigma m_{\nu}=0.06$eV, corresponding to the minimal value expected in the normal hierarchy scenario. There is clearly no fundamental reason to limit current analyses to this value and the neutrino mass should be let free to vary.
\item Neutrino effective number, $N_{\rm eff}$. Any particle that decouples from the primordial thermal plasma before the QCD transition could change the number of relativistic particles at recombination increasing $N_{\rm eff}$ from its standard value of $3.046$ (see e.g. \cite{baumaneff}). An increased value of $N_{\rm eff}$ can help in solving the Hubble constant tension (see e.g. \cite{R16}).
Reheating at energy scales close to the epoch of neutrino decoupling could on the contrary lower the value of $N_{\rm eff}$ \cite{deSalas:2015glj}.
\item Dark energy equation of state $w$. While current data are in agreement with a cosmological constant, the possibility of having a dark energy equation of state different from $-1$ is certainly open (see e.g. \cite{paper1}). Moreover, a time evolution for $w$ helps in solving the coincidence problem of why dark energy and dark matter have similar densities today. In this paper we consider two parametrizations, either $w$ constant with time or the Chevalier-Polarski-Linder parametrization (hereafter CPL) \cite{chevalier,linder}:
\begin{equation}
w(a)=w_0+(1-a)w_a
\end{equation}
where $a$ is the scale factor, $w_0$ is the equation of state today ($a=1$) and $w_a$ parametrizes its time evolution. This should be considered as a minimal extension since dark energy time dependences could be more complicated as, for example, in the case of rapid transitions. We consider dark energy perturbations following the approach of \cite{waynehu}.
\end{itemize}
In this paper we consider the following $10$ parameters extensions: $\Lambda$CDM+$\Omega_k$+$N_{\rm eff}$+$\Sigma m_{\nu}$+$w$ and $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$w_0$+$w_a$.
While we study extended models, for our simulated data we assume as a fiducial (true) model the standard $\Lambda$CDM model with parameters in agreement with the recent Planck constraints \cite{planck2015}: $\omega_b=0.02225$, $\omega_{c}= 0.1198$, $\tau=0.055$, $100\theta_{MC}=1.04077$, $\Sigma m_{\nu}=0.06$ eV and $n_s=0.9645$. The corresponding derived value of $H_0$ in this model is
$H_0=67.3$ km/s/Mpc. The theoretical models and the simulated data are computed with the latest version of the Boltzmann integrator CAMB \cite{camb}. Given a simulated dataset and a likelihood that compares data with theory, we extract the constraints on cosmological parameters using the Monte Carlo Markow Chain (MCMC) code {\sc CosmoMC}\footnote{\tt http://cosmologist.info}~\cite{Lewis:2002ah}.
\
\subsection{Forecasts for CMB}
We produce forecasts on cosmological parameters for future CMB experiments with a well established and common method (see e.g. \cite{core1,Capparelli:2017tyx,Renzi:2017cbg}). Under the assumption of the fiducial model described previously, we compute the theoretical CMB angular spectra for temperature, $C_{\ell}^{TT}$, $E$ and $B$ modes polarization $C_{\ell}^{EE}$ and $C_{\ell}^{BB}$, and cross temperature-polarization $C_{\ell}^{TE}$, using the Boltzmann code ~\cite{camb}.
Given an experiment with FWHM angular resolution $\theta$ and experimental sensitivity $w^{-1}$ (expressed in $[\mu K$-arcmin$]^2$), we can introduce an experimental noise for the temperature angular spectra of the form (see e.g. \cite{lesgourgues}):
\begin{equation}
N_\ell = w^{-1}\exp(\ell(\ell+1)\theta^2/8\ln2).
\end{equation}
A similar expression is used to describe the noise for the polarization spectra with $w_p^{-1}=2w^{-1}$ (one detector measures two polarization states).
We have then produced synthetic realisations of CMB data assuming different possible future CMB experiments with technical specifications as listed in Table~\ref{tab:spec}. In particular, we have considered a possible future CMB satellite experiments such as LiteBIRD \cite{litebird} and three possible configurations for ground-based telescopes as Stage-III 'wide' (S3wide), Stage-III 'deep' (S3deep) (see \cite{erminia}), and CMB-S4 (see e.g. \cite{stage4,Capparelli:2017tyx,Renzi:2017cbg}).
The simulated experimental spectra are then compared with the theoretical spectra
using a likelihood ${\cal L}$ given by
\begin{equation}
- 2 \ln {\cal L} = \sum_{l} (2l+1) f_{\rm sky} \left(
\frac{D}{|\bar{C}|} + \ln{\frac{|\bar{C}|}{|\hat{C}|}} - 3 \right),
\label{chieff}
\end{equation}
\noindent where $\hat{C}_l$ are the theoretical spectra plus noise, while $\bar{C}_l$ are the fiducial spectra plus noise (i.e. our simulated dataset). The quantities $|\bar{C}|$, $|\hat{C}|$ are :
\begin{eqnarray}
|\bar{C}| &=& \bar{C}_\ell^{TT}\bar{C}_\ell^{EE}\bar{C}_\ell^{BB} -
\left(\bar{C}_\ell^{TE}\right)^2\bar{C}_\ell^{BB} ~, \\
|\hat{C}| &=& \hat{C}_\ell^{TT}\hat{C}_\ell^{EE}\hat{C}_\ell^{BB} -
\left(\hat{C}_\ell^{TE}\right)^2\hat{C}_\ell^{BB}~,
\end{eqnarray}
where $D$ is defined as
\begin{eqnarray}
D &=&
\hat{C}_\ell^{TT}\bar{C}_\ell^{EE}\bar{C}_\ell^{BB} +
\bar{C}_\ell^{TT}\hat{C}_\ell^{EE}\bar{C}_\ell^{BB} +
\bar{C}_\ell^{TT}\bar{C}_\ell^{EE}\hat{C}_\ell^{BB} \nonumber\\
&&- \bar{C}_\ell^{TE}\left(\bar{C}_\ell^{TE}\hat{C}_\ell^{BB} +
2\hat{C}_\ell^{TE}\bar{C}_\ell^{BB} \right). \nonumber\\
\end{eqnarray}
In what follows we don't consider information from CMB lensing derived from trispectrum data.
\begin{table*}
\begin{center}
\begin{tabular}{lccccc}
\toprule
\horsp
Experiment \vertsp Beam \vertsp Power noise $w^{-1/2}$\vertsp $\ell_{max}$& $\ell_{min}$& $f_{sky}$\\
& &[\footnotesize$\mu$K-arcmin]& & &\\
\hline
\morehorsp
LiteBIRD & $30$' & $4.5$& $3000$&$2$&$0.7$\\
\morehorsp
S3deep & $1$' & $4$ & $3000$&$50$&$0.06$\\
\morehorsp
S3wide & $1.4$' & $8$ & $3000$&$50$&$0.4$\\
\morehorsp
CMB-S4 & $3$' & $1$& $3000$&$5,50$&$0.4$\\
\bottomrule
\end{tabular}
\end{center}
\caption{Specifications for the different experimental configurations considered in our paper. In case of polarization spectra the noise $w^{-1}$ is multiplied by a factor 2.}
\label{tab:spec}
\end{table*}
\
\subsection{Forecast for BAO}
For the future BAO dataset we consider the DESI experiment~\cite{Levi:2013gra}. If $D_V$ is the volume averaged distance, this is defined as:
\begin{equation}
D_V(z)\equiv \[\frac{\(1+z\)^2D_A(z)^2cz}{H(z)}\]^\frac{1}{3}
\end{equation}
\noindent where $D_A$ is the angular diameter distance and $H(z)$ the expansion rate.
Under the assumption of the fiducial model described previously, we compute the theoretical values of the ratio $r_s/D_V$, where $r_s$ is the sound horizon at the drag epoch when photons and baryons decouple, for the different redshifts in the range $z=[0.15-1.85]$ listed in Table~\ref{DESI}. Given the forecast uncertainties reported in~\cite{Font-Ribera:2013rwa} for $D_A/r_s$ and $H(z)$, we then compute the uncertainties on $r_s/D_V$ and we show them in Table~\ref{DESI}. The simulated BAO dataset is finally compared with the theoretical $r_s/D_V$ values through a Gaussian prior.
As a consistency test, we have checked that by using directly the $D_A/r_s$ value and the corresponding uncertainties reported in~\cite{Font-Ribera:2013rwa} instead of $r_s/D_V$, we obtain very similar results with constraints about $\sim 30 \%$ weaker on $H_0$ when combined with CMB-S4 data, in agreement with the results of \cite{addison}.
In principle it would be possible to forecast BAO data considering $D_A/r_s$ and $H(z)$ as independent measurements. However some small tension (around $1$ sigma level) is present between the current constraints from $D_A/r_s$ and $H(z)$ (see e.g. \cite{addison}, Figure 2 contours in the Top Left and Bottom Left panels for $\Omega_m\sim0.3$). It is clearly difficult to properly take into account a possible small tension between future $D_A/r_s$ and $H(z)$ measurements that could improve/reduce future BAO constraints. We therefore follow the approach of \cite{erminia} deriving the expected fractional uncertainties on $r_s/D_V$ for DESI from the fractional errors on $D_A/r_s$ and $H(z)$ forecasted in ~\cite{Font-Ribera:2013rwa}.
\begin{table}
\begin{center}
\begin{tabular}{lccc}
\toprule
\horsp
Redshift & $\frac{\sigma(r_s/D_V)}{r_s/D_V}$& $\sigma(r_s/D_V)$\\
\hline
\morehorsp
$0.15$&$2.57\%$&$0.00595$\\
\morehorsp
$0.25$&$1.71\%$&$0.00246$\\
\morehorsp
$0.35$&$1.32\%$&$0.00141$\\
\morehorsp
$0.45$&$1.08\%$&$0.00093$\\
\morehorsp
$0.55$&$0.91\%$&$0.00067$\\
\morehorsp
$0.65$&$0.79\%$&$0.00051$\\
\morehorsp
$0.75$&$0.70\%$&$0.00040$\\
\morehorsp
$0.85$&$0.68\%$&$0.00036$\\
\morehorsp
$0.95$&$0.75\%$&$0.00037$\\
\morehorsp
$1.05$&$0.77\%$&$0.00036$\\
\morehorsp
$1.15$&$0.76\%$&$0.00034$\\
\morehorsp
$1.25$&$0.76\%$&$0.00032$\\
\morehorsp
$1.35$&$0.83\%$&$0.00033$\\
\morehorsp
$1.45$&$0.96\%$&$0.00037$\\
\morehorsp
$1.55$&$1.21\%$&$0.00046$\\
\morehorsp
$1.65$&$1.89\%$&$0.00069$\\
\morehorsp
$1.75$&$2.91\%$&$0.00104$\\
\morehorsp
$1.85$&$3.87\%$&$0.00134$\\
\bottomrule
\end{tabular}
\end{center}
\caption{Specifications for the forecast DESI data, obtained by \cite{Font-Ribera:2013rwa}.}
\label{DESI}
\end{table}
\
\subsection{Forecast for standard sirens}
As stated in the introduction, in this paper we want to address the question of what kind of cosmological information can be obtained from GWSS systems within the coming decade (i.e. by $\sim2028$) when complementary measurements from CMB and BAO surveys will be available. We therefore focus our attention on GW experiments that could be completed in this time-scale: the Hanford-Livingston-Virgo (HLV) network of interferometers during the second year of operation at design sensitivity ($\sim2023$) and the the Hanford-Livingston-Virgo-Japan-India (HLVJI) network two years after the start of operations ($\sim 2026$)~\cite{2018LRR....21....3A}. We do not consider longer-term experiments such as the LISA \cite{lisa} or DECIGO \cite{decigo} missions or proposed third generation interferometers such as the Einstein Telescope \cite{ET} or the Cosmic Explore \cite{CE} that would presumably start operations no sooner than $2030$. Moreover, these experiments will be able to determine the luminosity distance of GWSS at higher redshift, opening the possibility to test the acceleration of the universe (i.e. the deceleration parameter), while here we only limit our discussion to the Hubble constant (although black holes standard sirens would probe these high redshifts earlier~\cite{2018arXiv180510270F}).
Considering HLV or HLVJI and assuming the optimistic case that all binary neutron star (BNS) systems have detected optical counterparts and associated redshift measurements, the major uncertainty on the projected constraint on $H_0$ from GWSS comes from the BNS detection rate. The current best estimate of the BNS rate is $R=1540^{+3200}_{-1220}\,$Gpc$^{-3}$yr$^{-1}$~\cite{bnsrate} (median and $90 \%$ credible interval)~\cite{GW170817}; it is very poorly constrained given that only one BNS event has been detected to date. Following \cite{holz} we forecast $4 \%$, $2\%$, and $1 \%$ uncertainties on the measurement of $H_0$ for the HLV network after two years at design sensitivity ($\sim2023$) and assuming lower, mean, and upper BNS rates of $R= 320\,$Gpc$^{-3}$yr$^{-1}$, $R=1540\,$Gpc$^{-3}$yr$^{-1}$, and $R=4740\,$Gpc$^{-3}$yr$^{-1}$. The corresponding accuracy for the HLVJI network operating after one year of operation ($\sim 2025$) reaches $3 \%$, $1.4 \%$, and $0.8 \%$ on $H_0$, while after two years it arrives at $2.8 \%$, $1.2 \%$, and $0.7 \%$ (see Figure~$3$ in \cite{holz}). By 2028 the HLVJI network would have an additional two years of operation, leading {\em very roughly}\/ to a factor of $\sqrt{2}$ improvement to $2 \%$, $0.85 \%$, and $0.5 \%$.
It is therefore possible that standard siren measurements will reach an accuracy of $1 \%$ by $2028$ (under the assumption that a majority of BNS mergers have detectable electromagnetic counterparts). Considering that our fiducial model has $H_0=67.3$ km/s/Mpc, we therefore assume a Gaussian prior of $H_0=67.3 \pm 0.673$ km/s/Mpc. In what follows we will refer to this (optimistic) prior as GWSS67.
On the other hand, we also consider the significantly more pessimistic $H_0$ prior of $4 \%$ ($H_0=67.3 \pm 2.7$ km/s/Mpc). This prior, just a factor of $\sim 4$ smaller than the current GW constraint based on a single event, is clearly extremely conservative but may happen if the BNS rate ends up on the low side (see e.g. \cite{GW170817, holz, rate1,rate2}). In what follows we will refer to this prior as PGWSS67.
These priors on $H_0$ are introduced by importance sampling on the models (samples) drawn from our MCMC simulations \cite{Lewis:2002ah}. In our case this translates into multiplying each sample weight by a Gaussian function, with mean and variance defined by the assumed $H_0$ prior, evaluated at the value of $H_0$ in the sample itself. For this to work it is only necessary that the obtained weights are significant for a large fraction of the re-weighted samples; this is a direct consequence of the requirement that the distribution from which the samples are drawn and the importance distribution are not too dissimilar.
\section{Results}
\subsection{$\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ Model}
\begin{table*}[!hbtp]
\begin{center}
\begin{tabular}{lcccc}
\toprule
\horsp
Parameter \vertsp LiteBIRD \vertsp S3deep \vertsp S3wide \vertsp CMB-S4 \\
\hline\hline
\morehorsp
$\Omega_\mathrm{b}h^2$ \vertsp ${\siround{0.02214}{5}}\pm{\siround{0.00023}{5}}$\vertsp${\siround{0.02222}{5}}\pm{\siround{0.00016}{5}}$\vertsp${\siround{0.0222}{5}}\pm{\siround{9e-05}{5}}$\vertsp${\siround{0.02219}{5}}\pm{\siround{5e-05}{5}}$\\
\morehorsp
$\Omega_\mathrm{c}h^2$ \vertsp ${\siround{0.1203}{4}}\pm{\siround{0.0042}{4}}$\vertsp${\siround{0.1199}{4}}\pm{\siround{0.003}{4}}$\vertsp${\siround{0.1198}{4}}\pm{\siround{0.0013}{4}}$\vertsp${\siround{0.1199}{4}}\pm{\siround{0.001}{4}}$\\
\morehorsp
$100\theta_\mathrm{MC}$ \vertsp ${\siround{1.04075}{5}}\pm{\siround{0.00078}{5}}$\vertsp${\siround{1.04065}{5}}\pm{\siround{0.00033}{5}}$\vertsp${\siround{1.04071}{5}}\pm{\siround{0.00016}{5}}$\vertsp${\siround{1.04071}{5}}\pm{\siround{0.00012}{5}}$\\
\morehorsp
$\tau$ \vertsp ${\siround{0.054}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.054}{3}}\pm{\siround{0.01}{3}}$\vertsp${\siround{0.053}{3}}\pm{\siround{0.01}{3}}$\vertsp${\siround{0.055}{3}}\pm{\siround{0.003}{3}}$\\
\morehorsp
$H_0$ \vertsp ${\siround{64}{2}}^{+\siround{8}{2}}_{-\siround{18}{2}}$\vertsp${\siround{59}{2}}^{+\siround{7}{2}}_{-\siround{19}{2}}$\vertsp${\siround{61}{2}}^{+\siround{7}{2}}_{-\siround{17}{2}}$\vertsp${\siround{60}{2}}^{+\siround{8}{2}}_{-\siround{11}{2}}$\\
\morehorsp
$\Omega_K$ \vertsp ${\siround{-0.014}{3}}^{+\siround{0.018}{3}}_{-\siround{0.005}{3}}$\vertsp${\siround{-0.027}{3}}^{+\siround{0.033}{3}}_{-\siround{0.01}{3}}$\vertsp${\siround{-0.016}{3}}^{+\siround{0.02}{3}}_{-\siround{0.006}{3}}$\vertsp${\siround{-0.012}{3}}^{+\siround{0.016}{3}}_{-\siround{0.004}{3}}$\\
\morehorsp
$\log(10^{10} A_\mathrm{s})$ \vertsp ${\siround{3.092}{3}}\pm{\siround{0.011}{3}}$\vertsp${\siround{3.09}{3}}\pm{\siround{0.021}{3}}$\vertsp${\siround{3.09}{3}}\pm{\siround{0.021}{3}}$\vertsp${\siround{3.093}{3}}\pm{\siround{0.006}{3}}$\\
\morehorsp
$n_\mathrm{s}$ \vertsp ${\siround{0.9629}{4}}^{+\siround{0.0073}{4}}_{-\siround{0.0074}{4}}$\vertsp${\siround{0.9656}{4}}\pm{\siround{0.0112}{4}}$\vertsp${\siround{0.965}{4}}^{+\siround{0.0048}{4}}_{-\siround{0.0046}{4}}$\vertsp${\siround{0.9648}{4}}\pm{\siround{0.0038}{4}}$\\
\morehorsp
$w$ \vertsp ${\siround{-1.069}{3}}^{+\siround{0.638}{3}}_{-\siround{0.297}{3}}$\vertsp${\siround{-0.896}{3}}^{+\siround{0.661}{3}}_{-\siround{0.279}{3}}$\vertsp${\siround{-0.911}{3}}^{+\siround{0.506}{3}}_{-\siround{0.243}{3}}$\vertsp${\siround{-0.846}{3}}^{+\siround{0.283}{3}}_{-\siround{0.234}{3}}$\\
\morehorsp
$N_{\rm eff}$ \vertsp ${\siround{3.069}{3}}^{+\siround{0.243}{3}}_{-\siround{0.246}{3}}$\vertsp${\siround{3.082}{3}}\pm{\siround{0.141}{3}}$\vertsp${\siround{3.063}{3}}\pm{\siround{0.07}{3}}$\vertsp${\siround{3.06}{3}}^{+\siround{0.046}{3}}_{-\siround{0.045}{3}}$\\
\morehorsp
$\Sigma m_\nu$\vertsp $< \siround{0.594}{3}$ eV\vertsp$< \siround{0.584}{3}$ eV\vertsp$< \siround{0.405}{3}$ eV\vertsp$< \siround{0.322}{3}$ eV\\\bottomrule
\end{tabular}
\caption{Forecasted constraints at $68 \%$ C.L. (upper limits at $95 \%$ C.L.) from future CMB experiments with specifications listed in Table~\ref{tab:spec} in an extended $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ $10$ parameters analysis. A $6$ parameters $\Lambda$CDM model is assumed as fiducial model. Parameters as $H_0$ and $w$ are practically unbounded. $\Omega_k$ and $\Sigma m_{\nu}$ are also weakly constrained.}
\label{CMBconstraints}
\end{center}
\end{table*}
\vspace{.1cm}
\begin{figure*}[!hbtp]
\includegraphics[width=.67\textwidth,keepaspectratio]{mixed-CMB.pdf}
\caption{Forecasted future constraints at $68 \%$ and $95 \%$ C.L. from future CMB data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ extended model. Clearly in this extended parameter space CMB data alone will be unable to significantly constrain geometrical parameters as $H_0$, $\Omega_k$ or $w$.}
\label{cmbalone}
\end{figure*}
We first forecast the constraints on cosmological parameters from future CMB data only, assuming the extended $10$ parameter model $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$. The constraints on cosmological parameters for the experimental configurations listed in Table~\ref{tab:spec} are reported in Table~\ref{CMBconstraints}, while 2D contour plots at $68 \%$ C.L. and $95 \%$ C.L. between the extra parameters are reported in Figure~\ref{cmbalone}.
We find that future experiments, including CMB-S4, will be unable to provide significant additional constraints on geometrical parameters such as $H_0$, $\Omega_k$, and $w$. This is due to the well known geometrical degeneracy that affects CMB observables (see, e.g.,~\cite{deg1,deg2,deg3}).
CMB-S4 will improve the constraints on $n_S$, $N_{\rm eff}$, $\Omega_bh^2$, and $\Omega_ch^2$ by a factor of $\sim 2$--5 with respect to LiteBIRD. These parameters are less affected by the geometrical degeneracy, and can thus be better constrained with an improvement in the angular resolution of the experiment. Constraints on neutrino masses will also only see marginal improvement (i.e. $\Sigma m_{\nu}<0.32$ eV at $95 \%$ C.L. for the strongest case from CMB-S4), which falls short of the sensitivity of $\Delta \Sigma m_{\nu} \sim 0.05\,$eV needed to test the inverted neutrino mass hierarchy at two standard deviations. The neutrino effective number will be, on the contrary, less affected and interesting constraints at the $\Delta N_{\rm eff} \sim 0.045$ level can be achieved with CMB-S4 even in the case of a very extended parameter space.
\begin{table*}[!hbtp]
\begin{center}
\begin{tabular}{lcccc}
\toprule
\horsp
Parameter \vertsp LiteBIRD+DESI \vertsp S3deep+DESI \vertsp S3wide+DESI \vertsp CMB-S4+DESI \\
\hline\hline
\morehorsp
$\Omega_\mathrm{b}h^2$ \vertsp ${\siround{0.02219}{5}}\pm{\siround{0.00022}{5}}$\vertsp${\siround{0.02219}{5}}\pm{\siround{0.00016}{5}}$\vertsp${\siround{0.02218}{5}}\pm{\siround{9e-05}{5}}$\vertsp${\siround{0.02218}{5}}\pm{\siround{5e-05}{5}}$\\
\morehorsp
$\Omega_\mathrm{c}h^2$ \vertsp ${\siround{0.1212}{4}}^{+\siround{0.0033}{4}}_{-\siround{0.0041}{4}}$\vertsp${\siround{0.1208}{4}}\pm{\siround{0.0027}{4}}$\vertsp${\siround{0.1199}{4}}\pm{\siround{0.0013}{4}}$\vertsp${\siround{0.1199}{4}}\pm{\siround{0.001}{4}}$\\
\morehorsp
$100\theta_\mathrm{MC}$ \vertsp ${\siround{1.04058}{5}}^{+\siround{0.00071}{5}}_{-\siround{0.0007}{5}}$\vertsp${\siround{1.04069}{5}}\pm{\siround{0.00031}{5}}$\vertsp${\siround{1.04075}{5}}\pm{\siround{0.00015}{5}}$\vertsp${\siround{1.04076}{5}}\pm{\siround{0.00011}{5}}$\\
\morehorsp
$\tau$ \vertsp ${\siround{0.055}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.057}{3}}\pm{\siround{0.009}{3}}$\vertsp${\siround{0.057}{3}}\pm{\siround{0.008}{3}}$\vertsp${\siround{0.055}{3}}^{+\siround{0.002}{3}}_{-\siround{0.003}{3}}$\\
\morehorsp
$H_0$ \vertsp ${\siround{67.8}{1}}^{+\siround{1.3}{1}}_{-\siround{1.5}{1}}$\vertsp${\siround{67.7}{1}}^{+\siround{1.2}{1}}_{-\siround{1.3}{1}}$\vertsp${\siround{67.4}{1}}^{+\siround{1.0}{1}}_{-\siround{1.2}{1}}$\vertsp${\siround{67.4}{1}}^{+\siround{1.0}{1}}_{-\siround{1.1}{1}}$\\
\morehorsp
$\Omega_K$ \vertsp ${\siround{0.0}{3}}^{+\siround{0.001}{3}}_{-\siround{0.002}{3}}$\vertsp${\siround{0.001}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.0}{3}}\pm{\siround{0.001}{3}}$\vertsp${\siround{0.0}{3}}\pm{\siround{0.001}{3}}$\\
\morehorsp
$\log(10^{10} A_\mathrm{s})$ \vertsp ${\siround{3.097}{3}}\pm{\siround{0.009}{3}}$\vertsp${\siround{3.101}{3}}\pm{\siround{0.018}{3}}$\vertsp${\siround{3.099}{3}}\pm{\siround{0.016}{3}}$\vertsp${\siround{3.095}{3}}^{+\siround{0.005}{3}}_{-\siround{0.006}{3}}$\\
\morehorsp
$n_\mathrm{s}$ \vertsp ${\siround{0.9656}{4}}^{+\siround{0.0069}{4}}_{-\siround{0.0068}{4}}$\vertsp${\siround{0.9637}{4}}\pm{\siround{0.0104}{4}}$\vertsp${\siround{0.9645}{4}}^{+\siround{0.0046}{4}}_{-\siround{0.0047}{4}}$\vertsp${\siround{0.9647}{4}}^{+\siround{0.0037}{4}}_{-\siround{0.0036}{4}}$\\
\morehorsp
$w$ \vertsp ${\siround{-1.013}{3}}^{+\siround{0.054}{3}}_{-\siround{0.047}{3}}$\vertsp${\siround{-1.022}{3}}^{+\siround{0.057}{3}}_{-\siround{0.047}{3}}$\vertsp${\siround{-1.01}{3}}^{+\siround{0.051}{3}}_{-\siround{0.045}{3}}$\vertsp${\siround{-1.005}{3}}^{+\siround{0.047}{3}}_{-\siround{0.043}{3}}$\\
\morehorsp
$N_{\rm eff}$ \vertsp ${\siround{3.118}{3}}^{+\siround{0.206}{3}}_{-\siround{0.237}{3}}$\vertsp${\siround{3.065}{3}}^{+\siround{0.136}{3}}_{-\siround{0.138}{3}}$\vertsp${\siround{3.049}{3}}\pm{\siround{0.067}{3}}$\vertsp${\siround{3.051}{3}}\pm{\siround{0.044}{3}}$\\
\morehorsp
$\Sigma m_\nu$\vertsp $< \siround{0.202}{3}$ eV\vertsp$< \siround{0.253}{3}$ eV\vertsp$< \siround{0.186}{3}$ eV\vertsp$< \siround{0.126}{3}$ eV\\
\bottomrule
\end{tabular}
\caption{Forecasted constraints at $68 \%$ C.L. (upper limits at $95 \%$ C.L.) from future CMB experiments with specifications listed in Table~\ref{tab:spec} plus information from the BAO DESI galaxy survey in an extended $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$, $10$ parameters, analysis. A $6$ parameters $\Lambda$CDM model is assumed as fiducial model. When comparing the results with those in the CMB alone case reported in Table~\ref{CMBconstraints} we can notice a significant improvement in geometrical parameters as $H_0$, $w$ and $\Omega_k$. Constraints on neutrino masses are also improved.}
\label{CMBDESIconstraints}
\end{center}
\end{table*}
\begin{figure*}[!hbtp]
\includegraphics[width=.67\textwidth,keepaspectratio]{mixed-CMB+DESI.pdf}
\caption{Forecasted constraints at $68 \%$ and $95 \%$ C.L. from CMB+DESI data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ extended model.}
\label{cmbdesi}
\end{figure*}
It is interesting to investigate how the inclusion of future BAO surveys, such as DESI, can break the geometrical degeneracy and improve the constraints derived from CMB data. Assuming the same $\Lambda$CDM fiducial model, we report the CMB+DESI constraints in Table~\ref{CMBDESIconstraints} and we show the 2D confidence level contours at $68 \%$ C.L. and $95 \%$ C.L. in Figure~\ref{cmbdesi}.
The geometrical parameters are constrained almost equally by all configurations, indicating that the additional constraining power arises from the inclusion of DESI. Curvature is now determined with a 0.1--0.2\% accuracy, while the equation of state can be determined with a $\sim 5 \%$ accuracy. It is interesting to note that a degeneracy is present between $\Omega_k$, $w$, and $\Sigma m_{\nu}$, i.e. the introduction of a neutrino mass limits the CMB+BAO constraints on curvature and $w$. In addition, after the inclusion of DESI, CMB-S4+DESI provides better constraints by a factor $\sim 2-4$ on parameters such as $n_S$ and $N_{\rm eff}$ with respect to LiteBIRD+DESI. The bounds on the sum of neutrino masses are however still affected by the remaining extra parameters (mostly by the anti-correlation with $w$ and the correlation with $\Omega_k$), resulting in a limit of $\Sigma m_{\nu} <0.126$ eV at $95 \%$ C.L. for the CMB-S4+DESI configuration and $\Sigma m_{\nu} <0.202$ eV at $95 \%$ C.L. for LiteBIRD+DESI. However the key result for our analysis is the constraint on the Hubble parameter. Again, between the several configurations we consider, CMB-S4+DESI provides the best constraint of $H_0=67.4^{+1.0}_{-1.1}\,$km/s/Mpc, i.e. an uncertainty on the value of the Hubble constant of the order of $\sim 1.5 \%$, while LiteBIRD+DESI gives $H_0=67.8^{+1.3}_{-1.5}\,$km/s/Mpc with an uncertainty of $\sim 2 \%$.
\begin{table*}[!hbtp]
\begin{center}
\begin{tabular}{lcccc}
\toprule
\horsp
Parameter \vertsp LiteBIRD+GWSS67 \vertsp S3deep+GWSS67 \vertsp S3wide+GWSS67 \vertsp CMB-S4+GWSS67 \\
\hline\hline
\morehorsp
$\Omega_\mathrm{b}h^2$ \vertsp ${\siround{0.02215}{5}}\pm{\siround{0.00023}{5}}$\vertsp${\siround{0.02221}{5}}\pm{\siround{0.00017}{5}}$\vertsp${\siround{0.0222}{5}}\pm{\siround{9e-05}{5}}$\vertsp${\siround{0.02219}{5}}\pm{\siround{5e-05}{5}}$\\
\morehorsp
$\Omega_\mathrm{c}h^2$ \vertsp ${\siround{0.1204}{4}}^{+\siround{0.0042}{4}}_{-\siround{0.0043}{4}}$\vertsp${\siround{0.1199}{4}}^{+\siround{0.0032}{4}}_{-\siround{0.003}{4}}$\vertsp${\siround{0.1198}{4}}^{+\siround{0.0014}{4}}_{-\siround{0.0013}{4}}$\vertsp${\siround{0.12}{4}}^{+\siround{0.001}{4}}_{-\siround{0.0009}{4}}$\\
\morehorsp
$100\theta_\mathrm{MC}$ \vertsp ${\siround{1.04075}{5}}\pm{\siround{0.0008}{5}}$\vertsp${\siround{1.04068}{5}}^{+\siround{0.00031}{5}}_{-\siround{0.00035}{5}}$\vertsp${\siround{1.04074}{5}}^{+\siround{0.00015}{5}}_{-\siround{0.00016}{5}}$\vertsp${\siround{1.04075}{5}}\pm{\siround{0.00011}{5}}$\\
\morehorsp
$\tau$ \vertsp ${\siround{0.055}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.054}{3}}\pm{\siround{0.01}{3}}$\vertsp${\siround{0.053}{3}}\pm{\siround{0.011}{3}}$\vertsp${\siround{0.055}{3}}^{+\siround{0.002}{3}}_{-\siround{0.003}{3}}$\\
\morehorsp
$H_0$ \vertsp ${\siround{67.3}{2}}^{+\siround{0.67}{2}}_{-\siround{0.68}{2}}$\vertsp${\siround{67.3}{2}}^{+\siround{0.65}{2}}_{-\siround{0.67}{2}}$\vertsp${\siround{67.26}{2}}^{+\siround{0.66}{2}}_{-\siround{0.63}{2}}$\vertsp${\siround{67.27}{2}}\pm{\siround{0.65}{2}}$\\
\morehorsp
$\Omega_K$ \vertsp ${\siround{-0.005}{3}}^{+\siround{0.007}{3}}_{-\siround{0.005}{3}}$\vertsp${\siround{-0.006}{3}}^{+\siround{0.007}{3}}_{-\siround{0.008}{3}}$\vertsp${\siround{-0.004}{3}}\pm{\siround{0.005}{3}}$\vertsp${\siround{-0.001}{3}}\pm{\siround{0.003}{3}}$\\
\morehorsp
$\log(10^{10} A_\mathrm{s})$ \vertsp ${\siround{3.093}{3}}\pm{\siround{0.01}{3}}$\vertsp${\siround{3.091}{3}}^{+\siround{0.022}{3}}_{-\siround{0.023}{3}}$\vertsp${\siround{3.09}{3}}\pm{\siround{0.022}{3}}$\vertsp${\siround{3.095}{3}}^{+\siround{0.005}{3}}_{-\siround{0.006}{3}}$\\
\morehorsp
$n_\mathrm{s}$ \vertsp ${\siround{0.9631}{4}}^{+\siround{0.0072}{4}}_{-\siround{0.0074}{4}}$\vertsp${\siround{0.9658}{4}}^{+\siround{0.0117}{4}}_{-\siround{0.0104}{4}}$\vertsp${\siround{0.9653}{4}}^{+\siround{0.0049}{4}}_{-\siround{0.0047}{4}}$\vertsp${\siround{0.9649}{4}}^{+\siround{0.0035}{4}}_{-\siround{0.0037}{4}}$\\
\morehorsp
$w$ \vertsp ${\siround{-1.199}{3}}^{+\siround{0.26}{3}}_{-\siround{0.112}{3}}$\vertsp${\siround{-1.208}{3}}^{+\siround{0.241}{3}}_{-\siround{0.142}{3}}$\vertsp${\siround{-1.1}{3}}^{+\siround{0.126}{3}}_{-\siround{0.086}{3}}$\vertsp${\siround{-1.032}{3}}^{+\siround{0.07}{3}}_{-\siround{0.046}{3}}$\\
\morehorsp
$N_{\rm eff}$ \vertsp ${\siround{3.073}{3}}^{+\siround{0.243}{3}}_{-\siround{0.255}{3}}$\vertsp${\siround{3.076}{3}}^{+\siround{0.147}{3}}_{-\siround{0.141}{3}}$\vertsp${\siround{3.059}{3}}\pm{\siround{0.07}{3}}$\vertsp${\siround{3.055}{3}}^{+\siround{0.044}{3}}_{-\siround{0.043}{3}}$\\
\morehorsp
$\Sigma m_\nu$\vertsp $< \siround{0.587}{3}$ eV\vertsp$< \siround{0.536}{3}$ eV\vertsp$< \siround{0.326}{3}$ eV\vertsp$< \siround{0.206}{3}$ eV\\\bottomrule
\end{tabular}
\caption{Forecasted constraints at $68 \%$ C.L. (upper limits at $95 \%$ C.L.) from CMB+GWSS67 data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ extended model.}
\label{CMBGW67constraints}
\end{center}
\end{table*}
\begin{figure*}[!hbtp]
\includegraphics[width=.67\textwidth,keepaspectratio]{mixed-CMB+GW67.pdf}
\caption{Forecasted constraints at $68 \%$ and $95 \%$ C.L. from CMB+GWSS67 data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ extended model.}
\label{cmbgw67}
\end{figure*}
As discussed in the previous section, a similar uncertainty can be reached by the HLVJI network after one year of observations ($\sim2025$) with a BNS detection rate of $R\ge 1540$ Gpc$^{-3}$yr$^{-1}$ or by HLV after two years of observations($\sim2023$) if the rate is $R\ge2800$ Gpc$^{-3}$yr$^{-1}$. For simplicity we have assumed that the standard siren accuracy on $H_0$ scales as $1/\sqrt{N_{\rm BNS}}$ where $N_{\rm BNS}$ is the number of observed BNS systems, which is a good approximation for $N\gtrsim20$~\cite{holz}. A first conclusion is that by 2025--2030 standard sirens may offer constraints on $H_0$ that are comparable in accuracy to those achievable from future CMB+BAO missions at a similar epoch.
Furthermore, given existing estimates of the BNS event rate, an even higher $H_0$ accuracy may be expected from GWSS.
In Table~\ref{CMBGW67constraints} and in Figure~\ref{cmbgw67} we report the future constraints achievable by a combination of the CMB data and a prior on the Hubble constant with a $1 \%$ accuracy (GWSS67). This GWSS67 prior, with respect to the CMB data alone, breaks the geometrical degeneracy and improves significantly the constraints on the corresponding parameters, now producing strong bounds on cosmological parameters such as curvature ($0.3 \%$ accuracy from CMB-S4+GWSS67) and $w$ ($7 \%$ accuracy from CMB-S4+GWSS67). The bound on neutrino masses is improved by $\sim 30 \%$, while there is no significant improvement on the remaining parameters ($N_{\rm eff}$, $n_S$, and the cold dark matter and baryon densities).
How would the inclusion of a GWSS measurement of $H_0$ impact cosmological constraints derived from a CMB+DESI? We answer to this question in Table~\ref{CMBDESIGW67constraints} and Figure~\ref{cmbdesigw670} where we report the constraints achievable from the full combined dataset.
We find that the combined analysis (in the case of CMB-S4) would constrain the Hubble constant with an accuracy of $\sim 0.5\,$km/s/Mpc, i.e. nearly a factor of two better than the CMB-S4+DESI case. A similar improvement is present with respect to LiteBIRD+DESI. Constraints on the dark energy equation of state are also significantly improved, by 30--40\%, reaching an accuracy of about $3\%$ with CMB-S4+DESI and $4 \%$ with LiteBIRD+DESI.
It is interesting to note that the constraints on $H_0$, $\Omega_k$, and $w$ coming from a combined analysis of DESI, GWSS, and a CMB mission such as LiteBIRD, S3deep, or S3wide, will be comparable or in some cases even better than the corresponding constraints coming from a CMB-S4+DESI dataset. For example, a $0.1\%$ accuracy on curvature or a $3 \%$ accuracy on $w$ can be reached by a S3wide+DESI+GWSS67 configuration instead of CMB-S4+DESI.
Alternatively, the GWSS measurement would also provide an interesting consistency check between different CMB+BAO datasets.
\begin{table*}[!hbtp]
\begin{center}
\begin{tabular}{lcccc}
\toprule
\horsp
Parameter \vertsp LiteBIRD+DESI+GWSS67 \vertsp S3deep+DESI+GWSS67 \vertsp S3wide+DESI+GWSS67 \vertsp CMB-S4+DESI+GWSS67\\
\hline\hline
\morehorsp
$\Omega_\mathrm{b}h^2$ \vertsp ${\siround{0.02218}{5}}\pm{\siround{0.00021}{5}}$\vertsp${\siround{0.02218}{5}}^{+\siround{0.00015}{5}}_{-\siround{0.00016}{5}}$\vertsp${\siround{0.02218}{5}}\pm{\siround{9e-05}{5}}$\vertsp${\siround{0.02218}{5}}\pm{\siround{5e-05}{5}}$\\
\morehorsp
$\Omega_\mathrm{c}h^2$ \vertsp ${\siround{0.1205}{4}}^{+\siround{0.0028}{4}}_{-\siround{0.0031}{4}}$\vertsp${\siround{0.1205}{4}}\pm{\siround{0.0024}{4}}$\vertsp${\siround{0.1199}{4}}\pm{\siround{0.0013}{4}}$\vertsp${\siround{0.1199}{4}}\pm{\siround{0.001}{4}}$\\
\morehorsp
$100\theta_\mathrm{MC}$ \vertsp ${\siround{1.04069}{5}}^{+\siround{0.00064}{5}}_{-\siround{0.00063}{5}}$\vertsp${\siround{1.04072}{5}}\pm{\siround{0.0003}{5}}$\vertsp${\siround{1.04075}{5}}\pm{\siround{0.00015}{5}}$\vertsp${\siround{1.04076}{5}}\pm{\siround{0.00011}{5}}$\\
\morehorsp
$\tau$ \vertsp ${\siround{0.055}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.057}{3}}\pm{\siround{0.009}{3}}$\vertsp${\siround{0.057}{3}}\pm{\siround{0.008}{3}}$\vertsp${\siround{0.055}{3}}^{+\siround{0.002}{3}}_{-\siround{0.003}{3}}$\\
\morehorsp
$H_0$ \vertsp ${\siround{67.37}{2}}^{+\siround{0.6}{2}}_{-\siround{0.61}{2}}$\vertsp${\siround{67.36}{2}}^{+\siround{0.58}{2}}_{-\siround{0.59}{2}}$\vertsp${\siround{67.32}{2}}\pm{\siround{0.57}{2}}$\vertsp${\siround{67.31}{2}}^{+\siround{0.54}{2}}_{-\siround{0.55}{2}}$\\
\morehorsp
$\Omega_K$ \vertsp ${\siround{0.0}{3}}^{+\siround{0.001}{3}}_{-\siround{0.002}{3}}$\vertsp${\siround{0.001}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.0}{3}}\pm{\siround{0.001}{3}}$\vertsp${\siround{0.0}{3}}\pm{\siround{0.001}{3}}$\\
\morehorsp
$\log(10^{10} A_\mathrm{s})$ \vertsp ${\siround{3.096}{3}}\pm{\siround{0.008}{3}}$\vertsp${\siround{3.1}{3}}\pm{\siround{0.018}{3}}$\vertsp${\siround{3.099}{3}}\pm{\siround{0.016}{3}}$\vertsp${\siround{3.095}{3}}^{+\siround{0.005}{3}}_{-\siround{0.006}{3}}$\\
\morehorsp
$n_\mathrm{s}$ \vertsp ${\siround{0.9648}{4}}\pm{\siround{0.0061}{4}}$\vertsp${\siround{0.9635}{4}}\pm{\siround{0.01}{4}}$\vertsp${\siround{0.9645}{4}}\pm{\siround{0.0046}{4}}$\vertsp${\siround{0.9648}{4}}\pm{\siround{0.0036}{4}}$\\
\morehorsp
$w$ \vertsp ${\siround{-1.003}{3}}^{+\siround{0.043}{3}}_{-\siround{0.039}{3}}$\vertsp${\siround{-1.009}{3}}^{+\siround{0.038}{3}}_{-\siround{0.035}{3}}$\vertsp${\siround{-1.007}{3}}\pm{\siround{0.03}{3}}$\vertsp${\siround{-1.003}{3}}\pm{\siround{0.028}{3}}$\\
\morehorsp
$N_{\rm eff}$ \vertsp ${\siround{3.076}{3}}^{+\siround{0.176}{3}}_{-\siround{0.178}{3}}$\vertsp${\siround{3.053}{3}}^{+\siround{0.124}{3}}_{-\siround{0.123}{3}}$\vertsp${\siround{3.048}{3}}^{+\siround{0.065}{3}}_{-\siround{0.066}{3}}$\vertsp${\siround{3.052}{3}}^{+\siround{0.043}{3}}_{-\siround{0.044}{3}}$\\
\morehorsp
$\Sigma m_\nu$\vertsp $< \siround{0.164}{3}$ eV\vertsp$< \siround{0.226}{3}$ eV\vertsp$< \siround{0.18}{3}$ eV\vertsp$< \siround{0.12}{3}$ eV\\\bottomrule
\end{tabular}
\caption{Forecasted constraints at $68 \%$ C.L. (upper limits at $95 \%$ C.L.) from CMB+DESI+GWSS67 data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ extended model.}
\label{CMBDESIGW67constraints}
\end{center}
\end{table*}
\begin{figure*}[!hbtp]
\includegraphics[width=.67\textwidth,keepaspectratio]{mixed-CMB+DESI+GWSS67.pdf}
\caption{Forecasted constraints at $68 \%$ and $95 \%$ C.L. from CMB+DESI+GWSS67 data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ extended model.}
\label{cmbdesigw670}
\end{figure*}
We also consider the possibility that future standard siren measurements of $H_0$ will confirm the current tension on the Hubble constant between CMB+BAO and local measurements from supernovae. It is interesting to evaluate at how many standard deviations a CMB+DESI measurement of $H_0$ will disagree with a GWSS determination of $H_0=73.30\pm0.73\,$km/s/Mpc.
From Table~\ref{CMBDESIconstraints}, we find that the standard siren measurement would be $4$ standard deviations from the expected LiteBIRD+DESI constraint, and at roughly $5$ standard deviations from the CMB-S4+DESI value. This is a significant improvement, since in an extended parameter space such the one we are considering the existing tension is at about $2$ standard deviations (see e.g. \cite{papero}).
Finally, let us consider a significantly more pessimistic GW prior on $H_0$ with a $\sim 4\%$ accuracy (PGWSS67).
In Table~\ref{CMBPGW67constraints} we report the constraints achievable from a combination of this prior with future CMB data. As expected, the constraints on curvature and $w$ are relaxed respect to the previous analyses of CMB+GWSS67 but only by a $\sim 10-20 \%$. In practice, the geometrical degeneracies between cosmological parameters present in CMB data only can be already sufficiently broken with a, pessimistic, PGWSS67 prior. An improvement of a factor four in the determination of $H_0$ will result in a, more modest, $10 \%$ improvement in the parameters. A first conclusion is therefore that in this theoretical framework, the GWSS67 and the PGWSS67 prior produce very similar constraints when combined with CMB data.
On the other hand, combining the PGWSS67 prior with CMB+DESI data has a small effect in improving the constraints on $w$. We have found that in this case the constraints on $w$ improve just by $\sim 5 \%$ while, as discussed above, the improvement in case of GWSS67 is larger than $\sim 20 \%$.
The $4 \%$ PGWSS67 prior will clearly provide little help in solving the current tension on the value of the Hubble parameter.
\begin{table*}[!hbtp]
\begin{center}
\begin{tabular}{lcccc}
\toprule
\horsp
Parameter \vertsp LiteBIRD+PGWSS67($4\%$) \vertsp S3deep+PGWSS67($4\%$) \vertsp S3wide+PGWSS67($4\%$) \vertsp S4+PGWSS67($4\%$) \\
\hline\hline
\horsp
$\Omega_\mathrm{b}h^2$ \vertsp ${\siround{0.02215}{5}}\pm{\siround{0.00023}{5}}$\vertsp${\siround{0.02222}{5}}\pm{\siround{0.00017}{5}}$\vertsp${\siround{0.02219}{5}}\pm{\siround{9e-05}{5}}$\vertsp${\siround{0.02219}{5}}\pm{\siround{5e-05}{5}}$\\
\morehorsp
$\Omega_\mathrm{c}h^2$ \vertsp ${\siround{0.1204}{4}}^{+\siround{0.004}{4}}_{-\siround{0.0044}{4}}$\vertsp${\siround{0.1199}{4}}^{+\siround{0.0031}{4}}_{-\siround{0.0029}{4}}$\vertsp${\siround{0.1198}{4}}^{+\siround{0.0014}{4}}_{-\siround{0.0013}{4}}$\vertsp${\siround{0.12}{4}}^{+\siround{0.001}{4}}_{-\siround{0.0011}{4}}$\\
\morehorsp
$100\theta_\mathrm{MC}$ \vertsp ${\siround{1.04073}{5}}^{+\siround{0.00079}{5}}_{-\siround{0.00078}{5}}$\vertsp${\siround{1.04068}{5}}^{+\siround{0.00034}{5}}_{-\siround{0.00033}{5}}$\vertsp${\siround{1.04074}{5}}\pm{\siround{0.00015}{5}}$\vertsp${\siround{1.04075}{5}}\pm{\siround{0.00012}{5}}$\\
\morehorsp
$\tau$ \vertsp ${\siround{0.055}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.054}{3}}\pm{\siround{0.01}{3}}$\vertsp${\siround{0.053}{3}}\pm{\siround{0.011}{3}}$\vertsp${\siround{0.055}{3}}^{+\siround{0.002}{3}}_{-\siround{0.003}{3}}$\\
\morehorsp
$H_0$ \vertsp ${\siround{67.03}{2}}\pm{\siround{2.68}{2}}$\vertsp${\siround{66.95}{2}}^{+\siround{2.65}{2}}_{-\siround{2.69}{2}}$\vertsp${\siround{66.86}{2}}^{+\siround{2.8}{2}}_{-\siround{2.69}{2}}$\vertsp${\siround{66.72}{2}}^{+\siround{2.52}{2}}_{-\siround{2.55}{2}}$\\
\morehorsp
$\Omega_K$ \vertsp ${\siround{-0.006}{3}}^{+\siround{0.007}{3}}_{-\siround{0.005}{3}}$\vertsp${\siround{-0.007}{3}}\pm{\siround{0.008}{3}}$\vertsp${\siround{-0.004}{3}}^{+\siround{0.005}{3}}_{-\siround{0.004}{3}}$\vertsp${\siround{-0.002}{3}}\pm{\siround{0.003}{3}}$\\
\morehorsp
$\log(10^{10} A_\mathrm{s})$ \vertsp ${\siround{3.093}{3}}\pm{\siround{0.01}{3}}$\vertsp${\siround{3.09}{3}}\pm{\siround{0.022}{3}}$\vertsp${\siround{3.09}{3}}\pm{\siround{0.021}{3}}$\vertsp${\siround{3.094}{3}}^{+\siround{0.005}{3}}_{-\siround{0.006}{3}}$\\
\morehorsp
$n_\mathrm{s}$ \vertsp ${\siround{0.9631}{4}}^{+\siround{0.0073}{4}}_{-\siround{0.0074}{4}}$\vertsp${\siround{0.9662}{4}}^{+\siround{0.0109}{4}}_{-\siround{0.0111}{4}}$\vertsp${\siround{0.9651}{4}}^{+\siround{0.0046}{4}}_{-\siround{0.0049}{4}}$\vertsp${\siround{0.9652}{4}}^{+\siround{0.004}{4}}_{-\siround{0.0039}{4}}$\\
\morehorsp
$w_0$ \vertsp ${\siround{-1.188}{3}}^{+\siround{0.274}{3}}_{-\siround{0.13}{3}}$\vertsp${\siround{-1.191}{3}}^{+\siround{0.254}{3}}_{-\siround{0.151}{3}}$\vertsp${\siround{-1.087}{3}}^{+\siround{0.148}{3}}_{-\siround{0.112}{3}}$\vertsp${\siround{-1.022}{3}}^{+\siround{0.088}{3}}_{-\siround{0.079}{3}}$\\
\morehorsp
$N_{\rm eff}$ \vertsp ${\siround{3.073}{3}}^{+\siround{0.244}{3}}_{-\siround{0.247}{3}}$\vertsp${\siround{3.078}{3}}^{+\siround{0.142}{3}}_{-\siround{0.14}{3}}$\vertsp${\siround{3.056}{3}}^{+\siround{0.068}{3}}_{-\siround{0.069}{3}}$\vertsp${\siround{3.058}{3}}^{+\siround{0.047}{3}}_{-\siround{0.048}{3}}$\\
\morehorsp
$m_\nu$\vertsp $< \siround{0.58}{3}$ eV\vertsp$< \siround{0.531}{3}$ eV \vertsp$< \siround{0.338}{3}$ eV\vertsp$< \siround{0.208}{3}$ eV\\\bottomrule
\end{tabular}
\caption{Forecasted constraints at $68 \%$ C.L. (upper limits at $95 \%$ C.L.) from CMB+PGWSS67 data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ extended model.}
\label{CMBPGW67constraints}
\end{center}
\end{table*}
\subsection{$\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$w_a$+$w_0$ Model}
\begin{table*}[!hbtp]
\begin{center}
\begin{tabular}{lcccc}
\toprule
\horsp
Parameter \vertsp LiteBIRD+DESI \vertsp S3wide+DESI \vertsp S3deep+DESI \vertsp CMB-S4+DESI \\
\hline\hline
\morehorsp
$\Omega_\mathrm{b}h^2$ \vertsp ${\siround{0.02214}{5}}\pm{\siround{0.00018}{5}}$\vertsp${\siround{0.02218}{5}}\pm{\siround{6e-05}{5}}$\vertsp${\siround{0.02217}{5}}\pm{\siround{0.00011}{5}}$\vertsp${\siround{0.02218}{5}}\pm{\siround{3e-05}{5}}$\\
\morehorsp
$\Omega_\mathrm{c}h^2$ \vertsp ${\siround{0.1201}{4}}\pm{\siround{0.0011}{4}}$\vertsp${\siround{0.1199}{4}}\pm{\siround{0.0009}{4}}$\vertsp${\siround{0.1207}{4}}^{+\siround{0.0018}{4}}_{-\siround{0.002}{4}}$\vertsp${\siround{0.1198}{4}}\pm{\siround{0.0008}{4}}$\\
\morehorsp
$100\theta_\mathrm{MC}$ \vertsp ${\siround{1.04072}{5}}\pm{\siround{0.00049}{5}}$\vertsp${\siround{1.04075}{5}}\pm{\siround{0.00013}{5}}$\vertsp${\siround{1.0407}{5}}\pm{\siround{0.00028}{5}}$\vertsp${\siround{1.04077}{5}}\pm{\siround{0.0001}{5}}$\\
\morehorsp
$\tau$ \vertsp ${\siround{0.055}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.057}{3}}\pm{\siround{0.009}{3}}$\vertsp${\siround{0.057}{3}}\pm{\siround{0.009}{3}}$\vertsp${\siround{0.055}{3}}^{+\siround{0.002}{3}}_{-\siround{0.003}{3}}$\\
\morehorsp
$H_0$ \vertsp ${\siround{66.15}{1}}\pm{\siround{2.25}{1}}$\vertsp${\siround{66.33}{1}}\pm{\siround{2.28}{1}}$\vertsp${\siround{66.35}{1}}\pm{\siround{2.42}{1}}$\vertsp${\siround{66.41}{1}}^{+\siround{2.18}{1}}_{-\siround{1.9}{1}}$\\
\morehorsp
$\Omega_K$ \vertsp ${\siround{0.0}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{-0.0}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.001}{3}}\pm{\siround{0.003}{3}}$\vertsp${\siround{-0.0}{3}}\pm{\siround{0.001}{3}}$\\
\morehorsp
$\log(10^{10} A_\mathrm{s})$ \vertsp ${\siround{3.095}{3}}\pm{\siround{0.004}{3}}$\vertsp${\siround{3.098}{3}}\pm{\siround{0.017}{3}}$\vertsp${\siround{3.1}{3}}\pm{\siround{0.018}{3}}$\vertsp${\siround{3.094}{3}}\pm{\siround{0.005}{3}}$\\
\morehorsp
$n_\mathrm{s}$ \vertsp ${\siround{0.9638}{4}}\pm{\siround{0.0042}{4}}$\vertsp${\siround{0.9644}{4}}\pm{\siround{0.0026}{4}}$\vertsp${\siround{0.9626}{4}}^{+\siround{0.006}{4}}_{-\siround{0.0059}{4}}$\vertsp${\siround{0.9645}{4}}\pm{\siround{0.0023}{4}}$\\
\morehorsp
$w_0$ \vertsp ${\siround{-0.859}{3}}^{+\siround{0.202}{3}}_{-\siround{0.259}{3}}$\vertsp${\siround{-0.883}{3}}^{+\siround{0.203}{3}}_{-\siround{0.252}{3}}$\vertsp${\siround{-0.872}{3}}^{+\siround{0.225}{3}}_{-\siround{0.269}{3}}$\vertsp${\siround{-0.901}{3}}^{+\siround{0.149}{3}}_{-\siround{0.228}{3}}$\\
\morehorsp
$w_a$ \vertsp ${\siround{-0.47}{3}}^{+\siround{0.795}{3}}_{-\siround{0.54}{3}}$\vertsp${\siround{-0.39}{3}}^{+\siround{0.749}{3}}_{-\siround{0.549}{3}}$\vertsp${\siround{-0.456}{3}}^{+\siround{0.818}{3}}_{-\siround{0.616}{3}}$\vertsp${\siround{-0.306}{3}}^{+\siround{0.661}{3}}_{-\siround{0.372}{3}}$\\
\morehorsp
$\Sigma m_\nu$\vertsp $< \siround{0.212}{3}$ eV\vertsp$< \siround{0.216}{3}$ eV\vertsp$< \siround{0.289}{3}$ eV\vertsp$< \siround{0.15}{3}$ eV\\\bottomrule
\end{tabular}
\caption{Forecasted constraints at $68 \%$ C.L. (upper limits at $95 \%$ C.L.) from CMB+DESI data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$w$+$w_a$ extended model. Note the significant increase in the error on $H_0$ (about a factor two) with respect to the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ scenario reported before.}
\label{CMBDESIconstraints2}
\end{center}
\end{table*}
\begin{figure*}[!hbtp]
\includegraphics[width=.67\textwidth,keepaspectratio]{mixed-CMBwa+DESI.pdf}
\caption{Forecasted constraints at $68 \%$ and $95 \%$ C.L. from CMB+DESI data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$w_0$+$w_a$ extended model.}
\label{cmbdesi2}
\end{figure*}
As shown in the previous section, the neutrino effective number $N_{\rm eff}$ will be measured with good accuracy even in extended parameter spaces. The main reason for this is due to the lack of the so-called early integrated Sachs Wolfe effect in polarization data. The inclusion of polarization helps in determining the amplitude of the EISW and $N_{\rm eff}$.
Since we are interested in evaluating the impact of a future GWSS measurement of $H_0$, it makes sense to further extend the number of geometric parameters. In what follows we substitute $N_{\rm eff}$ with $w_a$, considering therefore a dynamical dark energy equation of state described by a CPL form.
In Table~\ref{CMBDESIconstraints2} we report the constraints at $68 \%$ C.L. on cosmological parameters from the combination of future CMB and DESI data while in Figure~\ref{cmbdesi2} we report the corresponding 2D contours for the $68 \%$ and $95 \%$ confidence levels.
If we compare with the results in Table~\ref{CMBDESIconstraints2} and in Figure~\ref{cmbdesi2} with those previously obtained assuming $w=constant$ in Table~\ref{CMBDESIconstraints} and in Figure~\ref{cmbdesi} there is now a substantial increase (about a factor two!) in the error on $H_0$. Indeed, now the combination of CMB-S4+DESI data is able to constrain the Hubble constant to only $\sim 2\,$km/s/Mpc error, i.e. to a $\sim 3 \%$ accuracy. LiteBIRD+DESI constrains $H_0$ to $\sim 3.5 \%$ accuracy. These weaker constraints are due to the geometrical degeneracy between $H_0$, $w_a$, and $w_0$. The two dark energy parameters are now weakly determined, with uncertainties of the order of $\sim 20 \%$ for $w_0$ and $\sim 60$--70\% for $w_a$. $H_0$, $w_a$, and $w_0$ are also determined to similar accuracy by different CMB experiments, indicating that the constraining power in this case is coming primarily from DESI.
The constraint on $\Omega_k$ is virtually unchanged with respect to Table~\ref{CMBDESIconstraints}, and varies with the CMB experiment considered. The inclusion of $w_a$ weakens the future constraint on the sum of neutrino masses, $\Sigma m_{\nu}$. Other parameters, such as $n_S$, that are degenerate with $N_{\rm eff}$, are, on the contrary, now better constrained.
Given the strong degeneracy in the $w_0$--$w_a$ plane for these future experiments, it is clearly interesting to study the impact of a future GWSS determination of $H_0$. As discussed in the previous section, a $3 \%$ accuracy on $H_0$ can be reached by the HLV network after two years of operation if the BNS detection rate is $R>3500\,$Gpc$^{-3}$yr$^{-1}$, a value well inside current limits. The same accuracy can be achieved by the HLVJI network after just one year of observation even assuming the lowest BNS rate of $R=320\,$Gpc$^{-3}$yr$^{-1}$. We found that including a $3 \%$ GWSS prior to the CMB+DESI constraints reported in Table~\ref{CMBDESIconstraints2} the constraints on $H_0$ and on the dark energy parameters could be already improved at the level of $10-30\%$.
\begin{table*}[!hbtp]
\begin{center}
\begin{tabular}{lcccc}
\toprule
\horsp
Parameter \vertsp LiteBIRD+DESI+GWSS67 \vertsp S3wide+DESI+GWSS67 \vertsp S3deep+DESI+GWSS67 \vertsp CMB-S4+DESI+GWSS67 \\
\hline\hline
\morehorsp
$\Omega_\mathrm{b}h^2$ \vertsp ${\siround{0.02214}{5}}\pm{\siround{0.00017}{5}}$\vertsp${\siround{0.02218}{5}}\pm{\siround{5e-05}{5}}$\vertsp${\siround{0.02217}{5}}\pm{\siround{0.00012}{5}}$\vertsp${\siround{0.02218}{5}}\pm{\siround{3e-05}{5}}$\\
\morehorsp
$\Omega_\mathrm{c}h^2$ \vertsp ${\siround{0.1202}{4}}^{+\siround{0.001}{4}}_{-\siround{0.0011}{4}}$\vertsp${\siround{0.12}{4}}\pm{\siround{0.0009}{4}}$\vertsp${\siround{0.1207}{4}}^{+\siround{0.0017}{4}}_{-\siround{0.002}{4}}$\vertsp${\siround{0.1198}{4}}\pm{\siround{0.0008}{4}}$\\
\morehorsp
$100\theta_\mathrm{MC}$ \vertsp ${\siround{1.04074}{5}}\pm{\siround{0.00048}{5}}$\vertsp${\siround{1.04075}{5}}\pm{\siround{0.00013}{5}}$\vertsp${\siround{1.0407}{5}}\pm{\siround{0.00028}{5}}$\vertsp${\siround{1.04077}{5}}\pm{\siround{0.0001}{5}}$\\
\morehorsp
$\tau$ \vertsp ${\siround{0.055}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.057}{3}}\pm{\siround{0.008}{3}}$\vertsp${\siround{0.057}{3}}\pm{\siround{0.009}{3}}$\vertsp${\siround{0.055}{3}}\pm{\siround{0.002}{3}}$\\
\morehorsp
$H_0$ \vertsp ${\siround{67.21}{2}}^{+\siround{0.62}{2}}_{-\siround{0.63}{2}}$\vertsp${\siround{67.23}{2}}^{+\siround{0.67}{2}}_{-\siround{0.63}{2}}$\vertsp${\siround{67.24}{2}}\pm{\siround{0.64}{2}}$\vertsp${\siround{67.23}{2}}^{+\siround{0.63}{2}}_{-\siround{0.64}{2}}$\\
\morehorsp
$\Omega_K$ \vertsp ${\siround{0.0}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.0}{3}}\pm{\siround{0.001}{3}}$\vertsp${\siround{0.001}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{-0.0}{3}}\pm{\siround{0.001}{3}}$\\
\morehorsp
$\log(10^{10} A_\mathrm{s})$ \vertsp ${\siround{3.095}{3}}\pm{\siround{0.004}{3}}$\vertsp${\siround{3.098}{3}}^{+\siround{0.016}{3}}_{-\siround{0.017}{3}}$\vertsp${\siround{3.1}{3}}\pm{\siround{0.018}{3}}$\vertsp${\siround{3.095}{3}}\pm{\siround{0.005}{3}}$\\
\morehorsp
$n_\mathrm{s}$ \vertsp ${\siround{0.9638}{4}}\pm{\siround{0.0043}{4}}$\vertsp${\siround{0.9642}{4}}\pm{\siround{0.0026}{4}}$\vertsp${\siround{0.9625}{4}}\pm{\siround{0.0058}{4}}$\vertsp${\siround{0.9644}{4}}\pm{\siround{0.0022}{4}}$\\
\morehorsp
$w_0$ \vertsp ${\siround{-0.974}{3}}^{+\siround{0.078}{3}}_{-\siround{0.089}{3}}$\vertsp${\siround{-0.978}{3}}^{+\siround{0.081}{3}}_{-\siround{0.089}{3}}$\vertsp${\siround{-0.969}{3}}^{+\siround{0.084}{3}}_{-\siround{0.095}{3}}$\vertsp${\siround{-0.985}{3}}^{+\siround{0.066}{3}}_{-\siround{0.082}{3}}$\\
\morehorsp
$w_a$ \vertsp ${\siround{-0.147}{3}}^{+\siround{0.377}{3}}_{-\siround{0.282}{3}}$\vertsp${\siround{-0.127}{3}}^{+\siround{0.36}{3}}_{-\siround{0.304}{3}}$\vertsp${\siround{-0.188}{3}}^{+\siround{0.404}{3}}_{-\siround{0.32}{3}}$\vertsp${\siround{-0.08}{3}}^{+\siround{0.319}{3}}_{-\siround{0.225}{3}}$\\
\morehorsp
$\Sigma m_\nu$\vertsp $< \siround{0.196}{3}$ eV\vertsp$< \siround{0.205}{3}$ eV\vertsp$< \siround{0.278}{3}$ eV\vertsp$< \siround{0.14}{3}$ eV\\\bottomrule
\end{tabular}
\caption{Forecasted constraints at $68 \%$ C.L. (upper limits at $95 \%$ C.L.) from CMB+DESI+GWSS67 data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$w$+$w_a$ extended model. Note the significant improvement in accuracy on $H_0$ and on the dark energy parameters $w_0$ and $w_a$ with respect to the CMB+DESI case.}
\label{CMBDESIGW672}
\end{center}
\end{table*}
\begin{figure*}[!hbtp]
\includegraphics[width=.67\textwidth,keepaspectratio]{mixed-CMBwa+DESI+GW67.pdf}
\caption{Forecasted constraints at $68 \%$ and $95 \%$ C.L. from CMB+DESI+GWSS67 data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$w_0$+$w_a$ extended model.}
\label{cmbdesigw2}
\end{figure*}
However, a $\sim 1\%$ accuracy on $H_0$ is also directly attainable by future GWSS measurements, and it is interesting to discuss the impact of this improved determination on future combined cosmological parameter measurements.
We report the constraints on cosmological parameters for CMB+DESI+GWSS67 in Table~\ref{CMBDESIGW672} and the corresponding 2D confidence levels in Figure~\ref{cmbdesigw2}.
The measured value of the Hubble constant is practically identical to the assumed prior from the standard sirens (GWSS67), indicating that the standard siren measurements are contributing to the combined constraints on all related cosmological parameters.
In particular, the constraints on the dark energy parameters $w_0$ and $w_a$ are substantially improved, by a factor $\sim 1.6$--2.8, with the inclusion of the standard siren measurements.
Finally, in Table~\ref{CMBDESIPGW672} we report the expected constraints when combining future CMB data with a, pessimistic, PGWSS67 prior on the Hubble parameter. As we can see, including the PGWSS67 prior will improve the constraints on the dark energy parameters by $\sim 20-30 \%$ respect to CMB+DESI data. A $\sim 4 \%$ determination of the Hubble parameter can be therefore useful in this theoretical framework even when considering the CMB+DESI dataset.
However the constraints achievable with the PGWSS67 prior on $w_0$ will be about a factor two larger than those achievable with the GWSS67 prior.
\begin{table*}[!hbtp]
\begin{center}
\begin{tabular}{lcccc}
\toprule
\horsp
Parameter \vertsp LiteBIRD+DESI+PGWSS67\vertsp S3wide+DESI+PGWSS67\vertsp S3deep+DESI+PGWSS67\vertsp CMB-S4+DESI+PGWSS67\\
\hline\hline
\morehorsp
$\Omega_\mathrm{b}h^2$ \vertsp ${\siround{0.02214}{5}}\pm{\siround{0.00017}{5}}$\vertsp${\siround{0.02217}{5}}\pm{\siround{0.00011}{5}}$\vertsp${\siround{0.02218}{5}}\pm{\siround{6e-05}{5}}$\vertsp${\siround{0.02218}{5}}\pm{\siround{3e-05}{5}}$\\
\morehorsp
$\Omega_\mathrm{c}h^2$ \vertsp ${\siround{0.1201}{4}}^{+\siround{0.001}{4}}_{-\siround{0.0011}{4}}$\vertsp${\siround{0.1207}{4}}^{+\siround{0.0018}{4}}_{-\siround{0.002}{4}}$\vertsp${\siround{0.1199}{4}}\pm{\siround{0.0009}{4}}$\vertsp${\siround{0.1199}{4}}\pm{\siround{0.0008}{4}}$\\
\morehorsp
$100\theta_\mathrm{MC}$ \vertsp ${\siround{1.04073}{5}}\pm{\siround{0.00048}{5}}$\vertsp${\siround{1.0407}{5}}\pm{\siround{0.00028}{5}}$\vertsp${\siround{1.04075}{5}}\pm{\siround{0.00013}{5}}$\vertsp${\siround{1.04076}{5}}\pm{\siround{0.0001}{5}}$\\
\morehorsp
$\tau$ \vertsp ${\siround{0.055}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.057}{3}}\pm{\siround{0.009}{3}}$\vertsp${\siround{0.057}{3}}\pm{\siround{0.008}{3}}$\vertsp${\siround{0.055}{3}}^{+\siround{0.002}{3}}_{-\siround{0.003}{3}}$\\
\morehorsp
$H_0$ \vertsp ${\siround{66.64}{2}}^{+\siround{1.71}{2}}_{-\siround{1.72}{2}}$\vertsp${\siround{66.76}{2}}^{+\siround{1.82}{2}}_{-\siround{1.83}{2}}$\vertsp${\siround{66.74}{2}}^{+\siround{1.74}{2}}_{-\siround{1.72}{2}}$\vertsp${\siround{66.79}{2}}^{+\siround{1.69}{2}}_{-\siround{1.71}{2}}$\\
\morehorsp
$\Omega_K$ \vertsp ${\siround{0.0}{3}}\pm{\siround{0.002}{3}}$\vertsp${\siround{0.001}{3}}^{+\siround{0.002}{3}}_{-\siround{0.003}{3}}$\vertsp${\siround{0.0}{3}}\pm{\siround{0.001}{3}}$\vertsp${\siround{-0.0}{3}}\pm{\siround{0.001}{3}}$\\
\morehorsp
$\log(10^{10} A_\mathrm{s})$ \vertsp ${\siround{3.095}{3}}\pm{\siround{0.004}{3}}$\vertsp${\siround{3.1}{3}}\pm{\siround{0.018}{3}}$\vertsp${\siround{3.098}{3}}\pm{\siround{0.017}{3}}$\vertsp${\siround{3.095}{3}}\pm{\siround{0.005}{3}}$\\
\morehorsp
$n_\mathrm{s}$ \vertsp ${\siround{0.9638}{4}}\pm{\siround{0.0042}{4}}$\vertsp${\siround{0.9625}{4}}^{+\siround{0.0059}{4}}_{-\siround{0.0058}{4}}$\vertsp${\siround{0.9643}{4}}\pm{\siround{0.0026}{4}}$\vertsp${\siround{0.9643}{4}}\pm{\siround{0.0022}{4}}$\\
\morehorsp
$w_0$ \vertsp ${\siround{-0.912}{3}}^{+\siround{0.159}{3}}_{-\siround{0.195}{3}}$\vertsp${\siround{-0.918}{3}}^{+\siround{0.172}{3}}_{-\siround{0.204}{3}}$\vertsp${\siround{-0.926}{3}}^{+\siround{0.163}{3}}_{-\siround{0.19}{3}}$\vertsp${\siround{-0.938}{3}}^{+\siround{0.142}{3}}_{-\siround{0.182}{3}}$\\
\morehorsp
$w_a$ \vertsp ${\siround{-0.323}{3}}^{+\siround{0.619}{3}}_{-\siround{0.444}{3}}$\vertsp${\siround{-0.33}{3}}^{+\siround{0.643}{3}}_{-\siround{0.497}{3}}$\vertsp${\siround{-0.271}{3}}^{+\siround{0.588}{3}}_{-\siround{0.46}{3}}$\vertsp${\siround{-0.213}{3}}^{+\siround{0.548}{3}}_{-\siround{0.375}{3}}$\\
\morehorsp
$m_\nu$\vertsp $< \siround{0.205}{3}$ eV\vertsp$< \siround{0.284}{3}$ eV\vertsp$< \siround{0.211}{3}$ eV\vertsp$< \siround{0.148}{3}$ eV\\\bottomrule
\end{tabular}
\caption{Forecasted constraints at $68 \%$ C.L. (upper limits at $95 \%$ C.L.) from CMB+DESI+GWSS67 data for the experimental configurations in Table~\ref{tab:spec} in case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$w$+$w_a$ extended model. Note the significant improvement in accuracy on $H_0$ and on the dark energy parameters $w_0$ and $w_a$ with respect to the CMB+DESI case.}
\label{CMBDESIPGW672}
\end{center}
\end{table*}
\subsection{Figure of Merit}
\begin{table*}[!hbtp]
\begin{center}
\begin{tabular}{|c|c|cccc|}
\hline
\morehorsp
Model \vertsp Dataset \vertsp\ LiteBIRD \vertsp S3deep \vertsp S3wide \vertsp CMB-S4 \\
\hline
\hline
\morehorsp
$\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ & CMB & $5$ & $1$& $398$ &$29236$ \\
\horsp
\vertsp CMB$+$PGWSS67 & $110$ & $40$& $12732$ &$2.2\times10^{6}$ \\
\horsp
\vertsp CMB$+$GWSS67 & $262$ & $104$& $50929$ &$1.2\times10^{7}$ \\
\horsp
\vertsp CMB$+$DESI & $6659$ & $2415$& $383240$ & $3.74\times10^{7}$ \\
\horsp
\vertsp CMB$+$DESI$+$PGWSS67 & $7735$ & $2807$& $422008$ & $4.06\times10^{7}$\\
\horsp
\vertsp CMB$+$DESI$+$GWSS67 & $16928$ & $5484$& $752879$ & $7.39\times10^{7}$\\
\hline
\hline
\morehorsp
$\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$w_0$+$w_a$ & CMB & $7$ & $1$& $170$ &$9223$ \\
\horsp
\vertsp CMB$+$PGWSS67 & $111$ & $18$& $2732$ &$14402$ \\
\horsp
\vertsp CMB$+$GWSS67 & $291$ & $43$& $9231$ &$589791$ \\
\horsp
\vertsp CMB$+$DESI \vertsp$13335$ & $2394$& $227590$ &$1.04\times10^{7}$\\
\horsp
\vertsp CMB$+$DESI$+$PGWSS67 & $19458$ & $3577$& $323789$ &$1.6\times10^{7}$\\
\horsp
\vertsp CMB$+$DESI$+$GWSS67 & $57928$ & $11735$& $1.01\times10^{6}$ &$5.7\times10^{7}$\\
\hline
\end{tabular}
\end{center}
\caption{Improvement with respect to simulated CMB data of the global Figure of Merit for the two theoretical scenarios considered in the paper and for different combination of datasets. The FoM is normalized to the S3deep CMB alone case that provides the less constraining results.}
\label{tab:FOMs}
\end{table*}
It is interesting to quantify the improvement of a GWSS prior by comparing the overall Figure of Merit for the cases considered.
Given an experimental configuration and a set of $N$ parameters $p_i$ with $i=(1,...N)$, we can define the FoM from the covariance matrix of uncertainties on $p_i$ as (see e.g.
\cite{wmap9,core1}):
\begin{equation}
\rm FoM = (\det[\mathrm{cov} \ p_i\}])^{-1/2}
\end{equation}
\noindent that is proportional to the inverse of the volume of the constrained parameters space. It is important to stress that this FoM considers the whole parameter space and not just the dark energy parameters as in \cite{weinberg}.
In Table~\ref{tab:FOMs} we report the FoM for the two theoretical scenarios considered in this paper and for different combinations of datasets. The FoM are normalized to the S3deep, CMB only, value.
As we can see, in the case of $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ there is a significant improvement in FoM when the GWSS67 prior is included with the CMB data. The improvement is significant (between a factor $\sim 50$ and $\sim 400$) and larger in the case of the CMB-S4 dataset. A smaller but still significant improvement is present when the PGWSS67 prior is considered. This clearly shows that, once the geometrical degeneracies are broken by the introduction of the GWSS prior, there is a significantly improved parameter determination with this dataset.
It is interesting also to note that the S3wide configuration has a constraining power that is superior to LiteBIRD+GWSS67 and S3deep+GWSS67. When the DESI dataset is included there is an improvement by a factor $\sim 1000$ and $\sim 2400$. In this case the CMB dataset that would better benefit by the inclusion of the DESI data is S3deep. Both S3deep+DESI and LiteBIRD+DESI have a smaller FoM than S3wide+GWSS67, and S3wide+DESI has less constraining power than CMB-S4+GWSS67.
When further including the GWSS67 prior the improvement in FoM is about a factor 2--3 with respect to the CMB+DESI case, clearly showing that GWSS will be useful in further constraining the parameter space.
However, when considering the more pessimistic PGWSS67 prior the improvement with respect to the CMB+DESI case is just $\sim 10-20 \%$.
In the case of the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$w_0$+$w_a$ model the improvement in the FoM obtained by the inclusion of the GWSS67 prior in the case of the CMB data is about a factor of $\sim 50$. With the DESI dataset the improvement is a factor of $\sim 1000$--2400. As we can see these improvements are smaller if compared to the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ scenario, showing that in this case the parameter degeneracies are more severe. When the GWSS67 prior is included the improvement is about a factor $\sim 4$--6, larger if compared with the similar data combination for the $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ scenario. The combination of LiteBIRD, S3deep, and S3wide with DESI data has less constraining power than CMB-S4+GWSS67.
The inclusion of a PGWSS67 prior can improve by a $\sim 60 \%$ the FoM of CMB-S4 an CMB-S4+DESI.
\begin{figure*}[!hbtp]
\includegraphics[width=.47\textwidth,keepaspectratio]{LiteBIRD_FOMplot.pdf}
\includegraphics[width=.47\textwidth,keepaspectratio]{S3deep_FOMplot.pdf}
\includegraphics[width=.47\textwidth,keepaspectratio]{S3wide_FOMplot.pdf}
\includegraphics[width=.47\textwidth,keepaspectratio]{S4_FOMplot.pdf}
\caption{Figures of Merit for the theoretical models and experimental configurations considered in function of different priors on the Hubble parameter with a $4\%$, $3\%$, $2\%$, and $1\%$ accuracy respectively. The assumed CMB datasets are LiteBIRD (Top Left), S3deep (Top Right), S3wide (Bottom Left), and CMB-S4 (Bottom Right).}
\label{fomall}
\end{figure*}
Finally, in order to better visualize the impact of a future prior on $H_0$, we plot in Figure ~\ref{fomall} the values of the FoM in function of of $4$ different expected accuracies on the Hubble constant: $4 \%$, $3 \%$, $2 \%$, and $1 \$$.
We can firstly clearly see that the FoM will be in general larger in case of the "$w_0+w_a$" scenario with respect to the "$w_0+N_{\rm eff}$" for any experimental configuration (with the exception of LiteBIRD). The inclusion of an external prior on the Hubble parameter is therefore more efficient in improving the constraints in the case of a "$w_0+w_a$" model, where dynamical dark energy is considered. Secondly, while in the CMB only scenario an improvement in the accuracy of $H_0$ is always reflected in a substantial increase in the FoM, it seems that in the case of CMB+DESI and for the "$w_0+N_{\rm eff}$" model (the red lines in the figure) a significant increase is expected when moving to an accuracy below $2 \%$. An improved accuracy in $H_0$ from $4 \%$ to $2 \%$ produces larger improvements in the FoM for the CMB+DESI dataset in the case of the "$w_0+w_a$" scenario.
\section{Conclusions}
The recent observations of gravitational waves and electromagnetic emission produced by the merger of the binary neutron-star system GW170817 has introduced a complementary and direct method for measuring the Hubble constant. In the coming decade GW standard sirens are expected to produce constraints on $H_0$ with $\sim 1 \%$ accuracy. At the same time, improved constraints are expected from CMB experiments and from BAO surveys.
In an extended $\Lambda$CDM parameter space, where we have considered variations in curvature, neutrino mass, and the dark energy equation of state, we have found that a combination of future CMB and BAO data can constraint the Hubble constant at the level of 1.5--2\%.
A similar accuracy may be reached by the HLV network in the second year of observations if the the BNS rate is $R\ge2800$ Gpc$^{-3}$yr$^{-1}$, in agreement with current limits on $R$, or by the HLVJI network after one year of observations with a more conservative BNS detection rate of $R\ge 1540$ Gpc$^{-3}$yr$^{-1}$.
Gravitational wave standard sirens may reach a 1\% measurement of $H_0$ within the decade, which when combined with future CMB data would constrain
curvature to $0.3 \%$ and the dark energy equation of state to $\sim 5 \%$. A GWSS measurement of the Hubble constant would also improve the constraints on these geometrical parameters coming from future CMB+BAO data by by 30--40\%. In addition, the current 2$\sigma$ Hubble tension between CMB+BAO and supernova data could be strengthened to $5\sigma$ with the inclusion of standard siren constraints.
When we further include time variations in the dark energy equation of state, parameterizing its evolution with a CPL function, we find that future CMB+BAO data will constrain the Hubble constant to $\sim 3 \%$. This level of accuracy on $H_0$ can be independently reached by the HLV network of interferometers after the second year of operation if the BNS detection rate is $R>3500$ Gpc$^{-3}$yr$^{-1}$, a value again well inside current limits, or by the HLVJI network after one year of observations even considering a low BNS detection rate of $R=320$ Gpc$^{-3}$yr$^{-1}$. This standard siren measurement would therefore improve the CMB+BAO constraints on this model at the level of $10-30\%$.
Assuming a future $H_0$ accuracy of $\sim 1 \%$ from standard sirens, as to be expected within the decade, we find that the constraints on the dark energy equation of state parameters $w_0$ and $w_a$ from future CMB+BAO datasets can be improved by a factor 1.6--2.8. We conclude that standard siren measurements by the HLV and HLVJI gravitational-wave detector networks over the coming decade may significantly improve our understanding of cosmology.
We have also found that even a more pessimistic determination of $H_0$, with a a $\sim 4\%$ accuracy can significantly improve the constraints from CMB alone data in case of a $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$N_{\rm eff}$+$w$ model and from CMB alone and CMB+DESI data
in case of a $\Lambda$CDM+$\Omega_k$+$\Sigma m_{\nu}$+$w_0$+$w_a$
model.
Finally it is clearly worth mentioning that similar constraints on $H_0$ and dark energy parameters could come by combining CMB and BAO data with other complementary probes such as supernovae and cosmic shear (see e.g. \cite{aubourg,gwss11}). In this case future constraints from GWSS will play a crucial role in confirming these results and cross-validating the different approaches. In addition, these comparisons offer the exciting possibility of discovering new physics beyond the $\Lambda$-CDM scenario.
\acknowledgments
EDV acknowledges support from the European Research Council in the form of a Consolidator Grant with number 681431. AM thanks the University of Manchester and the Jodrell Bank Center for Astrophysics for hospitality. AM and FR are supported by TASP, iniziativa specifica INFN. DEH was partially supported by NSF grant PHY-1708081. He was also supported by the Kavli Institute for Cosmological Physics at the University of Chicago through NSF grant PHY-1125897 and an endowment from the Kavli Foundation. DEH also gratefully acknowledges support from the Marion and Stuart Rice Award. We thank Cristiano Palomba for useful comments.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
The gravitational waves (GWs) detected by LIGO-Virgo-KAGAR Collaboration have been widely applied to research on cosmology, astrophysics, and fundamental physics~\cite{LIGOScientific:2016lio,LIGOScientific:2016vpg,LIGOScientific:2017adf,LIGOScientific:2018cki,LIGOScientific:2018cki,LIGOScientific:2021aug,LIGOScientific:2021psn,LIGOScientific:2021sio}. After the third observing run (O3), the LIGO-Virgo-KAGAR collaboration released the third Gravitational-Wave Transient Catalog (GWTC-3), bringing the total number of GW events~\cite{LIGOScientific:2021djp} to 90. The events in GWTC-3 are dominated by the binary black holes (BBH), together with a few binary neutron stars (BNS) and neutron star–black hole binaries (NSBH). In particular, the observation of the GW170817 from a BNS merger~\cite{LIGOScientific:2017vwq} and its associated electromagnetic (EM) counterparts~\cite{LIGOScientific:2017ync,LIGOScientific:2017zic} announced the era of GW multi-messenger astronomy. GW170817 with its EM counterparts provided the first standard siren measurement of cosmic expansion history~\cite{LIGOScientific:2017adf}, which is independent from the traditional EM experiments such as the cosmic microwave background (CMB)~\cite{Planck:2018vyg}, baryon acoustic oscillations (BAO)~\cite{BOSS:2016wmc}, type Ia supernovae (SNe Ia)~\cite{Riess:2016jrr,Riess:2019cxk}, and strong gravitational lensing~\cite{Wong:2019kwg}.
GW standard siren is supposed to be one of the most promising probes (though currently not precise enough) to arbitrate the Hubble tension~\cite{Verde:2019ivm,Chen:2017rfc,Feeney:2018mkj,Borhanian:2020vyr} which arises from the $4.4\sigma$ discrepancy of the Hubble constant measurements between Planck~\cite{Planck:2018vyg} and SH0ES project~\cite{Riess:2019cxk}. Besides, GW standard sirens can be widely utilized for the study of cosmology, astrophysics, and fundamental physics~\cite{Dalal:2006qt,Cutler:2009qv,Sathyaprakash:2009xt,Zhao:2010sz,Cai:2016sby,Yang:2017bkv,Cai:2017aea,Belgacem:2017ihm,Belgacem:2019zzu,Yang:2021qge}. These applications of GW standard sirens are guaranteed by the fact that the luminosity distance can be directly inferred from the amplitude and shape of the waveform. Compared to the traditional standard candles--SNe Ia, the physics of GW standard sirens is very clear and there is no need for the calibration of distance~\cite{Schutz:1986gp}. However, the usage of standard sirens is drastically limited by the measurement of redshift of the GW sources, due to the mass-redshift degeneracy. Several techniques are proposed to obtain the redshift information. For sources with confirmed EM counterparts, the host galaxy and hence its redshift can be determined directly~\cite{Holz:2005df,Dalal:2006qt,Nissanke:2009kt}. These GWs associated with EM counterparts are dubbed as ``bright sirens''. For sources without the detection of EM counterparts, i.e., the ``dark sirens'', alternative techniques are needed to infer the source redshift. Many methods have been proposed such as adopting the astrophysically-motivated source mass distribution~\cite{Taylor:2012db,Farr:2019twy,You:2020wju,Mastrogiovanni:2021wsd}, counting all the potential host galaxies in the localized region and obtaining the statistical redshift information from the galaxy catalogs~\cite{Schutz:1986gp,DelPozzo:2011vcw,Nair:2018ign,LIGOScientific:2018gmd,DES:2019ccw,Gray:2019ksv,DES:2020nay,Borhanian:2020vyr,Finke:2021aom}, and cross-correlating between GWs and galaxies~\cite{Oguri:2016dgk,Mukherjee:2019wcg,Mukherjee:2020hyn,Bera:2020jhx,Mukherjee:2022afz}. Compared to the bright sirens, the constraints of cosmological parameters from dark sirens are much looser due to the undetermined redshift information. For using the galaxy catalogs, the large uncertainty of GW localization makes it very hard to pinpoint the true host galaxy and hence its redshift. In addition, the measurement of distance suffers from the degeneracy between distance and orbital inclination. While, for bright sirens, not only the host galaxy can be identified, but also the degeneracy can be broken with the help of the EM counterparts~\cite{Hotokezaka:2018dfi}. This makes the uncertainty of the Hubble constant from one dark siren much worse than that from one bright siren~\cite{LIGOScientific:2018gmd,DES:2019ccw,DES:2020nay}. However, the majority of dark sirens can compensate for this inferiority. For instance, the constraint of the Hubble constant from 46 selected dark sirens in GWTC-3 with GLADE+ K--band galaxy catalog information is comparable to that from bright siren GW170817~\cite{LIGOScientific:2021aug}. It also finds that the better sky localization of GW190814 makes this event more informative on the value of $H_0$ in comparison to the other GW events. Currently, due to the rarity of bright sirens (only one in total 90 events) and large sky errors of dark sirens (typically $10^{3}~\rm deg^2$), GW standard sirens of LIGO-Virgo-KAGRA are not precise enough to arbitrate the Hubble tension. Therefore, in order to improve the usage of GWs for the study of cosmic expansion history and other cosmological problems, one should resort to either more GW events with EM counterparts, namely bright sirens, or the precise localization of dark sirens. Considering the great challenge for the detection of EM counterparts and the very small fraction of bright sirens even with the future GW detector networks~\cite{Belgacem:2019tbw,Yang:2021qge}, the improvement of parameter estimation (in particular for the distance and localization) of dark sirens is very crucial.
In this paper, we would like to extend our research in~\cite{Yang:2022tig} which demonstrated that the eccentricity of long inspiralling compact binaries can significantly improve the distance inference and source localization of GWs. The non-negligible eccentricity of compact binaries that emit GWs is suggested in many investigations, which may contribute observational features in the sensitivity band of ground and space-based detectors~\cite{Antonini:2012ad,Samsing:2013kua,Thompson:2010dp,East:2012xq}. Different mechanisms of the dynamic formation of the compact binaries of black holes and neutron stars have been proposed to study their eccentricities~\cite{Rodriguez:2017pec,Samsing:2017xmd,Samsing:2017oij,Samsing:2018ykz,Wen:2002km,Pratten:2020fqn,OLeary:2008myb,Lee:2009ca,Lee:1994nq,Hong:2015aba}.
Orbital eccentricity is one of the most important features to distinguish between isolated and dynamical BBH formation scenarios~\cite{Nishizawa:2016jji,Nishizawa:2016eza,Breivik:2016ddj,Zevin:2021rtf}. Some studies indicate that a fraction of the binaries possesses eccentricities larger than 0.1 at 10 Hz~\cite{Wen:2002km,Silsbee:2016djf,Antonini:2017ash,Liu:2019gdc}. In addition to the implication of formation channels, the imprint of eccentricity in the waveform can also help improving the parameter estimation of GWs. The improvements of parameter estimation and source localization by eccentricity have been investigated in~\cite{Sun:2015bva,Ma:2017bux,Pan:2019anf} for the stellar-mass compact binaries with the ground-based detector networks and in~\cite{Mikoczi:2012qy} for the supermassive black hole binaries with space-borne LISA. For the stellar-mass binaries with ground-based detectors that are sensitive to high frequencies ($> 10$ Hz), the authors found the improvements in source localization increase with the eccentricity and mass of the binaries. For the case of the $100~M_{\odot}$ total mass BBH, the improvement factor is about 2 in general when eccentricity $e_0$ increases from 0.0 to 0.4. While for low-mass binaries, the improvement is negligible when total mass is smaller than $40~M_{\odot}$ and the localization is even worsened at some orientations when total mass is smaller than $5~M_{\odot}$~\cite{Pan:2019anf}. For the supermassive BBH observed by LISA ($10^{-4}-0.1$ Hz), the authors also found the source localization improves with increasing eccentricity and mass. In the case of ($\sim10^7~M_{\odot}$) supermassive BBH, the angular resolution is improved by $\sim 1$ order of magnitude for highly eccentric sources ($e=0.6$). Intriguingly, our recent research shows that in the mid-band ($0.1-10$ Hz), the source localization of the typical stellar-mass BBHs can achieve much more improvements from the nonvanishing eccentricities (as much as 1--3 orders of magnitude when $e_0=0.4$)~\cite{Yang:2022tig}. The distance inference can also be improved by more than 2 orders of magnitude in the near face-on orientations. These results suggest that eccentricity is of great significance for dark sirens as precise probes of cosmology. In this paper, we would like to present in some detail the parameter estimation of GWs emitted by the eccentric compact binaries in the mid-band. This is to complement our results in~\cite{Yang:2022tig}. Then the cosmological implications will be investigated, which is to extend our previous research.
The motivation for conducting this research in the mid-band is as follows. From current LIGO-Virgo-KAGRA detections, there is no strong evidence of eccentric compact binaries in the high-frequency band (>10 Hz)~\cite{LIGOScientific:2019dag,Romero-Shaw:2019itr,Nitz:2019spj,Wu:2020zwr}. The only exception is GW190521 which has been reported to be consistent with an eccentric BBH by separate teams~\cite{Romero-Shaw:2020thy,Gayathri:2020coq}. In~\cite{Romero-Shaw:2020thy}, the authors found GW190521 prefers a signal with eccentricity $e>0.1$ at 10 Hz, to a non-precessing, quasi-circular signal. However, for the lack of available waveforms which include both eccentricity and orbital precession, they found there is a degeneracy between non-spinning, moderately eccentric waveform and quasi-circular, precessing waveform. While, in~\cite{Gayathri:2020coq} the authors used the numerical relativity simulations which incorporate both procession and eccentricity, and found that GW190521 is consistent with a highly eccentric ($e=0.69^{+0.17}_{-0.22}$) merger at 90\% confidence level.
The scarcity of eccentric binaries in the LIGO/Virgo band can be ascribed to the damping of eccentricity (at leading order $\sim f^{-19/18}$) in the inspiral period of binaries, even if they are born with high eccentricity. It means that the probability of observing eccentric binaries is much higher in the mid-band.
Another great advantage in the mid-band is the long inspiral period (days to years) of the stellar-mass binaries. The motion of the space-borne detector can induce the modulation and Doppler effects in the phase of the waveform which yields important angular information. In addition, the effects of eccentricity can be accumulated over a long period and may lead to more improvements in parameter estimation. In the aspect of waveform modeling, the inpiral-only and frequency-domain waveform based on post-Newtonian approximation is accurate enough for the stellar-mass compact binaries in the mid-band. Finally, considering the nonsignificant improvement of localization by eccentricity in the LIGO/Virgo band~\cite{Pan:2019anf} (bear in mind that the typical localization of LIGO/Virgo binaries is around $10^2-10^3$ deg$^2$), the investigation of this issue in the mid-band is very essential.
In the mid-frequency band (0.1--10 Hz), the space-borne laser interferometers like DECIGO~\cite{Kawamura:2006up,Kawamura:2020pcg} and BBO~\cite{Harry:2006fi}, and the atom interferometers like MAGIS~\cite{Graham:2017pmn} and AEDGE~\cite{AEDGE:2019nxb} have been proposed. In this frequency band, the stellar-mass binary usually has a long inspiral period and the motion of the detectors could provide a very precise localization for the sources~\cite{Cutler:2009qv,Graham:2017lmg,Yang:2021xox,Liu:2022rvk,Yang:2022iwn}. In this paper, we adopt DECIGO as our fiducial detector in the mid-band. DECIGO consists of 4 clusters and each cluster has three satellites. Its pathfinder B-DECIGO with only one cluster is planned to be launched earlier than DECIGO in the mid-2030s.
We consider two scenarios of DECIGO for comparison. The first is the one cluster of DECIGO (hereafter we call it ``DECIGO-I'') with its design sensitivity $4\times 10^{-24}$ Hz$^{-1/2}$ at 1 Hz~\cite{Kawamura:2020pcg}. The second is the B-DECIGO whose sensitivity is around $6\times 10^{-23}$ Hz$^{-1/2}$ at 1 Hz. The main difference between DECIGO-I and B-DECIGO is the sensitivity, for which the former is 15 times better than the latter. One of the information we would like to provide in this research is how much we can benefit if B-DECIGO is upgraded to just one cluster of DECIGO. Our results can demonstrate the potential of DECIGO with only one cluster, which is more promising (e.g. for the launch time and cost) than the full configuration of DECIGO in the future.
The organization of this paper is as follows. In section~\ref{sec:typical} we focus on the parameter estimation of the GWs with various eccentricities in the mid-band. The main parameters we would like to investigate are luminosity distances and sky locations, from which we will also derive the 3-D localization volume for the GW sources. The improvements of the estimation for these parameters from eccentricity will be presented. In addition, the constraints of other parameters such as the chirp mass, inclination angle, and eccentricity will also be discussed. This is to complete and elaborate the results in our previous work~\cite{Yang:2022tig}. In section~\ref{sec:catalog}, we will simulate the catalogs of GWs with DECIGO-I and B-DECIGO based on current knowledge and on the assumption of various eccentricities. We focus on the golden dark sirens whose host galaxies can be uniquely identified. From the catalogs, we can assess the influence of eccentricity on the detection of golden dark sirens. In section~\ref{sec:cosmo}, we will discuss the implications of eccentricity for the applications of dark sirens in cosmology. Finally, we conclude our results in section~\ref{sec:conclusion} followed by some discussions.
\section{Parameter estimation of the typical binaries \label{sec:typical}}
\subsection{The eccentric waveform}
We adopt the non-spinning, inspiral-only EccentricFD waveform approximant available in {\sc LALSuite}~\cite{lalsuite} and use {\sc PyCBC}~\cite{alex_nitz_2022_6324278} to generate the eccentric waveform. EccentricFD corresponds to the enhanced post-circular (EPC) model~\cite{Huerta:2014eca}. To the zeroth order in the eccentricity, the model recovers the TaylorF2 post-Newtonian (PN) waveform at 3.5 PN order~\cite{Buonanno:2009zt}. To the zeroth PN order, the model recovers the PC expansion of~\cite{Yunes:2009yz}, including eccentricity corrections up to order $\mathcal{O}(e^8)$. In this eccentric waveform we have 11 parameters $P=\{\mathcal{M}_c,\eta,d_L,\iota,\theta,\phi,\psi,t_c,\phi_c,e_0,\beta\}$, with $\mathcal{M}_c$ the chirp mass, $\eta$ the symmetric mass ratio, $d_L$ the luminosity distance, ($\theta,~\phi$) the sky location of the source, $\psi$ the polarization angle, ($t_c,~\phi_c$) the time and phase at merger. $e_0$ and $\beta$ are the two additional parameters to the TaylorF2 waveform. The former is the initial eccentricity defined at the frequency $f_0$ and the latter is the azimuthal component of inclination angles (longitude of ascending nodes axis).
The eccentric waveform consists of multiple harmonics induced by the eccentricity of the orbit~\cite{Huerta:2014eca},
\begin{equation}
\tilde{h}(f)=-\sqrt{\frac{5}{384}}\frac{\mathcal{M}_c^{5/6}}{\pi^{2/3}d_L}f^{-7/6}\sum_{\ell=1}^{10}\xi_{\ell}\left(\frac{\ell}{2}\right)^{2/3}e^{-i\Psi_{\ell}} \,.
\end{equation}
The phase of each harmonic $\Psi_{\ell}$ is
\begin{equation}
\Psi_{\ell}=2\pi f t_c-\ell \phi_c+\left(\frac{\ell}{2}\right)^{8/3}\frac{3}{128\eta v_{\rm ecc}^5}\sum_{n=0}^{7}a_n v_{\rm ecc}^n \,.
\end{equation}
When $e_0=0$ it recovers the circular TaylorF2 with only the quadrupole mode ($\ell=2$). $\xi_{\ell}$ are functions of $e_0$ and angular parameters $P_{\rm ang}=\{\iota,~\theta,~\phi,~\psi,~\beta\}$~\cite{Yunes:2009yz}.
$v_{\rm ecc}$ is the modified velocity (relative to the orbit velocity $v=(\pi M f)^{1/3}$ when $e=0$) which is a function of eccentricity, $v_{\rm ecc}(f;e_0)=g(f;e_0)(\pi M f)^{1/3}$. The function $g(f;e_0)$ is expanded to $e_0^8$ and its specific form can be found in Eq. (13) of~\cite{Huerta:2014eca}.
$a_n$ is the corresponding coefficient of the 3.5 PN expansion~\cite{Buonanno:2009zt}. The waveform keeps up to 10 harmonics, which corresponds to a consistent expansion in the eccentricity to $\mathcal{O}(e^8)$ both in the amplitude and in the phase~\cite{Yunes:2009yz}. Eccentricity induces more harmonics to the waveform other than the dominant quadrupole mode. Multiple harmonics make the distance and angular parameters nontrivially coupled, enabling us to break the degeneracy between these parameters.
\subsection{The antenna response and observation of the harmonics}
In the mid-band, the motion of the space-borne detector in the long inspiral period should be taken into account. The frequency of each harmonic at a specific time (or orbital frequency $F$) is $\ell F$. In other words, the frequency of each harmonic $f$ corresponds to a different detector time, thus a different antenna response. Instead of the orbital frequency, we use the frequency of quadrupole mode $2F$ as the baseline frequency to derive the relation between time and GW frequency $t(f)$, by numerically solving the phase evolution of the eccentric orbits~\cite{Yunes:2009yz}. The orbits evolve faster with larger eccentricities (at a given orbital frequency). Thus the inspiral period (within the detector band) is shorter for a binary with greater eccentricity.
We first derive the antenna response functions $F_{+,\times}(t(f))$ in terms of the quadrupole frequency. Then for each harmonic, its corresponding antenna response functions should be $F_{+,\times}(t(2f/\ell))$. Here $f$ is the frequency of the harmonic $\ell$ and we transform it to the quadrupole frequency to derive the corresponding time and antenna response. Clearly, we can see the higher harmonics enter the detector band earlier, which could provide more angular information.
We follow~\cite{Rubbo:2003ap} for the modeling of the space-borne detectors, with the arm length $L=1000~(100)$ km for DECIGO-I (B-DECIGO). We set the initial observation time to be 1 year. Then the starting frequency of the quadrupole mode $f_{\rm start}$ can be calculated by setting $t_c-t(f_{\rm start})=1$ year. The starting frequency of each harmonic is $\ell f_{\rm start}/2$. So we should truncate the contribution of each harmonic before its starting frequency~\footnote{In the mid-band, we do not need to truncate the GWs after the inner-most stable circular orbit (ISCO) since the frequencies of the typical stellar-mass binaries at ISCO are much higher than the upper limit (10 Hz) of the detector band.},
\begin{equation}
\tilde{h}_{\rm 1~yr}(f)=\tilde{h}(f)\mathcal{H}(2f-\ell f_{\rm start}) \,,
\label{eq:h1yr}
\end{equation}
with the unit step function
\begin{equation}
\mathcal{H}(x)=
\begin{cases}
1 & {\rm if}~x\geq0 \,, \\
0 & {\rm otherwise} \,.
\end{cases}
\end{equation}
Now we use $\tilde{h}_{\rm 1~yr}(f)$ to present the GW signal during the 1-year observation.
\subsection{Parameter estimation using the Fisher information matrix }
To estimate the uncertainty and covariance of the parameters in the waveform, we adopt the Fisher matrix technique for GWs~\cite{Cutler:1994ys},
\begin{equation}
\Gamma_{ij}=\left(\frac{\partial h}{\partial P_i},\frac{\partial h}{\partial P_j}\right)\,,
\label{eq:Gamma}
\end{equation}
with $P_i$ being one of the 11 waveform parameters. Note in the circular case we only have 9 parameters excluding $e_0$ and $\beta$~\footnote{$\beta$ is meaningless in the circular case. We can still use eccentric waveform and set $e_0$ as a free parameter to constrain it in the circular case. However, in this paper, we focus on the improvement of parameter estimation of the eccentric case relative to the circular case. It means that we know the eccentricity {\it a prior} (though we still set eccentricity as a free parameter to be constrained in the eccentric case). We have checked whether including $e_0$ in the circular case has no obvious influence on our results.}.
The inner product is defined as
\begin{equation}
(a,b)=4\int_{f_{\rm min}}^{f_{\rm max}}\frac{\tilde{a}^*(f)\tilde{b}(f)+\tilde{b}^*(f)\tilde{a}(f)}{2 S_n(f)}df\,.
\label{eq:innerp}
\end{equation}
For the sensitivity $S_n(f)$ of B-DECIGO and DECIGO-I,
we use the fitting formula in~\cite{Yagi:2011wg} but rescale it according to the white paper of DECIGO~\cite{Kawamura:2020pcg}. We choose $f_{\rm min}=0.1$ Hz and $f_{\rm max}=10$ Hz, corresponding to the frequency band of DECIGO. Then the covariance matrix of the parameters is $C_{ij}=(\Gamma^{-1})_{ij}$, from which the uncertainty of each parameter $\Delta P_i=\sqrt{C_{ii}}$. The error of the sky localization is~\cite{Cutler:1997ta}
\begin{equation}
\Delta \Omega=2\pi |\sin(\theta)|\sqrt{C_{\theta\theta}C_{\phi\phi}-C_{\theta\phi}^2}\,.
\end{equation}
We calculate the partial derivatives $\partial \tilde{h}/\partial P_i$ numerically by $[\tilde{h}(f,P_i+dP_i)-\tilde{h}(f,P_i)]/dP_i$, with $dP_i=10^{-n}$. For each parameter, we need to optimize $n$ to make the derivative converge so that the Fisher matrix calculation is reliable. To check the robustness of our methodology, we first adopt EccentricFD waveform with $e_0=0$ and check its consistency with the analytical TaylorF2 waveform. We find that with a proper choice of $n$ for the numerical derivative of each parameter in Eq.~(\ref{eq:Gamma}), the Fisher matrix calculations from the waveform generated by {\sc PyCBC} and from the analytical TaylorF2 are very consistent with each other. This consistency check paves the way for the usage of EccentricFD waveform with a nonvanishing eccentricity.
\subsection{Mocking up typical binaries in GWTC-3}
We mock up five types of typical events from GWTC-3~\cite{LIGOScientific:2021djp}, i.e., a GW170817-like BNS with $(m_1,m_2)=(1.46,1.27)~M_{\odot}$, a GW200105-like neutron star–black hole binary (NSBH) with $(9.0,1.91)~M_{\odot}$, a GW191129-like light-mass BBH with $(10.7,6.7)~M_{\odot}$, a GW150914-like medium-mass BBH with $(35.6,30.6)~M_{\odot}$, and a GW190426-like heavy-mass BBH with $(106.9,76.6)~M_{\odot}$. Note the light, medium, and heavy masses are in the context of the mass range of the stellar-mass BBH in GWTC-3. The redshift and luminosity distance of each binary are also chosen to be consistent with the real event. Then the chirp mass $\mathcal{M}_c$ and symmetric mass ratio $\eta$ can be derived from the component masses. Note the parameters in the waveform are defined in the detector frame. To count the influence of the source orientation, we sample 1000 random sets of the angular parameters $P_{\rm ang}$ from the uniform and isotropic distribution. Note regarding the symmetry, we only sample the inclination angle $\iota\in [0, \pi/2]$, with the distribution $\cos\iota\sim \mathcal{U}[0,1]$. Since EPC waveform is tested valid with initial frequency up to 0.4~\cite{Huerta:2014eca}, we assign seven discrete initial eccentricities from 0 to 0.4 at $f_0=0.1$ Hz, i.e, $e_0=0$, 0.01, 0.05, 0.1, 0.2, 0.3, and 0.4. Without loss of generality, we fix the coalescence time and phase to be $t_c=\phi_c=0$. We then totally have $5\times 7\times 1000=35000$ cases.
Since we set a limit of the observation time $\sim 1$ year, we need to choose the proper starting frequency $f_{\rm start}$ for these typical binaries. By calculating $t_c-t(f)\sim 1$ year in the circular case, we set $f_{\rm start}$ to be 0.2, 0.1, 0.059, 0.026, and 0.0105 Hz for the typical BNS, NSBH, light BBH, medium BBH, and heavy BBH, respectively. Note in the same frequency range, the evolving time of the binary is shorter with a larger eccentricity. However, for a typical binary with each $e_0$, we set the same starting frequency. This means that in all cases the inspiral periods of the binaries can meet the limit of the observation time ($\leq 1$ year).
The signal-to-noise ratio (SNR) of these typical binaries can be calculated by
\begin{equation}
\rho^2=(h_{\rm 1yr},h_{\rm 1yr}) \,.
\end{equation}
Figure~\ref{fig:SNR} shows the SNR of the five typical binaries in the circular case with DECIGO-I. The SNR decreases with increasing the inclination angle (remind that $\iota$ is defined in the range $\cos\iota\in \mathcal{U}[0,1]$). The corresponding SNR with B-DECIGO can be approximately estimated by rescaling the SNR with DECIGO-I by a factor of $1/15$. Note that the SNR of the heaviest BBH GW190426 is not the largest due to its large distance. For a typical binary with different eccentricities, we find the SNR does not change significantly.
\begin{figure}
\includegraphics[width=0.45\textwidth]{SNR_e0_DECIGO-I}
\caption{The SNR of the five typical binaries with DECIGO-I when $e_0=0$.}
\label{fig:SNR}
\end{figure}
\subsection{The inference of luminosity distance and source localization}
We calculate the Fisher matrix for the five typical binaries and collect the results of all the 35000 cases. We define the ratio
\begin{equation}
R_{\Delta P}=\frac{\Delta P|_{e_0={\rm nonzero}}}{\Delta P|_{e_0=0}} \,,
\end{equation}
to indicate the improvement of the estimation of parameter $P$ induced by eccentricity relative to the quasi-circular case. $\Delta P$ is the error of the parameter $P$ derived from the Fisher matrix. If $R<1$, there is an improvement in the relevant parameter. A smaller $R$ indicates a larger improvement. On the contrary, $R>1$ means the eccentricity will worsen the relevant parameter estimation. To assess the dependence of parameter estimation on the source's angular parameters, we will show the scatter plots of the error of the parameters against the inclination angle (orientation) $\iota$ of the sources. The reason for choosing $\iota$ as the representative of the angular parameters is that we find the results are most sensitive to the inclination angle. To see this, figure~\ref{fig:SNR} shows that the SNR decreases as $\iota$ increases. So, a larger $\iota$ would degrade the parameter estimation. In addition, the distance inference heavily depends on the inclination angle due to the degeneracy between these two parameters. To summarize the parameter estimation among the 1000 orientations, we define the minimum, mean, and maximum values of the quantity $x$ in the 1000 orientations as $\min(x)$, $\mathbb{E}(x)$, and $\max(x)$, respectively.
In this section, we choose GW170817-like BNS and GW150914-like medium BBH as the representatives and show their results. We only choose the cases with $e_0=0$, 0.1, and 0.4 to give a concise look of the figures. The complementary results of other typical binaries are shown in the appendix~\ref{app:A}. In this section, we mainly focus on the estimation of distance and localization since they are the key parameters for the identification of the host galaxies of GW dark sirens and their cosmological applications.
In figures~\ref{fig:BNS_DI} and~\ref{fig:BBHmedium_DI} we show the errors of luminosity distance $\Delta d_L/d_L$, sky localization $\Delta \Omega$, and 3-dimensional (3-D) localization volume in 99\% confidence level (C.L.) $V_{99}$ against the orientation $\iota$ for GW170817-like BNS and GW150914-like medium BBH with DECIGO-I. We follow the method in~\cite{Yu:2020vyy} to transform $\Delta d_L$ and $\Delta \Omega$ (and the covariance between them) to the 99\% 3-D localization volume. We can clearly see the significant improvement of the distance inference by eccentricity in the near face-on (small $\iota$) orientations for both the BNS and medium BBH. As shown in the left panels of figures~\ref{fig:BNS_DI} and~\ref{fig:BBHmedium_DI}, the largest improvement is about 1.5 (2) orders of magnitude with $e_0=0.1~(0.4)$ for BNS and 2 (2.6) orders of magnitude with $e_0=0.1~(0.4)$ for medium BBH. The error of distance estimation is relatively smaller in larger $\iota$. For the BNS, the improvement of distance inference is almost negligible when $\iota$ is large. But for the medium BBH, there are still small improvements even in the near edge-on orientations. It suggests that the improvement is more distinct for a heavier binary with a larger eccentricity.
For the sky localization, we can see, in the middle panel of figure~\ref{fig:BNS_DI}, the improvement is almost negligible for BNS. In some orientations, the localization even gets worse, especially for the $e_0=0.1$ case. The $e_0=0.4$ case has a better performance but the improvement is still not very prominent. However, though BNS benefits little from eccentricity for its localization, it can be precisely localized even in the circular case due to its long inspiral period in the detector band.
While, for the medium BBH whose localization is much worse than that of BNS in the circular case (because of a much shorter inspiral in the detector band), a non-vanishing eccentricity can significantly improve the source localization in almost all orientations. The largest improvement is about 2 (3) orders of magnitude with $e_0=0.1$ (0.4). From the improvement of distance inference and source localization, we can expect the reduction of the 3-D localization volume which is shown in the right panels of figures~\ref{fig:BNS_DI} and~\ref{fig:BBHmedium_DI}. For BNS, the reduction of 3-D localization volume mainly comes from the improvement of distance inference in the near face-on orientations. While for the medium BBH, it benefits from the improvement of both distance inference and sky localization, which can reduce its 3-D localization volume more significantly. The largest improvement is about 2.5 (3.5) orders of magnitude with $e_0=0.1$ (0.4). If we assume the galaxy is uniformly distributed in the comoving volume and the number density $n_g= 0.01~\rm Mpc^{-3}$~\cite{Chen:2016tys}, then the threshold localization volume is $V_{\rm th}=100~\rm Mpc^3$. It means that the host galaxies of the GWs (dark sirens) whose localization volume $V_{99}\leq V_{\rm th}$ can be unambiguously identified. As shown in the right panel of figure~\ref{fig:BBHmedium_DI}, for the medium BBH with DECIGO-I, if $e_0=0$, the localization volume is larger than the threshold volume in most orientations. But a non-vanishing eccentricity can significantly improve its 3-D localization ($V_{99}< V_{\rm th}$ in almost all orientations) so that one can identify the host galaxy of the medium BBH.
\begin{figure*}
\includegraphics[width=0.3\textwidth]{ddL_BNS170817_DI}
\includegraphics[width=0.3\textwidth]{dOmega_BNS170817_DI}
\includegraphics[width=0.3\textwidth]{dvol_BNS170817_DI}
\caption{The distance inference, sky localization, and 3-D localization volume of GW170817-like BNS with DECIGO-I. We also show the ratio $R$ of each parameter in the bottom panel.}
\label{fig:BNS_DI}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.3\textwidth]{ddL_BBHmedium_DI}
\includegraphics[width=0.3\textwidth]{dOmega_BBHmedium_DI}
\includegraphics[width=0.3\textwidth]{dvol_BBHmedium_DI}
\caption{The distance inference, sky localization, and 3-D localization volume of GW150914-like medium BBH with DECIGO-I. Note the dashed horizontal line in the right panel denotes the threshold volume below which there is only one potential host galaxy, by assuming a uniform number density of galaxy $n_g=0.01~\rm Mpc^{-3}$.}
\label{fig:BBHmedium_DI}
\end{figure*}
As comparisons, the similar results of GW170817-like BNS and GW150914-like medium BBH with B-DECIGO are shown in figures~\ref{fig:BNS_BD} and~\ref{fig:BBHmedium_BD}. Since the main difference between B-DECIGO and DECIGO-I only lies in their sensitivity (the latter is 15 times better than the former), we can see the parameter estimations based on these two detectors are very similar and only differ by a constant factor. So the ratio $R$ of each parameter for B-DECIGO is the same as that for DECIGO-I (see the bottom panel of the figures). It means that the improvement of parameter estimation from eccentricity does not depend on the sensitivity of the detector. However, due to better sensitivity, we can see the distance inference and localization of BNS and medium BBH with DECIGO-I are more precise than with B-DECIGO.
\begin{figure*}
\includegraphics[width=0.3\textwidth]{ddL_BNS170817_BD}
\includegraphics[width=0.3\textwidth]{dOmega_BNS170817_BD}
\includegraphics[width=0.3\textwidth]{dvol_BNS170817_BD}
\caption{The same as figure~\ref{fig:BNS_DI}, but with B-DECIGO.}
\label{fig:BNS_BD}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.3\textwidth]{ddL_BBHmedium_BD}
\includegraphics[width=0.3\textwidth]{dOmega_BBHmedium_BD}
\includegraphics[width=0.3\textwidth]{dvol_BBHmedium_BD}
\caption{The same as figure~\ref{fig:BBHmedium_DI}, but with B-DECIGO.}
\label{fig:BBHmedium_BD}
\end{figure*}
The results show different features and trends for distance inference and sky localization. For distance inference, the errors become smaller in the larger inclination angle. On the contrary, the source localization is better in the smaller inclination angle. In addition, eccentricity can only significantly reduce the error of distance in the near face-on orientations (when the error is the largest). While for localization, it can always be improved in almost all orientations. Finally, the localization of small-mass binaries like BNS can not be significantly improved (sometimes even worsen) by eccentricity. But for distance inference, all binaries can benefit from eccentricity, especially in the near face-on orientations. Such different features can be explained as follows.
In the circular case, the large error of distance in the small inclination angle is due to the large degeneracy between distance and inclination angle when $\iota\sim 0$. The distance and inclination angle are simply coupled in the amplitude of the waveform as $h\sim \mathcal{A}_++\mathcal{A}_\times$,
where $\mathcal{A}_+\sim \frac{1}{d_L} \frac{1+\cos^2(\iota)}{2}$ and $\mathcal{A}_\times\sim\frac{1}{d_L}\cos(\iota)$. In order to identify the inclination of the binary system using the polarizations of the gravitational waves, we must distinguish the contributions of the plus ($+$) and cross ($\times$) polarizations. When the binary system is near face-on, the two amplitudes from plus and cross polarization have nearly identical contributions to the overall gravitational-wave amplitude. This is the main factor that leads to the strong degeneracy in the measurement of the distance and inclination when $\iota$ is small (see ~\cite{Usman:2018imj} for details). So, in the near face-on orientations, we can not measure the inclination angle precisely, as well as the distance. However, in the eccentric cases, the multiple harmonics make the distance and inclination angle ($\iota$ is written in the functions $\xi_{\ell}$) nontrivially coupled, allowing us to significantly break their degeneracy and hence improve the inference of both parameters. While, for a large inclination angle, the degeneracy between $d_L$ and $\iota$ is small. So the estimation of these two parameters is good enough and there is no further large room to improve it from eccentricity. As for the localization, there is no obvious degeneracy between the sky location parameters and inclination angle. From the Fisher matrix, the parameter estimation should be better for a higher SNR. From figure~\ref{fig:SNR}, we know the SNR is larger with a smaller $\iota$. So we can expect the localization (the estimation of ($\theta,~\phi$)) should be better when $\iota$ is smaller. The multiple harmonics induced by the eccentricity can add more information (e.g. the higher modes can enter the detector band much earlier) for the estimation of the sky location parameters, thus improving the localization regardless of the orientations. The 3-D localization ($V_{99}$) is just the combination of the distance inference ($\Delta d_L$) and sky localization ($\Delta \Omega$). Finally, the fact that eccentricity has little effect on the localization of small-mass binaries has also been reported in~\cite{Mikoczi:2012qy,Pan:2019anf}. We are not going to give an in-depth discussion here since it is beyond the scope of this paper.
To illustrate the improvement of distance inference and localization for these typical binaries with variable eccentricities, we show the largest improvement (the $\min(R)$ among the 1000 orientations) for each case in figure~\ref{fig:Rwe}. As we mentioned above, the values of $R$ with DECIGO-I and B-DECIGO should be the same and we only show one of them. Generally, a heavier binary with larger eccentricity can achieve more improvement for the distance inference and source localization. Note this can not be explained by the SNR of each binary. As suggested in figure~\ref{fig:SNR}, the heavier binary does not have to have a larger SNR. A simple explanation is that a larger eccentricity can make the non-quadrupole modes ($\ell\neq 2$) more prominent so that it can break the degeneracy of the parameters and provide more information for the parameter estimation. While, a heavier mass means a larger orbital velocity which can make the effects of eccentricity more distinct~\footnote{For instance, this can be suggested in the relation $v_{\rm ecc}(f;e_0)=g(f;e_0)(\pi M f)^{1/3}$. Note the fact that a higher-mass binary can achieve more improvement of localization from eccentricity has been indicated in previous work such as~\cite{Mikoczi:2012qy,Sun:2015bva,Ma:2017bux,Pan:2019anf}, but with different explanations.}. With $e_0=0.4$, the distance inference of these typical binaries (from BNS to heavy BBH) in the near face-on orientations can achieve 1.5--3 orders of magnitude improvements~\footnote{Note the largest improvement of distance inference in the near face-on orientations is sensitive to the sampling of the small $\iota$. The numbers here are different from that of~\cite{Yang:2022tig} because we use a different random seed for the sampling of $P_{\rm ang}$.}. For the typical BNS and NSBH, the improvement of localization is negligible. But for the typical BBH, the localization can achieve 1--3.5 orders of magnitude improvement. By combining the improvement of distance inference and sky localization, the 3-D localization of these typical binaries (from BNS to heavy BBH) can achieve 2--4.5 orders of magnitude improvement.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Rwe}
\caption{The largest improvement ($\min(R)$ among the 1000 orientations) of distance inference, localization, and 3-D localization volume for each binary with eccentricities from 0.01 to 0.4. The detector is either DECIGO-I or B-DECIGO.}
\label{fig:Rwe}
\end{figure}
We should note that there are some anomalies in figure~\ref{fig:Rwe}. For instance, the distance inference of BNS is a little better than that of NSBH. We also find that eccentricity has almost negligible effects on the localization of BNS and NSBH. It may even slightly worsen the localization in some cases. In addition, the localization of NSBH and light BBH achieve the largest improvement with $e_0=0.2$, but a larger eccentricity will reduce the improvement.
The reason for these nontrivial features in distance inference and source localization is as follows. On the one hand, eccentricity adds more harmonics to the waveform hence breaking the parameter degeneracy and improving the parameter estimation. The higher harmonics which enter the detector band earlier can also provide more angular information. On the other hand, in a specific frequency band with a higher eccentricity, the binary evolves faster thus the inspiral time (observation time) is shorter. This can downgrade the parameter estimation, especially for localization. We should also bear in mind that, in the eccentric case, there are two additional free parameters that make the parameter estimation be harder than in the circular case. In addition, with different starting frequencies, the frequency ranges of the multiple harmonics of these typical binaries covered in the detector band (0.1--10 Hz) are also different. These factors compete with each other and make the parameter inference differ from case to case.
In this section, we only choose the typical BNS and medium BBH with $e_0=0$, 0.1, and 0.4 as the representatives and show their distance inference and source localization as well as the corresponding improvement from eccentricity. The results for other typical binaries with different eccentricities can be inferred from the trends suggested in figure~\ref{fig:Rwe}. The full results such as the distance inference and localization of other typical binaries and the tables of the statistical results are summarized in the appendix~\ref{app:A}.
\subsection{The estimation of other parameters}
Apart from the distance and localization, the estimations of other parameters are also important. As we mentioned above, the determination of inclination angle is crucial for the distance inference hence the cosmological applications such as the measurement of the Hubble constant~\cite{LIGOScientific:2017adf,Hotokezaka:2018dfi}. In addition, it is also helpful for the modeling of associated EM counterpart like the narrow-beamed short gamma-ray burst~\cite{LIGOScientific:2017zic}. We have already shown that eccentricity can significantly improve distance inference in near face-on orientations. One can also expect that eccentricity has the same effect on the estimation of the inclination angle $\iota$. Besides, the chirp mass $\mathcal{M}_c$, symmetric mass ratio $\eta$, the time to coalescence $t_c$, and the eccentricity itself $e_0$ are also the key parameters for studying the properties of the binary and its astrophysical implications. Since these parameters are not the primary parameters we focus on in this paper, we only present some typical results for them. We choose BNS and medium BBH with $e_0=0$, 0.1, and 0.4 as the representatives and only show the results based on DECIGO-I (the parameter estimation with B-DECIGO can be inferred by rescaling the results with DECIGO-I based on the sensitivity of these two detectors).
Figure~\ref{fig:diota_BNS170817_DI} shows the error of inclination angle $\iota$ of GW170817-like BNS with DECIGO-I. As in the case of distance inference, we can see the significant improvement of the measurement of $\iota$ from eccentricity in the near face-on orientations. With $e_0=0.4$, the improvement can be as large as 2 orders of magnitude. The same feature in the inference of distance and inclination angle supports our argument above -- eccentricity can break the large degeneracy between distance and inclination angle in the near face-on orientations and thus improve their measurements at the same time. In the medium BBH case, the improvement of the inclination angle is relatively more significant than in the BNS case, which can be expected from the results of the distance inference.
\begin{figure}
\includegraphics[width=0.45\textwidth]{diota_BNS170817_DI}
\caption{The error of inclination angle $\iota$ of GW170817-like BNS with DECIGO-I. As in the case of distance inference, the large errors in the near face-on orientations are due to the huge degeneracy between distance and inclination angle. A nonvanishing eccentricity can alleviate this degeneracy hence improve the constraints.}
\label{fig:diota_BNS170817_DI}
\end{figure}
The errors of chirp mass for GW170817-like BNS and GW150914-like medium BBH are shown in figure~\ref{fig:dMc}. In the decihertz band, the chirp mass can be tightly constrained with $\Delta \mathcal{M}_c/\mathcal{M}_c\sim 10^{-6}-10^{-8}$. Similar to the case of distance inference, eccentricity has little effect on the estimation of chirp mass for small-mass binary like BNS. With $e_0=0.1$, it even slightly worsens the chirp mass estimation. The improvement is still negligible in the case $e_0=0.4$. However, for the larger-mass binary like the medium BBH, eccentricity can significantly improve the chirp mass estimation. As shown in the right panel of figure~\ref{fig:dMc}, the improvement is 1-2 orders of magnitude with $e_0=0.4$. Note the feature that eccentricity has little effect on small-mass binary and greater effects on larger-mass binary is also indicated in~\cite{Mikoczi:2012qy} for the massive BBH in the LISA band. For the estimation of symmetric mass ratio $\eta$, we find a similar result as for the chirp mass. For the medium BBH, $\Delta \eta \sim10^{-5}-10^{-6}$, but the improvement of $\eta$ from eccentricity is much smaller than that of chirp mass.
\begin{figure*}
\includegraphics[width=0.45\textwidth]{dMc_BNS170817_DI}
\includegraphics[width=0.45\textwidth]{dMc_BBHmedium_DI}
\caption{The estimation of chirp mass for GW170817-like BNS (left) and GW150914-like medium BBH (right) with DECIGO-I.}
\label{fig:dMc}
\end{figure*}
Figure~\ref{fig:dtcde0} shows an example of the estimation of coalescence time and initial eccentricity. We choose the medium BBH case with DECIGO-I as a representative. We can see an obvious improvement in the estimation of coalescence time from eccentricity. With $e_0=0.4$, the improvement can be as large as a factor of 10. We also find that in the BNS case, the improvement is relatively smaller but the estimation is more precise ($\min(\Delta t_c) \sim 10^{-3}$ s) due to the longer inspiral period. The estimation of eccentricity is shown in the right panel of figure~\ref{fig:dtcde0}. Since in the circular case ($e_0=0$), eccentricity is not a free parameter to be constrained, we choose $e_0=0.01$ to represent the small-eccentricity case. We can see the initial eccentricity can be very tightly constrained with $\Delta e_0\sim 10^{-6}-10^{-8}$ for $e_0=0.01$ to 0.4. A larger eccentricity can significantly improve the estimation of itself. We also find a similar result in the BNS case.
\begin{figure*}
\includegraphics[width=0.45\textwidth]{dtc_BBHmedium_DI}
\includegraphics[width=0.45\textwidth]{de0_BBHmedium_DI}
\caption{The estimation of coalescence time $t_c$ (left) and initial eccentricity $e_0$ (right) for GW150914-like medium BBH with DECIGO-I. Note for the estimation of eccentricity, we choose $e_0=0.01$ as a small-eccentricity case since $e_0$ is not a free parameter in the circular case.}
\label{fig:dtcde0}
\end{figure*}
In this section, we take a close look at the GW parameter estimation for the typical binaries in the circular and eccentric cases. We find that eccentricity can improve the estimation of almost all the waveform parameters. Generally, the improvement is more significant for a heavier binary with a larger eccentricity.
One notable point is that except for the distances and inclination angles, the parameter estimation is better in the smaller $\iota$ cases. As we have explained, the anomalous trend of the distance error against the inclination angle is due to large $d_L-\iota$ degeneracy when $\iota$ is small. As for other parameters, there is no obvious degeneracy with the inclination angle. A larger SNR in the smaller $\iota$ cases can provide a better parameter estimation. Though we only choose BNS and medium BBH with $e_0=0$, 0.1, and 0.4 as representatives, the results of other binaries and with various eccentricities can be inferred from the improvements suggested in figure~\ref{fig:Rwe}. In this paper, distance and localization are the key parameters we would like to investigate since they are crucial for dark sirens as accurate and precise probes of the Universe.
\section{Construction of the golden dark siren catalogs \label{sec:catalog}}
In the mid-band, the precise localization of BNS and NSBH, and the significant improvement of localization from eccentricity for BBH inspire us to investigate the possibility of unambiguously identifying their host galaxies. As shown in the right panel of figure~\ref{fig:BBHmedium_DI}, with the help of eccentricity, the GW150914-like BBH can be localized in a 3-D volume which is smaller than the threshold volume so that its host galaxy can be uniquely identified. For the lower-mass binaries like BNS and NSBH, though eccentricity has almost negligible effects on their localization, they have been precisely localized even without eccentricity (see figure~\ref{fig:BNS_DI}). So, in the mid-band with a non-vanishing eccentricity, it is possible for all three types of binaries (BNS, NSBH, and BBH) to be precisely localized so that their host galaxies can be uniquely identified.
In this section, we would like to mock up the GW catalogs with decihertz observatory like DECIGO. Based on our current knowledge, we forecast the population of GWs on the assumption of various eccentricities. We assume that the dark sirens like BNS, NSBH, and BBH, are not accompanied by the EM counterparts or the EM counterparts can not be detected. We focus on the well-localized dark sirens whose host galaxy can be uniquely identified (hereafter the golden dark sirens). The redshift of these golden dark sirens (from their host galaxies) can be unambiguously measured from the present galaxy catalog or follow-up spectroscopic observations. This makes golden dark sirens as good quality as bright sirens in terms of the redshift inference, thus can be used as precise probes in cosmology. We still assume two scenarios of DECIGO. We would like to compare these two configurations, to see the improvement of the population of GW detections (and also the golden dark sirens) if B-DECIGO is upgraded to DECIGO-I.
\subsection{Population of simulated GW detections}
We follow~\cite{Yang:2021xox,Yang:2022iwn} to simulate the GW detections and construct the catalogs of BNS, NSBH, and BBH. The merge rate per comoving volume at a specific redshift $R_m(z_m)$ is related to the formation rate of massive binaries and the time delay distribution $P(t_d,\tau)=\frac{1}{\tau}\exp(-t_d/\tau)$ with an e-fold time of $\tau=100$ Myr~\cite{Vitale:2018yhm},
\begin{equation}
R_m(z_m)=\int_{z_m}^{\infty}dz_f\frac{dt_f}{dz_f}R_f(z_f)P(t_d) \,.
\label{eq:Rm}
\end{equation}
Here $t_m$ (corresponding to redshift $z_m$) and $t_f$ are the look-back times when the systems merged and formed, $t_d=t_f-t_m$ is the time delay between the formation and merger, and $R_f$ is the formation rate of massive binaries. We assume the formation of compact binaries tracks the star formation rate. So $R_f$ is proportional to the Madau-Dickinson (MD) star formation rate~\cite{Madau:2014bja},
\begin{equation}
\psi_{\rm MD}=\psi_0\frac{(1+z)^{\alpha}}{1+[(1+z)/C]^{\beta}} \,,
\label{eq:psiMD}
\end{equation}
with parameters $\alpha=2.7$, $\beta=5.6$ and $C=2.9$. The normalization factor $\psi_0$ is determined by the local merger rates. We adopt the local merger rates of BNS, NSBH, and BBH inferred from GWTC-3, with $\mathcal{R}_{\rm BNS}=105.5^{+190.2}_{-83.9}~\rm Gpc^{-3}~\rm yr^{-1}$, $\mathcal{R}_{\rm NSBH}=45^{+75}_{-33}~\rm Gpc^{-3}~\rm yr^{-1}$, and $\mathcal{R}_{\rm BBH}=23.9^{+14.3}_{-8.6}~\rm Gpc^{-3}~\rm yr^{-1}$~\cite{LIGOScientific:2021psn}. Note the merger rate of NSBH is based on the assumption that the observed NSBH GW200105 and GW200115 are representatives of the population of NSBH. Then we convert the merger rate per comoving volume in the source frame to merger rate density per unit redshift in the observer frame through
\begin{equation}
R_z(z)=\frac{R_m(z)}{1+z}\frac{dV(z)}{dz} \,,
\label{eq:Rz}
\end{equation}
where $dV/dz$ is the comoving volume element.
Having the merger rates $R_z(z)$ for BNS, NSBH, and BBH, we can sample the redshift distribution of them respectively. In this paper, we just adopt the median $\mathcal{R}$ for the construction of the catalogs. We have 11 parameters in the waveform (for vanishing eccentricity there are 9 except $e_0$ and $\beta$). The luminosity distance $d_L$ is calculated from the sampled redshift by assuming a fiducial cosmological model $\Lambda$CDM with $H_0=67.72~\rm km~s^{-1}~Mpc^{-1}$ and $\Omega_m=0.3104$, corresponding to the mean values obtained from the latest \textit{Planck} experiment~\cite{Planck:2018vyg}. The sky localization ($\theta$, $\phi$), inclination angle $\iota$, polarization $\psi$, and $\beta$ are drawn from the uniform and isotropic distribution. Without loss of generality, we set the time and phase at coalescence to be $t_c=\phi_c=0$. As for the chirp mass and symmetric mass ratio, we consider different strategies for these three binary types. In the BNS case, we assume a uniform distribution of mass in [1, 2.5] $M_{\odot}$, which is consistent with the assumption for the prediction of the BNS merger rate in GWTC-3~\cite{LIGOScientific:2021psn}. In the NSBH case, since the merger rate is inferred by assuming the observed NSBH GW200105 and GW200115 are representatives of the population of NSBH, we just randomly choose the component mass to be one of these two events. As for the BBH case, we adopt the same strategy in~\cite{Yang:2021xox} but with the BBH population given by GWTC-3. We draw the distribution of component masses of simulated BBH based on the histogram of mass distribution from the real BBH detections in GWTC-3~\footnote{We first infer the histograms of primary mass $m_1$ and mass ratio $q$ from GWTC-3. The distribution of $m_1$ and $q$ for the simulated BBH are sampled accordingly. Then the second mass is just $m_2=m_1q$. We should make sure that $m_2\ge3~M_{\odot}$.}. The primary mass and mass ratio peak around $30-40~M_{\odot}$ and 0.7, respectively.
We set the initial observation time of B-DECIGO and DECIGO-I to be 1 year. The population of each type binary within 1 year in every redshift bin is sampled from $R_z(z)$. The starting frequency of each type binary is chosen to be the same as in section~\ref{sec:typical}. For BBH, since the peak primary mass is around $30-40~M_{\odot}$, we choose their starting frequencies to be that of the typical medium BBH. We consider five discrete eccentricities $e_0=0$, 0.1, 0.2, 0.3, and 0.4 and in each case, we assume all binaries have the same eccentricity. This is however not a realistic assumption and we will give a discussion on this later. For each assumption of eccentricity, we select the binaries with SNR>8 as the GW detections. We can assess the influence of eccentricity on the population of GW catalogs with DECIGO in 1-year observation time.
Figure~\ref{fig:hist_DI} shows the cumulative histograms of the simulated BNS, NSBH, and BBH detections by 1-year observation of DECIGO-I and on the assumption of various eccentricities. The numbers of the detections are accumulated quickly in the redshift [0, 5] and there are sparse events up to redshift 15, 18, and 20 for BNS, NSBH, and BBH, respectively. Note we set a cut-off at $z=20$ for all binaries. The total number of events is around $1.8\times 10^5$ for BNS, $1.5\times 10^5$ for NSBH, and $9\times 10^4$ for BBH. The smaller number of BBH is due to its lower predicted merger rate. For BNS and NSBH, a non-vanishing eccentricity can slightly decrease the number of detections. While for BBH, the decrease is almost negligible.
\begin{figure*}
\includegraphics[width=\textwidth]{hist_DI}
\caption{The cumulative histograms of the simulated detections on the assumption of various eccentricities for BNS (left), NSBH (middle), and BBH (right) by 1-year observation of DECIGO-I. Note in the BBH case the differences among different eccentricities are too small and hence invisible.}
\label{fig:hist_DI}
\end{figure*}
As a comparison, the cumulative histograms of BNS, NSBH, and BBH by 1-year observation of B-DECIGO are shown in figure~\ref{fig:hist_BD}. Since the sensitivity of B-DECIGO is inferior to that of DEICGO-I, we can see the numbers of detections are drastically decreased.
For BNS and NSBH, the largest redshift B-DECIGO can reach is around 0.25 and 0.6, respectively. The total number of detections is around one hundred and a few hundreds for BNS and NSBH. While, the horizon of BBH is much larger and can reach redshift more than 15. Though the merger rate of BBH is lower than that of BNS and NSBH, the total number of BBH detections can achieve around $3\times 10^4$. For all these three types of binaries, we can see the number of detections will be slightly decreased due to eccentricity. This can be explained by the fact that the eccentricity can shorten the inspiral period hence the observation time of the binary, causing a slight reduction of the SNR. Unlike BNS and BBH, the largest redshift of NSBH is also reduced by eccentricity (this also happens in the DECIGO-I case). This is due to the fixed component mass (either of GW200105 or GW200115) we choose when we construct the catalog of NSBH. We can not sample a larger mass to compensate for the decrease of the SNR by eccentricity. Thus the threshold redshift of NSBH in the circular case is not reachable in the eccentric case.
\begin{figure*}
\includegraphics[width=\textwidth]{hist_BD}
\caption{The same as figure~\ref{fig:hist_DI}, but with B-DECIGO. }
\label{fig:hist_BD}
\end{figure*}
\subsection{Golden dark sirens}
As we have mentioned, in the mid-band, eccentricity can make the localization of BNS, NSBH, and BBH so precise that their host galaxies can be uniquely identified. Now, we can demonstrate this argument and estimate how many such golden events can be observed by DEICGO in 1-year observing run from the catalogs in figures~\ref{fig:hist_DI} and~\ref{fig:hist_BD}.
Since we only focus on the well-localized events, we limit the catalogs to small redshifts $z<0.5$. For BNS, NSBH, and BBH with $z<0.5$, we transform the errors of distance and sky localization to the 3-D localization volumes. To estimate the numbers of potential host galaxies in the localization volume, we assume the galaxy is uniformly distributed in the comoving volume and the number density $n_g= 0.01~\rm Mpc^{-3}$. This number is derived by taking the Schechter function parameters in B-band $\phi_*=1.6\times 10^{-2} h^3 {\rm Mpc^{-3}}, \alpha=-1.07, L_*=1.2\times 10^{10} h^{-2} L_{B,\odot}$ and $h=0.7$, integrating down to 0.12 $L_*$ and comprising 86\% of the total luminosity~\cite{Chen:2016tys}. Then the threshold localization volume is $V_{\rm th}=100~\rm Mpc^3$. We choose the ones with 3-D localization volumes smaller than the threshold $V_{\rm th}$ as the golden dark sirens.
In figure~\ref{fig:V_loc_DI}, we show the 3-D localization volume of BNS, NSBH, and BBH detected by DECIGO-I within 1-year operation time. For the readability of the plots we only show the cases with $e_0=0$, 0.1, 0.2, and 0.4, which can present the main feature and tendency of the results (see figure~\ref{fig:Rwe}). For BNS and NSBH, there is no obvious improvement of the 3-D localization volume from eccentricity because the improvement only comes from the distance inference in the near face-on orientations. Though the improvement is negligible, BNS and NSBH are precisely localized and there are hundreds of golden events regardless of the eccentricity. The largest redshifts of golden dark BNS and NSBH are 0.25 and 0.35, respectively. While, in the BBH case, we can also detect a few golden dark sirens in the circular case below redshift 0.1. However, a nonvanishing eccentricity can significantly improve the localization so that much more golden dark BBH can be detected. The largest redshift of golden dark BBH is around 0.4 when $e_0=0.4$. We move the results for the distance inference and sky localization of BNS, NSBH, and BBH to appendix~\ref{app:B}.
\begin{figure*}
\includegraphics[width=\textwidth]{V_loc_DI}
\caption{The 3-D localization volume of BNS, and NSBH, and BBH detected by DECIGO-I within 1-year operation time. We set a cut-off at $z=0.5$. The dashed line is the threshold volume $V_{\rm th}=100~\rm Mpc^3$. }
\label{fig:V_loc_DI}
\end{figure*}
Figure~\ref{fig:V_loc_BD} shows the results of 3-D localization volume with B-DECIGO. As expected, in the B-DECIGO case, the numbers of golden dark sirens are much smaller. We can only detect a few golden BNS and NSBH up to redshift 0.05 and 0.1, respectively. For BBH, the golden event is unlikely in the circular case but a nonvanishing $e_0=0.2$ can make it possible to detect the golden dark BBH.
In table~\ref{tab:golden}, we summarize the number of golden dark sirens detected by DECIGO-I and B-DECIGO within 1-year observation time. As expected from the results in section~\ref{sec:typical}, eccentricity leads much more improvement for the localization of BBH than that of BNS and NSBH. With only one cluster of DECIGO running for 1 year in its design sensitivity, the observations of hundreds of golden BNS, NSBH, and tens of golden BBH are very promising. As a pathfinder in the near future, B-DECIGO can also observe a few golden BNS and NSBH. The detection of golden BBH is possible if it is eccentric. We can see the number of golden NSBH is relatively larger than that of BNS and BBH. This is due to the longer inspiral time of NSBHs (compared to BBH) and heavier mass (compared to BNS). The former provides good sky localization and the latter ensures well-inferred distance. Our results also suggest that, for NSBH and BBH, we can observe a little more golden events with $e_0=0.2$ than other eccentricities. This can be explained by the trends of the improvement of 3-D localization volume for NSBH and light BBH in figure~\ref{fig:Rwe}.
\begin{figure*}
\includegraphics[width=\textwidth]{V_loc_BD}
\caption{The same as figure~\ref{fig:V_loc_DI}, but with B-DECIGO.}
\label{fig:V_loc_BD}
\end{figure*}
\begin{table*}
\centering
\begin{tabular}{c|cccc|cccc|cccc}
\hline\hline
& \multicolumn{4}{c|}{Golden BNS} & \multicolumn{4}{c|}{Golden NSBH} & \multicolumn{4}{c}{Golden BBH} \\
\cline{2-13}
& $e_0=0$ & $e_0=0.1$ & $e_0=0.2$ & $e_0=0.4$ & $e_0=0$ & $e_0=0.1$ & $e_0=0.2$ & $e_0=0.4$ & $e_0=0$ & $e_0=0.1$ & $e_0=0.2$ & $e_0=0.4$ \\
\hline
DECIGO-I & 181 & 190 & 225 & 230 & 367 & 328 & 396 & 310 & 7 & 38 & 65 & 55 \\
\hline
B-DECIGO & 6 & 6 & 6 & 6 & 8 & 6 & 7 & 6 & 0 & 0 & 1 & 0 \\
\hline
\end{tabular}
\caption{The number of golden dark sirens for 1-year observation of DECIGO-I and B-DECIGO on the assumption of various eccentricities.}
\label{tab:golden}
\end{table*}
Here we would like to give a discussion on the assumptions of the eccentricity we made in the construction of the GW catalogs. In each case of $e_0$, we assume that the initial eccentricity is uniform for all the binaries. This is good for the comparison between different scenarios of initial eccentricity but not a realistic assumption. In principle, we should also sample the value of $e_0$ according to a given distribution, along with other waveform parameters when constructing the GW catalogs. However, the formation channel of compact binaries and the distribution of eccentricity at a specific frequency (in our case $f_0=0.1$ Hz) are still under debate and not clear~\cite{Wen:2002km,Kowalska:2010qg,Takatsy:2018euo}. Simulations suggest that eccentric mergers from strong gravitational encounters account for 10\% of the underlying population of BBH mergers in globular clusters, and about half of these having eccentricities larger than 0.1 at 10 Hz~\cite{Samsing:2013kua,Samsing:2017xmd,Samsing:2017rat} (and references therein). While, back to decihertz, the population of eccentric binaries and their eccentricities should be much larger. For the isolated formation scenario, even binaries born with high eccentricity could be fully circularized when entering the LIGO/Virgo band, by efficiently damping orbital eccentricity through angular momentum loss from GW emission. However, the eccentricity of the field binary at mid-band is still uncertain. Since eccentricity has small effects on the population of GW detections (see figures~\ref{fig:hist_BD} and \ref{fig:hist_DI}), the total number of events in the GW catalogs will not be largely altered. As for the number of golden dark sirens in a realistic situation (considering the distribution of eccentricity), we can estimate it by weighting the numbers of the circular and eccentric cases in table~\ref{tab:golden}, roughly as $N_{\rm gold}=\sum P_{e_0}N_{e_0}$. Here, $N_{e_0}$ is the number of golden dark sirens by assuming the same eccentricity $e_0$ for all binaries (the numbers in table~\ref{tab:golden}). $P_{e_0}$ is the actual fraction of the events which hold eccentricity $e_0$. Since we focus on the comparison of the results between different eccentricity scenarios, we prefer to present the results based on different $e_0$ separately.
When constructing the GW catalogs and estimating the numbers of golden dark sirens, we only adopt the median value of the predicted merger rates and neglect their uncertainties. The merger rates of BNS and NSBH are much more uncertain than that of BBH since we have lots of BBH detections and only a few BNS and NSBH detections in current LIGO-Virgo-KAGRA catalogs. This means the simulated BBH catalogs are more realistic than that of BNS and NSBH. We will give a discussion for this in section~\ref{sec:conclusion}. Also, considering the fact that eccentricity can significantly improve the localization for BBH only, we prefer concentrating on BBH golden dark sirens to show the implications of eccentric dark sirens on cosmology in section~\ref{sec:cosmo}.
\section{Cosmological implications \label{sec:cosmo}}
To make dark sirens accurate and precise cosmological probes, distance inference and source localization are two crucial factors. The GWs of long inspiraling compact binaries observed by the space-borne detector in the mid-band can provide much tighter constraints in these two aspects, thus the more precise measurement of the cosmological parameters, compared to the LIGO-Virgo band (see some examples~\cite{Yang:2021xox,Yang:2022iwn,Liu:2022rvk} and also in the millihertz band~\cite{Wang:2020dkc,Zhu:2021aat,Zhu:2021bpp}). Without EM counterparts, dark sirens in the mid-band still face the issues like large degeneracy between luminosity distance and inclination angle in the near face-on orientations, as well as the uncertain localization for the larger-mass BBH. In section~\ref{sec:typical} we demonstrate that the eccentricity, which is more likely to be non-vanishing in the mid-band than in the LIGO-Virgo band, can greatly alleviate these issues. The localization is more precise in the near face-on orientations where the eccentricity happens to significantly improve the distance inference. The precise localization for these compact binaries (BNS, NSBH, and BBH) makes it possible that we can identify their host galaxies without the help of the EM counterparts. Bear in mind that EM counterparts can help to identify the host galaxy~\cite{LIGOScientific:2017vwq,LIGOScientific:2017ync}, and also improve the constraint of inclination angle (and hence the distance inference)~\cite{Hotokezaka:2018dfi}. It means that eccentricity in the mid-band can make dark sirens equivalent to the bright sirens in the context of the distance and redshift measurement, or greatly reduce their uncertainties. These features can significantly improve the constraints of cosmological parameters/models from dark sirens. Also, the improvement in the estimation of other parameters like inclination angle and coalescence time can help for the follow-up search of EM counterparts and observation of GWs at high frequency.
In section~\ref{sec:catalog}, we construct the catalogs of GWs based on DECIGO-I and B-DECIGO. We demonstrate that a certain amount of golden dark BNS, NSBH, and BBH can be observed in the mid-band. In particular, a nonvanishing eccentricity can significantly increase the number of golden dark BBH. In this section, we would like to focus on these golden dark sirens and assess the improvement for the constraints of cosmological parameters/models from eccentricity, especially for BBH dark sirens. Since the sensitivity of B-DEICGO is not good enough, it can only observe $\mathcal{O}(1)$ golden dark sirens and the numbers are not greatly changed in the eccentric cases. Therefore, we only use the golden dark sirens with DECIGO-I to exemplify the improvement of cosmological constraints from eccentricity.
\subsection{The Hubble diagram of golden dark sirens}
To construct the Hubble diagram of these golden dark sirens, we assume the fiducial cosmological model to be $\Lambda$CDM with $H_0=67.72~\rm km~s^{-1}~Mpc^{-1}$ and $\Omega_{\rm m}=0.3104$, corresponding to the mean values obtained from the latest \textit{Planck} TT,TE,EE+lowE+lensing+BAO+Pantheon data combination~\cite{Planck:2018vyg}. We also fix the present CMB temperature $T_{\rm CMB}=2.7255~\rm K$, the sum of neutrino masses $\Sigma_{\nu}m_{\nu}=0.06~\rm eV$, and the effective extra relativistic degrees of freedom $N_{\rm eff}=3.046$, as in the {\it Planck} baseline analysis. We include the errors from weak lensing and the peculiar velocity of the source galaxy. For weak lensing, we adopt the analytically fitting formula~\cite{Hirata:2010ba,Tamanini:2016zlh}
\begin{equation}
\left(\frac{\Delta d_L(z)}{d_L(z)}\right)_{\rm lens}=0.066\left(\frac{1-(1+z)^{-0.25}}{0.25}\right)^{1.8} \,.
\end{equation}
We consider a delensing factor. Following~\cite{Speri:2020hwc} we adopt a phenomenological formula
\begin{equation}
F_{\rm delens}(z)=1-\frac{0.3}{\pi/2}\arctan(z/z_*) \,,
\end{equation}
where $z_*=0.073$. Then the final error from weak lensing is
\begin{equation}
\left(\frac{\Delta d_L(z)}{d_L(z)}\right)_{\rm delens}=F_{\rm delens}(z)\left(\frac{\Delta d_L(z)}{d_L(z)}\right)_{\rm lens} \,.
\end{equation}
For the peculiar velocity uncertainty, we use the fitting formula~\cite{Kocsis:2005vv},
\begin{equation}
\left(\frac{\Delta d_L(z)}{d_L(z)}\right)_{\rm pec}=\left[1+\frac{c(1+z)^2}{H(z)d_L(z)}\right]\frac{\sqrt{\langle v^2\rangle}}{c} \,,
\end{equation}
here we set the peculiar velocity value to be 500 km/s, in agreement with average values observed in galaxy catalogs. For the redshift measurement, we assume the host galaxies of all the golden dark sirens can be unambiguously identified and their redshifts are measured spectroscopically.
Figure~\ref{fig:HD} shows one realization of the Hubble diagram of the golden dark sirens observed by DECIGO-I in 1 year with $e_0=0$ and $e_0=0.2$. We have discarded three BNS events whose distance errors $\Delta d_L/d_L$ are larger than 50\% in the $e_0=0$ case since they are not useful in constraining cosmological parameters. While in $e_0=0.2$ case, all the events have distance errors smaller than $50\%$. In this section, we choose $e_0=0.2$ to represent the eccentric case. We can clearly see the golden BBHs have much smaller distance errors and they can reach higher redshifts (in the $e_0=0.2$ case), which means BBH should be more efficient than BNS and NSBH in measuring the cosmological parameters.
We have shown that BBH dark sirens benefit most from eccentricity. While BNS and NSBH dark sirens only get non-substantial help from eccentricity (e.g. for the distance inference in the near face-on orientations). Bear in mind that the possible associate EM counterparts can also help BNS and NSBH identify their host galaxies. Therefore, the improvement of cosmological constraints from eccentric dark sirens is mainly manifest in the case of BBH dark sirens. In addition, the predicted errors for the merger rates of BNS and NSBH are very uncertain, which makes the numbers of golden BNS and NSBH less affirmatory than that of BBH (see discussion in section~\ref{sec:conclusion}). Therefore, in this section, we stick to the BBH dark sirens in measuring the cosmological parameters and move the results of BNS and NSBH to appendix~\ref{app:C}.
Since the redshift of golden dark sirens mainly reside in the low redshift region where the dynamics of dark energy is not very sensitive, we only focus on the following two cosmological models.
\begin{figure*}
\includegraphics[width=0.45\textwidth]{HD_e0}
\includegraphics[width=0.45\textwidth]{HD_e02}
\caption{The Hubble diagram of the golden dark sirens observed by DECIGO-I in 1 year with $e_0=0$ (left) and $e_0=0.2$ (right). Note in the $e_0=0$ case, we discard three BNS events whose distance errors are larger than 50\%.}
\label{fig:HD}
\end{figure*}
\subsection{Constraints on $\Lambda$CDM model from BBH golden dark sirens}
In the baseline $\Lambda$CDM model with dark energy as a cosmological constant, there are two free parameters, namely, the Hubble constant $H_0$ and the matter density parameter $\Omega_{\rm m}$.
To infer the posteriors of the cosmological parameters, we run Markov-Chain Monte-Carlo (MCMC) using the package {\sc Cobaya}~\cite{Torrado:2020dgo,2019ascl.soft10019T}. The marginalized statistics of the parameters and the plots are produced by the Python package {\sc GetDist}~\cite{Lewis:2019xzd}.
When we sample the distance measurement in the Hubble diagram, the scatter of the random sampling should be taken into account. For each type of golden dark siren, we randomly sample 10 sets of Hubble diagrams (repeat the sampling 10 times). We run MCMC for each data set and choose the median result as the representative.
Figure~\ref{fig:LCDM_BBH} shows the constraints of the Hubble constant $H_0$ and matter density parameter $\Omega_{\rm m}$ by the golden dark BBH from 1-year observation of DECIGO-I. In the circular case, 7 golden dark BBH can constrain $H_0=67.80^{+1.50}_{-1.30}~\rm km~s^{-1}~Mpc^{-1}$. While in the eccentric case with $e_0=0.2$, 65 golden dark BBH can constrain $H_0=67.80\pm 0.46~\rm km~s^{-1}~Mpc^{-1}$. The precision of the Hubble constant measurement is improved from 2.06\% to 0.68\%. For the matter density parameter, $\Omega_{\rm m}=0.43^{+0.14}_{-0.41}$ with $e_0=0$ and $\Omega_{\rm m}=0.30^{+0.043}_{-0.051}$ with $e_0=0.2$. In the eccentric case, the golden dark BBH can reach higher redshifts, where $\Omega_{\rm m}$ is more sensitive. Therefore, eccentricity can greatly improve the precision of $\Omega_{\rm m}$ from 64\% to 16\%.
\begin{figure}
\includegraphics[width=0.45\textwidth]{LCDM_BBH}
\caption{The constraints of Hubble constant $H_0$ and matter density parameter $\Omega_{\rm m}$ with $\Lambda$CDM model from 1-year observation of the BBH golden dark sirens by DEICGO-I. Contours contain 68 \% and 95 \% of the probability. The dashed lines denote the fiducial value of the parameters.}
\label{fig:LCDM_BBH}
\end{figure}
\subsection{Constraints on modified gravity from BBH golden dark sirens}
The general test of general relativity (GR) by the propagation of GWs across cosmological distances has been investigated in recent research~\cite{Belgacem:2017ihm,Belgacem:2018lbp,Nishizawa:2017nef,Arai:2017hxj,Nishizawa:2019rra,LISACosmologyWorkingGroup:2019mwx,Belgacem:2019tbw,Belgacem:2019zzu,Mukherjee:2019wcg,DAgostino:2019hvh,Bonilla:2019mbm,Mukherjee:2020mha,Kalomenopoulos:2020klp,Mastrogiovanni:2020mvm,Mastrogiovanni:2020gua,Finke:2021aom}. In a generic modified gravity model the linearized evolution equation for GWs traveling on an FRW background in four-dimensional space-time is~\cite{LISACosmologyWorkingGroup:2019mwx}
\begin{equation}
\tilde{h}_A^{\prime\prime}+2[1-\delta(\eta)]\mathcal{H}\tilde{h}_A^{\prime}+[c_T^2(\eta)k^2+m_T^2(\eta)]\tilde{h}_A=\Pi_A \,,
\label{eq:GWpropa}
\end{equation}
where $\tilde{h}_A$ are the Fourier modes of the GW amplitude. $\mathcal{H}=a^{\prime}/a$ is the Hubble parameter in conformal time, the primes indicate derivatives with respect to conformal time $\eta$. $A=+,\times$ label the two polarizations of GWs. $\Pi_A$ is the source term, related to the anisotropic stress tensor. The function $\delta (\eta)$ modifies the frication term in the propagation equation. $c_T$ corresponds to the speed of gravitational waves. In theories of modified gravity (MG), the tensor mode can be massive, with $m_T$ being its mass. In GR, we have $\delta=0$, $c_T=c$, and $m_T=0$. The observation of GW170817/GRB170817A puts a very tight constraint on the speed of the gravitational wave, $(c_T-c)/c<\mathcal{O}(10^{-15})$~\cite{LIGOScientific:2017zic}.
In this paper, we only retain the deviations from GR induced by the friction term and set $c_T=c$ and $m_T=0$~\cite{Belgacem:2018lbp,Belgacem:2019tbw}. Then one can show the inferred ``GW luminosity distance'' in modified gravity theories is different from the traditional ``electromagnetic luminosity distance''~\cite{Belgacem:2017ihm,Belgacem:2018lbp},
\begin{equation}
d_L^{\rm gw}(z)=d_L^{\rm em}(z)\exp\left\{-\int_0^z\frac{dz^{\prime}}{1+z^{\prime}}\delta(z^{\prime})\right\} \,.
\end{equation}
To constrain the modified gravity theory (or to test GR), we need to constrain the $\delta$ function. In this section, we follow~\cite{Belgacem:2018lbp,Belgacem:2019tbw} and adopt the 2-parameter phenomenological parameterization for the deviation of GR,
\begin{equation}
\Xi(z)\equiv\frac{d_L^{\rm gw}(z)}{d_L^{\rm em}(z)}=\Xi_0+\frac{1-\Xi_0}{(1+z)^n} \,.
\label{eq:Xi}
\end{equation}
From GW and EM measurements one can therefore access the quantity $\delta(z)$, or equivalently $\Xi(z)$, a smoking gun of modified gravity. $\Xi_0=1$ recovers GR. This parametrization is very general and expected to fit the predictions from a large class of MG models~\cite{LISACosmologyWorkingGroup:2019mwx}. We set $n=2.5$ since it plays in general a lesser role in the parametrization~\cite{Belgacem:2018lbp}.
To constrain $\Xi_0$, we combine GWs from BBH golden dark sirens and EM data from CMB, BAO, and SNe Ia. We use CMB data from the latest {\it Planck} release~\cite{Planck:2018vyg}, that is, \textit{Planck} TT,TE,EE+lowE+lensing (hereafter {\it Planck}). For BAO, we adopt the isotropic constraints provided by 6dFGS at $z_{\rm eff}=0.106$~\cite{Beutler:2011hx}, SDSS-MGS DR7 at $z_{\rm eff}=0.15$~\cite{Ross:2014qpa}, and ``consensus'' BAOs in three redshift slices with effective redshifts $z_{\rm eff}$ = 0.38, 0.51, and 0.61~\cite{BOSS:2016apd,Vargas-Magana:2016imr,BOSS:2016hvq}. We use the Pantheon data~\cite{Pan-STARRS1:2017jku} as the latest compilation of SNe Ia. We also set the equation of state of dark energy $w_{\rm DE}$ as a free parameter. We incorporate the MG parameter $\Xi_0$ into {\sc Cobaya} and run MCMC to infer the posteriors.
Figure~\ref{fig:MG_BBH} shows the posteriors of the Hubble constant $H_0$, matter density parameter $\Omega_{\rm m}$, equation of state of dark energy $w_{\rm DE}$ and the MG parameter $\Xi_0$ inferred from the BBH golden dark sirens by 1-year observation of DECIGO-I, together with the EM anchors from {\it Planck}, BAO, and Pantheon data sets. With $e_0=0$, the golden dark BBH can constrain $\Xi_0=0.994\pm 0.062$. A nonvanishing eccentricity $e_0=0.2$ can improve the constraint to $\Xi_0=1.008\pm 0.016$. The precision of constraining MG effects through the phenomenological parameterization of GW propagation is improved from 6.2\% to 1.6\%. With more BBH golden dark sirens in the eccentric case, the constraints of other parameters are also slightly tighter. Since the MG parameter is more sensitive to the GW data, it can achieve more improvement than other parameters.
\begin{figure}
\includegraphics[width=0.45\textwidth]{MG_BBH}
\caption{Constraints on parameters of a phenomenological parameterization of modified GW propagation from 1-year observation of the BBH golden dark sirens by DEICGO-I combined with the EM experiments {\it Planck}, BAO, and Pantheon. Contours contain 68 \% and 95 \% of the probability. The dashed lines denote the fiducial value of $\Xi_0$.}
\label{fig:MG_BBH}
\end{figure}
In this section, we discuss the implications of eccentricity for dark sirens on cosmology. We estimate the number of golden dark sirens on the assumption of various eccentricities. We focus on the BBH golden dark sirens and investigate their constraints on cosmological parameters in the circular and eccentric cases. The golden dark BBH observed in a larger number and at higher redshifts with eccentricity can greatly improve the constraints of cosmological parameters and MG effects. Note in this section we only investigate the cases of golden dark BBH whose host galaxies can be uniquely identified. There are numerous dark BBH with multiple host galaxies and they can reach much higher redshifts. A statistical method should be adopted for the applications of these dark sirens on cosmology~\cite{LIGOScientific:2018gmd,DES:2019ccw,LIGOScientific:2021aug,Finke:2021aom}. These BBH dark sirens with significantly improved localization from eccentricity can also achieve large improvements in constraining cosmological parameters, especially for dynamics of dark energy and MG at high redshift. We leave this for future research.
\section{Conclusions and discussions \label{sec:conclusion}}
\subsection{A summary of the results}
In the first part of this paper, we investigate the parameter estimation of GWs emitted by eccentric compact binaries with decihertz observatory. The multiple harmonics induced by eccentricity can provide more information and break the degeneracy between parameters. We find eccentricity of compact binaries in the mid-band can greatly improve the parameter estimation of GWs. In particular, with $e_0=0.4$ at $f_0=0.1$ Hz, the typical binaries (BNS, NSBH, and BBH) can achieve 1.5-3 orders of magnitude improvement for the distance inference in the near face-on orientations and 1-3.5 orders of magnitude improvement for the localization. We find that other parameters like inclination angle, chirp mass, coalescence time, and eccentricity can also be estimated more precisely with a non-vanishing eccentricity. In this paper, we use DECIGO as an example detector in the mid-band. We consider two scenarios for the configuration of DECIGO. The first is the one cluster of DECIGO with its designed sensitivity, which we call DECIGO-I. The second is B-DECIGO which also has one cluster but with inferior sensitivity. The physical findings, i.e., the improvement of parameter estimation of GWs from eccentricity, do not rely on a specific detector.
In the second part, we construct the GW catalogs observed by DECIGO in 1 year. On the assumption of various eccentricities, we estimate the population of golden dark sirens whose localization is precise enough so that their host galaxies can be uniquely identified. We find that, with only one cluster of DECIGO running 1 year in its designed sensitivity, the observations of hundreds of golden BNS, NSBH, and several tens of golden BBH are very promising. A non-vanishing eccentricity can significantly enlarge the number of golden dark BBH. This can greatly improve the constraints of cosmological parameters and modified gravity from BBH dark sirens. For instance, the number of BBH golden dark sirens is 65 in $e_0=0.2$ case, compared to 7 in $e_0=0$ case. As a result, in the $\Lambda$CDM model, the constraints of Hubble constant and matter density parameter from BBH golden dark sirens can be improved from 2.06\% to 0.68\% and from 64\% to 16\%, respectively. Through the phenomenological parameterization of GW propagation, the constraint of modified gravity can be improved from 6.2\% to 1.6\%. While, due to the worse sensitivity, the results based on B-DECIGO are much inferior to that of DECIGO-I. The results in this paper can serve as a forecast for one cluster of DECIGO as well as for the early launched pathfinder B-DECIGO. These configurations are much more promising than the full configuration (four clusters) of DECIGO in the future.
Our result shows the remarkable significance of eccentricity for GWs on detection, data analysis, and physics including cosmology, astrophysics, and fundamental physics. The eccentric frequency-domain waveform adopted in this work is limited up to $e_0=0.4$. A ready-to-use more accurate and widely applicable eccentric waveform deserves extensive investigations by the community~\cite{Cao:2017ndf,Liu:2019jpg,Liu:2021pkr,Ramos-Buades:2021adz,Hinder:2017sxy,Moore:2019xkm,Joshi:2022ocr}, especially for GW detections in the mid-frequency band.
\subsection{The uncertainty of the golden dark sirens}
When constructing the GW catalogs and estimating the number of golden dark sirens, some assumptions have been made. We neglect the uncertainty of merger rates of $\mathcal{R}_{\rm BNS}=105.5^{+190.2}_{-83.9}~\rm Gpc^{-3}~\rm yr^{-1}$, $\mathcal{R}_{\rm NSBH}=45^{+75}_{-33}~\rm Gpc^{-3}~\rm yr^{-1}$, and $\mathcal{R}_{\rm BBH}=23.9^{+14.3}_{-8.6}~\rm Gpc^{-3}~\rm yr^{-1}$ and only adopt the median value. Due to the fewer events of BNS and NSBH in GWTC-3, their predicted event rates are more uncertain than that of BBH. In addition, the predicted merge rate of NSBH is based on the assumption that the observed NSBH GW200105 and GW200115 are representatives of the population of NSBH, which is not very realistic. We can roughly estimate the uncertainty of the numbers of golden dark sirens in table~\ref{tab:golden} using $\Delta N/N\sim\Delta \mathcal{R}/\mathcal{R}$~\footnote{Also note that the numbers of GW events in the catalogs are based on one realization of the sampling. However, the errors due to random sampling (Poisson) should be much smaller than the uncertainty of merger rates.}. The large uncertainty for the merger rates of BNS and NSBH makes the catalogs of golden dark BNS and NSBH very uncertain, as well as their applications on cosmology. The future observing run of LIGO-Virgo-KAGAR can provide a tighter prediction for the merger rates. Furthermore, from the observation perspective, it takes months to years to monitor the BNS and NSBH events in the mid-band, which is quite challenging in observing hundreds of events due to the time limits and overlap between signals. In contrast, the inspiral period of BBH is very short (hours to days), which makes it promising to search for tens of golden dark BBH in 1-year observation time. Considering the factors above, the results of BBH in our paper are more realistic and reliable than that of BNS and NSBH.
As we have mentioned, we estimate the number of golden dark sirens by assuming the same value of eccentricity for all the binaries. The realistic number should be a weighted summation of the numbers in circular and eccentric cases, according to the distribution of eccentricity for each type of binary in the mid-band. Thus the results of the cosmological constraints should also be averaged accordingly.
One can roughly estimate that $\Delta P_{\rm cosmo}\sim 1/\sqrt{N_{\rm gold}}$. For instance, even if only 10\% binaries have a nonvanishing eccentricity in the mid-band, e.g. $e_0=0.2$, the number of golden BBH can achieve $\sim 13$ with DECIGO-I. Then the Hubble constant in the $\Lambda$CDM model can be constrained at 1.5\% precision level. While, if eccentric BBH can take up 30\% of the total BBH in the mid-band, $H_0$ can be constrained at sub-percent level.
Another factor in the uncertainty of golden dark sirens is that we assume the galaxies are uniformly distributed in the Universe. However, the clustering and grouping feature of galaxies can make it much easier to infer the redshift information of GWs from the cluster and group of its host galaxy instead of the host galaxy itself~\cite{Yu:2020vyy}. This means that our estimation for the number of golden dark sirens can be very conservative.
\subsection{A caveat in the Fisher matrix}
Throughout this paper, we use the Fisher matrix to do the parameter estimation of GW waveform. Fisher matrix is a widely-used data analysis technique for the estimation of the waveform parameters and should be consistent with the more robust method like Markov chain Monte Carlo (MCMC) when the SNR of the signal is high enough (the strength of the signal or the sensitivity of the detector is good enough). Fisher matrix inherently assumes the errors (and the correlation) are Gaussian. Its nature is to assess the sensitivity of the waveform to the change of the parameters, from which one can get a bound of the error for each parameter. The estimation is more accurate (consistent with the full MCMC) for a larger SNR, in which case the parameters are tightly constrained. In this paper, the main purpose is to propose the idea that the eccentricity of the compact binary can significantly improve the parameter estimation of the GW waveform in the mid-band. Therefore we just stick to the theoretical (or physical) improvement of the parameter estimation of GW waveform from circular to eccentric cases, for which the Fisher matrix is adequate. In terms of the theoretical improvement, we say the improvement of the parameter estimation that the eccentric waveform ``should'' achieve (compared to circular waveform) in the ideal situation (Gaussian errors, adequate SNR). What we are interested in and would like to present in this paper is the physical improvement of parameter estimation from eccentricity in the waveform (by providing more information and breaking the degeneracy between parameters). This physical implication should not be limited by the condition of the detector (poor sensitivity and SNR, etc.). In this paper, we only adopt DECIGO as an example detector to demonstrate this improvement. We should bear in mind that if we conduct our data analysis with a more realistic technique such as MCMC which involves the prior information and the non-Gaussian posterior, especially when the SNR is low and the parameter is poorly constrained, the improvement could be inferior to the theoretical improvement derived from the Fisher matrix. However, the theoretical improvement we present in this paper can tell us that the constraints of the waveform parameters could be ``easier'' by that level if the binary is eccentric, no matter what techniques and detectors we adopt.
In section~\ref{sec:typical}, based on the example detectors, we compare the eccentric waveform with the circular one to show the improvement of the parameter estimation from eccentricity. This improvement is theoretical in the context of the Fisher matrix. Since the localization-relevant parameters are precisely constrained with the space-borne detector in the mid-band, the theoretical improvement of the localization derived from the Fisher matrix should be consistent with the full MCMC analysis. For the distance inference with large inclination angles, the error is very small and there is also no serious issue. While in the near face-on orientations, the distance error is quite large and there could be a discrepancy between the Fisher matrix and full MCMC which includes the prior information~\footnote{Note if the error of the parameter is even larger than the prior bound, the validity of the Fisher matrix which requires the Gaussian posteriors (even if we add a Gaussian prior of the bound to the Fisher matrix) should be in doubt.}.
However, in the case of DECIGO-I which has a higher sensitivity (hence a larger SNR and relatively smaller error), this discrepancy is quite small. Again, for the distance inference, we just report the theoretical improvement in the near face-on orientations. The issues we discuss here should not influence the highlight of our paper -- the significant improvement of localization from eccentricity.
In the second part of this paper, we focus on the construction of golden dark sirens (section~\ref{sec:catalog}) and the application of them on cosmology (section~\ref{sec:cosmo}). Since the distance and localization of golden dark sirens are most tightly constrained, they will not be influenced by the issues we discussed above.
Due to the lack of well-developed MCMC techniques for the mid-band GW parameter estimation in the literature, we just present the results solely based on the Fisher matrix. The more realistic MCMC approach of GW parameter estimation in the mid-band for the eccentric waveforms is to be developed and we leave this for future research.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,184
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This entry was posted on July 25, 2009 at 1:14 PM and is filed under Alaska, Democratic Party, Governor Sarah Palin, media bias, Republican, Rush Limbaugh, Sarah Palin. Tagged: Democratic Party, Limbaugh, Sarah Palin, staterun media. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
Sarah please lead us out of this mess. I don't want my mother cut off from medical care because of her age.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,902
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{"url":"https:\/\/physics.stackexchange.com\/questions\/303189\/construction-of-the-effective-lagrangian-in-chiral-pertubation-theory","text":"Construction of the effective Lagrangian in chiral pertubation theory\n\nThe QCD Lagrangian containing quarks and gluons has an approximate $SU(3)_\\text{L} \\times SU(3)_\\text{R}$ which is spontaneously broken by a non-vanishing quark condensate $\\langle \\bar{q} q \\rangle \\neq 0$ down to $SU(3)_\\text{V}$ and at the same time explicitly broken by quark masses. Former leads to eight Goldstones bosons, that acquire small masses due to the explicit symmetry breaking. Now, this Lagrangian is not really useful in the low energy regime, for instance below $\\Lambda_\\text{QCD}$, since there the hadrons describe the degrees of freedom. Thus at sufficiently low energies one only needs to take the Goldstone bosons $\\pi, K, \\eta$ into consideration in order to describe dynamics.\n\nWhat I am not understanding from here on is: In order to construct the effective Lagrangian containing the Goldstone boson fields (and external fields) one wants $\\mathcal{L}_\\text{eff}$ to be invariant regarding the chiral group $SU(3)_\\text{L} \\times SU(3)_\\text{R}$ instead of the vector subgroup $SU(3)_\\text{V}$. This bothers me, since the Goldstone bosons only \"exist\" once the chiral symmetry group is broken down to $SU(3)_\\text{V}$ which would mean that the chiral group is not a symmetry of the theory anymore. So, why isn't it enough to construct an effective Lagrangian that is invariant under $SU(3)_\\text{V}$?\n\nThere is a funny thing with spontaneous symmetry breaking: goldstone bosons appearing because of it realize non-linear representation of spontaneously broken symmetry. In Your case, although the Goldstone bosons (pseudo-scalar mesons) parametrize the coset $H \\simeq SU_{L}(3)\\times SU_{R}(3)\/SU_{V}(3)$, whose manifold is $\\simeq SU(3)$, they in fact realize the non-linear representation of the underlying $SU_{L}(3)\\times SU_{R}(3)$ symmetry group.\nYou can understand this in more or less straightforward way. First, \"extract\" the goldstones from the quark field, so that $$q(x) \\equiv (u,d,s)^{T} \\equiv U(x)\\tilde{q}(x),$$ where $$V(x) \\equiv \\text{exp}\\left( \\frac{i\\gamma_{5}\\epsilon_{a}(x)}{f_{\\pi}}\\right)$$ is the goldstones matrix (with fields $\\epsilon_{a}$ parametrizing the coordinates of the coset $H$ space), and $\\tilde{q}$ doesn't contain goldstones field. The linear combination of $\\epsilon_{a}$ are pseudo-scalar mesons.\nSince You know the transformation rule for $q$ under the global $SU_{L}(3)\\times SU_{R}(3)$ transformation, $$q(x) \\to q'(x) = \\text{exp}\\big(iP_{L}t_{a}c^{L}_{a} + iP_{R}t_{a}c^{R}_{a}\\big)q(x), \\quad P_{L\/R} = \\frac{1 \\mp \\gamma_{5}}{2}$$ You can determine the transformation rule for $\\epsilon_{a}(x)$. By using the fact that $V(x)$ represents the coset $SU_{L}(3)\\times SU_{R}(3)$, after slightly tedious but straightforward calculation You obtain $$U(x) \\equiv \\text{exp}\\big( \\frac{2it_{a}\\epsilon_{a}}{f_{\\pi}}\\big) \\to U'(x) e^{it_{a}c^{R}_{a}}U(x)e^{-it_{a}c^{L}_{a}},$$ i.e., the goldstones $\\epsilon_{a}$ form the representation $(3, \\bar{3})$ of the initial symmetry group $SU_{L}(3)\\times SU_{R}(3)$. The fact that they form non-linear representation comes from the non-trivial conditions $$UU^{\\dagger} = 1, \\quad \\text{det}U = 1$$","date":"2019-08-25 07:39:12","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\": 0.9590631127357483, \"perplexity\": 313.6966408206213}, \"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-2019-35\/segments\/1566027323221.23\/warc\/CC-MAIN-20190825062944-20190825084944-00009.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.springerprofessional.de\/calculus-ii\/13755930","text":"main-content\n\n## \u00dcber dieses Buch\n\nThe goal of this text is to help students leam to use calculus intelligently for solving a wide variety of mathematical and physical problems. This book is an outgrowth of our teaching of calculus at Berkeley, and the present edition incorporates many improvements based on our use of the first edition. We list below some of the key features of the book. Examples and Exercises The exercise sets have been carefully constructed to be of maximum use to the students. With few exceptions we adhere to the following policies. \u2022 The section exercises are graded into three consecutive groups: (a) The first exercises are routine, modelIed almost exactly on the exam\u00ad pIes; these are intended to give students confidence. (b) Next come exercises that are still based directly on the examples and text but which may have variations of wording or which combine different ideas; these are intended to train students to think for themselves. (c) The last exercises in each set are difficult. These are marked with a star (*) and some will challenge even the best students. Difficult does not necessarily mean theoretical; often a starred problem is an interesting application that requires insight into what calculus is really about. \u2022 The exercises come in groups of two and often four similar ones.\n\n## Inhaltsverzeichnis\n\n### Chapter 7. Basic Methods of Integration\n\nAbstract\nIn this chapter, we first collect in a more systematic way some of the integration formulas derived in Chapters 4\u20136. We then present the two most important general techniques: integration by substitution and integration by parts. As the techniques for evaluating integrals are developed, you will see that integration is a more subtle process than differentiation and that it takes practice to learn which method should be used in a given problem.\nJerrold Marsden, Alan Weinstein\n\n### Chapter 8. Differential Equations\n\nAbstract\nIn the first two sections of this chapter, we study two of the simplest and most important differential equations, which, describe oscillations, growth, and decay. A variation of these equations leads to the hyperbolic functions, which are important for integration and other applications. To end the chapter, we study two general classes of differential equations whose solutions can be expressed in terms of integrals. These equations, called separable and linear equations, occur in a number of interesting geometrical and physical examples. We shall continue our study of differential equations in Chapter 12 after we have learned more calculus.\nJerrold Marsden, Alan Weinstein\n\n### Chapter 9. Applications of Integration\n\nAbstract\nOur applications of integration in Chapter 4 were limited to area, distance-velocity, and rate problems. In this chapter, we will see how to use integrals to set up problems involving volumes, averages, centers of mass, work, energy, and power. The techniques developed in Chapter 7 make it possible to solve many of these problems completely.\nJerrold Marsden, Alan Weinstein\n\n### Chapter 10. Further Techniques and Applications of Integration\n\nAbstract\nBesides the basic methods of integration associated with reversing the differentiation rules, there are special methods for integrands of particular forms. Using these methods, we can solve some interesting length and area problems.\nJerrold Marsden, Alan Weinstein\n\n### Chapter 11. Limits, L\u2019H\u00f4pital\u2019s Rule, and Numerical Methods\n\nAbstract\nOur treatment of limits up to this point has been rather casual. Now, having learned some differential and integral calculus, you should be prepared to appreciate a more detailed study of limits.\nJerrold Marsden, Alan Weinstein\n\n### Chapter 12. Infinite Series\n\nAbstract\nThe decimal expansion $$\\tfrac{1}{3} = 0.3333 \\ldots$$ is a representation of $$\\tfrac{1}{3}$$ as an infinite sum $$\\tfrac{3}{{10}} + \\tfrac{3}{{100}} + \\tfrac{3}{{1000}} + \\tfrac{3}{{10,000}} + \\cdots$$. In this chapter, we will see how to represent numbers as infinite sums and to represent functions of x by infinite sums whose terms are monomials in x. For example, we will see that\n$$\\ln 2 = 1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\cdots$$\nand\n$$\\sin x = x - \\frac{{{{x}^{3}}}}{{1 \\cdot 2 \\cdot 3}} + \\frac{{{{x}^{5}}}}{{1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5}} - \\cdots .$$\nJerrold Marsden, Alan Weinstein\n\n### Backmatter\n\nWeitere Informationen","date":"2020-06-01 02:15:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 2, \"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\": 0.7730571627616882, \"perplexity\": 613.5404590371976}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347413901.34\/warc\/CC-MAIN-20200601005011-20200601035011-00117.warc.gz\"}"}
| null | null |
{"url":"https:\/\/math.stackexchange.com\/questions\/3705822\/finding-the-value-of-%CE%BB-to-make-an-integral-path-independent\/3705969","text":"# Finding the value of \u03bb to make an integral path-independent\n\nHow can I find \u03bb such that\n\n$$\\int_{(1,2)}^{(2,4)} \\left( \\frac{xy+\\lambda}{y}\\,dx + \\frac{2\\lambda y-x}{y^2}\\,dy \\right)$$\n\nis path independent, and what is the integral's value for that \u03bb?\n\nA line integral is path independent if the integrand can be written in terms of the gradient of a potential function $$U$$. So, let's do exactly that.\n\nLet $$U$$ be defined by $$\\nabla U \\cdot d\\vec{r} = \\frac{xy + \\lambda}y dx + \\frac{2 \\lambda y - x}{y^2} dy$$\n\nThis is our integrand. Supposing that this is the case, let \\begin{align}\\frac{\\partial U}{\\partial x} &= \\frac{xy + \\lambda}y \\\\ \\frac{\\partial U}{\\partial y} &= \\frac{2 \\lambda y - x}{y^2} \\end{align} If we integrate the first equation with respect to $$x$$, we obtain $$U = \\frac{x^2}2 + \\frac{\\lambda x}y + f(y)$$ for some function $$f$$. Then if we integrate the second equation with respect to $$y$$, we obtain $$U = 2 \\lambda \\ln{\\lvert y \\rvert} + \\frac{x}y + g(x)$$ for some function $$g$$.\n\nSetting these two equal, we find $$\\frac{x^2}2 + \\frac{\\lambda x}y + f(y) = 2 \\lambda \\ln{\\lvert y \\rvert} + \\frac{x}y + g(x)$$ so \\left\\{ \\begin{aligned} \\frac{x^2}2 &= g(x) \\\\ \\frac{\\lambda x}y &= \\frac{x}y \\\\ f(y) &= 2 \\lambda \\ln{\\lvert y \\rvert} \\end{aligned} \\right.\n\nFrom the second equation we find that $$\\lambda = 1$$, so $$g(x) = \\frac{x^2}2$$ and $$f(y) = 2 \\ln{\\lvert y \\rvert}$$.\n\nThus our potential function is, for some constant $$C$$, $$U = \\frac{x^2}2 + \\frac{x}y + 2 \\ln{\\lvert y \\rvert} + C$$\n\nTaking the gradient of this returns our original integrand (you can check this). So, $$\\lambda = 1$$.\n\nTo find the value of the integral, just evaluate the potential function at $$(1, 2)$$ and $$(2, 4)$$ and find their difference, just like evaluating a definite integral using an antiderivative at the bounds. You should get $$\\frac{3}2 + 2 \\ln 2$$.\n\nYou need to check if the Jacobian matrix of the function $$\\mathbf{F}(x,y)=\\begin{pmatrix}\\frac{xy+\\lambda}{y} \\\\ \\frac{2\\lambda y-x}{y^2}\\end{pmatrix}$$ is symmetric. This will tell you how you can choose $$\\lambda$$, your first question.\n\nOnce you have done that, look for a function $$\\varphi(x,y)$$ such that the gradient $$\\nabla\\varphi(x,y)$$ equals $$\\mathbf{F}$$. Such a $$\\varphi$$ is called a potential, and can only exist if the Jacobian matrix is symmetric. If you found such a potential, the answer to your second question is $$\\varphi(2,4)-\\varphi(1,2)$$.","date":"2021-12-02 04:51:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 29, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9999808073043823, \"perplexity\": 173.16803262556923}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964361064.69\/warc\/CC-MAIN-20211202024322-20211202054322-00610.warc.gz\"}"}
| null | null |
Q: Newbie Objective C developer question I have been looking everywhere for an answer to this question - perhaps I'm looking in the wrong places. Also, I'm brand new to Objective C although I have around 10 years of experience as a developer.
for this code:
[receiver makeGroup:group, memberOne, memberTwo, memberThree];
what would the method definition look like?
- (void)makeGroup:(Group *)g, (NSString *)memberOne, ...?
Thanks for any help you can provide. I know this is probably very simple...
Thanks,
R
A: It looks like you have a method that can take a variable number of arguments. If that's the case, the definition would look something like:
- (void)makeGroup:(Group *)g, ...;
Check out NSString's stringWithFormat: or NSArray's arrayWithObjects: methods for examples.
Edit: Upon further documentation reading, it seems that you are looking at the exact example that's in the Objective-C 2.0 documentation. The declaration you're looking for is right at the bottom of page 36.
A: You can receive an infinte number of arguments with an ellipsis (...). Check this for further details!
A: It would make more sense to have the members as a separate array argument, like -(void)makeGroup:(Group *)g members:(NSArray *)members. If you must do varargs (which is a pain), it should be written like -(void)makeGroup:(Group *)g members:(NSString *)firstMember, ....
Since I this is trying to figure out how an example method from the documentation would be declared, it would be like this:
- (void)makeGroup:(id)group, ...
Then you would start up the varags machinery with the group argument and use it to find the other arguments.
A: either you're looking for MrHen's answer if you're seeking to do your own class method or if you want to do them separately you write the following into your header file:
-(void)makeGroup:(Group *)g;
-(NSString *)memberOne;
A: EDIT: I answered the wrong question. Ignore this.
The correct way to do this is:
-(void)makeGroup:(Group *)g memberOne:(NSString *)memberOne memberTwo:(NSString *)memberTwo memberThree:(NSString *)memberThree {
...
}
The call will look like this:
[receiver makeGroup:group memberOne:memberOne memberTwo:memberTwo memberThree:memberThree];
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,910
|
from module_kits.vtk_kit.mixins import SimpleVTKClassModuleBase
import vtk
class vtkMCubesReader(SimpleVTKClassModuleBase):
def __init__(self, module_manager):
SimpleVTKClassModuleBase.__init__(
self, module_manager,
vtk.vtkMCubesReader(), 'Reading vtkMCubes.',
(), ('vtkMCubes',),
replaceDoc=True,
inputFunctions=None, outputFunctions=None)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,520
|
from __future__ import division
import pprint
import re
import six
from contextlib import contextmanager
from .exception import CustomError, CustomException
from ..const import EXC_TYPE, INSTRUMENT_TYPE, ACCOUNT_TYPE, UNDERLYING_SYMBOL_PATTERN, NIGHT_TRADING_NS
from ..utils.datetime_func import TimeRange
from ..utils.default_future_info import STOCK_TRADING_PERIOD, TRADING_PERIOD_DICT
from ..utils.i18n import gettext as _
def safe_round(value, ndigits=3):
if isinstance(value, float):
return round(value, ndigits)
return value
class Singleton(type):
_instances = {}
def __call__(cls, *args, **kwargs):
if cls not in cls._instances:
cls._instances[cls] = super(Singleton, cls).__call__(*args, **kwargs)
return cls._instances[cls]
class RqAttrDict(object):
'''
fuck attrdict
'''
def __init__(self, d=None):
self.__dict__ = d if d is not None else dict()
for k, v in list(six.iteritems(self.__dict__)):
if isinstance(v, dict):
self.__dict__[k] = RqAttrDict(v)
def __repr__(self):
return pprint.pformat(self.__dict__)
def __iter__(self):
for k, v in six.iteritems(self.__dict__):
yield k, v
def dummy_func(*args, **kwargs):
return None
def id_gen(start=1):
i = start
while True:
yield i
i += 1
class Nop(object):
def __init__(self):
pass
def nop(*args, **kw):
pass
def __getattr__(self, _):
return self.nop
def to_sector_name(s):
from ..model.instrument import SectorCode, SectorCodeItem
for _, v in six.iteritems(SectorCode.__dict__):
if isinstance(v, SectorCodeItem):
if v.cn == s or v.en == s or v.name == s:
return v.name
# not found
return s
def to_industry_code(s):
from ..model.instrument import IndustryCode, IndustryCodeItem
for _, v in six.iteritems(IndustryCode.__dict__):
if isinstance(v, IndustryCodeItem):
if v.name == s:
return v.code
return s
def create_custom_exception(exc_type, exc_val, exc_tb, strategy_filename):
try:
msg = str(exc_val)
except:
msg = ""
error = CustomError()
error.set_msg(msg)
error.set_exc(exc_type, exc_val, exc_tb)
import linecache
filename = ''
tb = exc_tb
while tb:
co = tb.tb_frame.f_code
filename = co.co_filename
if filename != strategy_filename:
tb = tb.tb_next
continue
lineno = tb.tb_lineno
func_name = co.co_name
code = linecache.getline(filename, lineno).strip()
error.add_stack_info(filename, lineno, func_name, code, tb.tb_frame.f_locals)
tb = tb.tb_next
if filename == strategy_filename:
error.error_type = EXC_TYPE.USER_EXC
user_exc = CustomException(error)
return user_exc
def run_when_strategy_not_hold(func):
from ..environment import Environment
from ..utils.logger import system_log
def wrapper(*args, **kwargs):
if not Environment.get_instance().is_strategy_hold:
return func(*args, **kwargs)
else:
system_log.debug(_("not run {}({}, {}) because strategy is hold").format(func, args, kwargs))
return wrapper
def merge_dicts(*dict_args):
result = {}
for d in dict_args:
result.update(d)
return result
def instrument_type_str2enum(type_str):
if type_str == "CS":
return INSTRUMENT_TYPE.CS
elif type_str == "Future":
return INSTRUMENT_TYPE.FUTURE
elif type_str == "Option":
return INSTRUMENT_TYPE.OPTION
elif type_str == "ETF":
return INSTRUMENT_TYPE.ETF
elif type_str == "LOF":
return INSTRUMENT_TYPE.LOF
elif type_str == "INDX":
return INSTRUMENT_TYPE.INDX
elif type_str == "FenjiMu":
return INSTRUMENT_TYPE.FENJI_MU
elif type_str == "FenjiA":
return INSTRUMENT_TYPE.FENJI_A
elif type_str == "FenjiB":
return INSTRUMENT_TYPE.FENJI_B
else:
raise NotImplementedError
INST_TYPE_IN_STOCK_ACCOUNT = [
INSTRUMENT_TYPE.CS,
INSTRUMENT_TYPE.ETF,
INSTRUMENT_TYPE.LOF,
INSTRUMENT_TYPE.INDX,
INSTRUMENT_TYPE.FENJI_MU,
INSTRUMENT_TYPE.FENJI_A,
INSTRUMENT_TYPE.FENJI_B
]
def get_account_type(order_book_id):
from ..execution_context import ExecutionContext
instrument = ExecutionContext.get_instrument(order_book_id)
enum_type = instrument.enum_type
if enum_type in INST_TYPE_IN_STOCK_ACCOUNT:
return ACCOUNT_TYPE.STOCK
elif enum_type == INSTRUMENT_TYPE.FUTURE:
return ACCOUNT_TYPE.FUTURE
else:
raise NotImplementedError
def exclude_benchmark_generator(accounts):
return {k: v for k, v in six.iteritems(accounts) if k != ACCOUNT_TYPE.BENCHMARK}
def get_upper_underlying_symbol(order_book_id):
p = re.compile(UNDERLYING_SYMBOL_PATTERN)
result = p.findall(order_book_id)
return result[0] if len(result) == 1 else None
def is_night_trading(universe):
for order_book_id in universe:
underlying_symbol = get_upper_underlying_symbol(order_book_id)
if underlying_symbol in NIGHT_TRADING_NS:
return True
return False
def merge_trading_period(trading_period):
result = []
for time_range in sorted(trading_period):
if result and result[-1].end >= time_range.start:
result[-1] = TimeRange(start=result[-1].start, end=max(result[-1].end, time_range.end))
else:
result.append(time_range)
return result
def get_trading_period(universe, account_list):
trading_period = []
if ACCOUNT_TYPE.STOCK in account_list:
trading_period += STOCK_TRADING_PERIOD
for order_book_id in universe:
if get_account_type(order_book_id) == ACCOUNT_TYPE.STOCK:
continue
underlying_symbol = get_upper_underlying_symbol(order_book_id)
trading_period += TRADING_PERIOD_DICT[underlying_symbol]
return merge_trading_period(trading_period)
def is_trading(dt, trading_period):
t = dt.time()
for time_range in trading_period:
if time_range.start <= t <= time_range.end:
return True
return False
@contextmanager
def run_with_user_log_disabled(disabled=True):
from .logger import user_log
if disabled:
user_log.disable()
try:
yield
finally:
if disabled:
user_log.enable()
def unwrapper(func):
f2 = func
while True:
f = f2
f2 = getattr(f2, "__wrapped__", None)
if f2 is None:
break
return f
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,696
|
<?xml version="1.0" encoding="utf-8"?>
<resources>
<!-- From: file:/usr/local/google/buildbot/repo_clients/https___googleplex-android.googlesource.com_a_platform_manifest.git/lmp-mr1-dev/frameworks/support/v7/appcompat/res/values-pt-rPT/strings.xml -->
<eat-comment/>
<string msgid="4600421777120114993" name="abc_action_bar_home_description">"Navegar para a página inicial"</string>
<string msgid="1397052879051804371" name="abc_action_bar_home_description_format">"%1$s, %2$s"</string>
<string msgid="6623331958280229229" name="abc_action_bar_home_subtitle_description_format">"%1$s, %2$s, %3$s"</string>
<string msgid="1594238315039666878" name="abc_action_bar_up_description">"Navegar para cima"</string>
<string msgid="3588849162933574182" name="abc_action_menu_overflow_description">"Mais opções"</string>
<string msgid="4076576682505996667" name="abc_action_mode_done">"Concluído"</string>
<string msgid="7468859129482906941" name="abc_activity_chooser_view_see_all">"Ver tudo"</string>
<string msgid="2031811694353399454" name="abc_activitychooserview_choose_application">"Escolher uma aplicação"</string>
<string msgid="3691816814315814921" name="abc_searchview_description_clear">"Limpar consulta"</string>
<string msgid="2550479030709304392" name="abc_searchview_description_query">"Consulta de pesquisa"</string>
<string msgid="8264924765203268293" name="abc_searchview_description_search">"Pesquisar"</string>
<string msgid="8928215447528550784" name="abc_searchview_description_submit">"Enviar consulta"</string>
<string msgid="893419373245838918" name="abc_searchview_description_voice">"Pesquisa por voz"</string>
<string msgid="3421042268587513524" name="abc_shareactionprovider_share_with">"Partilhar com"</string>
<string msgid="7165123711973476752" name="abc_shareactionprovider_share_with_application">"Partilhar com %s"</string>
<string msgid="1603543279005712093" name="abc_toolbar_collapse_description">"Reduzir"</string>
</resources>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 410
|
class CSFGENPage : public CDialog
{
DECLARE_DYNAMIC(CSFGENPage)
public:
CSFGENPage(CWnd* pParent = NULL); // standard constructor
virtual ~CSFGENPage();
// Dialog Data
enum { IDD = IDD_DIALOG_SFGEN };
protected:
virtual void DoDataExchange(CDataExchange* pDX); // DDX/DDV support
DECLARE_MESSAGE_MAP()
public:
void EnableQueryGroupBox(bool b);
void EnableCallTagGroupBox(bool b);
void EnableFSGroupBox(bool b);
void EnableConfigGroupBox(bool b);
afx_msg void OnBnClickedRadioQuery();
afx_msg void OnBnClickedRadioCalltag();
afx_msg void OnBnClickedRadioFs();
// afx_msg void OnBnClickedRadioConfig();
afx_msg void OnBnClickedButtonResetSfgen();
afx_msg void OnChildActivate();
afx_msg void OnBnClickedButtonQuerySfgen();
afx_msg void OnBnClickedCheckQuerySfgenBcast();
afx_msg void OnBnClickedButtonCallTagSfgen();
afx_msg void OnBnClickedCheckAnyidCall();
afx_msg void OnBnClickedButtonSetFsSfgen();
// afx_msg void OnBnClickedButtonGetFsSfgen();
// afx_msg void OnBnClickedButtonSetConfigSfgen();
// afx_msg void OnBnClickedCheckAnyid();
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 39
|
sponsored by Mary Gay Scanlon
Recognizing the roles and the contributions of Americas Certified Registered Nurse Anesthetists (CRNAs) and their critical role in providing quality health care for the public and the Nation's Armed Forces for more than 150 years, through multiple public health emergencies, and beyond. Jan 24 2023 Introduced In House
Supporting the goals and ideals of Korean American Day. Jan 13 2023 Introduced In House
Recognizing the 94th birthday of Martin Luther King, Jr., and the 40th anniversary of the Martin Luther King, Jr., Holiday. Jan 12 2023 Introduced In House
Opportunity To Address College Hunger Act Jan 12 2023 Introduced In House
TRUST in Congress Act Jan 12 2023 Introduced In House
HJRES 13 (118th Congress)
Proposing an amendment to the Constitution of the United States relating to the authority of Congress and the States to regulate contributions and expenditures intended to affect elections and to enact public financing systems for political campaigns. Jan 09 2023 Introduced In House
End Dark Money Act Jan 09 2023 Introduced In House
Commission to Study and Develop Reparation Proposals for African Americans Act Jan 09 2023 Introduced In House
Social Security Fairness Act of 2023 Jan 09 2023 Introduced In House
To designate the facility of the United States Postal Service located at 850 Walnut Street in McKeesport, Pennsylvania, as the First Sergeant Leonard A. Funk, Jr. Post Office Building. Nov 01 2021 Signed By President.
Faster Payments to Veterans' Survivors Act of 2022 Jul 01 2022 Signed By President.
To designate the facility of the United States Postal Service located at 345 South Main Street in Butler, Pennsylvania, as the "Andrew Gomer Williams Post Office Building". Apr 14 2022 Signed By President.
21st Century President Act May 17 2021 Signed By President.
Aviation Funding Stability Act of 2021 Jun 22 2021 Reported By The Committee On Transportation And Infrastructure. H. Rept. 117-665, Part I.
To provide for the admission and protection of refugees, asylum seekers, and other vulnerable individuals, to provide for the processing of refugees and asylum seekers in the Western Hemisphere, and to modify certain special immigrant visa programs, and for other purposes. Dec 22 2022 Introduced In House
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,190
|
\section{Introduction}
Face recognition has been a challenging problem for computer vision researchers during the past few years \cite{zhao2003face}, \cite{abate20072d}. Earlier hand-crafted feature representation and classical machine learning based methods were utilized for face recognition and analysis \cite{saber1998frontal}, \cite{wang2002facial}, \cite{deniz2003face}, \cite{ahonen2004face}, \cite{he2005face}, \cite{deniz2011face}, \cite{ldop}, \cite{chakraborty2018centre}, \cite{fdlbp}. Such hand-crafted feature-based methods often used to be influenced by the visual characteristics of the image, such as skin, texture, shape and facial parts features. These methods relied on traditional machine learning tools such as K-Nearest Neighbor (KNN), Support Vector Machine (SVM), Decision Tree, Random Forest, etc. for recognition.
After 2010, deep neural networks (DNNs) have emerged as a powerful learning mechanism in solving various tasks in computer vision, including face recognition. DNNs can learn the important features automatically from the image data in a hierarchical fashion at different levels of abstractness \cite{lecun2015deep}.
The deep learning mechanism has shown a huge improvement in 2012 for image recognition using convolutional neural networks (CNNs) \cite{alexnet}. Since then several variants of CNNs have been proposed for different applications of computer vision and image processing \cite{vgg}, \cite{resnet}, \cite{fasterrcnn}, \cite{imageretrieval}, \cite{feng2022iris}. The key success of deep learning models was observed due to the availability of powerful computational machines, such as Graphical Processing Units (GPUs) which can perform heavy parallel computation, and availability of large-scale datasets \cite{imagenet}.
The recent advancements in face recognition research, with the help of deep learning methods, can be observed from the outstanding performance of deep learning models \cite{srivastava2019performance}, such as VGGFace \cite{vggface}, \cite{vggface2} and FaceNet \cite{facenet}. So far, several deep learning models have been developed by the researchers for face recognition by introducing different CNNs, different objective functions, and different other strategies \cite{deepface}, \cite{sphereface}, \cite{arcface}, \cite{hmloss}, \cite{mishra2021multiscale}. A detailed discussion on the advancements of deep learning based face recogntion models can be found in a recent survey published by Wang and Deng in \cite{wang2021deep}.
The recent progress in deep learning based face recognition technology is heavily governed by the large-scale face datasets. Hence, several benchmark face datasets have been developed to drive the research and development activities in face recognition. The popular face datasets include Labelled Faces in Wild \cite{lfw}, CASIA WebFace \cite{casiawebface}, VGGFace \cite{vggface}, VGGFace2 \cite{vggface2}, UMDFace \cite{umdface}, Microsoft Celeb (MS-Celeb-1M) \cite{msceleb} and WebFace \cite{webface}. We present a detailed comparison of the different face datasets in terms of the size, annotations and different characteristics in Table \ref{tab:challengeComparison} and Table \ref{tab:datasetComparison}. Some other face datasets include multi-face dataset \cite{dubey2018multi}, YouTube Face Dataset \cite{wolf2011face}, CelebFaces Attributes Dataset (CelebA) \cite{liu2015faceattributes}, etc. These existing datasets were generally developed by mining the images of celebrities from the public domain. The major focus of these datasets is on more and more samples. Hence, some of these datasets are also noisy and does not contain the scenarios where users take their pictures by themselves using their smart phones, i.e., selfie images.
Some reasearchers have tried to come across the datasets which include the selfie images, such as Selfie Dataset\footnote{https://www.crcv.ucf.edu/data/Selfie/} by Kalayeh et al. \cite{kalayeh2015take}, DocFace \cite{shi2018docface} \& DocFace+ \cite{shi2019docface+} by Shi and Jan, and SelfieCity\footnote{https://selfiecity.net/}. The images in Selfie Dataset \cite{kalayeh2015take} is collected from Instagram and annotated with different attributes. Hence the Selfie Dataset \cite{kalayeh2015take} is used for analyzing ``How to Take a Good Selfie" instead of face recognition. However, we aim to develop a wild selfie dataset, which poses a great challenge by avoiding any environmental constrains while taking the selfies. Further, the proposed dataset should be useful for face recognition in selfie images. The DocFace performs the matching of ID photos with live face photos/selfies \cite{shi2018docface}. The DocFace+ supports multiple ID/selfie per class for ID-Selfie matching \cite{shi2019docface+}.
The SelfieCity dataset is collected from Instagram and labelled based on the Cities. This dataset is also not available publicly for the research purposes.
The existing selfie datasets are either not available publicly or not suitable for face recognition in selfie images. Thus, through this paper, we propose a selfie dataset for face recognition. Following are the contributions of this paper:
\begin{enumerate}
\item A wild selfie dataset (WSD) containing more than 45K images from 42 individuals is proposed to facilitate face recognition in selfie images captured using smartphone cameras.
\item The proposed dataset poses a great challenge in terms of severe variability in image characteristics, such as aspect ratio, partial faces, occlusion, illumination, scale, expression, viewpoint, blur, reflection and augmented reality.
\item We test the performance of existing deep learning based face recognition models to show the complexity of the proposed WSD dataset.
\end{enumerate}
The rest of the paper is presented by detailing the proposed dataset in Section 2, experimental setup in Section 3, results in Section 4, and conclusion in Section 5.
\begin{figure*}[!t]
\centering
\includegraphics[width=\textwidth,height=15cm]{full_collage_image.png}
\caption{Sample images depicting the challenges in the proposed WSD dataset. The rows from top to bottom include the images having effect of AR Filters, Mirrored Reflections, Blurred Images, Partial Faces, Occlusions, Illumination Changes, Scaling, Different Expressions and Emotions, Different Alignments, View Point Variation and Different Aspect Ratios.}
\label{fig:wsdExamples}
\end{figure*}
\section{Proposed Wild Selfie Dataset}
This section describes the proposed Wild Selfie Dataset (WSD) including the selfie image collection process, dataset characteristics, different challenges and a comparison with the existing face datasets. The challenges in WSD include combinations of diverse and difficult real-world selfie image scenarios that humans click selfies in, like augmented reality (AR) filters, blurred and occluded images, mirrored reflections, etc. The sample images depicting different challenges in the dataset are shown in Fig. \ref{fig:wsdExamples}. The dataset comprises of a total of 45,424 selfie images from 42 participants including 24 females and 18 males of 18-31 years age group. The average number of samples in each category is 1,082 with minimum and maximum number of samples in a category as 518 and 2,634, respectively. The labelling is performed manually in terms of the bounding box annotations, subject class levels, and gender of subjects. The dataset is divided into training and test sets with 40,862 and 4,562 images, respectively. The category-wise number of training and test images is summarized in Fig. \ref{fig:train-test}. It can be observed that the dataset is splitted in the training and test sets uniformly.
\subsection{WSD Dataset Creation}
The procedure of creation of the proposed WSD Dataset consists of the following steps: Raw data collection, Pre-processing raw data, Near duplicate elimination, Face detection and bounding box annotation, Manual Filtering and Human Annotation, and Labelling the faces. Next we describe each step in detail.
\subsubsection{Raw Data Collection}
Participants were asked to contribute to the Wild Selfie Dataset by submitting selfie images and self-recorded videos. A selfie is a photograph that a person takes of himself/ herself. Selfie images may be captured through the front or rear camera of the phone, with the aid of a selfie stick, or using laptop cameras, provided the captured images include the human face. The selfie videos were strictly restricted to be taken through the front camera of a smartphone which captured the face of the subject. The agreement has been signed with the participants to use their images for non-commercial research and development purposes.
\subsubsection{Pre-Processing Raw Data}
After collection of the raw data of selfie images and videos, the unsupported file formats and corrupted images are removed from the dataset. Then, the images are extracted from video frames. The multimedia framework FFmpeg\footnote{FFmpeg Developers: http://ffmpeg.org/} is used to extract images from each video. For each individual video, different number of images are extracted depending on expression, illumination, surrounding background and other variations.
\subsubsection{Near Duplicate Elimination}
After extracting images from video frames, all data are in the form of images. At this stage duplicate elimination is imperative so when a model is trained on the dataset, it is not affected by undue bias. A duplicate image in the Wild Selfie Dataset collection is defined as one which is an exact (duplicate) or almost exact (near duplicate) pixel-to-pixel match to an identical image or is obtained from another image through cropping or rescaling techniques. An earnest attempt to remove duplicates in the WSD has been made by us. There exist image pairs in the final WSD dataset, that are similar, but not the exact same. A clear distinction is visible in variations that may be present in the face position, orientation, illumination or expression of the person.
\subsubsection{Face Detection and Bounding Box Annotation}
Following near duplicate image removal, unique selfies of each participant are obtained. The annotation of the WSD Dataset is performed by generating the face bounding boxes. The Dlib \cite{dlib09} general-purpose cross-platform software library is used to detect the faces and to obtain the top-left and bottom-right facial bounding box coordinates. The coordinates are used to compute the width and height of the bounding box. Thus, the final bounding box annotations include the top-left coordinates (X, Y), the width (W), and the height (H).
Upon human evaluation of the generated bounding boxes, we find the following issues: 1) not all bounding boxes contain faces, 2) some bounding boxes include the faces having severe occlusion with others, and 3) some faces remain undetected by Dlib. Thus, manual checking and annotations are also performed, as described next, to correct the above issues.
\subsubsection{Manual Filtering and Human Annotation}
As mentioned above, the manual checking and human annotation are performed to fix the discrepancies caused by Dlib face detection. For each image, the bounding boxes generated through Dlib are manually verified and corrected using the following three criteria: 1) if the image contains a bounding box having a face of the corresponding person then the same is retained, 2) if the image contains a bounding box having a face of a person other than the labeled person, then that bounding box is removed and the bounding box of the corresponding individual is then manually computed, and 3) if Dlib is unable to detect a face in the selfie then the bounding box of the corresponding participant is manually computed.
\subsubsection{Labelling the Faces}
We also provide the face identities of the subjects, for use in face recognition task. In order to make the labels independent of actutal identity of the persons involved, we provide the face identities using the unique codes, such as WSDXX, where XX is from 01 to 42 for all 42 persons participated in data collection. The gender of all participants are also labelled.
\begin{figure*}[!t]
\centering
\includegraphics[trim={7mm 7mm 7mm 20mm},clip, width=\columnwidth]{Training.pdf}
\hspace{3mm}
\includegraphics[trim={7mm 7mm 7mm 20mm},clip, width=\columnwidth]{Test.pdf}
\caption{The number of samples in different categories of WSD dataset under training and test sets.}
\label{fig:train-test}
\end{figure*}
\begin{figure*}[!t]
\centering
\includegraphics[width=0.328\textwidth]{Yaw.png}
\includegraphics[width=0.328\textwidth]{Pitch.png}
\includegraphics[width=0.328\textwidth]{Roll.png}
\caption{Histogram of Yaw angles (left), Pitch angles (middle), and Roll angles (right) of faces in the WSD dataset.}
\label{fig:yaw-pitch-roll}
\end{figure*}
\subsection{Dataset Distribution}
To analyse the camera movement, the distribution of the WSD dataset is calculated through head pose estimation. The head pose is determined by the position of the human face and it's allignment. A camera can be rotated in three ways: side-to-side, up-and-down, and around the optical axis with three corresponding angles, i.e., Yaw, Pitch, and Roll, respectively. OpenCV tool \cite{opencv_library} and 6 key landmark points are used to estimate the 3D head pose of images. The six landmarks used are: Left-Eye-Left-Corner, Right-Eye-Right-Corner, Mouth-Left-Corner, Mouth-Right-Corner, Nose-Center-Tip, and Chin-Center-Tip. Yaw angles vary around the perpendicular axis. Yaw positive values depict the right side positions and negative values depict the left side positions. Pitch angles vary across the lateral axis. Pitch positive values depict the upward direction and negative values depict the downward direction. Roll angles vary around the longitudinal axis. Roll positive values depict clockwise rotation and negative values depict counter-clockwise rotation. The yaw, pitch and roll distributions of the proposed WSD dataset are illustrated in Fig. \ref{fig:yaw-pitch-roll}. The yaw distribution shows that more people are inclined towards a slight right profile, though variation is throughout visible. The pitch distribution shows that people tilt their head more towards the bottom in comparison to top while clicking the pictures. The roll distribution shows that the left head rotation is present heavily than the right counterpart. It is also interesting to note that no image head is exactly straight.
\subsection{WSD Dataset Characteristics and Challenges}
\indent The proposed WSD dataset poses multiple not-available-before challenges in a single dataset that mimic real-world selfie data. The major challenges included in WSD dataset are AR (Augmented Reality) filters, Mirrored reflections, Blurred images, Partial faces, Occlusions, Illumination changes, Scaling variations, Different expressions and emotions, Variation in alignments, View Point variation, and Different aspect ratios. The following subsections describe in detail on how the WSD dataset defines and incorporates the challenges mentioned above.
\begin{table*}[!t]
\centering
\caption{Comparison of challenges in the proposed WSD dataset vs the existing face datasets. Here, \textcolor{OliveGreen}{\cmark}: Present, \textcolor{red}{\xmark}: Absent, and \textcolor{mustard}{\textbf{P}}: Partial.}
\begin{tabular}{ | m{0.17\textwidth} | m{0.05\textwidth} | m{0.06\textwidth} | m{0.08\textwidth} | m{0.06\textwidth} | m{0.06\textwidth} | m{0.06\textwidth} | m{0.047\textwidth} | m{0.048\textwidth} | m{0.06\textwidth} | m{0.04\textwidth} |}
\hline
\multicolumn{11}{|c|}{A comparison of challenges among different face datasets} \\
\hline
Dataset Name & View-point Variation & Different Alignments & Different Expressions \& Emotions & Scaling Changes & Illu-mination Changes & Occlu-sions & Partial Faces & Blur Images & Mirror Reflections & AR Filters\\
\hline
LFW \cite{lfw} & \textcolor{OliveGreen}{\cmark} & \textcolor{red}{\xmark} & \textcolor{mustard}{\textbf{P}} & \textcolor{red}{\xmark} & \textcolor{mustard}{\textbf{P}} & \textcolor{red}{\xmark} &
\textcolor{red}{\xmark} &
\textcolor{red}{\xmark} &
\textcolor{red}{\xmark} &
\textcolor{red}{\xmark}\\
\hline
CASIA WebFace \cite{casiawebface} & \textcolor{OliveGreen}{\cmark} & \textcolor{red}{\xmark} &
\textcolor{red}{\xmark} &
\textcolor{red}{\xmark} & \textcolor{mustard}{\textbf{P}} & \textcolor{red}{\xmark} &
\textcolor{red}{\xmark} &
\textcolor{red}{\xmark} &
\textcolor{red}{\xmark} &
\textcolor{red}{\xmark}\\
\hline
VGG Face \cite{vggface} &
\textcolor{OliveGreen}{\cmark} & \textcolor{OliveGreen}{\cmark} & \textcolor{OliveGreen}{\cmark} & \textcolor{red}{\xmark} & \textcolor{mustard}{\textbf{P}} & \textcolor{red}{\xmark} &
\textcolor{mustard}{\textbf{P}} &
\textcolor{red}{\xmark} &
\textcolor{red}{\xmark} &
\textcolor{red}{\xmark}\\
\hline
UMDFace \cite{umdface} & \textcolor{OliveGreen}{\cmark} & \textcolor{OliveGreen}{\cmark} & \textcolor{OliveGreen}{\cmark} & \textcolor{OliveGreen}{\cmark} & \textcolor{OliveGreen}{\cmark} &
\textcolor{mustard}{\textbf{P}} &
\textcolor{mustard}{\textbf{P}} &
\textcolor{red}{\xmark} &
\textcolor{red}{\xmark} &
\textcolor{red}{\xmark}\\
\hline
WSD & \textcolor{OliveGreen}{\cmark} & \textcolor{OliveGreen}{\cmark} & \textcolor{OliveGreen}{\cmark} & \textcolor{OliveGreen}{\cmark} & \textcolor{OliveGreen}{\cmark} &
\textcolor{OliveGreen}{\cmark} &
\textcolor{OliveGreen}{\cmark} &
\textcolor{OliveGreen}{\cmark} &
\textcolor{OliveGreen}{\cmark} &
\textcolor{OliveGreen}{\cmark} \\
\hline
\end{tabular}
\label{tab:challengeComparison}
\end{table*}
\begin{table*}[!t]
\caption{A comparison of the proposed WSD dataset with existing face datasets.}
\centering
\begin{tabular}{ | m{0.17\textwidth} | m{0.09\textwidth} | m{0.08\textwidth} | m{0.09\textwidth} | m{0.45\textwidth} |}
\hline
\multicolumn{5}{|c|}{Dataset Statistics Comparison Table} \\
\hline
Dataset Name & Number of Subjects & Number of Images & Availability & Annotation Properties\\
\hline
LFW \cite{lfw} & 5,749 & 13,233 & Public & Several Annotation Attributes\\
\hline
CelebFaces \cite{celebfaces} & 10,177 & 202,599 & Private & 5 Landmarks and 40 Binary Attributes\\
\hline
CASIA WebFace \cite{casiawebface} & 10,575 & 494,414 & Public & -\\
\hline
VGG Face \cite{vggface} & 2,622 & 2,600,000 & Public & Face bounding boxes and Coarse pose\\
\hline
UMDFace \cite{umdface} & 8,501 & 367,920 & Public & Face bounding boxes, 21 keypoints, Gender and 3D pose\\
\hline
Proposed WSD & 42 & 45,426 & Public & Selfie images, Class labels, Face bounding boxes, and Gender of subjects\\
\hline
\end{tabular}
\label{tab:datasetComparison}
\end{table*}
\subsubsection{AR Filters}
Augmented Reality (AR) is the technology which superimposes a computer-generated image on the user's view of the real-world. There are many smartphone inbuilt applications that provide AR filters like Snapchat, Instagram, B612, etc. that are used by the contributors of the WSD dataset. The AR filters transform the user's image using a wide range of accessories and combinations of special effects, e.g., the Snapchat application transforms the human face into a dog or zombie using animated tricks. While people play with filters for entertainment, the human face still remains visible, and can be both detected and recognized. This makes the dataset more challenging and represents the images mimicing the selfie clicking behaviour of young population.
\subsubsection{Mirrored Reflections}
In the WSD dataset, there are two kinds of mirrored selfies. The first one is due to the pictures taken while standing in front of a mirror by the rear camera of smartphones. The second one is due to the capturing of images through the front camera of smartphones with the selfie containing face reflections.
\subsubsection{Blurred Images}
There are two kinds of blurred images present in the WSD dataset. While some blurred pictures are clicked with the moving camera, others are captured with blur filters `on' in the smartphone. The faces in the selfies are detectable and easily recognizable by the human beings.
\subsubsection{Partial Faces}
Many selfies present in WSD contain only partial visible faces and do not inlude the complete face. Some selfies contain only the left/right half or only the top/bottom portion of the face. A human glance can be easily detected and recognised in these images. Some images are also tilted with multiple alignments in addition to partial faces, adding the diversity and severe challenges in the WSD dataset.
\subsubsection{Occlusions}
The occlusions and obstructions are present in several images of the WSD dataset. Multiple variations of such obstructions such as a facial regions being covered by hand, pen covering the nose tip, AR filters hiding the mouth region, etc. have been included in WSD.
\subsubsection{Illumination Changes}
The WSD dataset incorporates multiple natural lighting conditions in an unconstrained way. Selfies are clicked during dawn, noon, afternoon, dusk, late evening, and even at night. In some images, even more challenging scenarios such as lighting variation from natural light falling on faces from slits through vents or window panes, are present. Collected images have both natural and filtered artificial lighting present.
\subsubsection{Scaling Variations}
Participators clicked selfies in different poses consisting of different sizes of their faces, by changing the camera positions and angles with respect to their faces. Some are full close-up shots covering the facial region from the eyebrows to the chin, while others are shots from far containing the full face along with high degree of background regions.
\subsubsection{Different Expressions and Emotions}
The images in the WSD dataset contain different expressions and emotions of the subjets. Some facial sentiments present include sad, angry, happy, sleepy, excited, irritated, pouting, sulking, crying, smiling, fearful, laughing, yawning, winking, disgusting, and surprised.
\subsubsection{Different Poses and Alignments}
The geometric alignment of selfies also differ greatly in the WSD dataset. As mentioned above, Roll angles in the dataset are diversely varied with no image being strictly vertically aligned leading to different poses and alignments in the WSD dataset.
\subsubsection{View Point Variation}
Multiple profiles of each candidate are included in the WSD dataset as analyzed above in terms of the Yaw characteristics which represent the variations in the view point variations.
\subsubsection{Different Aspect Ratios}
Aspect ratio is the ratio of the width to the height of an image. Since all candidates used the front or back camera of their smartphones and/or mobile devices to click the selfies and videos, the aspect ratios obtained are diverse. While data collection, no restriction on aspect ratio is kept. Some common aspect ratios in the WSD dataset are 1:1, 5:4, 4:3, 3:2, and 16:9. We use the dataset after resizing the images to a fixed size.
\subsection{Comparison with Existing Datasets}
Table \ref{tab:challengeComparison} compares the proposed WSD dataset with the existing face datasets, on the basis of the various challenges present in WSD. We can observe from the existing face datasets that though some challenges like profile variation are present in all the existing datasets, few challenges like scaling, occlusion and partial faces together are present only in a few datasets. The proposed WSD dataset is the only dataset to contain selfies with blurred images, mirrored reflections and AR filters. All the challenges present in the proposed dataset make it highly relevant depicting the real-world scenario. Table \ref{tab:datasetComparison} presents a statistical comparison of the proposed WSD dataset with the existing face datasets. Though the number of subjects and samples are less in the proposed dataset, the images in WSD are collected by the subjects themselves in an unconstrained enviorenement. Whereas the existing datasets are mostly crawled from internet. The existing face datasets do not contain the selfie images which pose its own challenges. Moreover, the proposed dataset contains only the selfie images which have become poupular among young population to capture the images. Further, as mentioned earlier, there exist no publicly available selfie dataset for face detection and recognition in selfie images.
\section{Methodology and Experimental Setup}
In this paper, we adopt the standard face detection (i.e., face region localization) and face recognition methodology for the experiments. Specifically, we consider the existing deep learning based face detection and face recogniton approaches to judge their performance on the proposed WSD dataset. The results of two face detection models and three face recognition models are computed by training the models using the WSD training set and evaluating using the WSD test set. All the networks are trained/fine-tuned for about 100 epochs on our WSD training set using Tesla T4 GPU and evaluated on our WSD test set.
Before using the WSD for face recognition task, we perform face detection using WSD dataset. The state-of-the-art You Only Look Once - v3 (YOLOv3) \cite{yolov3} and Multi-task Cascade Convolutional Neural Network (MTCNN) \cite{mtcnn} models are used to test the face detection performance on the proposed WSD dataset. The YOLOv3 model \cite{yolov3} is a real-time object detection CNN model by utilizing a fast Darknet module as compared to previous versions of YOLO. The MTCNN model \cite{mtcnn} consists of three separate convolutional networks, including P-net, R-net and O-net. The MTCNN model first resizes the input to create the image pyramid followed by non-maximum suppression and bounding box regression for every scaled image to find the face regions in the image. In both cases of YOLOv3 and MTCNN, we use the pretrained models and fine-tune it using our WSD training set and evaluate the model performance on the WSD test set in terms of the mean average precision (mAP). For YOLOv3, we use batch size of 16, learning rate of 0.0001, and batch normalization after each convolution layer for regularization. In case of MTCNN model, we use batch size of 32, learning rate of 0.001, and batch normalization after each convolution layer for regularization.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{DetectionFailedCases.png}
\caption{The sample face images for which both the YOLOv3 and MTCNN fail to detect the faces. The score for the corresponding image to have a face using both the models is mentioned.}
\label{fig:detection_failed}
\end{figure*}
Next we perform the experiments for face recognition on the proposed selfie dataset with three state-of-the-art face recognition models, namely VGGFace \cite{vggface}, VGGFace2 \cite{vggface2} and FaceNet \cite{facenet}. The VGGFace model \cite{vggface} uses CNN descriptors for face recognition. The CNN descriptors are obtained using the VGG-Very-Deep-16 CNN architecture. The VGGFace model is trained on VGGFace dataset consisting of 2,622 identities over 2.6 million images. We use the pretrained VGGFace model and fine-tune it using the WSD training set to evaluate the performance of VGGFace on the WSD test set. In case of VGGFace model, training/fine-tuning on the WSD dataset is done by stochastic gradient descent with 0.9 momentum coefficient using mini-batches of 64 samples and learning rate of 0.01. The VGGFace model is regularised using dropout and weight decay factor of $5 \times 10^-4$. The VGGFace2 model \cite{vggface2} also uses the CNN descriptors for face recognition similar to VGGFace, however, the CNN descriptors are computed using ResNet-50 model \cite{resnet} rather than VGG CNN architecture. The VGGFace2 model is trained on VGGFace2 dataset that consists of 9,131 identities and 3.31 million images. We use the pretrained VGGFace2 model and fine-tune it using the WSD training set to evaluate its performance on the WSD test set. The training configuration of VGGFace2 model is similar to VGGFace, except the Adamax optimizer with learning rate of 0.001 is used for VGGFace2 model. FaceNet model \cite{facenet} uses face embeddings generated using deep CNN like Inception network followed by Triplet loss layer for face recognition and verification. We evaluate the performance of the FaceNet model on the proposed WSD test set by fine tuning the face embeddings of FaceNet model using the WSD training set with triplet loss. In case of FaceNet model, a batch size of 16, learning rate of 0.001 and L2 Regularization are used while fine-tuning face embeddings.
\section{Experimental Results}
In this section, we provide the results obtained using the state-of-the-art deep learning models for face detection and face recognition on the proposed dataset. We also make a comparison of WSD with the benchmark face datasets.
\begin{table}[t]
\caption{Face detection results (in \% of accuracy) using YOLOv3 \cite{yolov3} and MTCNN \cite{mtcnn} models on WSD dataset in terms of mean average precision (mAP) metric. The existing results on FDDB \cite{fddb} and WIDER FACE \cite{widerface} datasets are also included for a comparison.}
\centering
\begin{tabular}{| p{0.474\columnwidth} | p{0.18\columnwidth} | p{0.18\columnwidth} |}
\hline
\textbf{Dataset} & \textbf{YOLOv3} & \textbf{MTCNN} \\\hline
FDDB \cite{fddb} & 93.90 \cite{li2020face} & 90.20 \cite{li2020face}\\\hline
Wider Face (Easy) \cite{widerface} & 87.60 \cite{tuli2020novel} & 85.10 \cite{mtcnn}\\\hline
Wider Face (Medium) \cite{widerface} & 85.80 \cite{tuli2020novel} & 82.00 \cite{mtcnn}\\\hline
Wider Face (Hard) \cite{widerface} & 72.90 \cite{tuli2020novel} & 60.70 \cite{mtcnn}\\\hline
\textbf{WSD (Ours)} & 96.92 & 95.77\\\hline
\end{tabular}
\label{tab:detectionResults}
\end{table}
\subsection{Face Detection Results}
Table \ref{tab:detectionResults} shows the face detection results on WSD and other face datasets, in terms of mAP by using the YOLOv3 \cite{yolov3} and MTCNN \cite{mtcnn} CNN models. The results on the WSD dataset are depicted in the first row. The publicaly available results on benchmark face detection datasets, such as ace Detection Data Set and Benchmark (FDDB) \cite{fddb} and WIDER FACE \cite{widerface} are also included in the table for a comparison. It can be noticed that the performance of both the models on the WSD dataset is higher as compared to FDDB and WIDER FACE dataset. This is due to the nature of images in the proposed and existing datasets. Specifically, the selfie images are generally captured from a camera placed within a limited distance from the subject. This leads to significantly higher facial regions in selfie images as compared to the non-selfie images. Moreover, the majority of the images in the proposed WSD dataset consist of images with single face. Hence, the face detection in the proposed dataset is easier as compared to large-scale non-selfie face datasets. However, the real challenges in the proposed dataset are observed for face recognition as described next.
\begin{figure*}[!t]
\centering
\includegraphics[width=\textwidth]{ClassificationFailedCases.png}
\caption{Few sample of failed cases, where all the models misclassify the face to some other person. The correct class with score and incorrect class with score are mentioned in Green and Red color, respectively.}
\label{fig:classification_failed}
\end{figure*}
We report some failure cases where both the YOLOv3 and MTCNN models are not able to detect the face regions in Fig. \ref{fig:detection_failed} alongwith the score of having the face. It can be seen that the score to detect a face in the images is very low using both the models, which indicate the complexity of the images for face detection. Upon analysis with the failure cases, we notice that the selfies with severe background illumination changes are not being detected, irrespective of whether the face is clearly visible or challenging. In addition to this, people having occlusions near the eye are harder to detect. Partial images are another diffuculty to detect and in many cases, images containing AR filters had incorrect detections. In such cases bounding box does not contain the face, but other regions. These are the challenges for the face detection task in selfie images.
\begin{table}[!t]
\caption{Face recognition results (in \% of accuracy) using VGGFace, VGGFace2 and FaceNet models on WSD dataset in terms of validation accuracy metric. The existing results on LFW \cite{lfw}, CASIA WebFace \cite{casiawebface}, VGGFace2 \cite{vggface2}, YouTube Faces \cite{wolf2011face} and IJB-A \cite{ijba}, IJB-B \cite{ijbb}, and IJB-C \cite{ijbc} face datasets are also included for a comparison.}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{| p{0.307\columnwidth} | p{0.18\columnwidth} | p{0.18\columnwidth} | p{0.26\columnwidth} |}
\hline
Dataset & VGGFace & VGGFace2 & FaceNet \\\hline
LFW & 98.95 \cite{vggface} & - & 99.63 \cite{facenet}\\\hline
CASIA WebFace & 90.7 \cite{zhang2017local} & - & 99.05 \cite{william2019face}, \cite{xu2020lightweight}\\\hline
VGGFace2 & 89.4 \cite{vggface2} & 96.1 \cite{vggface2} & 99.65 \cite{william2019face}, \cite{xu2020lightweight}\\\hline
YouTube Faces & 97.3 \cite{vggface} & - & 95.12 \cite{facenet} \\\hline
IJB-A & 95.4 \cite{vggface2} & 98.0 \cite{vggface2} & - \\\hline
IJB-B & 85.0 \cite{vggface2} & 93.8 \cite{vggface2} & - \\\hline
IJB-C & - & 95.0 \cite{vggface2} & - \\\hline
\textbf{WSD (Ours)} & 88.53 & 92.78 & 93.98\\\hline
\end{tabular}
}
\label{tab:recognitionresults}
\end{table}
\subsection{Face Recognition Results}
Table \ref{tab:recognitionresults} shows the results of applying VGGFace \cite{vggface}, VGGFace2 \cite{vggface2} and FaceNet \cite{facenet} face recognition techniques on different face reginition datasets including the WSD. The first row depicts the face recognition results on the proposed WSD dataset. In order to perform the results comparison, the existing results on the benchmark datasets, such as LFW \cite{lfw}, CASIA WebFace \cite{casiawebface}, VGGFace2 \cite{vggface2}, YouTube Faces \cite{wolf2011face}, Iarpa Janus Benchmark-A (IJB-A) \cite{ijba}, IJB-B \cite{ijbb} and IJB-C \cite{ijbc}, are also reported. It is quite clear from the face recognition results that the performance of the face recognition models is significantly lower on the proposed WSD dataset as compared to the existing face datasets. The lower face recognition performance on the proposed WSD dataset is due to the inherent challenges present in the dataset, such as illumination changes, AR filters, occlusion, scale changes, blur, pose variations, and many more.
Sample images are shown in Fig. \ref{fig:classification_failed} where all the classification models fail to classify the faces into correct class. The correct class alongwith score for each model is shown in Green color. Whereas, the incorrectly recognized classes alongwith score for each model is shown in Red color. It can be seen that the class score for correct class is very low for all the models. On the other hand, the class score for incorrect class is very high. This shows the complexity level of the images present in the developed dataset. Overall, the experimental results confirm that the proposed WSD dataset is challenging for face recognition in selfie images.
\section{Conclusion and Future Directions}
A wild selfie dataset (WSD) is proposed in this paper, for face recognition in selfie images. The proposed dataset includes the face images of mostly youngsters with more female subjects than male subjects. The WSD dataset is very challenging in several terms, such as images containing effects of AR filters, mirrored reflections, blur, partial faces, occlusions, illumination changes, scaling, different expressions and emotions, different alignments, view point variation and different aspect ratios. The proposed WSD dataset is tested for face detection and face recogntion tasks. The face detection results are computed using YOLOv3 and MTCNN models. It is found that the face detection results are satisfactory on the WSD dataset as the selfie images are captured from a limited distance. However, the presence of severe illumination changes and occlusions pose significant challenges to the face detection models. The face recognition results are computed using VGGFace, VGGFace2 and FaceNet models. We notice that the performance of the face recogntion models is significantly lower as compared to the existing face datasets. The face recognition results confirm the complexity of the proposed WSD dataset, consisting of real challenges. The proposed WSD dataset can significantly boost the advancements in the face recognotion technology to better tackle the challenges posed while capturing the selfie images in real scenario.
{\small
\bibliographystyle{ieee_fullname}
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"redpajama_set_name": "RedPajamaArXiv"
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Scriptio continua
Scriptio continua (Latin for "continuous script"), also known as scriptura continua or scripta continua, is a style of writing without spaces, or other marks between the words or sentences. The form also lacks punctuation, diacritics, or distinguished letter case. In the West, the oldest Greek and Latin inscriptions used word dividers to separate words in sentences; however, Classical Greek and late Classical Latin both employed scriptio continua as the norm.[1][2]
Although scriptio continua is evidenced in most Classic Greek and Classic Latin manuscripts, different writing styles are depicted in documents that date back even further. Classical Latin did often use the interpunct, especially in monuments and inscriptions.
The earliest texts in Classical Greek that used the Greek alphabet, as opposed to Linear B, were formatted in a constant string of capital letters from right to left. Later, this evolved to "boustrophedon," which included lines written in alternating directions. It was only later on that the Romans adapted the Etruscan alphabet to write Latin and, in the process, switched from using points to divide words to the Greek practice of scriptio continua.[3]
Before (and after) the advent of the codex (book), Latin and Greek script was written on scrolls by enslaved scribes. The role of the scribe was to simply record everything he or she heard, in order to create documentation. Because speech is continuous there was no need to add spaces. Typically, the reader of the text was a trained performer, who would have already memorized the content and breaks of the script. During these reading performances, the scroll acted as a cue sheet, and therefore did not require in-depth reading.
While the lack of word parsing forced the reader to distinguish elements of the script without a visual aid, it also presented him with more freedom to interpret the text. The reader had the liberty to insert pauses and dictate tone, making the act of reading a significantly more subjective activity than it is today. However, the lack of spacing also led to some ambiguity because a minor discrepancy in word parsing could give the text a different meaning. For example, a phrase written in scriptio continua as collectamexiliopubem may be interpreted as collectam ex Ilio pubem, meaning "a people gathered from Troy", or collectam exilio pubem, "a people gathered for exile". Thus, readers had to be much more cognizant of the context to which the text referred.[4]
Over time, the current system of rapid silent reading for information replaced the older, slower, and more dramatic performance based reading,[5] and word dividers and punctuation became more beneficial to text.[6] Though paleographers disagree about the chronological decline of scriptio continua throughout the world, it is generally accepted that the addition of spaces first appeared in Irish and Anglo-Saxon Bibles and Gospels from the seventh and eighth centuries.[7]:21 Subsequently, an increasing number of European texts adopted conventional spacing, and within the thirteenth and fourteenth centuries, all European texts were written with word separation.[7]:120-121
When word separation became the standard system, it was seen as a simplification of Roman culture because it undermined the metric and rhythmic fluency generated through scriptio continua. In contrast, paleographers today identify the extinction of scriptio continua as a critical factor in augmenting the widespread absorption of knowledge in the Pre-Modern Era. By saving the reader the taxing process of interpreting pauses and breaks, the inclusion of spaces enables the brain to comprehend written text more rapidly. Furthermore, the brain has a greater capacity to profoundly synthesize text and commit a greater portion of information to memory.[7]:16-17
Scriptio continua is still in use in Thai script, other Southeast Asian abugidas: (Burmese, Khmer, Javanese, Balinese, Sundanese script), Lao, and in languages that use Chinese characters (Chinese and Japanese). However, modern vernacular Chinese differentiates itself from ancient scriptio continua through its use of punctuation, although this method of separation was borrowed from the West only about a century ago. Before this, the only forms of punctuation found in Chinese writings were marks to denote quotes, proper nouns, and emphasis. Modern Tibetic languages also employ a form of scriptio continua; while they punctuate syllables, they do not use spacing between units of meaning.
Latin text in scriptio continua with typical capital letters, taken from Cicero's De finibus bonorum et malorum:
With ancient Latin punctuation is:
NEQVE•PORRO•QVISQVAM•EST•QVI•DOLOREM•IPSVM•QVIA•DOLOR•SIT•AMET•CONSECTETVR•ADIPISCI•VELIT
Greek text in scriptio continua with typical capital letters, taken from Hesiod's Theogony:
ΜΟΥΣΑΩΝΕΛΙΚΩΝΙΑΔΩΝΑΡΧΩΜΕΘΑΕΙΔΕΙΝΑΙΘΕΛΙΚΩΝΟΣΕΧΟΥΣΙΝΟΡΟΣΜΕΓΑΤΕΖΑΘΕΟΝΤΕΚΑΙΠΕΡΙΚΡΗΝΗΙΟΕΙΔΕΑΠΟΣΣΑΠΑΛΟΙΣΙΝΟΡΧΕΥΝΤΑΙΚΑΙΒΩΜΟΝ
A form of scriptio continua has become common in internet e-mail addresses and domain names where, because the "space" character is invalid, the address for a website for "Example Fake Website" is written as examplefakewebsite.com – without spaces between the separate words. However, the "underscore" or "dash" characters are often used as stand-ins for the "space" character when its use would be invalid and their use would not be.
The Chinese did not encounter the problem of incorporating spaces into their text because, unlike most orthographic systems, characters were written by combining a series of letters, so each already represented a word or morpheme.[3] On top of that, Chinese also lacked any form of punctuation until the 20th century as a result of interaction with Western civilizations.[10]
Like Chinese, Japanese implements extensive use of Chinese characters, called kanji in Japanese. However, due to the radical differences between the Chinese and Japanese languages, writing Japanese exclusively in kanji would make it extremely difficult to read.[11] This can be seen in texts that predate the modern kana system, in which Japanese was written entirely in kanji and man'yōgana, the latter of which are characters written solely to indicate a word's pronunciation as opposed to its meaning. For that reason, different syllabary systems called kana were developed to differentiate phonetic graphemes from ideographic ones.
Modern Japanese is typically written using three different types of graphemes, the first being kanji and the latter two being kana systems, the cursive hiragana and the angular katakana. While spaces are not normally used in writing, boundaries between words are often quickly perceived by Japanese speakers since kana are usually visually distinct from kanji. Japanese speakers also know that certain words, morphemes, and parts of speech are typically written using one of the three systems. Kanji is typically used for words of Japanese and Chinese origin as well as content words (i.e. nouns, verbs, adjectives, adverbs). Hiragana is typically used for native Japanese words, as well as commonly known words, phrases, and particles, as well as inflections of content words like verbs, adjectives, and adverbs. Katakana is typically used for loanwords from languages other than Chinese, onomatopoeia, and emphasized words.
Like Chinese, Japanese lacked any sort of punctuation until interaction with Western civilizations became more common. Punctuation was adopted during the Meiji Period.
Modern Thai script, which was said to have been created by King Ram Khamhaeng in 1283, does not contain any spaces between words, but spaces only indicate the clear endings of clauses or sentences.[12]
Below is a sample sentence of Thai written first without spaces between words (with Thai romanization in parentheses), second written in Thai with spaces between words (also with Thai romanization in parentheses), then finally translated into English.
An example of the first line of the Universal Declaration of Human Rights in Javanese script,[14] and when they are divided (in some modern writings) by spaces and dash sign, which looks different.
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Silvio Horta, 'Ugly Betty' Creator, Found Dead in Apparent Suicide
Stuart Oldham and Will Thorne
Variety 7 January 2020
Silvio Horta, creator of ABC's popular series "Ugly Betty," was found dead in a Miami motel room on Tuesday in an apparent suicide. He was 45.
Sources tell Variety that Horta died of a self-inflicted gunshot wound. A representative for Horta confirmed his death, but declined to comment on the nature of it.
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The American version of "Ugly Betty," which starred America Ferrera, ran for four seasons on ABC from 2006 to 2010 and was adapted from the hit Colombian telenovela "Yo soy Betty, la fea." Horta served as showrunner and head writer on the comedy-drama.
Ferrera posted on Instagram about his death, saying "His talent and creativity brought me and so many others such joy and light."
The show centered on Betty Suarez, a young, smart Mexican American woman from Queens who lands a job at an upscale fashion magazine in Manhattan. During its run, the show won two Golden Globes, one for best comedy series and one for Ferrera. The original series was created in 1999 by Fernando Gaitan and ran until 2001.
Horta, who was born in Miami to Cuban parents and majored in film at NYU, had a project with Mary J. Blige in development at Fox in 2018. The prospective series, titled "Move," was inspired by the life of famed choreographer and creative director Laurieann Gibson, who created routines for the likes of Beyonce, Michael Jackson, Alicia Keys and Lady Gaga.
His big break came in 1998 when he wrote the screenplay for horror-thriller "Urban Legend." The script was turned into a successful feature starring Jared Leto and "Dune" star Alicia Witt.
Horta's moved into TV in the earlly 2000s with the Sci Fi series "The Chronicle," which ran for a single season and starred Octavia Spencer, among others. In 2003, Horta continued his science fiction run with the UPN series "Jake 2.0." The series, which also only lasted one season, revolved around a computer expert (Christopher Graham) who, while working for the NSA, was accidentally infected by nanobots, giving him superhuman powers.
Horta's most recent credit was for writing the 2015 TV movie "The Curse of the Fuentes Women."
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Rochelle and Marvin Humes share adorable first pics of their kids
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Q: How to group by time for XTS object in R? I have this a time series xts object in R.
Basically the time series duration is a few months, I want to know the trend for different time points.
I want to get the median or mean for different time points.
library(xts)
library(lubridate)
Time <- seq(ymd_hms("2019-01-01 00:00:00"), ymd_hms("2019-03-29 23:59:59"), "hour")
length(Time)
Data <- rnorm(2112, 1, 5)
Time_Series <- xts(x = Data , order.by = Time)
Take this code as example.
How can I get the mean for the data at time 00:00:00? similarly the mean of data in 01:00:00, 02:00:00, 03:00:00 ...
Thank you for your helping in advance!
A: This one-liner uses aggregate.zoo producing a zoo object whose time is the hour. No additional packages are used.
aggregate(Time_Series, hour, mean)
giving:
0 0.4237426
1 1.8814963
2 1.2917437
3 1.4307028
4 1.3691019
5 0.3762082
6 1.3866948
# ...snip...
Note that the data in the question is not reproducible since set.seed was not used so this just shows what the output looks like.
A: One way would be to convert the time series to dataframe and get mean by hour.
library(dplyr)
library(lubridate)
Time_Series %>%
zoo::fortify.zoo() %>%
group_by(hour = hour(Index)) %>%
summarise(mn = mean(Data))
# A tibble: 24 x 2
# hour mn
# <int> <dbl>
# 1 0 1.53
# 2 1 0.414
# 3 2 1.24
# 4 3 1.07
# 5 4 1.32
# 6 5 1.34
# 7 6 0.998
# 8 7 -0.615
# 9 8 0.924
#10 9 0.484
# … with 14 more rows
A method with base R would be using aggregate
df <- zoo::fortify.zoo(Time_Series)
df$hour <- format(df$Index, "%H")
aggregate(Data~hour, df, mean)
data
set.seed(23)
Time <- seq(ymd_hms("2019-01-01 00:00:00"), ymd_hms("2019-03-29 23:59:59"), "hour")
Data <- rnorm(2112, 1, 5)
Time_Series <- xts(x = Data , order.by = Time)
names(Time_Series) <- "Data"
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{
"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
The Khovanov-Kuperberg algebras $K^\epsilon$ were introduced in 2012 by Mackaay -- Pan -- Tubbenhauer \cite{2012arXiv1206.2118M} and the author \cite{LHR1} in order to give an algebraic definition of $\ensuremath{\mathfrak{sl}}_3$-homology for tangles.
In \cite{2012arXiv1206.2118M}, the split Grothendieck group of the category $K^\epsilon$-$\mathsf{pmod}$ is computed. This shows, as in the $\ensuremath{\mathfrak{sl}}_2$ case, that we have a categorification of $\hom_{U_q(\ensuremath{\mathfrak{sl}}_3)}(\CC, V^{\otimes \epsilon})$ (here $\epsilon$ is an admissible sequence of signs).\marginpar{sens: $\CC$ avant ou après ?}
The proof of this result is far from being easy\footnote{In the $\ensuremath{\mathfrak{sl}}_2$ case this is quite direct thanks to a Schur lemma argument.}, furthermore, the non-elliptic webs, the good candidates to correspond
to the indecomposable modules, fails to play this role, the story starts to become dramatically more complicated than in the $\ensuremath{\mathfrak{sl}}_2$ case.
This paper have a down-to-earth approach and gives a fully combinatorial characterisation of indecomposable web-modules.
\subsubsection*{Acknowledgments} The author wishes to thank Christian Blanchet for suggesting the subject.
\label{sec:acknoledgment}
\section{The Khovanov--Kuperberg algebras}
\label{sec:khov-kuperb-algebr}
\subsection{The 2-category of web-tangles}
\label{sec:webs-foams}
\subsubsection{Webs}
In the following $\epsilon=(\epsilon^1,\dots,\epsilon^n)$ (or $\epsilon_0$, $\epsilon_1$ etc.) will always be a finite sequence of signs, its \emph{length} $n$ will be denoted by $l(\epsilon)$, such an $\epsilon$ will be \emph{admissible} if $\sum_{i=1}^{l(\epsilon)}\epsilon^i$ is divisible by 3.
\label{sec:webs}
\label{sec:two-category-web}
\begin{dfn}[Kuperberg, \cite{MR1403861}]\label{dfn:closed-web}
A \emph{closed web} is a plane trivalent oriented finite graph (with possibly some vertexless loops and multiple edges) such that every vertex is either a sink or a source.
\end{dfn}
\begin{figure}[h]
\centering
\begin{tikzpicture}[yscale= 0.8, xscale= 0.8]
\input{./sw_ex_closed_web}
\end{tikzpicture}
\caption{Example of a closed web.}
\label{fig:example-closed-web}
\end{figure}
\begin{req}\label{req:basic-on-web}
The orientation condition is equivalent to say that the graph is bipartite (by sinks and sources).
\begin{prop}
A closed web contains at least a square, a digon or a vertexless circle.
\end{prop}\label{prop:closed2elliptic}
\begin{proof}
It is enough to consider $w$ a connected web. A connected web is always 2-connected (because of the flow), hence it makes sense to use the Euler characteristic. Suppose that $w$ is not a circle. We have:
\[\#F - \#E + \#V = 2, \] but we have $3\#V=2\#E$. And if we denote by $F_i$ the set of faces with $i$ sides we have:
\[\sum_{i>0}i\#i\mathcal{F}_i = 2\#E.\]
All together, this gives:
\[
\sum_{i>0}F_i - \frac{i}6 F_i =2,
\]
and this proves that some faces have strictly less than 6 sides.
\end{proof}
\end{req}
\begin{prop}[Kuperberg\cite{MR1403861}] \label{prop:Kup}
There exists one and only one map $\kup{\cdot}$ from closed webs to Laurent polynomials in $q$ which is invariant by isotopy, multiplicative with respect to disjoint union and which satisfies the following local relations~:
\begin{align*}
\kup{\websquare[0.4]} &= \kup{\webtwovert[0.4]} + \kup{\webtwohori[0.4]}, \\
\kup{\webbigon[0.4]\,} &= [2] \cdot \kup{\webvert[0.4]\,},\\
\kup{\webcircle[0.4]} &= \kup{\webcirclereverse[0.4]} = [3],
\end{align*}
where $[n]\eqdef\frac{q^n-q^{-n}}{q-q^{-1}}.$ We call this polynomial the \emph{Kuperberg bracket}. It's easy to check that the Kuperberg bracket of a web is symmetric in $q$ and $q^{-1}$.
\end{prop}
\begin{proof}
Uniqueness comes from remark \ref{req:basic-on-web}. The existence follows from the representation theoretic point of view developed in \cite{MR1403861}. Note that a non-quantified version of this result is in \cite{MR1172374}.
\end{proof}
\begin{dfn}
The \emph{degree} of a symmetric Laurent polynomial $P(q)=\sum_{i\in \ZZ}a_iq^{i}$ is $\max_{i\in \ZZ}\{i \textrm{ such that }a_i\neq 0\}$.
\end{dfn}
\begin{dfn}\label{dfn:webtangle}
A \emph{$(\epsilon_0,\epsilon_1)$-web-tangle} $w$ is an intersection of a closed web $w'$ with $[0,1]\times [0,1]$ such that :
\begin{itemize}
\item there exists $\eta_0 \in ]0,1]$ such that: \\ \(w\cap [0,1] \times [0,\eta_0] = \{\frac{1}{2l(\epsilon_0)}, \frac{1}{2l(\epsilon_0)} + \frac{1}{l(\epsilon_0)}, \frac{1}{2l(\epsilon_0)} + \frac{2}{l(\epsilon_0)}, \dots, \frac{1}{2l(\epsilon_0)} + \frac{l(\epsilon_0)-1}{l(\epsilon_0)} \}\times [0,\eta_0],\)
\item there exists $\eta_1 \in [0,1[$ such that: \\ \(w\cap [0,1] \times [\eta_1,1] = \{\frac{1}{2l(\epsilon_1)}, \frac{1}{2l(\epsilon_1)} + \frac{1}{l(\epsilon_1)}, \frac{1}{2l(\epsilon_1)} + \frac{2}{l(\epsilon_1)}, \dots, \frac{1}{2l(\epsilon_1)} + \frac{l(\epsilon_1)-1}{l(\epsilon_1)} \}\times [\eta_1,1],\)
\item the orientations of the edges of $w$, match $-\epsilon_0$ and $+\epsilon_1$ (see figure \ref{fig:exampl_webtangle} for conventions).
\end{itemize}
When $\epsilon_1$ is the empty sequence, then we'll speak of \emph{$\epsilon_0$-webs}. And if $w$ is an $\epsilon$-web we will say that $\epsilon$ is the \emph{boundary} of $w$.
\end{dfn}
If $w_1$ is a $(\epsilon_0,\epsilon_1)$-web-tangle and $w_2$ is a $(\epsilon_1, \epsilon_2)$-web-tangle we define $w_1w_2$ to be the $(\epsilon_0,\epsilon_2)$-web-tangle obtained by gluing $w_1$ and $w_2$ along $\epsilon_1$ and resizing. Note that this can be thought as a composition if we think about a $(\epsilon,\epsilon')$-web-tangle as a morphism from $\epsilon'$ to $\epsilon$ (i.~e.~ the web-tangles should be read as morphisms from top to bottom). The \emph{mirror image} of a $(\epsilon_0,\epsilon_1)$-web-tangle $w$ is mirror image of $w$ with respect to $\RR\times \{\frac12\}$ with all orientations reversed. This is a $(\epsilon_1,\epsilon_0)$-web-tangle and we denote it by $\bar{w}$. If $w$ is a $(\epsilon,\epsilon)$-web-tangle the \emph{closure} of $w$ is the closed web obtained by connecting the top and the bottom by simple arcs (this is like a braid closure). We denote it by $\mathrm{tr}(w)$.
\begin{figure}[h]
\centering
\begin{tikzpicture}[yscale= 0.5, xscale= 0.5]
\input{./sw_ex_webtangle}
\end{tikzpicture}
\caption{Two examples of $(\epsilon_0,\epsilon_1)$-web-tangles with $\epsilon_0 = (-,-,-)$ and $\epsilon_1=(-,-,+,+)$ and the mirror image of the second one.}
\label{fig:exampl_webtangle}
\end{figure}
\begin{dfn}
A web-tangle with no circle, no digon and no square is said to be \emph{non-elliptic}. The non-elliptic web-tangles are the minimal ones in the sense that they cannot be reduced by the relations of proposition \ref{prop:Kup}.
\end{dfn}
\begin{prop}[Kuperberg, \cite{MR1403861}] \label{prop:NEFinite}
For any given couple $(\epsilon_0,\epsilon_1)$ of sequences of signs there are finitely many non-elliptic $(\epsilon_0,\epsilon_1)$-web-tangles.
\end{prop}
\begin{req}
From the combinatorial flow modulo 3, we obtain that there exist some $(\epsilon_0,\epsilon_1)$-webs if and only if the sequence $-\epsilon_0$ concatenated with $\epsilon_1$ is admissible.
\end{req}
\subsubsection{Foams} All material here comes from \cite{MR2100691}.
\label{sec:foams}
\begin{dfn}
A \emph{pre-foam} is a smooth oriented compact surface $\Sigma$ (its connected component are called \emph{facets}) together with the following data~:
\begin{itemize}
\item A partition of the connected components of the boundary into cyclically ordered 3-sets and for each 3-set $(C_1,C_2,C_3)$, three orientation preserving diffeomorphisms $\phi_1:C_2\to C_3$, $\phi_2:C_3\to C_1$ and $\phi_3:C_1\to C_2$ such that $\phi_3 \circ \phi_2 \circ \phi_1 = \mathrm{id}_{C_2}$.
\item A function from the set of facets to the set of non-negative integers (this gives the number of \emph{dots} on each facet).
\end{itemize}
The \emph{CW-complex associated with a pre-foam} is the 2-dimensional CW-complex $\Sigma$ quotiented by the diffeomorphisms so that the three circles of one 3-set are identified and become just one called a \emph{singular circle}.
The \emph{degree} of a pre-foam $f$ is equal to $-2\chi(\Sigma')$ where $\chi$ is the Euler characteristic, $\Sigma'$ is the CW-complex associated with $f$ with the dots punctured out (i.~e.~ a dot increases the degree by 2).
\end{dfn}
\begin{req}
The CW-complex has two local models depending on whether we are on a singular circle or not. If a point $x$ is not on a singular circle, then it has a neighborhood diffeomorphic to a 2-dimensional disk, else it has a neighborhood diffeomorphic to a Y shape times an interval (see figure \ref{fig:yshape}).
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=1]
\input{./sw_yshape}
\end{tikzpicture}
\caption{Singularities of a pre-foam}
\label{fig:yshape}
\end{figure}
\end{req}
\begin{dfn}
A \emph{closed foam} is the image of an embedding of the CW-complex associated with a pre-foam such that the cyclic orders of the pre-foam are compatible with the left-hand rule in $\RR^3$ with respect to the orientation of the singular circles\footnote{We mean here that if, next to a singular circle, with the forefinger of the left hand we go from face 1 to face 2 to face 3 the thumb points to indicate the orientation of the singular circle (induced by orientations of facets). This is not quite canonical, physicists use more the right-hand rule, however this is the convention used in \cite{MR2100691}.}. The \emph{degree} of a closed foam is the degree of the underlying pre-foam.
\end{dfn}
\begin{dfn}\label{dfn:wwfoam}
If $w_b$ and $w_t$ are $(\epsilon_0,\epsilon_1)$-web-tangles, a \emph{$(w_b,w_t)$-foam} $f$ is the intersection of a foam $f'$ with $\RR\times [0,1]\times[0,1]$ such that
\begin{itemize}
\item there exists $\eta_0 \in ]0,1]$ such that $f\cap \RR \times [0,\eta_0]\times [0,1] = \{\frac{1}{2l(\epsilon_0)}, \frac{1}{2l(\epsilon_0)} + \frac{1}{l(\epsilon_0)}, \frac{1}{2l(\epsilon_0)} + \frac{2}{l(\epsilon_0)}, \dots, \frac{1}{2l(\epsilon_0)} + \frac{l(\epsilon_0)-1}{l(\epsilon_0)} \}\times [0,\eta_0]\times [0,1]$,
\item there exists $\eta_1 \in [0,1[$ such that $f\cap \RR \times [\eta_1,1]\times [0,1] = \{\frac{1}{2l(\epsilon_1)}, \frac{1}{2l(\epsilon_1)} + \frac{1}{l(\epsilon_1)}, \frac{1}{2l(\epsilon_1)} + \frac{2}{l(\epsilon_1)}, \dots, \frac{1}{2l(\epsilon_1)} + \frac{l(\epsilon_1)-1}{l(\epsilon_1)}\}\times [\eta_1,1]\times [0,1]$,
\item there exists $\eta_b \in ]0,1]$ such that $f\cap \RR \times [0,1 ]\times [0, \eta_b] = w_b \times [0, \eta_b]$,
\item there exists $\eta_t \in [0,1[$ such that $f\cap \RR \times [0,1 ]\times [\eta_t, 1] = w_t \times [\eta_t,1]$,
\end{itemize}
with compatibility of orientations of the facets of $f$ with the orientation of $w_t$ and the reversed orientation of $w_b$.
The \emph{degree} of a $(w_b,w_t)$-foam $f$ is equal to $\chi(w_b)+\chi(w_t)-2\chi(\Sigma)$ where $\Sigma$ is the underlying CW-complex associated with $f$ with the dots punctured out.
\end{dfn}
If $f_b$ is a $(w_b,w_m)$-foam and $f_t$ is a $(w_m, w_t)$-foam we define $f_bf_t$ to be the $(w_b,w_t)$ foam obtained by gluing $f_b$ and $f_t$ along $w_m$ and resizing. This operation may be thought as a composition if we think of a $(w_1,w_2)$-foam as a morphism from $w_2$ to $w_1$ i.~e.~ from the top to the bottom. This composition map is degree preserving. Like for the webs, we define the \emph{mirror image} of a $(w_1,w_2)$-foam $f$ to be the $(w_2,w_1)$-foam which is the mirror image of $f$ with respect to $\RR\times\RR\times \{\frac12\}$ with all orientations reversed. We denote it by $\bar{f}$.
\begin{dfn}
If $\epsilon_0=\epsilon_1=\emptyset$ and $w$ is a closed web, then a $(\emptyset,w)$-foam is simply called \emph{foam} or \emph{$w$-foam} when one wants to focus on the boundary of the foam.
\end{dfn}
All these data together lead to the definition of a monoidal 2-category.
\begin{dfn}
The 2-category $\mathcal{WT}$ is the monoidal\footnote{Here we choose a rather strict point of view and hence the monoidal structure is strict (we consider everything up to isotopy), but it is possible to define the notion in a non-strict context, and the same data gives us a monoidal bicategory.} 2-category given by the following data~:
\begin{itemize}
\item The objects are finite sequences of signs,
\item The 1-morphism from $\epsilon_1$ to $\epsilon_0$ are isotopy classes (with fixed boundary) of $(\epsilon_0,\epsilon_1)$-web-tangles,
\item The 2-morphism from $\widehat{w_t}$ to $\widehat{w_b}$ are $\mathbb Q$-linear combinations of isotopy classes of $(w_b,w_t)$-foams, where\ \ $\widehat{\cdot}$\ \ stands for the ``isotopy class of''. The 2-morphisms come with a grading, the composition respects the degree.
\end{itemize}
The monoidal structure is given by concatenation of sequences at the $0$-level, and disjoint union of vertical strands or disks (with corners) at the $1$ and $2$ levels.
\end{dfn}
\subsection{Khovanov's TQFT for web-tangles}
\label{sec:khovanov-tqft-web}
In \cite{MR2100691}, Khovanov defines a numerical invariant for pre-foams and this allows him to construct a TQFT $\mathcal{F}$ from the category $\hom_{\mathcal{WT}}(\emptyset,\emptyset)$ to the category of graded $\mathbb Q$-modules, (via a universal construction à la BHMV
\cite{MR1362791}). This TQFT is graded (this comes from the fact that pre-foams with non-zero degree are evaluated to zero), and satisfies the following local relations (brackets indicate grading shifts)~:
\begin{align*}
\mathcal{F}\left(\websquare[0.4]\,\right) &=
\mathcal{F}\left({\webtwovert[0.4]}\,\right) \oplus
\mathcal{F}\left({\webtwohori[0.4]}\,\right), \\
\mathcal{F}\left({\webbigon[0.4]}\,\right) &=
\mathcal{F}\left({\webvert[0.4]}\,\right)\{-1\}\oplus \mathcal{F}\left({\webvert[0.4]}\right)\{1\},\\
\mathcal{F}\left({\webcircle[0.4]\,}\right) &=
\mathcal{F}\left({\webcirclereverse[0.4]}\,\right) = \mathbb Q\{-2\} \oplus \mathbb Q \oplus\mathbb Q\{2\}.
\end{align*}
These relations show that $\mathcal{F}$ is a categorified counterpart of the Kuperberg bracket. We sketch the construction below.
\begin{dfn}
We denote by $\mathcal{A}$ the Frobenius algebra $\ZZ[X]/(X^3)$ with trace $\tau$ given by:
\[\tau(X^2)=-1, \quad \tau(X)=0, \quad \tau(1)=0.\]
We equip $\mathcal{A}$ with a graduation by setting $\deg(1)=-2$, $\deg(X)=0$ and $\deg(X^2)=2$. With these settings, the multiplication has degree 2 and the trace has degree -2. The co-multiplication is determined by the multiplication and the trace and we have :
\begin{align*}
&\Delta(1) = -1\otimes X^2 - X\otimes X - X^2\otimes 1 \\
&\Delta(X) = -X\otimes X^2 - X^2\otimes X \\
&\Delta(X^2) = -X^2\otimes X^2
\end{align*}
\end{dfn}
This Frobenius algebra gives us a 1+1 TQFT (this is well-known, see~\cite{MR2037238} for details), we denote it by $\mathcal{F}$~: the circle is sent to $\mathcal{A}$, a cup to the unity, a cap to the trace, and a pair of pants either to multiplication or co-multiplication. A dot on a surface will represent multiplication by $X$ so that $\mathcal{F}$ extends to the category of oriented dotted (1+1)-cobordisms.
We then have a surgery formula given by figure~\ref{fig:surg}.
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale=0.5]
\input{./sw_surg}
\end{tikzpicture}
\caption{The surgery formula for the TQFT $\mathcal{F}$.}
\label{fig:surg}
\end{figure}
This TQFT gives of course a numercial invariant for closed dotted oriented surfaces. If one defines numerical values for the differently dotted theta pre-foams (the theta pre-foam consists of 3 disks with trivial diffeomorphisms between their boundary see figure \ref{fig:thetapre}) then by applying the surgery formula, one is able to compute a numerical value for pre-foams.
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\input{./sw_thetaprefoam}
\end{tikzpicture}
\caption{The dotless theta pre-foam.}
\label{fig:thetapre}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale = 0.7]
\input{./sw_thetaeval}
\end{tikzpicture}
\caption{The evaluations of dotted theta foams, the evaluation is unchanged when one cyclically permutes the faces. All the configurations which cannot be obtained from these by cyclic permutation are sent to $0$ by $\mathcal{F}$.}
\label{fig:thetaeval}
\end{figure}
In \cite{MR2100691}, Khovanov shows that setting the evaluations of the dotted theta foams as shown on figure \ref{fig:thetaeval}, leads to a well defined numerical invariant $\mathcal{F}$ for pre-foams. This numerical invariant gives the opportunity to build a (closed web, $(\cdot,\cdot)$-foams)-TQFT~: for a given web $w$, consider the $\mathbb Q$-module generated by all the $(w,\emptyset)$-foams, and mod this space out by the kernel of the bilinear map $(f,g)\mapsto \mathcal{F}(\bar{f}g)$. Note that $\bar{f}g$ is a closed foam. Khovanov showed that the obtained graded vector spaces are finite dimensional with graded dimensions given by the Kuperberg formulae, and he showed that we have the local relations described on figure~\ref{fig:localrel}.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.25]
\input{./sw_bubblesrel}
\end{tikzpicture}
\vspace{0.7cm}
\begin{tikzpicture}[scale=0.24]
\input{./sw_bamboorel}
\end{tikzpicture}
\vspace{0.7cm}
\begin{tikzpicture}[scale=0.4]
\input{./sw_digonrel}
\end{tikzpicture}
\vspace{0.7cm}
\begin{tikzpicture}[scale=0.32]
\input{./sw_squarerel}
\end{tikzpicture}
\vspace{0.7cm}
\begin{tikzpicture}[scale=0.5]
\input{./sw_dotsmigrel}
\end{tikzpicture}
\caption{Local relations for 2-morphism in $\mathbb{WT}$. The first 3 lines are called bubbles relations, the 2 next are called bamboo relations, the one after digon relation, then we have the square relation and the 3 last ones are the dots migration relations.}
\label{fig:localrel}
\end{figure}
This method allows us to define a new graded 2-category $\mathbb{WT}$. Its objects and its 1-morphisms are the ones of the 2-category $\mathcal{WT}$ while its 2-morphisms-spaces are the ones of $\mathcal{WT}$ mod out like in the last paragraph. One should notice that a $(w_b,w_t)$-foam can always be deformed into a $(\mathrm{tr}(\bar{w_b}w_t),\emptyset)$-foam and vice-versa. Khovanov's results restated in this language give that if $w_b$ and $w_t$ are $(\epsilon_0,\epsilon_1)$-web-tangles, the graded dimension of $\hom_{\mathbb{WT}}(w_t,w_b)$ is given by $\kup{\mathrm{tr}(\bar{w_b}w_t)}\cdot q^{l(\epsilon_0)+l(\epsilon_1)}$. Note that when $\epsilon_1=\emptyset$, there is no need to take the closure, because $w_b\bar{w_t}$ is already a closed web. The shift by $l(\epsilon_0)+l(\epsilon_1)$ comes from the fact that $\chi(\mathrm{tr}(\bar{w_b}w_t)) = \chi(w_t)+\chi(w_b) - (l(\epsilon_0)+l(\epsilon_1))$.
\begin{prop}\label{prop:relFR}
We consider the set \FR{} of local relations which consists of:
\begin{itemize}
\item the surgery relation,
\item the evaluations of the dotted spheres and of the dotted theta-foams,
\item the square relations and the digon relations (see figure~\ref{fig:localrel}).
\end{itemize}
We call them the \emph{foam relations} or relations \FR, then for any closed web $w$ $\mathcal{F}(w)$ is isomorphic to $\mathcal{G}(w)$ modded out by \FR.
\end{prop}
\subsection{The Kuperberg-Khovanov algebra $K^\epsilon$}
\label{sec:algebra-k_s}
We want to extend the Khovanov TQFT to the 0-dimensional objects i.~e.~ to build a 2-functor from the 2-category $\mathcal{WT}$ to the 2-category of algebras. We follow the methodology of \cite{MR1928174} and we start by defining the image of the $0$-objects~: they are the algebras $K^\epsilon$. This can be compared with \cite{2012arXiv1206.2118M}.
\begin{dfn}
Let $\epsilon$ be an admissible finite sequence of signs. We define $\tilde{K}^\epsilon$ to be the full sub-category of $\hom_{\mathbb{WT}}(\emptyset,\epsilon)$ whose objects are non-elliptic $\epsilon$-webs. This is a graded $\mathbb Q$-algebroid. We recall that a $k$-algebroid is a $k$-linear category. This can be seen as an algebra by setting~:
\[K^\epsilon = \bigoplus_{(w_b,w_t)\in (\mathrm{ob}(\tilde{K}^\epsilon))^2} \hom_{\mathbb{WT}}(w_b,w_t)\]
and the multiplication on $K^\epsilon$ is given by the composition of morphisms in $\tilde{K}_\epsilon$ whenever it's possible and by zero when it's not. We will denote $\ensuremath{\otimes}[_{w_1}]{K}{^{\epsilon}_{w_2}}\eqdef \hom_{\mathbb{WT}}(w_2,w_1)$. This is a unitary algebra because of proposition \ref{prop:NEFinite}. The unite element is $\sum_{w\in \mathrm{ob}(\tilde{K}_\epsilon)} 1_w$. Suppose $\epsilon$ is fixed, for $w$ a non-elliptic $\epsilon$-web, we define $P_w$ to be the left $K^\epsilon$-module~:
\[
P_w=\bigoplus_{w'\in\mathrm{ob}(\tilde{K}_\epsilon)}\hom_{\mathbb{WT}}(w,w') = \bigoplus_{w'\in\mathrm{ob}(\tilde{K}_\epsilon)} \ensuremath{\otimes}[_{w'}]{K}{^\epsilon_{w}}.
\]The structure of module is given by composition on the left.
\end{dfn}
For a given $\epsilon$, the modules $P_w$ are all projective and we have the following decomposition in the category of left $K^\epsilon$-modules~:
\[
K^\epsilon=\bigoplus_{w\in \mathrm{ob}(\tilde{K}_\epsilon)} P_w.
\]
\begin{prop}
Let $\epsilon$ be an admissible sequence of signs, and $w_1$ and $w_2$ two non-elliptic $\epsilon$-webs, then the graded dimension of $\hom_{K^\epsilon}(P_{w_1}, P_{w_2})$ is given by $\kup{(\bar{w_1}w_2)}\cdot q^{l(\epsilon)}$.
\end{prop}
\begin{proof}
An element of $\hom_{K^\epsilon}(P_{w_1}, P_{w_2})$ is completely determined by the image of $1_{w_1}$ and this element can be sent on any element of $\hom_{\mathbb{WT}}(P_{w_2}, P_{w_1})$, and $\mathop{\mathrm{dim}}\nolimits_q(\hom_{\mathbb{WT}}(P_{w_1}, P_{w_2}))=\kup{(\bar{w_1}w_2)}\cdot q^{l(\epsilon)}$.
\end{proof}
In what follows we will prove that some modules are indecomposable, they all have finite dimension over $\mathbb Q$ hence it's enough to show that their rings of endomorphisms contain no non-trivial idempotents. It appears that an idempotent must have degree zero, so we have the following lemma~:
\begin{lem}\label{lem:monic2indec}
If $w$ is a non-elliptic $\epsilon$-web such that $\kup{\bar{w}w}$ is monic of degree $l(\epsilon)$, then the graded $K^\epsilon$-module $P_w$ is indecomposable.
\end{lem}
\begin{proof}
This follows from previous discussion~: if $\hom_{K^\epsilon}(P_w,P_w)$ contained a-non trivial idempotent, there would be at least two linearly independent elements of degree 0, but $\mathop{\mathrm{dim}}\nolimits((\hom_{\mathbb{WT}}(P_{w}, P_{w})_0) = a_{-l(\epsilon)}$ if we write $\kup{\bar{w}w}=\sum_{i\in \ZZ}a_iq^i$ but as $\kup{\bar{w}w}$ is symmetric (in $q$ and $q^{-1}$) of degree $l(\epsilon)$ and monic, $a_{-l(\epsilon)}$ is equal to 1 and this is a contradiction.
\end{proof}
We have a similar lemma to prove that two modules are not isomorphic.
\begin{lem}\label{lem:0monic2noiso}
If $w_1$ and $w_2$ are two non-elliptic $\epsilon$-webs such that $\kup{\bar{w_1}w_2}$ has degree strictly smaller than $l(\epsilon)$, then the graded $K^\epsilon$-modules $P_{w_1}$ and $P_{w_2}$ are not isomorphic.
\end{lem}
\begin{proof}
If they were isomorphic, there would exist two morphisms $f$ and $g$ such that $f\circ g=1_{P_{w_1}}$ and therefore $f\circ g$ would have degree zero. The hypothesis made implies that $f$ and $g$ (because $\kup{\bar{w_1}w_2} = \kup{\bar{w_2}w_1}$) have positive degree so that the degree of their composition is as well positive.
\end{proof}
\begin{req}
The way we constructed the algebra $K^\epsilon$ is very similar to the construction of $H^n$ in \cite{MR1928174}. Using the same method we can finish the construction of a $0+1+1$ TQFT~:
\begin{itemize}
\item If $\epsilon$ is an admissible sequence of signs, then $\mathcal{F}(\epsilon) = K^\epsilon$.
\item If $w$ is a $(\epsilon_1,\epsilon_2)$-web-tangle with $\epsilon_1$ and $\epsilon_2$ admissible, then
\[\mathcal{F}(w) = \bigoplus_{\substack{u\in \mathrm{ob}(\tilde{K}^{\epsilon_1}) \\ v\in \mathrm{ob}(\tilde{K}^{\epsilon_2})}} \mathcal{F}(\bar{u}wv),
\] and it has a structure of graded $K^{\epsilon_1}$-module-$K^{\epsilon_2}$. Note that if $w$ is a non-elliptic $\epsilon$-web, then $\mathcal{F}(w)=P_w$.
\item If $w$ and $w'$ are two $(\epsilon_1,\epsilon_2)$-web-tangles, and $f$ is a $(w,w')$-foam, then we set
\[
\mathcal{F}(f) = \sum_{\substack{u\in \mathrm{ob}(\tilde{K}^{\epsilon_1}) \\ v\in \mathrm{ob}(\tilde{K}^{\epsilon_2})}} \mathcal{F}(\ensuremath{\otimes}[_{\bar{u}}]{f}{_v}),
\] where $\ensuremath{\otimes}[_{\bar{u}}]{f}{_v}$ is the foam $f$ with glued on its sides $\bar{u}\times[0,1]$ and $v\times [0,1]$. This is a map of graded $K^{\epsilon_1}$-modules-$K^{\epsilon_2}$.
\end{itemize}
We encourage the reader to have a look at this beautiful construction for the $sl_2$ case in \cite{MR1928174}.
\end{req}
In the $\ensuremath{\mathfrak{sl}}_2$ case the classification of projective indecomposable modules is fairly easy, and a analogous result, would state in our context that the projective indecomposable modules are exactly the modules associated with non-elliptic webs. However we have:
\begin{prop}[\cite{MR2457839}, see\cite{LHRThese} for details]\label{prop:Pwdec}
Let $\epsilon$ be the sequence of signs: $(+,-,-,+,+,-,-,+,+,-,-,+)$, and let $w$ and $w_0$ be the two $\epsilon$-webs given by figure~\ref{fig:thewebw}. Then the web-module $P_w$ is decomposable and admits $P_{w_0}$ as a direct factor.
\end{prop}
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale= 0.7]
\input{./sw_thewebw}
\end{tikzpicture}
\caption{The $\epsilon$-webs $w$ (on the left) and $w_0$ (on the right), to fit in formal context of the 2-category one should stretch the outside edges to horizontal line below the whole picture, we draw it this way to enjoy more symmetry. To simplify we didn't draw the arrows.}
\label{fig:thewebw}
\end{figure}
\input{rgchapter4}
\bibliographystyle{alpha}
\chapter[Characterisation of indecomposable web-modules]{A characterisation of indecomposable web-modules}
\label{cha:red-graphs}
In \cite{LHR1}, one gives a sufficient condition for a web-module to be indecomposable. All the argumentation relies on the computation of the dimension of the space of the degree 0 endomorphisms of web-modules: in fact, when for a web $w$, this space has dimension 1, then the web-module $P_w$ is indecomposable. Translated in terms of Kuperberg bracket, it says (see as well lemma~\ref{lem:monic2indec}):
\begin{quotation}
\noindent\emph{If $w$ is an $\epsilon$-web such that $\kup{\overline{w}w}$ is monic of degree $l(\epsilon)$, then the $K^\epsilon$-module $P_w$ is indecomposable.}
\end{quotation}
The aim of this chapter is to prove the converse. This will give the following characterisation of indecomposable web-modules:
\begin{thm}
Let $w$ be an $\epsilon$-web. The $K^\epsilon$-module $P_w$ is indecomposable if and only if $\kup{\overline{w}w}$ is monic of degree $l(\epsilon)$. Furthermore if the $K^\epsilon$-module $P_w$ is decomposable it contains another web-module as a direct factor.
\end{thm}
The proof relies on some combinatorial tools called red graphs. In a first part we give an explicit construction (in terms of foams) of a non-trivial idempotent associated to a red graph. In a second part we show that when an $\epsilon$-web $w$ is such that $\kup{\overline{w}w}$ is not monic of degree $l(\epsilon)$, then it contains a red graph.
\section{Red graphs}
\label{sec:redgraph}
\subsection{Definitions}
\label{sec:defRG}
The red graphs are sub-graphs of the dual graphs webs, we recall here the definition of a dual graph. For an introduction to graph theory we refer to~\cite{MR0256911} and \cite{MR2368647}.\marginpar{trouver une bonne référence pour la theorie des graphe topologique}
\begin{dfn}
Let $G$ be a plane graph (with possibly some vertex-less loops), we define \emph{the dual graph $D(G)$ of $G$} to be the abstract graph given as follows:
\begin{itemize}
\item The set of vertices $V(D(G))$ of $D(G)$ is in one-one correspondence with the set of connected components of $\RR^2\setminus G$ (including the unbounded connected component). Such connected component are called \emph{faces}.
\item The set of edges of $D(G)$ is in one-one correspondence with the set of edges of $G$ (in this construction, vertex-less loops are not seen as edges). If an edge $e$ of $G$ is adjacent to the faces $f$ and $g$ (note that $f$ may be equal to $g$ if $e$ is a bridge), then the corresponding edge $e'$ in $D(G)$ joins $f'$ and $g'$, the vertices of $D(G)$ corresponding to $f$ and $g$.
\end{itemize}
\end{dfn}
Note that in general the faces need not to be diffeomorphic to disks. It is easy to see that the dual graph of a plane graph is planar: we place one vertex inside each face, and we draw an edge $e'$ corresponding to $e$ so that it crosses $e$ exactly once and it crosses no other edges of $G$. Such an embedding of $D(G)$ is a \emph{plane dual} of the graph $G$ (see figure~\ref{fig:exdualgrpah}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\input{./th_exdualgraph}
\end{tikzpicture}
\caption{In black an $\epsilon$-web $w$ and in blue the dual graph of $w$. The dotted edges are all meant to belong to $D(w)$ and to reach the vertex $u$ corresponding to the unbounded component of $\RR^2\setminus w$.}
\label{fig:exdualgrpah}
\end{figure}
\begin{dfn}\label{dfn:red-graph}
Let $w$ be an $\epsilon$-web, a \emph{red graph} for $w$ is a non-empty subgraph $G$ of $D(w)$ such that:
\begin{enumerate}[(i)]
\item\label{item:dfnRG1} All faces belonging to $V(G)$ are diffeomorphic to disks. In particular, the unbounded face is not in $V(G)$.
\item\label{item:dfnRG2} If $f_1$, $f_2$ and $f_3$ are three faces of $w$ which share together a vertex, then at least one of the three does not belong to $V(G)$.
\item\label{item:dfnRG3} If $f_1$ and $f_2$ belongs to $V(G)$ then every edge of $D(w)$ between $f_1$ and $f_2$ belongs to $E(G)$, i.~e.~ $G$ is an induced subgraph of $D(w)$.
\end{enumerate}
If $f$ is a vertex of $G$ we define $\ensuremath{\mathrm{ed}}(f)$, \emph{the external degree of $f$}, by the formula: \[ \ensuremath{\mathrm{ed}}(f) = \deg_{D(w)}(f) - 2\deg_G(f).\]
\end{dfn}
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1]
\input{./th_exRG}
\end{tikzpicture}
\caption{Example of a red graph.}
\label{fig:exRG}
\end{figure}
\begin{req}\label{rk:evenextdeg}
Note that the external degree of a face $f$ is always an even number because $w$ being bipartite, all cycles are of even length and hence $\deg_{D(w)}$ is even.
\end{req}
Let $G$ be a red graph for $w$, then if on the web we colour the faces which belongs to $V(G)$, then the external degree of a face $f$ in $V(G)$ is the number of half-edges of $w$ which touch the face $f$ and lie in the uncoloured region. These half-edges are called the \emph{grey half-edges} of $f$ in $G$ or of $G$ when we consider the set of all grey half-edges of all vertices of $G$. See figure~\ref{fig:halfgrey}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=0.6]
\input{./th_halfgrayedge}
\end{tikzpicture}
\caption{Interpetation of the external degree in terms of grey half-edges. On the left, a portion of a web $w$ with a red graph; on the right, the same portion of $w$ with the vertices of $G$ orange-coloured. The external degree of $f$ is the number of half edges touching $f$ which are not orange. In our case $\ensuremath{\mathrm{ed}}(f)=2$.}
\label{fig:halfgrey}
\end{figure}
An oriented red graph is a red graph together with an orientation, \emph{a priori} there is no restriction on the orientations, but as we shall see just a few of them will be relevant to consider.
\begin{dfn}\label{dfn:index-red-graph}
Let $w$ be an $\epsilon$-web, $G$ be a red graph for $w$ and $o$ an orientation for $G$, we define the level $i_o(f)$ (or $i(f)$ when this is not ambiguous) of a vertex $f$ of $G$ by the formula:
\begin{align*}
i_o(f) &\eqdef 2 - \frac{1}{2}\ensuremath{\mathrm{ed}}(f) - \#\{\textrm{edges of $G$ pointing to $f$}\} \\
&= 2- \frac{\deg_{D(w)}}2 + \#\{\textrm{edges of $G$ pointing away from $f$}\}
\end{align*}
and the level $I(G)$ of $G$ is the sum of levels of all vertices of $G$.
\end{dfn}
\begin{req}\label{req:formindex}
The level is an integer because of remark~\ref{rk:evenextdeg}.
Note that the level of $G$ does not depend on the orientation of $G$ and we have the formula:\[ I(G) = 2\#V(G) - \#E(G) - \frac{1}{2}\sum _{f\in v(G)}\ensuremath{\mathrm{ed}}(f).\]
\end{req}
\begin{dfn}\label{dfn:RG-admissible}
A red graph is \emph{admissible} if one can choose an orientation such that for each vertex $f$ of $G$ we have: $i(f)\geq 0$. Such an orientation is called \emph{a fitting orientation}. An admissible red graph $G$ for $w$ is \emph{exact} if $I(G)=0$.
\marginpar{For a web $w$ we define $M(w) = \max I(G)$ where the maximum is taken over all admissible red graph $G$ for $w$.???}
\end{dfn}
\begin{dfn}\label{dfn:paired-RG}\marginpar{should we put the orientation ?}
Let $w$ be an $\epsilon$-web and $G$ be a red graph for $w$. A \emph{pairing} of $G$ is a partition of the grey half-edges of $G$ into subsets of 2 elements such that for any subset the two half-edges touch the same face $f$, and one points to $f$ and the other one points away from $f$. A red graph together with a pairing is called a \emph{paired red graph.}
\end{dfn}
\begin{dfn}
A red graph $G$ in an $\epsilon$-web $w$ is \emph{fair} (resp.~{} \emph{nice}) if for every vertex $f$ of $G$ we have $\ensuremath{\mathrm{ed}}(f)\leq 4$ (resp.~{} $\ensuremath{\mathrm{ed}}(f)\leq 2$).
\end{dfn}
\begin{lem}\label{lem:RGadm2fair}
If $G$ is an admissible red graph in an $\epsilon$-web $w$, then $G$ is fair.
\end{lem}
\begin{proof}
It follows directly from the definition of the level.
\end{proof}
\begin{cor}\label{cor:RGinNEhaveVs}
Let $w$ be a non-elliptic $\epsilon$ web, then if $G$ is an admissible red graph for $w$ then it has at least two edges.
\end{cor}
\begin{proof}
If G would contain just one vertex $f$, this would have external degree greater or equal to 6, contradicting lemma~\ref{lem:RGadm2fair}. We can actually show that such a red graph contains at least 6 vertices (see corollary~\ref{cor:no-tree} and proposition~\ref{prop:largecycle}).
\end{proof}
\begin{req}\label{req:number-of-pairing}
If a red graph $G$ is nice, there is only one possible pairing. If it is fair the number of pairing is $2^n$ where $n$ denote the number of vertices with external degree equal to 4.
If on a picture one draws together a web $w$ and a red graph $G$ for $w$, one can encode a pairing of $G$ on the picture by joining\footnote{We impose that $w$ intersect the dashed lines only at their ends.} with dashed line the paired half-edges. Note that if $G$ is fair it's always possible to draw disjoint dashed lines (see figure~\ref{fig:pairings} for an example).
\end{req}
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=0.7]
\input{./th_pairing}
\end{tikzpicture}
\caption{A web $w$, a red graph $G$ and the two possible pairings for $G$.}
\label{fig:pairings}
\end{figure}
The rest of the chapter (respectively in section~\ref{sec:RG2idem} and \ref{sec:kup2RG}) will be devoted to show the following two theorems:
\begin{thm}\label{thm:RG2idempotent}
To every exact paired red graph of $w$ we can associate a non trivial idempotent of $Hom_{K^\epsilon}(P_w,P_w)$. Further more the direct factor associated with the idempotent is a web-module.
\end{thm}
\begin{thm}\label{thm:on-monic-2-RG}
Let $w$ be a non-elliptic $\epsilon$-web, then if $\kup{\overline{w}w}$ is non-monic or have degree bigger than $l(\epsilon)$, then there exists an exact red graph for $w$, therefore the $K^\epsilon$-module $P_w$ is decomposable.
\end{thm}
\subsection{Combinatorics on red graphs}
\label{sec:comb-red-graphs}
On the one hand, the admissibility of a red graph relies on the local non-negativity of the level for some orientation, on the other hand the global level $I$ does not depend on the orientation. However, it turns out that the existence of admissible red graph $G$ for an $\epsilon$-web $w$ can be understood thanks to $I$ in some sense:
\begin{prop}\label{prop:2admissible}
Let $w$ be an $\epsilon$-web, suppose that there exists $G$ a red graph for $w$ such that $I(G)\geq 0$, then there exists an admissible red graph $\widetilde{G}$ for $w$ such that $I(\widetilde{G})\geq I(G)$.\marginpar{mettre non-empty dans la defnition de red graph}
\end{prop}
\begin{proof} If $G$ is already admissible, there is nothing to show, hence we suppose that $G$ is not admissible. Among all the orientations for $G$, we choose one such that $\sum_{f\in V(G)} |i(f)|$ is minimal, we denote is by $o$. From now on $G$ is endowed with this orientation. As $G$ is not admissible there exists some vertices with negative level and some with positive level.
We first show that there is no oriented path from a vertex $f_p$ with $i_o(f_p)> 0$ to a vertex $f_n$ with $i_o(f_n)<0$. Suppose there exists $\gamma$ such a path. Let us inspect $o'$ the orientation which is the same as $o$ expect along the path $\gamma$ where it is reversed. For all vertices $f$ of $G$ but $f_p$ and $f_n$, we have $i_o(f)=i_{o'}(f)$ (for all the vertices inside the path, the position of the edges pointing to them is changed, but not their number), and we have:
\[
i_{o'}(f_p) = i_o(f_p)-1 \quad i_{o'}(f_n) = i_o(f_n)+1.
\] But then $\sum_{f\in V(G)} |i_{o'}(f)|$ would be strictly smaller than $\sum_{f\in V(G)} |i_o(f)|$ and this contradicts that $o$ is minimal.
We consider $(\widetilde{G}, \tilde{o})$ the induced oriented sub-graph of $(G,o)$ with set of vertices $V(\widetilde{G})$ equal to the vertices of $G$ which can be reach from a vertex with positive level by an oriented path. This set is not empty since it contains the vertices with positive degree. It contains no vertex with negative degree. For all vertices of $\widetilde{G}$, we have:
\begin{align*} i_{\tilde{o}}(f) &= 2 - \frac{\deg_{D(w)}(f)}2 + \#\{\textrm{edges of $G$ pointing away from $f$ in $\widetilde{G}$}\} \\ &=
2 - \frac{\deg_{D(w)}(f)}2 + \#\{\textrm{edges of $G$ pointing away from $f$ in $G$}\} \\
&= i_o(f).\end{align*}
The second equality holds because if $f$ is in $V(\widetilde{G})$ all the edges in $E(G)\setminus E(G')$. $G$ which are not in $\widetilde{G}$ point to $f$ by definition of $\widetilde{G}$. This shows that $\widetilde{G}$ is admissible and $I(\widetilde{G})>I(G)$.
\end{proof}
\begin{lem}\label{lem:RGinNE2niceRG}
Let $w$ be a non-elliptic web, suppose that it contains a red graph of level $k$, then it contains an admissible nice red graph of level at least $k$.
\end{lem}
\begin{proof}
We consider $G$ a red graph of $w$ of level $k$. Thanks to lemma \ref{prop:2admissible} we can suppose that it is admissible. We can take a minimal red graph $G$ for the property of being of level at least $k$ and admissible. The graph $G$ is endowed with a fitting orientation. Now suppose that it is not nice, it means that there exists a vertex $v$ of $G$ which have exterior degree equal to 4. But $G$ being admissible all the edges of $G$ adjacent to $v$ point out of $v$, so that we can remove $v$ i.~e.~ we can consider then induced sub-graph $G'$ with all the vertex of $G$ but $v$ with the induced orientation. Then it is admissible, with the same level, hence $G$ is not minimal, contradiction.
\end{proof}
For a non-elliptic $\epsilon$-web, the existence of an exact red graph may appear as an exceptional situation between the case where there is no admissible red graph and the case where all admissible red graphs are non-exact. The aim of the rest of this section is to show the proposition~\ref{prop:exist-exact} which indicates that this is not the case. On the way we state some small results which are not directly useful for the proof but may alight what red-graphs look like.
\begin{prop}\label{prop:exist-exact}
Let $w$ be a non-elliptic $\epsilon$-web. If there exists an admissible red graph for $w$ then there exists an exact red graph for $w$.
\end{prop}
\begin{dfn}\label{dfn:subred}
Let $w$ be an $\epsilon$-web, and $G$ and $G'$ two admissible red graphs for $w$. We say that $G'$ is a \emph{red sub-graph} of $G$ if $V(G')\subset V(G')$. We denote by $\ensuremath{\mathcal{G}}(G)$ the set of all admissible red sub-graphs. It is endowed with the order given by the inclusion of sets of vertices. We say that $G$ is \emph{minimal} if $\ensuremath{\mathcal{G}}(G) = \{G\}$.
\end{dfn}
Note that a red sub-graph is an induced sub-graph and that a minimal red-graph is connected.
\begin{lem}\label{lem:no_cut}
Let $w$ be an $\epsilon$-web and $G$ a minimal admissible red graph endowed with a fitting orientation. There is no non-trivial partition of $V(G)$ into two sets $V_1$ and $V_2$ such that for each vertex $v_1$ in $V_1$ and each vertex $v_2$ in $V_2$ every edge between $v_1$ and $v_2$ is oriented from $v_1$ to $v_2$.
\end{lem}
\begin{proof}
If there were a such a partition, we could consider the red sub-graph $G'$ with $V(G') =V_2$. For every vertex in $V_2$ the level is the same in $G$ and in $G'$ and hence, $G'$ would be admissible and $G$ would not be minimal.
\end{proof}
\begin{cor}
\label{cor:noleaf4minimal} Let $w$ be a $\epsilon$-web and $G$ a minimal admissible red graph for $w$, then the graph $G$ has no leaf\footnote{We mean vertex of degree 1.}. Therefore if it has 2 or more vertices, then it is not a tree.
\end{cor}
\begin{proof}
Indeed, if $v$ were a leaf of $G$, the vertex $v$ would be either a sink or a source, hence $V(G)\setminus\{v\}$ and $\{v\}$ would partitioned $V(G)$ in a way forbidden by lemma~\ref{lem:no_cut}.
\end{proof}
\begin{cor}\label{cor:no-tree}
If $G$ is an admissible red graph for a non-elliptic $\epsilon$-web $w$, then $G$ is not a tree.
\end{cor}
\begin{proof}
Consider a minimal red sub-graph of $G$. Thanks to corollaries~\ref{cor:RGinNEhaveVs} and \ref{cor:noleaf4minimal}, it is not a tree, hence $G$ is not a tree.
\end{proof}
\begin{lem}
Let $w$ be an $\epsilon$-web and $G$ a minimal red graph for $w$. If $G$ has more than 2 vertices, then it is nice.
\end{lem}
\begin{proof}
Suppose that we have a vertex $v$ of $G$ with external degree equal to 4. Consider a fitting orientation for $G$. All edges of $G$ adjacent to $v$ must point out, otherwise the degree of $v$ would be negative. So $v$ would be a sink and, thanks to lemma~\ref{lem:no_cut}, this is not possible.
\end{proof}
\begin{lem}\label{lem:strong-connected}
Let $w$ be a non-elliptic $\epsilon$-web and $G$ a minimal admissible red graph. If the red graph $G$ is endowed with a fitting orientation, then it is strongly connected.
\end{lem}
The terms \emph{weakly connected} and \emph{strongly connected} are classical in graph theory the first means that the underlying unoriented graph is connected in the usual sense. The second that for any pair of vertices $v_1$ and $v_2$, there exists an oriented path from $v_1$ to $v_2$ and an oriented path from $v_2$ to $v_1$.
\begin{proof}
Let $v$ be a vertex of $G$, consider the subset $V_v$ of $V(G)$ which contains the vertices of $G$ reachable from $v$ by an oriented path. The sets $V_v$ and $V(G)\setminus V_v$ form a partition of $V(G)$ which must be trivial because of lemma~\ref{lem:no_cut}, but $v$ is in $V_v$ therefore $V_v=V(G)$, this is true for any vertex $v$, and this shows that $G$ is strongly connected.
\end{proof}
\begin{prop}\label{prop:largecycle}
If $G$ is a red graph for a non-elliptic $\epsilon$-web $w$, then any (not-oriented) simple cycle has at least 6 vertices.
\end{prop}
\begin{proof}Take $C$ a non-trivial simple cycle in $G$. \marginpar{We can suppose that $C$ is minimal. A red graph contains no triangle, this shows that some faces of $w$ must be nested in the cycle $C$.} We consider the collection of faces of $w$ nested by $C$ (this is non empty thanks to condition~(\ref{item:dfnRG3}) of the definition of red graphs). This defines a plane graph $H$. We define $H'$ to be the graph $H$ with the bivalent vertices smoothed (we mean here that if locally $H$ looks like $\vcenter{\hbox{\tikz{\draw (0,0)-- (1,0); \filldraw (0.5,0) circle (1pt);}}}$, then $H'$ looks like $\vcenter{\hbox{\tikz{\draw (0,0) -- (1,0);}}}$). An example of this construction is depicted on figure~\ref{fig:exlargecycle}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=0.95]
\input{./rg_exlargecycle}
\end{tikzpicture}
\caption{On the left the $\epsilon$-web $w$ and the red graph $G$, in the middle the graph $H$, and on the right, the graph $H'$.}
\label{fig:exlargecycle}
\end{figure}
The $\epsilon$-web $w$ being non-elliptic, each face of $H$ has at least $6$ sides. We compute the Euler characteristic of $H'$:
\[\chi(H')= \#F(H')-\#E(H')+\#V(H') =2. \]
As in proposition~\ref{prop:closed2elliptic}, this gives us $\sum_{i\in \NN} F_i(H')(1-\frac i6) =2$ where $F_i(H)$ is the number of faces of $H'$ with $i$ sides. Restricting the sum to $i\leq 5$ and considering $F'_i$ the number of bounded faces, we have:
\[\sum_{i=0}^5 F'_i(H')(6 - i) \geq 6.\]
But the bounded faces of $H'$ with less that 6 sides come from bounded faces of $H$ which have at least 6 sides. The number $n$ of bivalent vertices in $H$ is therefore greater than or equal to $\sum_{i=0}^5 F'_i(H')(6 - i)$ i.~e.~ greater than or equal to 6. But $n$ is as well the length of the cycle $C$. \marginpar{faire un exemple ?}
\end{proof}
Note that a cycle in a red graph can have an odd length (as in the example of figure~\ref{fig:exlargecycle}).
\begin{lem}
Let $G$ be a minimal admissible red graph for a non-elliptic $\epsilon$-web $w$. Then $G$ has at least one vertex with degree 2.
\end{lem}
\begin{proof}
Suppose that all vertices of $G$ have degree greater or equal to 3, then the graph $G$ would contain a face with less than 5 sides (this is the same argument than in proposition~\ref{prop:closed2elliptic} which tells that a closed web contains a circle, a digon or a square). But this contradicts lemma~\ref{prop:largecycle}. \marginpar{terminology: oriented graph or digraph ?}
\end{proof}
The proposition~\ref{prop:exist-exact} is a direct consequence of the following lemma:
\begin{lem}\label{lem:minimal2exact}
Let $w$ be a non-elliptic $\epsilon$-web and $G$ a minimal admissible red graph for $w$. Then $G$ is exact.
\end{lem}
\begin{proof}
We endow $G$ with $o$ a fitting orientation. Suppose $G$ is not exact, then we can find a vertex $f$ with $i_o(f)>0$.
We first consider the case where $\deg(f)=2$. The $\epsilon$-web $w$ being non-elliptic, $\ensuremath{\mathrm{ed}}(f)\geq 2$. This shows that the two edges adjacent to $f$ point away from $f$, hence, $f$ is a sink and this contradicts lemma~\ref{lem:no_cut}.
Now, let us consider the general case. Let $f'$ be a vertex with degree 2. The lemma \ref{lem:strong-connected} implies that there exists $\gamma$ an oriented path from $f$ to $f'$. Let us reverse the orientations of the edges of $\gamma$. We denote by $o'$ this new orientation. Then we have $i_{o'}(f)=i_{o}(f)-1\geq 0$ and $i_{o'}(f')=i_G(f')+1\geq 1$. The levels of all other edges are not changed, hence $o'$ is a fitting orientation, and we are back in the first situation (where $f'$ plays the role of $f$).
\end{proof}
\section{Idempotents from red graphs}
\label{sec:RG2idem}
\input{rgchapter4bis}
\subsection{On the identity foam}
\label{sec:identity-foam}
\begin{dfn}
Let $w$ be an $\epsilon$-web, and $f$ a $(w,w)$-foam, we say that $f$ is \emph{reduced} if every facet of $f$ is diffeomorphic to a disk and if $f$ contains no singular circle (i.~e.~ only singular arcs). In particular this implies that every facet of $f$ meets $w\times \{0\}$ or $w\times\{1\}$.
\end{dfn}
The aim of this section is to prove the following proposition:
\begin{prop}\label{prop:idfoam}
Let $w$ be a non-elliptic $\epsilon$-web. If $f$ is a reduced $(w,w)$-foam which is equivalent (under the foam relations \FR{}) to a non-zero multiple of $w\times [0,1]$, then the underlying pre-foam is diffeomorphic to $w\times [0,1]$ and contains no dot.
\end{prop}
For this purpose we begin with a few technical lemmas:
\begin{lem} \label{lem:foamwithdots}
Let $w$ be a closed web and $e$ an edge of $w$. Then there exists $f$ a $(\emptyset, w)$-foam which is not equivalent to 0 such that the facet of $f$ touching the edge $e$ contains at least one dot.
\end{lem}
\begin{proof}
We prove the lemma by induction on the number of edges of the web $w$.
It is enough to consider the case $w$ connected because the functor $\mathcal{F}$ is monoidal. If the web $w$ is a circle this is clear, since a cap with one dot on it is not equivalent to 0. If $w$ is the theta web, then this is clear as well, since the half theta foam with one dot on the facet meeting $e$ is not equivalent to 0.
Else, there exists a square or digon in $w$ somewhere far from $e$. Let us denote $w'$ the web similar to $w$ but with the digon replaced by a single strand or the square smoothed in one way or the other. By induction we can find an $(\emptyset, w')$-foam $f'$ non-equivalent to 0 with one dot on the facet touching $e$.
Next to the strand or the smoothed square, we consider a digon move or a square move (move upside down the pictures of figure~\ref{fig:smdmc}). Seen as a $(w',w)$-foam it induces an injective map. Therefore, the composition of $f'$ with this $(w',w)$-foam is not equivalent to 0 and has one dot on the facet touching $e$.
\end{proof}
\begin{notation}
Let $w$ be an $\epsilon$-web, and $e$ be an edge of $w$. We denote by $f(w,e)$ the $(\emptyset,\overline{w}w)$-foam which is diffeomorphic to $w\times [0,1]$ with one dot on the facet $e\times [0,1]$. We denote by $f(w,\emptyset)$ the $(\emptyset,\overline{w} w)$-foam which is diffeomorphic to $w\times [0,1]$ with no dot on it.
\end{notation}
\begin{cor}
Let $w$ be an $\epsilon$-web, and $e$ an edge of $w$, then $f(w,e)$ is non-equivalent to 0.
\end{cor}
\begin{proof}
From lemma~\ref{lem:foamwithdots} we know that for any $w$, there exists a $(w,w)$-foam which is non equivalent to 0 and is the product of $f(w,e)$ with another $(w,w)$-foam. This proves that the $(w,w)$-foam $f(w,e)$ is not equivalent to 0.
\end{proof}
\begin{dfn}
If $w$ is an $\epsilon$-web. We say that it contains a $\lambda$ (resp.~a $\cap$, resp.~a $H$) if next to the border $w$ looks like one of the pictures of figure~\ref{fig:dfnUHY}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\input{./sw_dfnUHY}[scale=0.6]
\end{tikzpicture}
\caption{From left to right: a $\lambda$, a $\cap$ and a $H$.}
\label{fig:dfnUHY}
\end{figure}
\end{dfn}
\begin{lem}\label{lem:UHYinNE}
Every non-elliptic $\epsilon$-web contains at least a $\lambda$, a $\cap$ or an $H$.
\end{lem}
\begin{proof}
The closed web $\bar{w}w$ contains a circle a digon or a square, and this happens only if $w$ contains a $\cap$ a $\lambda$ or a $H$.
\end{proof}
\begin{req}
In fact, one can ``build'' every non-elliptic web with this three elementary webs. This is done via the ``growth algorithm'' (see~\cite{MR1684195}).
\end{req}
\begin{lem}\label{lem:NEwebwithdots}
Let $w$ be a non-elliptic $\epsilon$-web. Then the elements of $\left( f(w,e)\right)_{e\in E(w)}$ are pairwise non-equivalent (but they may be linearly dependant).
\end{lem}
\begin{proof}[Sketch of the proof]
We proceed by induction on the number of edges of $w$. The initiation is straightforward since if $w$ has only one edge there is nothing to prove. We can distinguish several case thanks to lemma~\ref{lem:UHYinNE}:
If $w$ contains a $\cap$, we denote by $e$ the edge of this $\cap$, and by $w'$ the $\epsilon'$-web similar to $w$ but with the cap removed. Suppose that $e_1=e$, then $e_2\neq e$ and, then the $(\emptyset,\overline{w}w)$-foams $f(w,e_1)$ and $f(w,e_2)$ are different because if we cap the cup (we mean $e\times I$) by a cap with one dot on it, on the one hand we obtain a $(\emptyset,\overline{w'}w')$-foam equivalent to $0$ and on the other hand a $(\emptyset,\overline{w'}w')$-foam equivalent to $f(w',\emptyset)$. Thanks to lemma~\ref{lem:foamwithdots},
we know that this last $(\emptyset,\overline{w'}w')$-foam is not equivalent to 0. If $e_1$ and $e_2$ are different from $e$, it is clear as well, because $f(w, e_1)$ and $f(w,e_2)$ can be seen as compositions of $f(w', e_1)$ and $f(w',e_2)$ with a birth (seen as a $(\overline{w'}w', \overline{w}w)$-foam) which is known to correspond to injective map.
This is the same kind of argument for the two other cases. The digon relations and the square relations instead of the sphere relations.
\end{proof}
\begin{lem}\label{lem:red1touch2wI}
Let $w$ be an $\epsilon$-web and $f$ a reduced $(w,w)$-foam $f$. Suppose that every facet touches $w\times\{0\}$ on at most one edge, and touches $w\times\{1\}$ on at most one edge, then it is isotopic to $w\times [0,1]$.
\end{lem}
\begin{proof}
The proof is inductive on the number of vertices of $w$. If $w$ is a collection of arcs, the foam $f$ has no singular arc. As $f$ is supposed to be reduced, it has no singular circle. Therefore it is a collection of disks which corresponds to the arcs of $w$, and this proves the result in this case.
We suppose now that $w$ has at least one vertex. Let us pick a vertex $v$ which is a neighbour (via an edge that we call $e$) of the boundary $\epsilon$ of $w$. We claim that the singular arc $\alpha$ starting at $v\times\{0\}$ must end at $v\times\{1\}$.
Indeed, the arc $\alpha$ cannot end on $w\times\{0\}$, for otherwise, the facet $f$ touching $e$ would touch another edge of $w$. Therefore the arc $\alpha$ ends on $w\times\{1\}$.
For exactly the same reasons, it has to end on $v\times\{1\}$, so that the facet which touches $e\times\{0\}$ is isotopic to $e\times I$, now we can remove a neighbourhood of this facet and we are back in the same situation with a $\epsilon'$-web with less vertices, and this concludes.
\end{proof}
\begin{proof}[Proof of proposition~\ref{prop:idfoam}]
We consider $w$ a non-elliptic $\epsilon$-web. Let $f$ be a reduced $(w,w)$-foam such that $f$ is equivalent to $w\times I$ up to a non-trivial scalar. Because of lemma~\ref{lem:NEwebwithdots}, the foam $f$ satisfies the hypotheses of lemma~\ref{lem:red1touch2wI}, so that $f$ is isotopic to $w\times [0,1]$.
\end{proof}
We conjecture that the proposition~\ref{prop:idfoam} still holds without the non-ellipticity hypothesis. However the proof has to be changed since lemma \ref{lem:NEwebwithdots} cannot be extend to elliptic webs (consider the facets around a digon).
\begin{cor}
If $w$ is a non-elliptic $\epsilon$-web and $w'$ is an $\epsilon$-web with strictly less vertices than $w$, then if $f$ is a $(w,w')$-foam and $g$ is a $(w',w)$-foam, then the $(w,w)$-foam $fg$ cannot be equal to a scalar times the identity.
\end{cor}
\begin{landscape}
\begin{figure}
\begin{tikzpicture} [scale = 0.7]
\begin{scope}
\input{./th_exdecofmodules}
\end{scope}
\node at (2.9, 0) {$\simeq \, P \, \oplus$};
\begin{scope}[xshift = 5.8cm]
\input{./th_exdecofmodules1}
\end{scope}
\begin{scope}[xshift = 11.9cm]
\input{./th_exdecofmodules1}
\end{scope}
\node at (8.7, 0) {$\{-1\}\,\,\,\,\oplus$};
\node at (14.5,0) {$\{+1\}$};
\begin{scope}[yshift =-4.5cm, xshift = -2cm]
\node at (2, 0) {$\oplus$};
\begin{scope}[xshift = 4.4cm]
\input{./th_exdecofmodules2}
\end{scope}
\node at (6.6, 0) {$\oplus$};
\begin{scope}[xshift = 8.8cm, yscale= -1]
\input{./th_exdecofmodules2}
\end{scope}
\node at (11, 0) {$\oplus$};
\begin{scope}[xshift = 13.2cm, yscale= -1]
\input{./th_exdecofmodules3}
\end{scope}
\node at (15.4, 0) {$\oplus$};
\begin{scope}[xshift = 17.6cm, yscale= 1]
\input{./th_exdecofmodules4}
\end{scope}
\node at (19.8, 0) {$\oplus$};
\begin{scope}[xshift = 22.0cm, yscale= -1]
\input{./th_exdecofmodules4}
\end{scope}
\node at (24.2, 0) {$\oplus$};
\begin{scope}[xshift = 26.4cm, yscale= 1]
\input{./th_exdecofmodules5}
\end{scope}
\end{scope}
\begin{scope}[yshift =-9cm, xshift = -2cm]
\node at (2, 0) {$\oplus$};
\begin{scope}[xshift = 4.4cm, xscale = -1]
\input{./th_exdecofmodules2}
\end{scope}
\node at (6.6, 0) {$\oplus$};
\begin{scope}[xshift = 8.8cm, yscale= -1, xscale = -1]
\input{./th_exdecofmodules2}
\end{scope}
\node at (11, 0) {$\oplus$};
\begin{scope}[xshift = 13.2cm, yscale= -1, xscale = -1]
\input{./th_exdecofmodules3}
\end{scope}
\node at (15.4, 0) {$\oplus$};
\begin{scope}[xshift = 17.6cm, yscale= 1, xscale = -1]
\input{./th_exdecofmodules4}
\end{scope}
\node at (19.8, 0) {$\oplus$};
\begin{scope}[xshift = 22.0cm, yscale= -1, xscale = -1]
\input{./th_exdecofmodules4}
\end{scope}
\node at (24.2, 0) {$\oplus$};
\begin{scope}[xshift = 26.4cm, yscale= 1, xscale = -1]
\input{./th_exdecofmodules5}
\end{scope}
\end{scope}
\end{tikzpicture}
\hspace{1cm}
\caption{Example of a decomposition of a web-module into indecomposable modules. All direct factors which are web-modules are obtained through idempotents associated with red graphs. The module $P$ is not a web-module but is a projective indecomposable module.}
\end{figure}
\end{landscape}
\section{Characterisation of indecomposable web-modules}
\label{sec:kup2RG}
\subsection{General View}
\label{sec:general-view}
The lemma~\ref{lem:monic2indec} states that the indecomposability of a web-modules $P_w$ can be deduced from the Laurent polynomial $\kup{\overline{w}w}$. In this section we will show a reciprocal statement. We first need a definition:
\begin{dfn}
Let $\epsilon$ be an admissible sequence of signs of length $n$, an $\epsilon$-web $w$ is said to be \emph{virtually indecomposable} if $\kup{\overline{w}w}$ is a monic symmetric Laurent polynomial of degree $n$.
An $\epsilon$-web which is not virtually indecomposable is \emph{virtually decomposable}. If $w$ is a virtually decomposable $\epsilon$-web, we define the \emph{level of $w$} to be the integer $\frac12 (\deg \kup{\overline{w}w} -n)$.
\end{dfn}
Despite of its fractional definition, the level is an integer.
With this definition, lemma~\ref{lem:monic2indec} can be rewritten:
\begin{lem}
\label{lem:VI2indec}
If $w$ is a virtually indecomposable $\epsilon$-web, then $M(w)$ is an indecomposable $K^\epsilon$-module.
\end{lem}
The purpose in this section is to prove a reciprocal statement in order to have:
\begin{thm}\label{thm:Charac}
Let $\epsilon$ be an admissible sequence of signs of length $n$, and $w$ an $\epsilon$-web. Then the $K^\epsilon$-module $P_w$ is indecomposable if and only if $w$ is virtually indecomposable.
\end{thm}
\begin{req}
Note that we do not suppose that $w$ is non-elliptic, but as a matter of fact, if $w$ is elliptic then $\kup{\overline{w}w}$ is not monic of degree $n$ and the module $P_w$ is decomposable.
\end{req}
To prove the unknown direction of theorem~\ref{thm:Charac} we use red graphs developed in the previous section and will show a more precise version of the theorem:
\begin{thm}\label{thm:thmwithRG}
If $w$ is a non-elliptic virtually decomposable $\epsilon$-web of level $k$, then $w$ contains an admissible red graph of level $k$, hence $\mathrm{End}_{K^\epsilon}(P_w)$ contains a non-trivial idempotent and $P_w$ is decomposable.
\end{thm}
\begin{proof}[Proof of theorem \ref{thm:Charac} assuming theorem \ref{thm:thmwithRG}] Let $w$ be a virtually decomposable $\epsilon$-web and let us denote by $k$ its level. From theorem~\ref{thm:thmwithRG} we know that there exists a red graph $G''$ of level $k$. But then, thanks to proposition~\ref{prop:2admissible}, there exists $G'$ a sub red graph of $G''$ which is admissible. And finally, the proposition \ref{prop:exist-exact} shows the existence of an exact red graph $G$ in $w$. We can apply theorem \ref{thm:RG2idempotent} to $G$ and this tells that $P_w$ is decomposable.
\end{proof}
The proof of theorem \ref{thm:thmwithRG} is a recursion on the number of edges of the web $w$. But for the recursion to work, we need to handle elliptic webs as well. We will actually show the following:
\begin{prop}\label{prop:techRG}
\begin{enumerate}
\item\label{it:ptRG1} If $w$ is a $\partial$-connected $\epsilon$-web which is virtually decomposable of level $k\geq 1$ then there exists $S$ a stack of nice red graphs for $w$ of level greater or equal to $k$ such that $w_S$ is $\partial$-connected.
\item\label{it:ptRG2} If $w$ is a $\partial$-connected $\epsilon$-web which is virtually decomposable of level $k\geq 1$, contains no digon and contains exactly one square which is supposed to be adjacent the unbounded face then there exists a nice red graph $G$ in $w$ of level greater or equal to $k$ such that $w_G$ is $\partial$-connected.
\item\label{it:ptRG3} If $w$ is a non-elliptic $\epsilon$-web which is virtually decomposable of level $k\geq 0$ then there exists a nice red graph $G$ in $w$ of level greater of equal to $k$ such that $w_G$ is $\partial$-connected.
\end{enumerate}
\end{prop}
Before proving the proposition we need to introduce \emph{stacks of red graphs} (see below), and the notion of $\partial$-connectedness (see section \ref{sec:part-conn}).
Then we will prove the proposition~\ref{prop:techRG} thanks to a technical lemma (lemma \ref{lem:tech}) which will be proven in section~\ref{sec:proof-lemmatech} after an alternative glance on red graphs (section~\ref{sec:new-approach-to-red-graph}).
\begin{req}
It is easy to see that a non-elliptic superficial $\epsilon$-web contains no red graphs of non-negative level, hence this result is strictly stronger than the result of~\cite{LHR1}.
\end{req}
\begin{dfn}
Let $w$ be an $\epsilon$-web, \emph{a stack of red graphs $S=(G_1, G_2,\dots, G_l)$ for $w$} is a finite sequence of paired red graphs such that $G_1$ is a red graph of $w_1\eqdef w$, $G_2$ is a red graph of $w_2\eqdef w_{G_1}$, $G_3$ is a red graph of $w_3\eqdef (w_{G_1})_{G_2} = (w_2)_{G_2}$ etc. We denote $(\cdots((w_{G_1})_{G_2})\cdots)_{G_l}$ by $w_S$ and we denote $l$ by $l(S)$ and we say that it is \emph{the length of $S$}. We define the level of a stack to be the sum of the levels of the red graphs of the stack.\marginpar{change level into level and $k$-irregular into $k$ -virtually decomposable}
\end{dfn}
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\input{./th_stack}
\end{tikzpicture}
\caption{A stack of red graphs of length 2.}
\label{fig:stack}
\end{figure}
\begin{dfn}
A stack of red graphs is \emph{nice} if all its red graphs are nice. Note that in this case the pairing condition on red graphs is empty.
\end{dfn}
\subsection{The $\partial$-connectedness}
\label{sec:part-conn}
\begin{dfn}
An $\epsilon$-web is \emph{$\partial$-connected} if every connected component of $w$ touches the border.
\end{dfn}
A direct consequence is that a $\partial$-connected $\epsilon$-web contains no circle.
\begin{lem}
A non-elliptic $\epsilon$-web is $\partial$-connected.
\end{lem}
\begin{proof}
An $\epsilon$-web which is not $\partial$-connected has a closed connected component, this connected component contains at least a circle, a digon or a square and hence is elliptic.
\end{proof}
\begin{lem}
Let $w$ be a $\partial$-connected $\epsilon$-web with a digon, the web $\epsilon$-web $w'$ equal to $w$ except that the digon reduced (see figure~\ref{fig:reddig}) is still $\partial$-connected. In other words $\partial$-connectedness is preserved by digon-reduction.
\end{lem}
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\input{./th_reddig}
\end{tikzpicture}
\caption{On the left $w$, on the right $w'$.}
\label{fig:reddig}
\end{figure}
\begin{proof}
This is clear because every path in $w$ can be projected onto a path in $w'$.
\end{proof}
Note that $\partial$-connectedness is not preserved by square reduction, see for example figure~\ref{fig:sq2partial}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.2]
\input{./th_sq2partial}
\end{tikzpicture}
\caption{The $\partial$-connectedness is not preserved by square reduction.}
\label{fig:sq2partial}
\end{figure}
However we have the following lemma:
\begin{lem} If $w$ is a $\partial$-connected $\epsilon$-web which contains a square $S$ then one of the two $\epsilon$-webs obtained from $w$ by a reduction of $S$ (see figure~\ref{fig:tworeduction}) is $\partial$-connected.
\end{lem}
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[scale=0.7]
\input{./th_tworeduction}
\end{tikzpicture}
\caption{On the right the $\epsilon$-web $w$ with the square $S$, on the middle and on the right, the two reductions of the square $S$.}
\label{fig:tworeduction}
\end{figure}
\begin{proof} Consider the oriented graph $\tilde{w}$ obtained from $w$ by removing the square $S$ and the 4 half-edges adjacent to it (see figure~\ref{fig:sqremoved}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale= 0.7]
\input{./th_square-removed}
\end{tikzpicture}
\caption{On the left $w$, on the right $\tilde{w}$.}
\label{fig:sqremoved}
\end{figure}
We obtain a graph with 4 less cubic vertices than $w$ and 4 more vertices of degree 1 than $w$. We call $E_S$ the cyclically ordered set of the 4 vertices of $\tilde{w}$ of degree 1 next to the removed square $S$. The orientations of the vertices in $E_S$ are $(+,-,+,-)$. Note that in $\tilde{w}$, the flow modulo 3 is preserved everywhere, so that the sum of orientation of vertices of degree 1 of any connected component must be equal to 0 modulo 3. Suppose now that there is a connected component $t$ of $\tilde{w}$ which has all its vertices of degree 1 in $E_S$, the flow condition implies that either all vertices of $E_S$ are vertices of $t$ or exactly two consecutive vertices of $E_S$ are vertices of $t$, or that $t$ has no vertex of degree 1. The first situation cannot happen because by adding the square to $t$ we would construct a free connected component of $w$ which is supposed to be $\partial$-connected, the last situation neither for the same reason. So the only thing that can happen is the second situation. If there were two different connected components $t_1$ and $t_2$ of $\tilde{w}$ such that $t_1$ and $t_2$ have all their vertices of degree 1 in $E_S$, then adding the square to $t_1\cup t_2$ would lead to a free connected component of $w$, so their is at most one connected component of $\tilde{w}$ with all this vertex of degree 1 in $E_S$ call this vertices $e_+$ and $e_-$, and call $e'_+$ and $e'_-$ the two other vertices of $E_S$ (the indices gives the orientation). If we choose $w'$ to be the $\epsilon$-web corresponding to the smoothing which connects $e_+$ with $e'_-$ and $e_-$ with $e'_+$, then $w'$ is $\partial$-connected.
\end{proof}
\begin{dfn}
Let $w$ be a $\partial$-connected $\epsilon$-web and $S$ a square in $w$. The square $S$ is \emph{a $\partial$-square} if the two $\epsilon$-webs $w_{=}$ and $w_{||}$ obtained from $w$ by the two reductions by the square $S$ are $\partial$-connected.
\end{dfn}
\begin{lem}\label{lem:pc2ps}
If $w$ is a $\partial$-connected web, then either it is non-elliptic, or it contains either a digon or a $\partial$-square.
\end{lem}
\begin{proof}
Suppose that $w$ is not non-elliptic. As $w$ is $\partial$-connected it contains no circle. If must contains at least a digon or a square, if it contains a digon we are done, so suppose $w$ contains no digon. We should show that at least one square is a $\partial$-square. Suppose that there is no $\partial$-square, it means that for every square $S$, there is a reduction such that the $\epsilon$-web resulting $w_{s(S)}$ obtained by replacing $w$ by the reduction has a free connected component $t_S$. Let us consider a square $S_0$ such that $t_{S_0}$ is as small as possible (in terms of number of vertices for example). The web $t_{S_0}$ is closed and connected, so that either it is a circle, or it contains a digon or at least two square. If $t_{S_0}$ is a circle then $w$ contains a digon just next to the square $S_0$, and we excluded this case (see figure \ref{fig:circle2digon}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale= 0.7]
\input{./th_circle2digon}
\end{tikzpicture}
\caption{on the left $w_{S_0}$, on the right $w$. If $t_{S_0}$ is a circle, then $w$ contains a digon.}
\label{fig:circle2digon}
\end{figure}
If it contains a digon, the digon must be next to where $S_0$ was smoothed else the digon would already be in $w$. It appears hence that the digon comes from a square $S_1$ in $w$ ($S_1$ is adjacent to $S_0$), and $t_{S_1}$ has two vertices less than $T_{S_0}$ which is excluded (see figure~\ref{fig:digon2square}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=0.7]
\input{./th_digon2square}
\end{tikzpicture}
\caption{On the left $w_{S_0}$, on the right $w$. If $t_{S_0}$ contains a digon then $w$ contains a square adjacent to $S_0$.}
\label{fig:digon2square}
\end{figure}
The closed web $t_{S_0}$ contains at least two squares so that we can pick up one, we denote it by $S'$, which is far from $S_0$ and hence comes from a square in $w$. Now at least one of the two smoothings of the square $S'$ must disconnect $t_{s_0}$ else the square $S'$ would be a $\partial$-square in $w$. But as it disconnects $t_{S_{0}}$, $t_{S'}$ is a strict sub graph of $t_{S_0}$, and this contradict the minimality of $S_0$. And this concludes that $w$ must contain a $\partial$-square.
\end{proof}
\subsection{Proof of proposition~\ref{prop:techRG}}
\label{sec:proof-propkey}
In this section we prove the proposition~\ref{prop:techRG} admitting the following technical lemma:
\begin{lem}\label{lem:tech}
Let $w$ be a $\partial$-connected $\epsilon$-web which contains, no digon and one square which touches the unbounded face. Let $G$ be a nice red graphs of $w$ and $G'$ a nice red graph of $w_{G}$ such that $w_G$ and $w_{G'}$ are $\partial$-connected, then there exists $G''$ a red graph of $w$ such that $(w_{G})_{G'} = w_{G''}$ and the level of $G''$ is bigger or equal to the level of $G$ plus the level of $G'$.
\end{lem}
This lemma says that under certain condition one can ``flatten'' two red graphs.
\begin{proof}[Proof of proposition~\ref{prop:techRG}]
As we announced this will be done by recursion on the number of edges of $w$.
We supposed than \ref{it:ptRG1}, \ref{it:ptRG2} and \ref{it:ptRG3} hold for all $\epsilon$-webs with strictly less than $n$ edges, and we
consider an $\epsilon$-web with $n$ edges. Note that whenever $w$ is non-elliptic the statement \ref{it:ptRG3} is stronger than the statement \ref{it:ptRG1}, so that we won't prove \ref{it:ptRG1} in this case.
We first prove \ref{it:ptRG1}:
If $w$ contains a digon, then we apply the result on $w'$ the $\epsilon$-web similar to $w$ but with the digon reduced (i.~e.~ replaced by a single strand).
The red graph $G$ which consist of only one edge (the digon) and no edge is nice and has level equal to 1 (see figure~\ref{fig:rganddig}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale = 0.7]
\input{./sw_fig8}
\end{tikzpicture}
\caption{On the left $w'$, on the right $w$ with the red graph $G$.}
\label{fig:rganddig}
\end{figure}
If $w'$ is not virtually decomposable or virtually decomposable of level 0, then $w$ is virtually decomposable of level 1. In this case, the stack with only one red graph equal to $G$ is convenient and we are done.
Else we know that $w'$ is of level $k-1$ and that there exists a nice stack of red graphs $S'$ of level $k-1$ in $w'$ and we consider the stack $S$ equal to the concatenation of $G$ with $S'$, it is a nice stack of red graphs of level $k$ and we are done.
Suppose now that the $\epsilon$-web $w$ contains no digon, but a square, then it contains a $\partial$-square (see lemma \ref{lem:pc2ps}). Suppose that the level of $w$ is $k\geq 1$ (else there is nothing to show), then at least one of the two reductions is virtually decomposable of level $k$ (this is a Cauchy-Schwartz inequality see~\cite[Section 1.1]{LHRThese} for details). Then we consider $w'$ the $\epsilon$-web obtained by a reduction of the square so that it is of level $k$. From the induction hypothesis we know that there exists a stack of red graphs $S'$ in $w'$ of level $k$. If all the red graphs of $S'$ are far from the location of the square, then we can transform the stack $S'$ into a stack of $w$ with the same level. Else, we consider $G'$ the first red graph of $S'$ which is close from the square location and according to the situation we define $G$ by the moves given on figure~\ref{fig:S2Sp}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=0.6]
\begin{scope}[xshift = 0cm, yshift= 8cm]
\input{./sw_fig1}
\end{scope}
\begin{scope}[xshift = 0cm, yshift= 4cm]
\input{./sw_fig2}
\end{scope}
\begin{scope}[xshift = 6cm, yshift= 0cm]
\input{./sw_fig3}
\end{scope}
\begin{scope}[xshift = 12cm, yshift= 8cm]
\input{./sw_fig4}
\end{scope}
\begin{scope}[xshift = 12cm, yshift= 4cm]
\input{./sw_fig5}
\end{scope}
\end{tikzpicture}
\caption{Transformations of $G'$ to obtain $G$.}
\label{fig:S2Sp}
\end{figure}
Replacing $G'$ by $G$ we can transform, the stack $S'$ into a stack for the $\epsilon$-web $w$.
We now prove \ref{it:ptRG2}.
From what we just did, we know that $w$ contains a nice stack of red graphs of level $k$. Among all the nice stacks of red graphs of $w$ with level greater or equal to $k$, we choose one with a minimal length, we call it $S$. If its length were greater or equal to $2$, then lemma \ref{lem:tech} would tell us that we could take the first two red graphs and replace them by just one red graph with a level bigger or equal to the sum of their two levels, so that $S$ would not be minimal, this prove that $S$ has length $1$, therefore, $w$ contains a nice red graph of level at least $k$.
We now prove \ref{it:ptRG3}.
The border of $w$ contains at least a $\cap$, a $\lambda$, or an $H$ (see figure~\ref{fig:dfnUHY}). In the two first cases, we can consider $w'$ the $\epsilon$-web with the $\cap$ removed or the $\lambda$ replaced by a single strand, then $w'$ is non-elliptic and virtually decomposable of level $k$ and there exists a nice red graph in $w'$ of level at least $k$, this red graph can be seen as a red graph of $w$, and we are done. If the border of $w$ contains no $\lambda$ and no $\cap$, then it must contains an $H$. There are two ways to reduce the $H$ (see figure~\ref{fig:Hreduced}). At least one of the two following situation happens:
$w_{||}$ is virtually decomposable of level $k$ or $w_{\_}$ is virtually decomposable of level $k+1$.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\input{./sw_Hreduced}
\end{tikzpicture}
\caption{The $H$ of $w$ (on the left) and its two reductions: $w_{||}$ (on the middle) and $w_=$ (on the right).}
\label{fig:Hreduced}
\end{figure}
In the first situation, one can do the same reasoning as before:
$w_{||}$ being non-elliptic, the induction hypothesis gives that we can find a nice red graph of level at least $k$ in $w_{||}$, this red graph can be seen as a red graph of $w$ and we are done.
In the second situation, we consider $w_{\_}$, we can apply the induction hypothesis to $w_{\_}$ (we are either in case \ref{it:ptRG2} or in case \ref{it:ptRG3}), so we can find a nice red graph of level at least $k+1$, coming back to $H$ this gives us a red graph of level at least $k$ (but maybe not nice), and we can conclude via the lemma~\ref{lem:RGinNE2niceRG}.
\end{proof}
\input{rgchapter4ter}
\subsection{Foam diagrams}
\label{sec:foam-diagrams}
\begin{dfn}
Let $w$ be an $\epsilon$-web, a \emph{foam diagram} $\kappa$ for $w$ consists of the following data:
\begin{itemize}
\item the $\epsilon$-web $w$,
\item a fair paired red graph $G$,
\item a function $\delta$ (called a \emph{dot function for $w$}) from $E(w)$ the set of edges of $w$ to $\NN$ the set of non-negative integers. This function will be represented by the appropriate number of dots on each edge of $w$.
\end{itemize}
With a foam diagram $\kappa$ we associate $f(\kappa)$ the $(w_G,w_G)$-foam given by $p_G\circ s_w(\delta) \circ i_G$, where $s_w(\delta)$ is $\ensuremath{\mathrm{id}}_w=w\times [0,1]$ the identity foam of $w$ with on every facet $e\times [0,1]$ (with $e\in E(w)$) exactly $\delta(e)$ dots. The $(w_G,w_G)$-foam $f(\kappa)$ is equal to $p_G\circ i_G$, with dots encoded by $\delta$. A foam diagram will be represented by the $\epsilon$-web drawn together with the red graph, and with some dots added on the edges of the $\epsilon$-web in order to encode $\delta$.
\end{dfn}
We will often assimilate $\kappa=(w,G,\delta)$ with $f(\kappa)$ and it will be seen as an element of $\hom_{K^\epsilon}(P_{w_G},P_{w_G})$. We can rewrite
some of the relations depicted on figure~\ref{fig:localrel} in terms of foam diagrams:
\begin{prop}\label{prop:relfoamdiag} The following relations on foams associated with foam diagrams hold:
\begin{itemize}
\item The 3-dots relation:
\[
\begin{tikzpicture}
\input{./fd_3dots}
\end{tikzpicture}
\]
\item The sphere relations:
\[
\begin{tikzpicture}
\input{./fd_spheres}
\end{tikzpicture}
\]\marginpar{dashed instead of dotted}
\item The digon relations:
\[
\begin{tikzpicture}
\input{./fd_digon}
\end{tikzpicture}
\]
\[
\begin{tikzpicture}
\input{./fd_digon2}
\end{tikzpicture}
\]
\item The square relations:
\[
\begin{tikzpicture}
\input{./fd_square}
\end{tikzpicture}
\]
\item The E-relation:
\[
\begin{tikzpicture}
\input{./fd_Erel}
\end{tikzpicture}
\]
\end{itemize}
The dashed lines indicate the pairing, and when the orientation of the $\epsilon$-web is not depicted the relation holds for any orientation.
\end{prop}
\begin{proof}
This is equivalent to some of the relations depicted on figure~\ref{fig:localrel}.
\end{proof}
\begin{lem}\label{lem:fd2idwdots}
Let $w$ be an $\epsilon$-web and $\kappa = (w,G,\delta)$ a foam diagram, with $G$ a fair paired red graph. Then $f(\kappa)$ is equivalent to a $\ZZ$-linear combination of $s_{w_G}(\delta_i)= f((w_G,\emptyset, \delta_i))$ for $\delta_i$ some dots functions for $w_G$.
\end{lem}\marginpar{Signaler qu'on travail avec la TQFT pour $R= \mathbb Q$}
\begin{proof}
Thanks to the E-relation of proposition~\ref{prop:relfoamdiag}, one can express $f(\kappa)$ as a $\ZZ$-linear combination of $f((w_j,G_j,\delta_j))$ where the $G_j$'s are red graphs without any edge. Tanks to the sphere, the digon and square relations of proposition~\ref{prop:relfoamdiag}, each $f((w_j,G_j,\delta_j))$ is equivalent either to 0 or to $\pm f(w_G,\emptyset, \delta'_j)$. This proves the lemma.
\end{proof}
\begin{lem}\label{lem:fde2id}
Let $w$ be an $\epsilon$-web and $\kappa = (w,G,\delta)$ a foam diagram, with $G$ exact, then $f(\kappa)$ is equivalent to a multiple of $w_G\times [0,1]$.
\end{lem}
\begin{proof}
From the previous lemma we know that $f(\kappa)$ is equivalent to a $\ZZ$-linear combination of $w_G\times [0,1]$ with some dots on it. We will see that the foam $f(\kappa)$ has the same degree as the foam $w_G\times [0,1]$. This will prove the lemma because adding a dot on a foam increases its degree by 2.
To compute the degree of $f(\kappa)$ we see it as a composition of elementary foams thanks to its definition:
\begin{align*}
\deg f(\kappa) =& \deg(w\times [0,1]) + 2\left( 2\#V(G) - \left(\#E(G)+ \frac{\#\{\textrm{grey half-edges of $G$}\}}{2} \right) \right) \\
=& |\partial w| + 2\cdot 0 \\
=& \deg w_G \times [0,1].
\end{align*}
The first equality is due to the decomposition pointed out in remark ~\ref{req:ssandcap} and because an unzip (or a zip) has degree -1 and a cap (or a cup) has degree +2. The factor 2 is due to the fact $f(\kappa)$ is the composition of $i_G$ and $p_G$. The second one follows from the exactness of $G$.
\end{proof}
To prove the proposition~\ref{prop:pciisid}, we just need to show that in the situation of the last lemma, the multiple is not equal to zero. In order to evaluate this multiple, we extend foam diagrams to (partially) oriented paired red graphs by the local relation indicated on figure~\ref{fig:fd2oRG}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\input{./fd_orRG}
\end{tikzpicture}
\caption{Extension of foam diagrams to oriented red graphs. }
\label{fig:fd2oRG}
\end{figure}
By ``partially oriented'' we mean that some edges may be oriented some may not. If $G$ is partially oriented, and $\kappa$ is a foam diagram with red graph $G$, we say that $\kappa'$ is the \emph{classical foam diagram} associated with $\kappa$ if it obtained from $\kappa$ by applying the relation of figure~\ref{fig:fd2oRG} on every oriented edges. Note that $\kappa$ and $\kappa'$ represent the same foam.
\begin{dfn}
If $w$ is an $\epsilon$-web, $G$ a red graph for $w$ and $o$ a partial orientation of $G$ we define $\gamma(o)$ to be equal to ${\#\{\textrm{negative edges of $G$}\}}$. A \emph{negative} (or \emph{positive}) edge is an oriented edge of the red graph, and it's negativity (or positivity) is given by figure~\ref{fig:nepe}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\input{./fd_posnegedge}
\end{tikzpicture}
\caption{On the left, a positive edge. On the right, a negative edge.}
\label{fig:nepe}
\end{figure}
\end{dfn}
\begin{lem}\label{lem:sumoforientation}
Let $w$ be an $\epsilon$-web and $G$ a partially oriented red graph with $e$ a non-oriented edge of $G$, then we have the following equality of foams:
\[ \begin{tikzpicture}
\input{./fd_orientedge}
\end{tikzpicture} \]
If $G$ is an un-oriented red graph for $w$ and $\delta$ a dots function for $w$, then:
\[f(w,G,\delta) =\sum_{o} (-1)^{\gamma(o)}f(w, G_o, \delta),\]
where $G_o$ stands for $G$ endowed with the orientation $o$, and $o$ runs through all the $2^{\#E(G)}$ complete orientations of $G$.
\end{lem}
\begin{proof}
The first equality is the translation of the E-relation (see proposition~\ref{prop:relfoamdiag}) in terms of foam diagrams of partially oriented red graphs. The second formula is the expansion of the first one to all edges of $G$.
\end{proof}
\begin{lem}\label{lem:nonfitting20}
If $w$ is an $\epsilon$-web, $G$ an exact paired red graph for $w$, $o$ a non-fitting orientation for $G$ and $\delta$ the null dot function on $w$, then the $(w_g,w_G)$-foam $f(w, G_o, \delta)$ is equivalent to 0.
\end{lem}
\begin{proof}
The orientation $o$ is a non-fitting orientation. Hence, there is at least one vertex $v$ of $G$ so that $i_o(v)>0$. There are two different situations, either $i_o(v)=1$ or $i_o(v)=2$. Using the definition of a foam diagrams for oriented red graphs (figure~\ref{fig:fd2oRG}), we deduce that $\kappa'$ the classical foam diagram associated with $f(w, G_o, \delta)$ looks around $v$ like one of the three following situations:
\[
\begin{tikzpicture}
\input{./fd_0lowdegree}
\end{tikzpicture}
\]
The sphere relations and the digon relations provided by proposition~\ref{prop:relfoamdiag} we see that the foam $f(w, G_o, \delta)$ is equivalent 0.
\end{proof}
\begin{lem}\label{lem:fitting2id}
If $w$ is an $\epsilon$-web, $G$ an exact paired red graph for $w$, $o$ a fitting orientation for $G$ and $\delta$ the null dots function on $w$, then the $(w_G,w_G)$-foam $f(w, G_o, \delta)$ is equivalent to $(-1)^{\mu(o)} w_G\times I$, where $\mu(o)=\#V(G) + \#\{\textrm{positive digons of $G_o$}\}$ (see definition figure~\ref{fig:5localsituation}).
\end{lem}
\begin{proof}
Let $\kappa'=(w',G',\delta')$ be the classical foam diagram associated with $(w, G_o,\delta)$.
The red graph $G'$ has no edge. Locally, the foam diagram $\kappa'$ corresponds to one of the 5 situation depicted on figure~\ref{fig:5localsituation}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\input{./fd_5situations}
\end{tikzpicture}
\caption{The 5 different local situations of a foam diagram $\kappa'$ next to a vertex of $G'$. On the second line, the digon on the left is \emph{positive} and the digon on the right is \emph{negative}.}
\label{fig:5localsituation}
\end{figure}
But now using some relations of proposition~\ref{prop:relfoamdiag} we can remove all the vertices of $G'$, we see that $f(w,G_o, \delta)$ is equivalent to $(-1)^{\#V(G')-\#\{\textrm{positive digons}\}}$ because the positive digon is the only one with no minus sign in the relations of prop~\ref{prop:relfoamdiag}. This proves the result because $V(G)=V(G')$.
\end{proof}
\begin{lem}
If $w$ is an $\epsilon$-web, $G$ an exact paired red graph for $w$ and $o_1$ and $o_2$ two fitting orientations for $G$, then $\mu(o_1) + \gamma(o_1) = \mu(o_2) + \gamma(o_2) $. \marginpar{ toujours nul ?}
\end{lem}
\begin{proof}
We consider $\kappa'_1=(w',G',\delta_1)$ and $\kappa'_2(w',G',\delta_2)$ the two classical foam diagrams corresponding to $(w,G_{o_1},\delta)$ and $(w,G_{o_2},\delta)$, with $\delta$ the null dots function for $w$.
The red graph $G'$ has no edge, and the local situation are depicted on figure~\ref{fig:5localsituation}. Consider $v$ a vertex of $G'$, then a side of the face of $w$ corresponding to $v$ is either clockwise or counterclockwise oriented (with respect to this face). From the definition of $\gamma$ we obtain that for $i=1,2$, $\gamma(o_i)$ is equal to the number of dots in $\kappa'_i$ on clockwise oriented edges in $w'$. The dots functions $\delta_1$ and $\delta_2$ differs only next to the digons, so that $\gamma(o_1) - \gamma(o_2)$ is equal to the number of negative digons in $\kappa'_1$ minus the number of negative digons in $\kappa'_2$. So that we have:
\begin{align*}
\gamma(o_1) - \gamma(o_2) &= \mu(o_2) -\mu(o_1) \\
\gamma(o_1)+ \mu(o_1) &= \gamma(o_2)+\mu(o_2).
\end{align*}
\end{proof}
\begin{proof}[Proof of proposition~\ref{prop:pciisid}.]
The foam $p_G\circ i_G$ is equal to $f(w,G,\delta)$ with $\delta$ the null dot function on $w$. From the lemmas~\ref{lem:sumoforientation}, \ref{lem:nonfitting20} and \ref{lem:fitting2id} we have that:
\begin{align*}
f(w,G,\delta)& = \sum_{\textrm{$o$ fitting orientation of $G$}} (-1)^{\gamma(o)} f(w, G_o, \delta) \\
=& \sum_{\textrm{$o$ fitting orientation of $G$}} (-1)^{\gamma(o) + \mu(o)} w_G\times [0,1]\\
= & \pm\#\{\textrm{fitting orientations of $G$}\} w_G\times [0,1].
\end{align*}
The red graph $G$ is supposed to be exact. This means in particular that the set of fitting orientation is not empty. So that $p_G\circ i_G$ is a non-trivial multiple of $\ensuremath{\mathrm{id}}_{w_G}=w_G\times [0,1]$.
\end{proof}
\begin{proof}[Proof of theorem \ref{thm:RG2idempotent}]
From the proposition~\ref{prop:pciisid}, we know that there exists a non zero integer $\lambda_G$ such that $p_G\circ i_G= \lambda_G w_G$. Hence, $\frac{1}{\lambda_G}i_G\circ p_G$ is an idempotent. It's clear that it's non-zero. It is quite intuitive that it is not equivalent to the identity foam, for a proper proof, see proposition~\ref{prop:idfoam}.
\end{proof}
\subsection{A new approach to red graphs.}
\label{sec:new-approach-to-red-graph}
In this section we give an alternative approach to red graphs: instead of starting with a web and simplifying it with a red graph we construct a red graph from a web and a simplification of this web. For this we need a property of webs that we did not use so far.
\begin{prop}
Let $w$ be a closed web, then it admits a (canonical) face-3-colouring with the unbounded face coloured $c\in \ZZ/3\ZZ$. We call this colouring \emph{the face-colouring of base $c$} of $w$. When $c$ is not mentioned it is meant to be $0$.
\end{prop}
\begin{proof}
We will colour connected components of $\RR^2\setminus w$ with elements of $\ZZ/3\ZZ$.
We can consider the only unbounded component $U$ of $\RR^2\setminus w$. We colour it by $c$, then for each other connected component $f$, we consider $p$ an oriented path from a point inside $U$ to a point inside $f$, which crosses the $w$ transversely, we then define the colour of $f$ to be the sum (modulo 3) of the signs of the intersection of the path $p$ with $w$ (see figure~\ref{fig:signcol} for signs convention). This does not depend on the path because in $w$ the flow is always preserved modulo 3. And, by definition, two adjacent faces are separated by an edge, so that they do not have the same colour.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\input{./th_signcol}
\end{tikzpicture}
\caption{On the left a positive crossing, on the right a negative one. The path is dashed and the web is solid.}
\label{fig:signcol}
\end{figure}
\end{proof}
\begin{cor}
Let $w$ be an $\epsilon$-web, then the connected component of $\RR\times \RR_+ \setminus w$ admits a (canonical) 3-colouring with the unbounded connected component coloured by $c$. We call this colouring \emph{the face-colouring of base $c$} of $w$.
\end{cor}
\begin{proof}
We complete $w$ with $\overline{w}$ and we use the previous proposition to obtain a colouring of the faces. This gives us a canonical colouring for $\RR\times \RR_+ \setminus w$.
\end{proof}
Note that in this corollary it is important to consider the connected component of $\RR\times \RR_+ \setminus w$ instead of the connected component of $\RR^2 \setminus w$. Let us formalise this in a definition.
\begin{dfn}
If $w$ is an $\epsilon$-web, the \emph{regions} of $w$ are the connected components of $\RR\times \RR_+ \setminus w$. The \emph{faces} of $w$ are the regions which do not intersect $\RR \times \{0\}$.
\end{dfn}
\begin{dfn}
Let $w$ be an $\epsilon$-web, an $\epsilon$-web $w'$ is a
\emph{simplification of $w$} if
\begin{itemize}
\item the set of vertices of $w'$ is
included in the set of vertices of $w$,
\item every edge $e$ of $w'$ is
divide into an odd number of intervals
$([a_i,a_{i+1}])_{i\in[0,2k]}$ such that for every $i$ in $[0, k]$,
$[a_{2i},a_{2i+1}]$ is an edge of $w$ (with matching orientations)
and for every $i$ in $[0, k-1]$, $[a_{2i+1},a_{2i+2}]$ lies in the
faces of $w$ opposite to $[a_{2i}, a_{2i+1}]$ with respect to $a_{2i+1}$ (see figure~\ref{fig:simplificationedge}).
\end{itemize}
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=0.7]
\input{./th_simplificationedge}
\end{tikzpicture}
\caption{Local picture around $a_k$. The edge of $w'$ is orange and large, while the $\epsilon$-web $w$ is black and thin.}
\label{fig:simplificationedge}
\end{figure}
\end{dfn}
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale= 0.8]
\input{./sw_thewebwsimplification}
\end{tikzpicture}
\caption{The $\epsilon$-web $w$ (on black) and $w_0$ (in orange) of proposition~\ref{prop:Pwdec} seen in terms of simplification.}
\label{fig:exsimplification}
\end{figure}
\begin{lem}\label{lem:coherent-col}
Let $w$ be a $\epsilon$-web and $w'$ a $\partial$-connected simplification of $w$ If $e$ is an edge of $w$ which is as well a (part of an) edge of $w'$, then in the face-colourings of base $c$ of $w$ and $w'$, the the regions adjacent to $e$ in $w$ and in $w'$ are coloured in the same way.
\end{lem}
\begin{proof}
This is an easy recursion on how $e$ is far from the border.
\end{proof}
Note that in this definition the embedding of $w'$ with respect to $w$ is very important.
\begin{dfn}
Let $w$ be an $\epsilon$-web and $w'$ a simplification of $w$. We consider the face-colourings of $w$ and $w'$. A face $f$ of $w$ lies in one or several regions of $w'$. This face $f$ is \emph{essential with respect to $w'$} if all regions of $w'$ it intersects do not have the same colour as $f$.
\end{dfn}
\begin{req}
We could have write this definition with region of $w$ instead of faces, but it is easy to see that a region of which is not a face $w$ is never essential.
\end{req}
\begin{lem}\label{lem:intersect2essential}
Let $w$ be a $\partial$-connected $\epsilon$-web and $w'$ a $\partial$-connected simplification of $w$. If a face $f$ of $w$ is not essential with respect to $w'$ then it intersects only one region of $w'$.
\end{lem}
\begin{proof}
Consider a face $f$ of $w$ which intersects more than one region of $w'$. We will prove that it is essential with respect to $w'$. Consider an edge $e'$ of $w'$ which intersects $f$ (there is at least one by hypothesis), when we look next to the border of $f$ next to $e'$ we find a vertex $v$ of $w$ (see figure~\ref{fig:localfacefp}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=0.7]
\input{./th_localfacefp}
\end{tikzpicture}
\caption{A part of the face $f'$ next to an edge $e'$ of $w'$. Above $v$ the colours of $w$ and $w'$ are coherent thanks to lemma~\ref{lem:coherent-col}.}
\label{fig:localfacefp}
\end{figure}
We want to prove that none of the faces of $w'$ which are adjacent to $e'$ has the same colour as the face $f$. This follows from the lemma \ref{lem:coherent-col}, and from the fact that the part of $e'$ above $v$ is an edge of $w$ (see figure~\ref{fig:localfacefp}).
\end{proof}
\begin{cor}\label{cor:how-are-essential-faces}
Let $w$ be a $\partial$-connected $\epsilon$-web and $w'$ a $\partial$-connected simplification of $w$. If a face $f$ of $w$ intersects a region of $w'$ which has the same colour, it is not essential.
\end{cor}
\begin{prop}
Let $w$ be a $\partial$-connected $\epsilon$-web (this implies that every face of $w$ is diffeomorphic to a disk) and $w'$ a $\partial$-connected simplification of $w$. Then there exists a (canonical) paired red graph $G$ such that $w'$ is equal to $w_G$. We denote it by $G_{w\to w'}$.
\end{prop}
\marginpar{face vs connected component of...}
\begin{proof}
We consider the canonical colourings of the faces of $w$ and $w'$. The red graph $G$ is the induced sub-graph of $w^\star$ (the dual graph of $w$) whose vertices are essential faces of $w$ with respect to $w'$. The pairing is given by the edges of $w'$. We need to prove first that this is indeed a red graph, and in a second step that $w_G = w'$.
We consider a vertex $v$ of $w$ and the 3 regions next to it. We want to prove that at least one of the 3 regions is not essential with respect to $w'$. If the vertex $v$ is a vertex of $w'$ then lemma~\ref{lem:coherent-col} and corollary \ref{cor:how-are-essential-faces} give that none of the three regions is essential. Else, $v$ either lies inside an edge of $w'$ or it lies in a face of $w'$ (see figure~\ref{fig:3possvertex}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=0.7]
\input{./th_3possvertex}
\end{tikzpicture}
\caption{The three configurations for the vertex $v$ of $w$: it is a vertex of $w'$ (on the left), it lies inside an edge of $w'$ (on the middle), it lies inside a region of $w'$ (on the right).}
\label{fig:3possvertex}
\end{figure}
Consider the first situation: one of the 3 regions intersects two different regions of $w'$ hence it is essential thanks to lemma~\ref{lem:intersect2essential}, the two others are not thanks to corollary~\ref{cor:how-are-essential-faces}.
In the last situation, the 3 regions have different colours so that one of them has the same colour than the colour of the region of $w'$ where $v$ lies in, this region is therefore not essential (corollary \ref{cor:how-are-essential-faces}). This shows that $G$ is a red graph (we said nothing about the admissibility).
Let us now show that $w'=w_G$. We consider a collection $(N_{f})_{f\in V(G)}$ of regular neighbourhoods of essential faces of $w$ with respect to $w'$. Let us first show that for every essential face $f$ of $w$, if $N_f$ is a regular neighbourhood of $f$, the restriction of $w_G$ and $w'$ matchs. As $f$ is essential it is a vertex of $G$. Then the restriction of $w_G$ to $N_f$ is just a collection of strands joining the border to the border, just as $w'$.
In $\RR\times \RR_+\setminus \left(\bigcup_{f\in V(G)}N_f\right)$ the $\epsilon$-webs $w'$ and $w_G$ are both equal to $w$. This complete the proof.
\marginpar{illustration with $w$ and $w_0$}
\end{proof}
Note that $G_{w\to w'}$ depends on how $w'$ is embedded to see it as a simplification of $w$.
\begin{dfn}
Let $w$ a $\epsilon$-web and $w'$ a simplification of $w$, then the simplification is \emph{nice}, if for every region $r$ of $w$, $r\cap w'$ is either the empty set or connected.
\end{dfn}
We have the natural lemma:
\begin{lem}
Let $w$ be a $\partial$-connected $\epsilon$-web and $w'$ a $\partial$-connected simplification of $w$. The simplification is nice if and only if the red graph $G_{w\to w'}$ is nice
\end{lem}
\begin{proof}
Thanks to lemma \ref{lem:intersect2essential}, only essential faces of $w$ with respect to $w'$ can have non trivial intersection with $w'$, and for an essential face $f$, twice the number of connected component of $f\cap w'$ is equal to the exterior degree of the vertex of $G_{w\to w'}$ corresponding to $f$.
\end{proof}
\begin{lem}\label{lem:essfacesmatters}
If $w$ is a $\partial$-connected $\epsilon$-web, and $w'$ is a $\partial$-connected simplification of $w$. Then the level of $G_{w\to w'}$ is given by the following formula:
\[
i(G_{w\to w'}) = 2\#\{\textrm{essential faces of $w$ wrt. $w'$}\} - \frac{\#V(w) - \# V(w')}{2}.
\]
\end{lem}
This shows that the embedding of $w'$ influences the level of $G_{w\to w'}$ only on the number of essential faces of $w$ with respect to $w'$
\begin{proof}
The level of a red graph $G$ is given by:
\[
i(G) = 2\#V(G) - \#E(G) - \sum_{f\in V(G)}\frac{\ensuremath{\mathrm{ed}}(f)}{2}.
\]
By definition of $G_{w\to w'}$, we have:
\[ \{\textrm{essential faces of $w$ wrt. $w'$}\} = V(G_{w\to w'}).\]
The only thing to realise is that
we have:
\[
2\left(\#E(G)_{w\to w'} + \sum_{f\in V(G_{w\to w'})}\frac{\ensuremath{\mathrm{ed}}(f)}{2}\right) = \#V(w) - \# V(w'),
\] and this follows from the definition of $w_{G_{w\to w'}}=w'$.
\end{proof}
\begin{dfn}\label{dfn:avoidfill}
If $f$ is a face of $w$, $w'$ a simplification of $w$ and $r$ a region of $w'$, we say that $f$ avoids $r$ if $f\cap r=\emptyset$ or if the boundary of $r$ in each connected component of $f\cap r$ joins two consecutive vertices of $f$ (see figure \ref{fig:avoiding}). In the first case we say that $f$ avoid $r$ trivially.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\input{./th_nontrivavoid}
\end{tikzpicture}
\caption{The local picture of a face $f$ (in white) of $w$ (in black) non-trivially avoiding a region $r$ (in yellow) of $w'$ (in orange).}
\label{fig:avoiding}
\end{figure}
If $f$ is an essential face of $w$ with respect to $w'$ and $r$ is a region of $w'$, we say that \emph{$f$ fills $r$}, if $f$ does not avoid $r$. If $F'$ is a set of region of $w'$ we say that \emph{$f$ fills (resp.~{} avoids) $F'$} if it fills at least one region of $F'$ (resp.~{} avoids all the regions of $F'$). We define:
\[
n(f,F') \eqdef \#\{r\in F' \,\textrm{such that $f$ fills $r$}\}.
\]
If $G'$ is a red graph of $w'$, we write $n(f,G')$ for $n(f,V(G'))$.
\end{dfn}
With the same notations, and with $F$ a set of face of $w$, we have the following equality:
\begin{align}
\#F = \#\{\textrm{faces $f$ of $F$ avoiding $F'$}\} + \sum_{f'\in F'}\sum_{\substack{f \in F \\ \textrm{$f$ fills $f'$}}} \frac1{n(f,F')}. \label{eq:sumofcontrib}
\end{align}
\begin{lem}
Let $w$ be a $\partial$-connected $\epsilon$-web and $w'$ a nice $\partial$-connected simplification of $w$. Let $F'$ be a collection of faces of $w'$, then for every face $f$ of $w$, we have: $n(f, F') \leq 2$.
\end{lem}
\begin{proof}
This is clear since $f\cap w'$ consists of at most one strand, so that it intersects at most 2 faces of $F'$.
\end{proof}
\begin{req}\label{rqe:hex2bign}
Let $w$ be a $\epsilon$-web, $w'$ a nice $\partial$-connected simplification of $w$ and $f$ an essential face of $w$ with respect to $w'$. Suppose that $f$ has at least 6 sides of $w$. Suppose furthermore that it intersects two regions $r_1$ and $r_2$ of $w'$, then either it (non-trivially) avoids one of them, either it fills both of them. If $f$ avoids $r_2$ then at least two neighbours (in $G_{w\to w'})$ of $f$ fill $r_1$ (see picture~\ref{fig:fillhex}).
If on the contrary $f$ has just one neighbour which fills $r_1$, then $f$ fills $r_2$. Under this condition, for any collection $F'$ of regions of $w'$ with $\{r_1, r_2\} \subseteq F'$ we have: $n(f, F')=2$.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\input{./th_fillhex}
\end{tikzpicture}
\caption{On the left $f$ avoids $r_2$, on the right it fills $r_1$ and $r_2$.}
\label{fig:fillhex}
\end{figure}
\end{req}
\begin{dfn}\label{dfn:sigma}
We set $\displaystyle{\sigma(f', F\to F') \eqdef \sum_{\substack{f \in F \\ \textrm{$f$ fills $f'$}}} \frac1{n(f,F')} }$. If $G$ is a red graph for $w$ and $G'$ a red graph for $w'$ we write $\sigma(f', G\to G' )$ for $\sigma(f', V(G)\to V(G'))$.
\end{dfn}
\subsection{Proof of lemma~\ref{lem:tech}}
\label{sec:proof-lemmatech}
In this section we use the point of view developed in section~\ref{sec:new-approach-to-red-graph} to prove the lemma~\ref{lem:tech}. We restate it with this new vocabulary:
\begin{lem}\label{lem:techNV}
Let $w$ be an a $\partial$-connected $\epsilon$-web which contains no digon and exactly one square. We suppose furthermore that this square touches the unbounded face. Let $G$ be a nice red graphs of $w$ and $G'$ a nice red graph of $w'=w_{G}$, then there exists $\widetilde{w}$ a nice simplification of $w$ such that:
\begin{enumerate}[A)]
\item\label{it:condA} the $\epsilon$-webs $(w_{G})_{G'}$ and $\widetilde{w}$ are isotopic,
\item\label{it:condB} the following equality holds:
\[
\#V( \widetilde{G}) \geq \# V(G) + \# V(G'),
\]
\end{enumerate}
where $\widetilde{G}$ denote the red graph $G_{w\to \widetilde{w}}$.
\end{lem}
\begin{proof}
Because of the condition~\ref{it:condA}, we already know the isotopy class of the web $\widetilde{w}$. To describe it completely, we only need to specify how $\widetilde{w}$ is embedded. For each face $f'$ of $w'$ which is a vertex of $G'$, let us denote $N_{f'}$ a regular neighbourhood of $f'$. We consider $U$ the complementary of $\bigcup_{f'} N_{f'}$. Provided this is done in a coherent fashion, it's enough to specify how $\widetilde{w}$ looks like in $U$ and in $N_{f'}$ for each face $f'$ of $w'$.
If $f'$ is a face of $w'$ which is in $G'$, we consider two different cases:
\begin{enumerate}
\item\label{it:face0} the face $f'$ corresponds to a vertex of $G'$ with exterior degree equal to 0,
\item\label{it:face2}the face $f'$ corresponds to a vertex of $G'$ with exterior degree equal to 2.
\end{enumerate}
These are the only cases to consider since $G'$ is nice.
Let us denote by $w''$ the $\epsilon$-web $(w')_{G'}$. We want $\widetilde{w}$ and $w''$ to be isotopic. So let us look at $w''\cap U$ and at $w''\cap N_{f'}$ in the two cases.
Around $U$, the $\epsilon$-web $w'$ does not ``see'' the red graph $G'$, so that $U\cap w''= U\cap w'_{G'} = U\cap w'$.
If the face $f'$ has exterior degree equal to 0 (case~\ref{it:face0}), then we have: $N_{f'}\cap w'' = U\cap w'_{G'}= \emptyset$. \marginpar{example ?}
If the face $f'$ has exterior degree equal to 2 (case~\ref{it:face2}), then we have: $N_{f'}\cap w'' = U\cap w'_{G'}$ is a single strands cutting $N_{f'}$ into two parts. \marginpar{example}
We embed $\widetilde{w}$ such that $U\cap \widetilde{w}$ and $U\cap w''$ are equal and for each face $f'$ corresponding to a vertex of $G'$,
$N_{f'}\cap \widetilde{w}$ and $N_{f'}\cap w''$ are isotopic (relatively to the boundary).
We claim that if $f'$ is a vertex with external degree equal to 0 then:
\begin{align}\begin{cases}\sigma (f', \widetilde{G}\to G') \geq \sigma(f', G \to G') +\frac12 & \textrm {if $S \subseteq N_{f'}$,} \\
\sigma (f', \widetilde{G}\to G') \geq \sigma(f', G \to G') +1 & \textrm {if $S\nsubseteq N_{f'}$,}
\end{cases}\label{eq:sigmaface0}
\end{align}
where $S$ is the square of $w$.
The restriction\footnote{We only consider the vertices of $G$ which fill $f'$.} $G_{f'}$ of $G'$ to $f'$ is a graph which satisfies the following conditions:
\begin{itemize}
\item it is bi-coloured (because the vertices of $G$ are essential faces of $w$ with respect to $w'$),
\item it is naturally embedded in a disk because $N_{f'}$ is diffeomorphic to a disk,
\item the degree of the vertices inside the disks have degree at least three (because the only possible square of $w$ touches the border) and the vertices on the border (these are the one which intersect an other region of $w'$) have degree at least 1.
\end{itemize}
The regions of $G_{f'}$ and the vertices of one of the two colours of $G_{f'}$ become vertices of $\widetilde{G}$ (see example depicted on figure~\ref{fig:exampleed0}). To proves the inequality (\ref{eq:sigmaface0}), one should carefully count regions of $G_{f'}$. There are two different cases in (\ref{eq:sigmaface0}) because the remark~\ref{rqe:hex2bign} do not apply to the square.
Hence we can apply the lemma~\ref{lem:techtech0} which proves (\ref{eq:sigmaface0}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale= 1.4]
\input{./rg_exampleed0}
\end{tikzpicture}
\caption{Example of the procedure to define $\widetilde{G}$ when the exterior degree of $f'$ is equal to 0.}
\label{fig:exampleed0}
\end{figure}
The only thing remaining to specify is how $\widetilde{w}$ looks in $N_{f'}$ where $f'$ is a vertex of $G'$ of exterior degree equal to 2. Note that we need to embed $\widetilde{w}$ so that it is a nice simplification of $w$. We claim that it is always possible to find such an embedding so that the inequality (\ref{eq:sigmaface0}) is satisfied. In this case the graph $G_{f'}$ is in the same situation as before, but is important to notice that the faces of $G_{f'}$ have at least 6 sides (this is a consequence of proposition~\ref{prop:largecycle}).
The vertices of $\widetilde{G}$ are the regions of $G_{f'}$ and the vertices of $G_{f'}$ of one of the two colours on one side of the strand and the vertices of $G_{f'}$ of the other colour on the other side of the strand (see figure~\ref{fig:exampleed0} for an example). Hence in order to show that the inequality (\ref{eq:sigmaface0}) holds, one should carefully count the regions and the vertices of $G_{f'}$, this is done by lemma~\ref{sec:case-with-exterior2}.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale =1.4]
\input{./rg_exampleed2}
\end{tikzpicture}
\caption{Example of the procedure to define $\widetilde{G}$ when the exterior degree of $f'$ is equal to 2.}
\label{fig:exampleed2}
\end{figure}
So now we have a simplification $\widetilde{w}$ of $w$, such that the graph $\widetilde{G}=G_{w\to \widetilde{w}}$ satisfies (\ref{prop:largecycle}) for each region $f'$ of $G'$. The square $S$ of $w$ is in at most one $N_{f'}$ so that if we sum (\ref{prop:largecycle}) for all the vertices of $f'$, we obtain:
\[
\sum_{f'\in F'} \sigma(f', \widetilde{G}\to G') \geq \sum_{f'\in F'} \sigma(f', G \to G') + \#V(G') - \frac12,
\]
and using (\ref{eq:sumofcontrib}), we have:
\[
\#V(\widetilde{G}) \geq \#V(G') + \#V(G) - \frac12,
\]
but $\#V(\widetilde{G})$ being an integer we have $V(\widetilde{G}) \geq \#V(G') + \#V(G)$
\end{proof}
\subsection{Proof of combinatorial lemmas}
\label{sec:proof-techn-lemm}
This proof is dedicated to the two technical lemmas used in the last section. We first introduce the ad-hoc objects and then state and prove the lemmas.
\begin{dfn}
A \emph{$D$-graph} is a graph $G$ embedded in the disk $D^2$. The set of vertices $V(G)$ is partitioned in two sets: $V^\partial(G)$ contains the vertices lying on $\partial D^2$, while $V^\textrm{in}$ contains the others. The set $F(G)$ of connected components of $D^2\setminus G$ is partitioned into two sets $F^\textrm{in}$ contains the connected component included in $\mathring{D^2}$, while $F^\partial$ contains the others.
A $D$-graph is said to be \emph{non-elliptic} if:
\begin{itemize}
\item every vertex $v$ of $V^\textrm{in}$ has degree greater or equal to 3,
\item every vertex $v$ of $V^\partial$ has degree greater or equal than 1,
\item the faces of $F^\mathrm{in}$ are of size at least 6.
\end{itemize}
A \emph{coloured $D$-graph} is a $D$-graph $G$ together with:
\begin{itemize}
\item a vertex-2-colouring (by $\mathrm{\color{darkgreen}green}$ and $\mathrm{\color{darkblue}blue}$) of the vertices of $G$ (this implies that $G$ is bipartite),
\item a subdivision of $\mathcal{D}^2$ into two intervals (we allow one interval to be the empty set and the other one to be the full circle, in this case we say that $G$ is \emph{circled-coloured}): a $\mathrm{\color{darkgreen}green}$ one and a $\mathrm{\color{darkblue}blue}$ one (denoted by $I_\mathrm{\color{darkblue}blue}$ and $I_\mathrm{\color{darkgreen}green}$) when they are real intervals we defines $x$ and $y$ to be the two intersection points of $I_\mathrm{\color{darkgreen}green}$ and $I_\mathrm{\color{darkblue}blue}$ with the convention that when one scans $\partial D^2$ clockwise, one see $x$, then $I_\mathrm{\color{darkgreen}green}$, then $y$ and finally $I_\mathrm{\color{darkblue}blue}$.
\end{itemize}
The vertices of $V^\partial$ are supposed neither on $x$ nor on $y$. The colour of a vertex is not supposed to fit the colour of the interval it lies on. We set $V_{\mathrm{\color{darkgreen}green}}$ (resp.~{} $V_\mathrm{\color{darkblue}blue}$) the set of $\mathrm{\color{darkgreen}green}$, (resp.~{} $\mathrm{\color{darkblue}blue}$) vertices, and
$V_{\mathrm{\color{darkgreen}green}}^\partial$, $V_{\mathrm{\color{darkgreen}green}}^\mathrm{in}$, $V_{\mathrm{\color{darkblue}blue}}^\partial$ and $V_{\mathrm{\color{darkblue}blue}}^\mathrm{in}$ in the obvious way.
If $G$ is a coloured $D$-graph, and $v$ is a vertex of $V^\partial$ of we set:
\[
n(v) = \begin{cases} 2 & \textrm{if $v$ has degree 1 and the colour of $v$ fits the colour of the interval,} \\ 1 & \textrm{else.}
\end{cases}
\]
If $v$ is a vertex of $V^\mathrm{in}$, we set $n(v)=1$. Note that this definition of $n$ is a translation of the $n$ of the previous section (see remark~\ref{rqe:hex2bign}).
\end{dfn}
\subsubsection{Case with exterior degree equal to 0}
\label{sec:case-with-exterior0}
\begin{lem} \label{lem:techtech0}
Let $G$ be a non-elliptic circled-coloured $D$-graph (with the circle coloured by a colour $c$), then:
\[\#F \geq 1 + \sum_{v\in V_c} \frac{1}{n(v)}.
\]
\end{lem}
\begin{proof}By symmetry, we may suppose that $c=\mathrm{\color{darkgreen}green}$.
To show this we consider the graph $H$ obtain by gluing to copies of $G$ along the boundary of $D^2$ this is naturally embedded in the sphere. We write the Euler characteristic:
\begin{align}
\#F(H) - \#E(H) + \#V(H) = 1 + \#C(H), \label{eq:eulerchar}
\end{align}
where $C(H)$ is the set of connected components of $H$. We have the following equalities:
\begin{align*}
\#F(H) &= 2\#F^{\textrm{in}}(G) + \#F^{\partial}(G),\\
\#F^{\partial}(G) &= {\#V^{\partial}(G)} + 1 -\#C(H), \\
\#E(H) &= 2 \#E(G) = \sum_{v\in V(G)} \deg(v) = 2\sum_{v\in V_{\mathrm{\color{darkgreen}green}}(G)} \deg(v),\\
\#V(H) &= 2\#V^{\textrm{in}}(G) + \#V^{\partial}(G)
\end{align*}
So that we can rewrite (\ref{eq:eulerchar}):
\begin{align}
2\#F^{\textrm{in}}(G) + 2\#F^{\partial}(G) +2\#V^{\textrm{in}}(G) = 2 + 2 \#E(G). \label{eq:EX2}
\end{align}
Now we use the what we know about degrees of the vertices:
\begin{align*}
\#E(G)&\geq \frac{3}2 \# V^{\textrm{in}}(G)+ \frac12 \#V^{\partial,1}(G) +1 \#V^{\partial,>1}(G), \\
\#E(G)&\geq 3 \# V_{\mathrm{\color{darkgreen}green}}^{\textrm{in}}(G)+ \#V_{\mathrm{\color{darkgreen}green}}^{\partial,1}(G) +2 \#V_{\mathrm{\color{darkgreen}green}}^{\partial,>1}(G).
\end{align*}
Where $V^{\partial,1}$ (resp.~{} $V^{\partial,>1}$) denotes the subset of $V^\partial$ with degree equal to $1$ (resp.~{} with degree strictly bigger than 1).
If we sum $\frac23$ of the first inequality and $\frac13$ of the second one, and inject this in~(\ref{eq:EX2}) we obtain:
\begin{align*}
\#F(G) + \#V^{\textrm{in}}(G) &\geq 1+ \#E(G) \\
\#F(G) + \#V^{\textrm{in}}(G) &\geq V^{\textrm{in}}(G)+ \frac13 \#V^{\partial,1}(G) +\frac23 \#V^{\partial,>1}(G) \\
& \quad + \# V_{\mathrm{\color{darkgreen}green}}^{\textrm{in}}(G) + \frac13\#V_{\mathrm{\color{darkgreen}green}}^{\partial,1}(G) + \frac{2}3 \#V_{\mathrm{\color{darkgreen}green}}^{\partial,>1}(G) \\
\#F(G) &\geq \# V_{\mathrm{\color{darkgreen}green}}^{\textrm{in}}(G)+ \frac23 \#V_{\mathrm{\color{darkgreen}green}}^{\partial,1}(G) +\frac43 \#V_{\mathrm{\color{darkgreen}green}}^{\partial,>1}(G) \\
&\quad + \frac13\#V_{\mathrm{\color{darkblue}blue}}^{\partial,1}(G) + \frac{2}3 \#V_{\mathrm{\color{darkblue}blue}}^{\partial,>1}(G) \\
&\geq \# V_{\mathrm{\color{darkgreen}green}}^{\textrm{in}}(G)+ \frac12 \#V_{\mathrm{\color{darkgreen}green}}^{\partial,1}(G) + \#V_{\mathrm{\color{darkgreen}green}}^{\partial,>1}(G) \\
& \geq \sum_{v\in V_\mathrm{\color{darkgreen}green}} \frac{1}{n(v)}.
\end{align*}
\end{proof}
\subsubsection{Case with exterior degree equal to 2}
\label{sec:case-with-exterior2}
\begin{lem}\label{lem:bg1UYH}
If $G$ is a non-elliptic $D$-graph, then all the faces of $F$ are diffeomorphic to disks, and if it is non-empty, then at least one of the following situations happens:
\begin{enumerate}[(1)]
\item\label{it:bg1} the set $V^{\partial,>1}$ is non empty,
\item\label{it:cap} there exists, two $\cap$'s (see figure~\ref{fig:UDgraphUHY}) (if $G$ consists of only one edge, the two $\cap$'s are actually the same one counted two times because it can be seen as a $\cap$ on its two sides),
\item\label{it:3YH} there exists three $\lambda$'s or $H$'s (see figure~\ref{fig:UDgraphUHY}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.1]
\input{./th_DgraphUHY}
\end{tikzpicture}
\caption{From left to right: a $\cap$, a $\lambda$ and an $H$. The circle $\partial D^2$ is thick and grey, the $D$-graph is thin and black. Note that the vertices inside $D^2$ may have degree bigger than 3.}
\label{fig:UDgraphUHY}
\end{figure}
\end{enumerate}
\end{lem}
\begin{proof}
This is the same Euler characteristic-argument that we used in lemma~\ref{lem:UHYinNE}.
\end{proof}
\begin{dfn}
A cut in a (not circled-) coloured $D$-graph is a simple oriented path $\gamma:[0,1] \to D$ such that:
\begin{itemize}
\item we have $\gamma(0)=x$ and $\gamma(1)=y$, therefore $I_\mathrm{\color{darkgreen}green}$ is on the left and $I_\mathrm{\color{darkblue}blue}$ is on the right\footnote{We use the convention that the left and right side are determined when one scans $\gamma$ from $x$ to $y$.}. (see figure \ref{fig:cut}),
\item for every face $f$ of $G$, $f\cap \gamma$ is connected,
\item the path $\gamma$ crosses $G$ either transversely at edges joining a $\mathrm{\color{darkgreen}green}$ vertex on left and a $\mathrm{\color{darkblue}blue}$ vertex on the right, or at vertices of $V^\partial$ whose colours do not fit with the intervals they lie on.
\end{itemize}
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.1]
\input{./th_cut.tex}
\end{tikzpicture}
\caption{A cut in a coloured $D$-graph (note that $G$ is elliptic).}
\label{fig:cut}
\end{figure}
If $\gamma$ is a cut we denote by $\ensuremath{\otimes}[^{l(\gamma)}]{V}{}(G)$ and $\ensuremath{\otimes}[^{r(\gamma)}]{V}{}(G)$ the vertices located on the left (resp.~{} on the right) of $\gamma$. (The vertices located on $\gamma$ are meant to be both on the left and on the right).
\end{dfn}
\begin{lem}\label{lem:techtech2}
Let $G$ be is a non-elliptic (not circled-) coloured $D$-graph, then there exists a cut $\gamma$ such that:
\[\#F(G) \geq 1+ \sum_{v\in \ensuremath{\otimes}[^{l(\gamma)}]{V}{_\mathrm{\color{darkgreen}green}}} \frac1{n(v)} + \sum_{v\in \ensuremath{\otimes}[^{r(\gamma)}]{V}{_\mathrm{\color{darkblue}blue}}}\frac1{n(v)}. \]
\end{lem}
\begin{proof}
The proof is done by induction on $s(G)\eqdef 3\# E(G) +4\# V^{\partial,>1}(G)$. If this quantity is equal to zero then the $D$-graph is empty, then we choose $\gamma$ to be any simple arc joining $x$ to $y$, and the lemma says $1\geq 1$ which is true.
We set:
\[
C(G,\gamma)\eqdef\sum_{v\in \ensuremath{\otimes}[^{l(\gamma)}]{V}{_\mathrm{\color{darkgreen}green}}} \frac1{n(v)} + \sum_{v\in \ensuremath{\otimes}[^{r(\gamma)}]{V}{_\mathrm{\color{darkblue}blue}}}\frac1{n(v)}.
\]
It is enough to check the situations (\ref{it:bg1}), (\ref{it:cap}) and (\ref{it:3YH}) described in lemma \ref{lem:bg1UYH}.
\paragraph*{Situation (\ref{it:bg1})}
Let us denote by $v$ a vertex of $V^{\partial, >1}$. There are two cases: the colour of $v$ fits with the colours of the intervals it lies on or not.
If the colours fit, say both are $\mathrm{\color{darkgreen}green}$, we consider $G'$ the same coloured $D$-graph as $G$ but with $v$ split into $v_1$, $v_2$ \dots $v_{\deg(v)}$ all in $V^{\partial,1}(G')$ (see figure~\ref{fig:GGpsit1}). We have $s(G')= S(G)-4< s(G)$ and $G'$ non-elliptic, therefore we can apply the induction hypothesis. We can find a cut $\gamma'$ with $\#F(G')\geq 1+ C(G',\gamma')$. Note that $\gamma'$ does not cross any $v'$, so that we can lift $\gamma'$ in the $D$-graph $G$. This gives us $\gamma$. We have:
\begin{align*}
C(G,\gamma)&= C(G',\gamma') + \frac1{n(v)} - \sum_{k=1}^{\deg(v)}\frac1{n(v_k)} \\
&= C(G',\gamma') + 1 - \frac{\deg(v)}2 \\
&\geq C(G',\gamma').
\end{align*}
On the other hand $\#F(G)= \#F(G')$ so that we have $\#F(G)\geq 1+ C(G,\gamma)$.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.1]
\input{./th_GGpsit1}
\end{tikzpicture}
\caption{Local picture of $G$ (on the left) and $G'$ (on the right) around, when $v$ is $\mathrm{\color{darkgreen}green}$ and lies on $I_\mathrm{\color{darkgreen}green}$.}
\label{fig:GGpsit1}
\end{figure}
If the colours do not fit (say $v$ is $\mathrm{\color{darkblue}blue}$), we construct $G'$ a coloured $D$-graph which is similar to $G$ every where but next to $v$. The vertex $v$ is pushed in $D^2$ (we denote it by $v'$) and we add a new vertex $v''$ on $\partial D^2$ and an edge $e$ joining $v'$ and $v''$. The coloured $D$-graph $G'$ is non-elliptic and $s(G')= s(G)-4+3<s(G)$ so that we can apply the induction hypothesis and find a cut $\gamma'$ with $\#F(G')\geq 1+ C(G',\gamma')$.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.1]
\input{./th_GGpsit1b}
\end{tikzpicture}
\caption{Local picture of $G$ (on the left) and $G'$ (on the right) around $v$, when $v$ is $\mathrm{\color{darkblue}blue}$ and lies on $I_\mathrm{\color{darkgreen}green}$.}
\label{fig:GGpsit1b}
\end{figure}
If $\gamma'$ does not cross $e$ we can lift $\gamma'$ in $G$ (this gives us $\gamma$). We have
\begin{align*}C(G,\gamma)= C(G',\gamma')+ \frac{1}{n(v)} - \frac{1}{n(v')} =C(G',\gamma')+ 1- 1 = C(G',\gamma').
\end{align*}
On the other hand, we have $F(G)= F(G')$, so that $\#F(G)\geq 1+ C(G,\gamma)$.
Consider now the case where $\gamma'$ crosses $e$. Then we consider the cut $\gamma$ of $G$ which is the same as $\gamma$ far from $v$, and which around $v$ crosses $G$ in $v$ (see figure~\ref{fig:GGpsitc}). We have:
\begin{align*}
C(G,\gamma)&= C(G',\gamma')+ \frac{1}{n(v)} - \frac{1}{n(v')} - \frac{1}{n(v'')}\\
C(G,\gamma)&= C(G',\gamma') +1-1-\frac12\\
C(G,\gamma)&\geq C(G',\gamma').
\end{align*}
But $\#F(G)= \#F(G')$, so that we have $\#F(G)\geq 1+ C(G,\gamma)$.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.1]
\input{./th_GGpsit1c}
\end{tikzpicture}
\caption{How to transform $\gamma'$ into $\gamma$.}
\label{fig:GGpsitc}
\end{figure}
\paragraph*{Situation (\ref{it:cap})}
We now suppose that $G$ contains two $\cap$'s. Let us denote by $v_g$ (resp.~{} $v_b$) the $\mathrm{\color{darkgreen}green}$ (resp.~{} $\mathrm{\color{darkblue}blue}$) vertex of the $\cap$ and by $e$ the edge of the cap. There are different possible configurations depending where $x$ and $y$ lies. As there are at least two caps, we may suppose $y$ is far from the $\cap$.
There are 3 different configurations (see figure~\ref{fig:3configsit2}):
\begin{itemize}
\item the point $x$ is far from the $\cap$,
\item the point $x$ is in the $\cap$ and $v_g\in I_\mathrm{\color{darkgreen}green}$ and $v_b \in I_\mathrm{\color{darkblue}blue}$,
\item the point $x$ is in the $\cap$ and $v_g\in I_\mathrm{\color{darkblue}blue}$ and $v_b \in I_\mathrm{\color{darkgreen}green}$,
\end{itemize}
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.1]
\input{./th_3configsit2}
\end{tikzpicture}
\caption{The three possible configurations.}
\label{fig:3configsit2}
\end{figure}
We consider $G'$ the coloured $D$-graph similar to $G$ except that the $\cap$ is removed. The coloured $D$-graph $G'$ is non-elliptic and $s(G')= s(G)- 3 <s(G)$ so that we can apply the induction hypothesis and find a cut $\gamma'$ with $\#F(G')\geq 1+ C(G',\gamma')$.
Let us suppose first that $x$ is far from the $\cap$, then $v_b$ and $v_g$ both lie either on $I_\mathrm{\color{darkgreen}green}$ or on $I_\mathrm{\color{darkblue}blue}$. By symmetry we may consider that they both lie on $I_\mathrm{\color{darkgreen}green}$. We can lift $\gamma'$ in $G$ (this gives $\gamma$) so that it does not meet the $\cap$. We have:
\begin{align*}
C(G,\gamma)&= C(G',\gamma')+ \frac{1}{n(v_g)} \\
&= C(G',\gamma')+ \frac12.
\end{align*}
But $\#F(G) = \#F(G')+1$, hence $\#F(G)\geq 1+ C(G,\gamma)$.
Suppose now that the point $x$ is in the $\cap$ and $v_g\in I_\mathrm{\color{darkgreen}green}$ and $v_b \in I_\mathrm{\color{darkblue}blue}$. We can lift $\gamma'$ in $G$ so that it crosses $e$ (see figure~\ref{fig:ggpsit2b}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.1]
\input{./th_ggpsit2b}
\end{tikzpicture}
\caption{How to transform $\gamma'$ into $\gamma$.}
\label{fig:ggpsit2b}
\end{figure}
We have:
\begin{align*}
C(G,\gamma)&= C(G',\gamma')+ \frac{1}{n(v_g)} +\frac{1}{n(v_b)} \\
&= C(G',\gamma')+ \frac12+\frac12 \\
&= C(G',\gamma')+1.
\end{align*}
But $\#F(G) = \#F(G')+1$, hence $\#F(G)\geq 1+ C(G,\gamma)$.
Suppose now that the point $x$ is in the $\cap$ and $v_g\in I_\mathrm{\color{darkblue}blue}$ and $v_b \in I_\mathrm{\color{darkgreen}green}$. We can lift $\gamma'$ in $G$ so that it crosses\footnote{We could have chosen to cross $v_b$.} $v_g$ (see figure~\ref{fig:ggpsit2c}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.1]
\input{./th_ggpsit2c}
\end{tikzpicture}
\caption{How to transform $\gamma'$ into $\gamma$.}
\label{fig:ggpsit2c}
\end{figure}
We have:
\begin{align*}
C(G,\gamma)&= C(G',\gamma')+ \frac{1}{n(v_g)} \\
&= C(G',\gamma')+1.
\end{align*}
But $\#F(G) = \#F(G') +1 $, hence $\#F(G)\geq 1+ C(G,\gamma)$.
\paragraph{Situation (\ref{it:3YH})}
\label{sec:situation-3}
We suppose now that there are three $\lambda$'s or $H$'s. One can suppose that a $\lambda$ or an $H$ is far from $x$ and from $y$.
Consider first that there is a $\lambda$ far from $x$ and $y$. Let us denote by $v_1$ and $v_2$ the two vertices of the $\lambda$ which belongs to $V^\partial(G)$, by $v$ the vertex of the $\lambda$ which is in $V^\textrm{in}(G)$ and by $e_1$ (resp.~{} $e_2$) the edge joining $v$ to $v_1$ (resp.~{} $v_2$). We consider $G'$ the $D$-graph where the $\lambda$ is replaced by a single strand: the edges $e_1$ and $e_2$ and the vertices $v_1$ and $v_2$ are suppressed. The vertex $v$ is moved to $\partial D^2$ (and renamed $v'$). This is depicted on figure~\ref{fig:2confY}. The coloured $D$-graph $G'$ is non-elliptic and $s(G')<s(G)$ so that we can apply the induction hypothesis and find a cut $\gamma'$ with $\#F(G')\geq 1+ C(G',\gamma')$.
The vertices $v_1$ and $v_2$ have the same colour, by symmetry we may suppose that they are both $\mathrm{\color{darkgreen}green}$. It implies that $v$ and $v'$ are both $\mathrm{\color{darkblue}blue}$.
There are two different configurations:
\begin{itemize}
\item the vertices $v_1$ and $v_2$ lie on $I_\mathrm{\color{darkgreen}green}$,
\item the vertices $v_1$ and $v_2$ lie on $I_\mathrm{\color{darkblue}blue}$.
\end{itemize}
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1]
\input{./rg_2confY}
\end{tikzpicture}
\caption{On the center, the two possible configurations for a $\lambda$, on the sides, the $D$-graphs $G'$ obtained from $G$.}
\label{fig:2confY}
\end{figure}
Let us first suppose that the vertices $v_1$ and $v_2$ lie on $I_\mathrm{\color{darkgreen}green}$. If the cut $\gamma'$ does not cross $v'$ then we can canonically lift it in $G$. This gives us $\gamma$. We have:
\begin{align*}
C(G,\gamma)&= C(G',\gamma')+ \frac{1}{n(v_1)} + \frac{1}{n(v_2)} \\
&= C(G',\gamma')+\frac12 + \frac12.
\end{align*}
But $\#F(G) =\# F(G')+1$, hence $\#F(G)\geq 1+ C(G,\gamma)$.
If the cut $\gamma'$ crosses $v'$, we lift $\gamma'$ in $G$ so that it crosses $e_1$ and $e_2$ (see figure~\ref{fig:ggpsit2c}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.1]
\input{./th_ggpYsit2}
\end{tikzpicture}
\caption{How to transform $\gamma'$ into $\gamma$.}
\label{fig:ggpYsit2}
\end{figure}
In this case we have:
\begin{align*}
C(G,\gamma)&= C(G',\gamma')+ + \frac{1}{n(v)} - \frac1{n(v')} + \frac{1}{n(v_1)} + \frac{1}{n(v_2)} \\
&= C(G',\gamma')+1 -1+ \frac12 + \frac12.
\end{align*}
Hence, $\#F(G)\geq 1+ C(G,\gamma)$.
Now suppose that the vertices $v_1$ and $v_2$ lie on $I_\mathrm{\color{darkblue}blue}$, this implies that $\gamma'$ does not meet $v'$, so that we can lift $\gamma'$ canonically in $G$, this gives us $\gamma$, we have:
\begin{align*}
C(G,\gamma)&= C(G',\gamma')+ \frac{1}{n(v)} - \frac{1}{n(v')} \\
&= C(G',\gamma')+1-1.
\end{align*}
Hence $\#F(G)\geq 1+ C(G,\gamma)$.
We finally consider a $H$ far from $x$ and $y$. We take notation of the figure~\ref{fig:GGsitH} to denote vertices and edges of the $H$, we consider $G'$ the $D$-graph where the the $H$ is simplified (see figure~\ref{fig:GGsitH} for details and notation). The coloured $D$-graph $G'$ is non-elliptic and $s(G')= s(G) -3\times 3 + 2\times 4<s(G)$ so that we can apply the induction hypothesis and find a cut $\gamma'$ with $\#F(G')\geq 1+ C(G',\gamma')$.
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.4]
\input{./th_GGsitH}
\end{tikzpicture}
\caption{How to transform $G$ into $G'$.}
\label{fig:GGsitH}
\end{figure}
Up to symmetry there is only one configuration, therefore we may suppose that $v_1$ is $\mathrm{\color{darkgreen}green}$ and lies on $I_\mathrm{\color{darkgreen}green}$. This implies that $v_2$ and $v_3$ are $\mathrm{\color{darkblue}blue}$ and that $v_4$ is $\mathrm{\color{darkgreen}green}$. Because of the colour condition, the cut $\gamma'$ does not cross $v'_4$ and may cross $v'_3$. If it does not cross $v'_3$, one can canonically lift $\gamma'$ in $G'$ and we have:
\begin{align*}
C(G,\gamma)&= C(G',\gamma')+ \frac{1}{n(v_1)} + \frac{1}{n(v_4)} - \frac{1}{n(v'_4)} \\
&\geq C(G',\gamma')+\frac12 +1 -1. \\
&\geq C(G',\gamma') +\frac12.
\end{align*}
But $\#F(G) =\# F(G')+1$, hence $\#F(G)\geq 1+ C(G,\gamma)$.
If the cut $\gamma'$ crosses $v'_3$, on lift it in $\ensuremath{\mathcal{G}}$ so that it crosses $e_1$ and $e_2$ (see figure~\ref{fig:GGsitHb}).
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.4]
\input{./th_GGsitHb}
\end{tikzpicture}
\caption{How to transform $\gamma'$ into $\gamma$.}
\label{fig:GGsitHb}
\end{figure}
So that we have:
\begin{align*}
C(G,\gamma)&= C(G',\gamma')+ \frac{1}{n(v_1)} + \frac{1}{n(v_3)} - \frac{1}{n(v'_3)} + \frac{1}{n(v_4)} - \frac{1}{n(v'_4)} \\
&\geq C(G',\gamma')+\frac12 +1 -1 + 1 -\frac12. \\
&\geq C(G',\gamma') +1.
\end{align*}
But $\#F(G) =\# F(G')+1$, hence $\#F(G)\geq 1+ C(G,\gamma)$.
\paragraph{Conclusion}
\label{sec:conclusion}
For all situations, using the induction hypothesis we can construct a cut $\gamma$ such that: $\#F(G)\geq 1+ C(G,\gamma)$. This proves the lemma.
\end{proof}
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"redpajama_set_name": "RedPajamaArXiv"
}
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<?php
/*
* To change this license header, choose License Headers in Project Properties.
* To change this template file, choose Tools | Templates
* and open the template in the editor.
*/
/**
* Description of ConfirmStockin
*
* @author 志鹏
*/
class StockTransferConfirmStockin extends WorkflowAbstract {
//put your code here
}
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{
"redpajama_set_name": "RedPajamaGithub"
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Q: Why is the string group not a Lie group? The string group $String(n)$ is by definition a 3-connected cover of $Spin(n)$. This definition determines the homotopy type of the string group.
[In a previous version of this question I screwed up the definition and caused some confusion, see the comments below.]
A common argument is saying that "the string group cannot be a Lie group because it has vanishing $\pi_3$". This is obviously not a complete argument because $(\mathbb{R},+)$ is a nice Lie group with vanishing $\pi_3$.
What is the correct statement about Lie group structures on the string group, and how does one prove it?
A: To follow up, there is now an infinite-dimensional Lie group model of String:
*
*Thomas Nikolaus, Christoph Sachse, Christoph Wockel, A Smooth Model for the String Group, Int. Math. Res. Not. 16 (2013) 3678-3721, doi:10.1093/imrn/rns154, arXiv:1104.4288.
A: As David Roberts is saying it's conceivable the string group could be represented by an infinite dimension manifold. I'm totally agnostic on that, but as I interpret the question it's asking why it's not equivalent (as an H-space?) to a non-compact finite dimensional Lie group (David Roberts also explains that for a compact simply connected Lie group we always have $\pi_3$ non vanishing). I think though the underlying space has cohomology in infinitely many dimensions. Let me illustrate this in the case of $\mathrm{String}(3)$. So we have a Serre spectral sequence for the fibration $K(\mathbb{Z},2)\to \mathrm{String}(3) \to S^3$. Now thinking of $\mathbb{Z}[x]$ as the cohomology ring of $K(\mathbb{Z},2)$, the differential has to be $d:x \mapsto e$, the generator for the cohomology of $S^3$. So using the Leibnitz rule, $x^2\mapsto 2x\otimes e$, $x^3 \mapsto 3x^2\otimes e$... etc. This means that $H^5(\mathrm{String}(3))= \mathbb{Z}/2\mathbb{Z}$, $H^7(\mathrm{String}(3))=\mathbb{Z}/3\mathbb{Z}$... etc
A: The result is that a compact, connected simple Lie group $G$ has $\pi_3(G) = \mathbb{Z}$. Simple covering space or subgroups arguments should get you to $\mathrm{SO}(n)$ which is all that matters. For that matter start with the 1-connected $\mathrm{Spin}(n)$.
[OK, a short train ride later, now I'm home from work. To continue...]
The fibre of the 3-connected cover is a 2-type, and in the case of $\mathrm{Spin}(n)$ this is a $K(\mathbb{Z},2)$, so at the very least, $\mathrm{String}(n)$ can't be finite-dimensional. If one could construct a primitive[1] $PU(\mathcal{H})$-bundle on $\mathrm{Spin}(n)$ whose Dixmier-Douady classs was the generator $\langle -,[-,]\rangle \in H^3(\mathrm{Spin}(n),\mathbb{Z})$, then you would have an infinite-dimensional Lie group model for $\mathrm{String}(G)$ (here $\mathcal{H}$ is a infinite-dimensional separable Hilbert space, $PU(\mathcal{H})$ is then a smooth model for $K(\mathbb{Z},2)$, if we take the norm topology, making it a Banach Lie group).
([1] Primitive in the sense that for the group operations $G\times G\to G$ and $(-)^{-1}:G\to G$ there are bundle maps covering them.)
I don't know if this is possible or not, but I'm sure this idea has occurred to someone before, and since we haven't seen it, there might be a reason (well, I haven't seen it and everyone goes on about $\mathrm{String}_G$ only being a topological group).
|
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{"url":"https:\/\/tex.stackexchange.com\/questions\/490411\/how-can-i-draw-this-spiral-with-vertical-sides-in-tikz","text":"# How can I draw this spiral (with vertical sides) in tikz?\n\nHow can I draw this spiral (with vertical sides) in tikz?\n\nThe only way I would know how, would be to copy and paste code to draw semicircles of different radius, followed by connecting the semicircles one at a time, which would take a very long time.\n\n\u2022 Welcome to TeX.SX. Questions about how to draw specific graphics that just post an image of the desired result are really not reasonable questions to ask on the site. Please post a minimal compilable document showing that you've tried to produce the image and then people will be happy to help you with any specific problems you may have. See minimal working example (MWE) for what needs to go into such a document. \u2013\u00a0L. F. May 12 at 5:45\n\nI hope nobody gets mad at me for reopening the question but I feel that for the first question one may make exceptions. Instead of copying you could use foreach.\n\n\\documentclass[tikz,border=3.14mm]{standalone}\n\\begin{document}\n\\begin{tikzpicture}\n\\def\\vlength{3}\n\\def\\hstep{0.6}\n\\def\\Xmax{9}\n\\fill[gray!40] ({-(\\Xmax+3)*\\hstep},-\\vlength)\nrectangle (\\hstep\/2,\\vlength);\n\\draw[thick,dashed] (-1*\\hstep,\\vlength)\n-- (-1*\\hstep,-\\vlength)\nforeach \\X in {1,...,\\Xmax}\n{\\ifodd\\X\n-- ({\\hstep*(\\X+1)},\\vlength)\n\\else\n-- ({-\\hstep*(\\X+1)},-\\vlength)\n\\fi}\n-- ({-\\hstep*(\\Xmax+2)},-1*\\vlength);\n\\draw[thick] (-1*\\hstep,\\vlength)\n-- (0,-\\vlength)\n(\\hstep,-1*\\vlength)\nforeach \\X in {1,...,\\Xmax}\n{\\ifodd\\X\n-- (\\hstep*\\X,\\vlength)\n\\else\n-- ({-\\hstep*\\X},-1*\\vlength)\n\\fi} -- ({-\\hstep*(\\Xmax+1)},-1*\\vlength)\n\\end{tikzpicture}\n\\end{document}\n\n\nOr with pgf keys.\n\n\\documentclass[tikz,border=3.14mm]{standalone}\n\\begin{document}\n\\begin{tikzpicture}[hstep\/.initial=0.6,vlength\/.initial=3,\nXmax\/.store in=\\Xmax,Xmax=9]\n\\fill[gray!40] ({-(\\Xmax+3)*\\pgfkeysvalueof{\/tikz\/hstep}},-\\pgfkeysvalueof{\/tikz\/vlength})\nrectangle (\\pgfkeysvalueof{\/tikz\/hstep}\/2,\\pgfkeysvalueof{\/tikz\/vlength});\n\\draw[thick,dashed] (-1*\\pgfkeysvalueof{\/tikz\/hstep},\\pgfkeysvalueof{\/tikz\/vlength})\n-- (-1*\\pgfkeysvalueof{\/tikz\/hstep},-\\pgfkeysvalueof{\/tikz\/vlength})\nforeach \\X in {1,...,\\Xmax}\n{\\ifodd\\X\n-- ({\\pgfkeysvalueof{\/tikz\/hstep}*(\\X+1)},\\pgfkeysvalueof{\/tikz\/vlength})\n\\else\n-- ({-\\pgfkeysvalueof{\/tikz\/hstep}*(\\X+1)},-\\pgfkeysvalueof{\/tikz\/vlength})\n\\fi}\n-- ({-\\pgfkeysvalueof{\/tikz\/hstep}*(\\Xmax+2)},-1*\\pgfkeysvalueof{\/tikz\/vlength});\n\\draw[thick] (-1*\\pgfkeysvalueof{\/tikz\/hstep},\\pgfkeysvalueof{\/tikz\/vlength})\n-- (0,-\\pgfkeysvalueof{\/tikz\/vlength})\n(\\pgfkeysvalueof{\/tikz\/hstep},-1*\\pgfkeysvalueof{\/tikz\/vlength})\nforeach \\X in {1,...,\\Xmax}\n{\\ifodd\\X\n-- (\\pgfkeysvalueof{\/tikz\/hstep}*\\X,\\pgfkeysvalueof{\/tikz\/vlength})","date":"2019-09-20 15:59:38","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\": 0.8608313798904419, \"perplexity\": 3039.8225187773696}, \"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-2019-39\/segments\/1568514574050.69\/warc\/CC-MAIN-20190920155311-20190920181311-00079.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.physicsforums.com\/threads\/linear-regression-linear-least-squares-least-squares-non-linear-least-squares.95985\/","text":"# Linear Regression, Linear Least Squares, Least Squares, Non-linear Least Squares\n\n1. Oct 21, 2005\n\n### hotvette\n\nIt seems to me that Linear Regression and Linear Least Squares are often used interchangeably, but I believe there to be subtle differences between the two. From what I can tell (for simplicity let's assume the uncertainity is in y only), Linear Regression refers to the general case of fitting a straight line to a set of data, but the method of determining optimal fit can be most anything (e.g. sum of vertical differences, sum of absolute value of vertical differences, max vertical difference, sum of square of vertical differences, etc.), whereas Linear Least Squares refers to a specific measure of optimal fit, namely, sum of the square of vertical differences.\n\nActually, it seems to me that Linear Least Squares doesn't necessarily mean that you are fitting a straight line to the data, it just means that the modelling function is linear in the unknowns (e.g. $y = ax^2 + bx + c$ is linear in a, b, and c). Perhaps it is established convention that Linear Least Squares does, in fact, refer to fitting a straight line, whereas Least Squares is the more general case?\n\nLastly, Non-linear Least Squares refers to cases where the modelling function is not linear in the unknowns (e.g. $y = e^{-ax^b}$, where a,b are sought).\n\nIs my understanding correct on this?\n\nLast edited: Oct 21, 2005\n2. Oct 21, 2005\n\n### Dr Transport\n\nYou can \"linearize\" $$y = e^{-ax^b}$$ so that you are fitting an $$-x^b$$ as opposed to an exponential by fitting $$\\ln(y) = -ax^b$$. This is done all the time, maybe not linear, but a much easier function to fit in the long run. It is still non-linear in the strictest sense.\n\nYou are correct in saying that least squares is a more general case. In theory you can fit any polynomial, exponential, logarithmic, etc....by using least squares.\n\nLast edited: Oct 21, 2005\n3. Oct 22, 2005\n\n### Dr Transport\n\nIn poking around some of my numerical analysis texts I was reminded of the Levenberg-Marquardt method of fitting curves to data. It seems to be one of the more robust nonlinear least squares methods.\n\nhttp:\/\/www.library.cornell.edu\/nr\/bookcpdf\/c15-5.pdf [Broken]\n\nTake a look....\n\nLast edited by a moderator: May 2, 2017\n4. Oct 27, 2005\n\n### hotvette\n\nThanks. I agree re $y = e^{-ax^b}$. Bad choice on my part. I've read several web articles on Levenberg-Marquardt Method but don't seem to quite follow. I've seen what appears to be 2 versions, one for general unconstrained optimization where you are minimizing an objective function, which for least squares would be $\\epsilon^2 = \\Sigma(f(x_i)-y_i)^2$.\n\n$$[x^{k+1}] = [x^k] - \\alpha [H(x^k) + \\beta ]^{-1}[J(x^k)]$$\n\nWhere x is the unknown parameter list, $H(x^k)$ is the Hessian matrix of second derivatives of the objective function $\\epsilon^2$ with respect to the unknown parameters, $\\alpha, \\beta$ are parameters that control the stability of the iterative solution, and $J(x^k)$ is the Jacobian of first derivatives of the objective function with respect to the unknown parameters. I've actually been successful in using this for non-linear least squares problems, but convergence is extremely sensitive to $\\alpha, \\beta$. In this version, the values for y are used within the objective function. Thus, we have a single equation in n unknowns (depending on the complexity of the fitting function) that we are trying to minimize.\n\nI've also seen articles talking specifically to using Levenberg-Marquardt for non-linear least squares, using a solution technique NOT requiring 2nd derivatives and is completely analogous to the linear least squares solution:\n\n$$a^{k+1} = a^k - \\alpha {[J^TJ + \\beta I]^{-1}J^Tf(x)}$$\n\nWhere a is the unknown parameters and J is the Jacobian of $y_i = f_i(x_i)$ with respect to the unknown parameters. In this form, the 2nd derivative isn't used and I've seen comments to the effect that the 2nd derivatives can lead to unstable situations. This version I can't get to work at all and I suspect there is something wrong with my interpretation.\n\nI'd appreciate some help with where I'm going astray with the 2nd version. It seems strange to me that one implementation uses 2nd derivatives and the other doesn't. Many thanks!\n\nLast edited: Oct 27, 2005\n5. Oct 27, 2005\n\n### hotvette\n\nI found the discrepancy. The $f(x)$ in:\n\n$$a^{k+1} = a^k - \\alpha {[J^TJ + \\beta I]^{-1}J^Tf(x)}$$\n\nis really $f(x)-y$. I was just using $y$. I used it successfully and the convergence wasn't nearly as dependent on $\\alpha$ and $\\beta$ as it was using the first method. I'm amazed that 2 different ways of supposedly using the same method can produce such different results, not in the final answer, but in the complexity of setup (i.e. computing 2nd derivatives vs not) and stability of the solution. I guess that comment I read about 2nd derivatives contributing to instability was right. The net seems to be that the non-linear least squares version of Levenberg-Marquardt is much more stable than the general version for unconstrained optimization. Amazing.\n\nLast edited: Oct 27, 2005","date":"2018-02-25 16:06:40","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\": 0.8158004879951477, \"perplexity\": 356.42268051224124}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891816647.80\/warc\/CC-MAIN-20180225150214-20180225170214-00463.warc.gz\"}"}
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\section{Introduction}\label{sec_intro}
\IEEEPARstart{W}{ith} incredible advances in \tip{AI} fields, there is an increasing demand for low-power audio \tip{IoT} devices that process human speech on the device without data transmission to the cloud. These smart devices are required to ensure always-on operation, real-time response, small form factor, and longer battery lifetime. As such, an ultra-low-power wake-up functionality is being highlighted with rapidly growing popularity because it allows hierarchical power gating of increasingly complex tasks for audio \tip{IoT} nodes. \tip{KWS} and \tip{VAD} are widely used user-interactive methods to wake-up smart devices. \tip{KWS} is used to detect predefined keywords in an audio stream while \tip{VAD} detects when a human voice is present.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_intro.jpg}
\caption{Processing stages for \tip{KWS} in an audio \tip{IoT} device.}\label{fig:intro}
\end{center}
\vspace{-5mm}
\end{figure}
Fig. \ref{fig:intro} shows the typical processing stages for \tip{KWS}. The user says a keyword into the microphone of an edge device such as a remote control or wireless earbud. The microphone output is further processed by a \tip{FEx} which generates frequency-selective \tip{FVs} that are continuously streamed to a \tip{DNN}-based classifier. The classifier outputs the probability scores of different keywords. \tip{IoT} devices benefit from a tiny form factor and the use of a small battery such as a coin cell, e.g., for smart tags. Generally a $<$100\,$\mu$W system-level power is desirable including not only the \tip{KWS} IC itself but also the microphone and other system components. Moreover, a low-latency response is desired considering a \tip{KWS}-driven hierarchical processing system used in an interactive environment. For example, a study on the perception of self-generated speech showed that a delay exceeding 20\,ms becomes disturbing for users \cite{stone1999tolerable}.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{Figures/Fig_software.jpg}
\caption{Architecture of the KWS software model (left) and simulated KWS accuracy (right).}\label{fig:software}
\end{center}
\vspace{-5mm}
\end{figure*}
A 12-class \tip{KWS} IC that includes the whole processing chain starting from the \tip{ADC} to the \tip{DNN} classifier \cite{giraldo2019kws} reported that the \tip{FEx} is the most power-hungry stage accounting for 40\% of the power dissipation in the entire IC. To reduce the power budget of edge devices thereby facilitating longer battery lifetime or smaller battery size, various circuit design techniques have been proposed for both \tip{KWS} and \tip{VAD} ICs. However, most of them traded off between power and latency. In \cite{oh2019vad}, a 142\,nW \tip{VAD} IC using sequential mixer-based \tip{FEx} was proposed where the operational principle is similar to that used for bio-impedance sensors \cite{kim2020bioz, ha2019bioz}. However, this sequential frequency scanning is too undersampled for the \tip{KWS} and results in a 512\,ms latency for \tip{VAD}. In \cite{shan2021kws}, a serialized digital \tip{FEx} was used in the \tip{KWS} IC where the processing stages are pipelined. Although this IC consumed only 510\,nW, its latency was limited to 64\,ms and it needed an off-chip 16-bit \tip{ADC}, had only 2KB memory for binary \tip{CNN} thereby its accuracy was only reported for 5 keywords.
Another approach is the use of an analog voltage-domain \tip{FEx} which exploits low-power analog circuits to achieve both low-power and low-latency responses. The processing chain of the analog \tip{FEx} typically consists of a multi-channel \tip{BPF}, a \tip{HWR} or a \tip{FWR}, and an \tip{ADC}. Here, the speed requirement of \tip{ADC} is highly relaxed to 10\,ms-to-100\,ms (10\,Hz-to-100\,Hz), which corresponds to the size of frame shift in audio signal processing. This is possible because the output of rectifier represents the magnitude response of the input speech and thus it is a low-frequency signal. Previous works that used a voltage-domain analog \tip{FEx} and a back-end classifier to implement \tip{VAD} \cite{badami2016vad, yang2019vad} and \tip{KWS} \cite{wang2021kws} tasks, reported 205\,nW-to-1\,$\mu$W power dissipation and 10\,ms-to-100\,ms latency. However, voltage-domain analog \tip{FEx} is unfriendly for CMOS technology scaling, thereby the power efficiency of analog approaches are predicted to be degraded in advanced nanometer-scale processes. This is because $V_\text{DD}$ is scaling down faster than $V_\text{TH}$, thus voltage-domain signals have less headroom. Reduced headroom results in reduced maximal signal swing, which in turn, reduces the \tip{DR} that is critical for keeping \tip{KWS} accuracy high across a range of audio amplitude levels. Furthermore, the intrinsic gain ($g_\text{m}r_\text{o}$) of the transistors is also degraded, leading to the DC gain reduction in analog feedback loops. This issue can be mitigated with a larger transistor length, gain boosting, or multi-stage amplifiers; however, these approaches come with costs in area, power, and bandwidth.
To this end, we propose a time-domain analog \tip{FEx} that exploits the scaling-friendly nature of the ring-oscillator. It is the first silicon-verified ring-oscillator-based audio \tip{FEx} reported to date. When integrated with an on-chip \tip{RNN} classifier, the resulting IC demonstrates power-efficient \tip{KWS} capability. The \tip{FEx} circuits extensively use time-domain signal representation techniques including \tip{PWM} and \tip{PFM}, therefore it does not suffer from headroom degradation and its associated signal swing loss issue. In other words, it is more suitable for low-supply implementation than voltage-domain designs. The ring-oscillator-based circuit utilizes its infinite DC gain characteristic when configured as a time-domain integrator \cite{drost2012ringfilter}. As such, the transfer function of time-domain \tip{FEx} circuits such as \tip{BPF} are not affected by the degradation of the intrinsic gain of transistors. Overall, the proposed \tip{KWS} IC consumes 23\,$\mu$W and {has only} 12.4\,ms {inference} latency on a 12-class \tip{GSCD}~\cite{gscd}.
There have been similar approaches to implement the oscillator-based \tips{BPF} for audio \tip{IoT} applications \cite{Gutierrez2019vcofilter, Goux2020ilofilter}. However, none of them proposed a clear design strategy to implement a time-domain rectifier or demonstrated an audio classification task using the fabricated oscillator-based \tips{BPF}.
This article is an extension of a previous work presented in \cite{kim2022kws}. The integrated chip also includes a switched-capacitor energy harvester circuit, a voltage reference, and a low-dropout regulator. However, {in this paper,} we focus on the new circuit techniques of the \tip{KWS} core. The paper is organized as follows. Section~\ref{sec_software} presents the software modeling of the \tip{KWS} modules in this work. Section~\ref{sec_KWS_IC} covers the description for the overall architecture and design details of the implemented circuits and Section\,\,\ref{sec_measurement} presents measurement results and performance summary of the prototype chip. Section\,\,\ref{sec_conclusion} concludes this work.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{Figures/Fig_overall.jpg}
\caption{Overall architecture of the proposed KWS IC.}\label{fig:overall}
\end{center}
\vspace{-5mm}
\end{figure*}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_timing.jpg}
\caption{Timing diagram of the GRU-FC classifier computation according to the input feature vectors.}\label{fig:timing}
\end{center}
\vspace{-5mm}
\end{figure}
\section{Software Modeling}\label{sec_software}
The architecture of our \tip{KWS} IC was developed based on prior silicon cochlea and edge audio-inference ICs \cite{yang2019vad, badami2016vad, wang2021kws, kim2022review, lyon1988cochlea}. We implemented a Python model of the \tip{KWS} IC including the analog \tip{FEx} as shown in Fig.~\ref{fig:software}. Our model implements a bank of \tips{BPF} (second-order butterworth filter) inspired by modeling of biological cochlea \cite{lyon1988cochlea}, a \tip{FWR} ($|x|$), an averaging block (low-pass filter), a subsampler, and a quantizer. The subsampler was added to realize the relaxed speed requirement of the quantizer as discussed in Section~\ref{sec_intro}. As the \tip{GSCD} samples have a 16\,kHz sampling rate, the number of averaged samples and the rate of subsampling operation were selected to match the target frame shift size (16\,ms in our work) of the audio \tip{FV}. In contrast to prior analog \tips{FEx} \cite{yang2019vad, badami2016vad}, we added additional \tip{FV} processing stages before the \tip{FV} is fed to the classifier. These stages consist of 1) a logarithmic compression stage inspired by the adaptive gain compression mechanism of biological cochleas and 2) an input normalization stage which is widely used in \tip{DNN} models, both of which help to improve the \tip{KWS} accuracy on \tip{GSCD}. We chose a \tip{GRU}-based \tip{RNN} classifier for the last stage of our \tip{KWS}, as it has been frequently used in automatic speech recognition tasks \cite{amodei2016DeepSpeech}.
Fig.~\ref{fig:software} shows the accuracy of the software simulation starting from the baseline model which does not include the compressor and normalizer stages. As seen on the right graph of Fig.~\ref{fig:software}, the baseline model achieved 77.89\% which increases to 91.35\% KWS accuracy on the 12-class \tip{GSCD} test set with the addition of the 2~stages. The following design parameters were chosen for our software model: First, we used a 16-channel \tip{BPF} which was also used in previous works \cite{yang2019vad, badami2016vad, wang2021kws}, and a $Q$-factor of 2 for the \tips{BPF}. Second, the center frequencies of the bank of \tips{BPF} are distributed according to the Mel scale (from 100\,Hz to 8\,kHz). The 8\,kHz value is also the bandwidth of the analog front-end presented in \cite{giraldo2019kws}. Note that we oversampled the input speech 2$\times$ (from 16\,kHz to 32\,kHz sampling rate), to avoid the 8\,kHz center frequency overlapping with Nyquist frequency (8\,kHz with a 16\,kHz sampling rate). Third, a 12-bit quantizer (before logarithmic compression in Fig.~\ref{fig:software}), Fourth, a 16\,ms frame shift, the same value used in \cite{giraldo2019kws, shan2021kws}, Fifth, a 10-bit output logarithmic compressor and a 14-bit normalized feature vector ($\text{FV}_\text{Norm}$) that is fed to a 2-layer 48-hidden-unit \tip{GRU} and an \tip{FC} layer. Sixth, 14-bit and 8-bit quantizations were applied to the activations and weights respectively. The baseline model accuracy shown in Fig.~\ref{fig:software} would be higher if using floating-point activations because the 14-bit quantization (6-bit integral part and 8-bit fractional part) cannot cover the dynamic range of the 12-bit unsigned quantizer output.
\section{KWS IC with Time-Domain Analog FEx}\label{sec_KWS_IC}
The overall architecture of our \tip{KWS} IC is shown in Fig. \ref{fig:overall}. It is composed of an analog front-end and a digital back-end. The analog front-end is designed to match our software model. The first stage of the analog front-end is a \tip{FLL}-based \tip{VTC} (Section~\ref{sec_vtc}). It features a nested analog \tip{FLL} circuit that linearizes the voltage-to-frequency response of the \tip{VCO}. A voltage-domain audio input from a \tip{SE} \tip{Mic} is converted into a time-domain multi-phase \tip{PWM} output through the \tip{VTC}. The second stage of the analog front-end is a 16-channel rectifying \tip{BPF} (Rec-\tip{BPF}). Each channel has a time-domain second-order \tip{BPF} (Section~\ref{sec_bpf}) featuring an inherent \tip{FWR} functionality. The output of each \tip{BPF} channel are \tip{PWM} signals and they are further converted into \tip{PFM} signals through a \tip{SRO}-based rate-encoder.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{Figures/Fig_vtc.jpg}
\caption{0.5 V supply single-ended (SE) input \tip{FLL}-based \tip{VTC}.}\label{fig:vtc}
\end{center}
\vspace{-5mm}
\end{figure*}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_vtc_sub-osc_sim.jpg}
\caption{Simulation result of the supply-temperature compensation in sub-\tip{OSC}.}\label{fig:vtc_sub-osc_sim}
\end{center}
\vspace{-5mm}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_vtc_sim.jpg}
\caption{Simulation result of the \tip{FLL}-based \tip{VTC}.}\label{fig:vtc_sim}
\end{center}
\vspace{-5mm}
\end{figure}
In the digital back-end, the \tip{PFM} signals are fed into a digital differentiator ($1-z^{-1}$). Here, the signal path from the \tip{SRO} to the digital differentiator builds a first-order $\Delta\Sigma$ \tip{TDC} \cite{elshazly2014sro} which corresponds to the quantizer in our software model. The output of the digital differentiator is further processed through subsequent stages including a decimation filter which performs the averaging and subsampling operation in our software model. It also includes an offset subtractor ($\beta$) which removes the free-running frequency component of the \tip{SRO}, a per-channel gain calibrator ($\alpha$) which corrects the inter-channel gain mismatch, a logarithmic \tip{LUT}, and an input normalizer. Both $\mu$ and $\sigma$ shown in Fig.~\ref{fig:overall} are respectively the mean and the standard deviation of the output of logarithmic \tip{LUT} ($\text{FV}_\text{Log}$ in Fig.~\ref{fig:overall}) from our chip with the \tip{GSCD} training set. With the normalizer, $\mu$ is subtracted from the $\text{FV}_\text{Log}$ and the resulting subtracted output is multiplied by a value $1/\sigma$. The resulting output of the $\text{FV}_\text{Norm}$ is a 16-channel signed 14-bit \tip{FV} which is generated every 16\,ms of frame shift as shown in Fig.~\ref{fig:timing}. For each \tip{FV}, a 2-layer \tip{GRU} \tip{RNN} and 1-layer \tip{FC} digital accelerator outputs the most probable keyword over 12 classes with a 12.4\,ms latency (see Fig.~\ref{fig:timing}).
\subsection{Voltage-to-Time Converter}\label{sec_vtc}
Previous \tip{VAD} and \tip{KWS} ICs have used mainly a differential output microphone interface \cite{giraldo2019kws, yang2019vad, wang2021kws}. However, commercial off-the-shelf differential output \tip{MEMS} microphones typically consume $>$100 $\mu$W. To realize a system-level low-power audio \tip{IoT} device, a low-power \tip{SE}-interface \tip{MEMS} microphone is preferred because it consumes as little as $\sim$10\,$\mu$W (e.g., InvenSense ICS-40310 \cite{ics-40310}). But this approach makes it difficult to obtain good linearity because \tip{SE} signals do not reject even-order harmonics. In general, a linear \tip{FEx} is preferred as it makes training the back-end \tip{DNN} classifier easier and enhances the spectral purity of an audio signal with minimal harmonics and intermodulation distortions. Particularly, the design of a \tip{SE}-input ring-oscillator-based \tip{VTC} circuit becomes even more difficult because \tips{VCO} exhibit poor linearity compared to a conventional voltage-domain \tips{OTA}.
In this work, we propose to use a nested analog \tip{FLL} around the ring-\tip{VCO} to enhance the linearity of the \tip{VTC}. Fig. \ref{fig:vtc} shows the architecture and transistor-level schematic of the \tip{FLL}-based \tip{VTC}. The fundamental design principle is adopted from the ring-oscillator-based \tip{LPF} presented in \cite{drost2012ringfilter}, however capacitive coupling is used with $C_\text{IN}$ to isolate the DC bias of the \tip{VTC} from the microphone. Therefore, its core operation is similar to the capacitively-coupled voltage amplifier \cite{Harrison2003CCIA}, but the \tip{VTC} circuit converts the input voltage into the multi-phase \tip{PWM} output instead of voltage. A pseudo-differential architecture is implemented using a dual-\tip{VCO} structure along with the phase detector \cite{drost2012ringfilter}. One input port of the \tip{VTC} is connected to the \tip{SE} microphone and the other input port is tied to ground. The 15-array phase detector receives a 15-phase frequency-modulated signal out of the \tips{VCO} and generates a 15-phase \tip{PWM} output which represents the phase difference of the \tips{VCO}. Note that exploiting the multi-phase \tip{PWM} scheme pushes spurious \tip{PWM} tones to higher frequency range without necessitating a higher running frequency of the \tip{VCO} \cite{drost2012ringfilter}. The outputs of the phase detector are buffered with two inverters and used to close the feedback loop through a 15-array thermometer-coded capacitive \tip{DAC}. Since the input node of the \tip{VCO} acts as a virtual-ground, the generated multi-phase \tip{PWM} signal becomes a time-domain approximated input voltage where the amplitude is encoded into the duty-cycle of \tip{PWM}. A variation-tolerant pseudo-resistor \cite{djekic2018pseudores} with a voltage reference $V_\text{B}$ sets the common-mode DC bias voltage of the \tips{VCO}.
The \tip{FLL}-based \tip{VCO} includes a single-branch current comparator \cite{jang2018pll} to operate the analog \tip{FLL}. As shown in the schematic diagram of the \tip{FLL}-\tip{VCO} in Fig. \ref{fig:vtc}, an input current generator drives the input voltage to $R_\text{REF}$ to generate a low-side current signal $I_\text{IN}=V_\text{IN}/R_\text{REF}$ where $A_{2}$ amplifier is designed to have a 34\,dB gain. A high-side current $I_\text{SC}=15V_\text{X}C_\text{REF}f_\text{FLL}$ flows through a 15-phase switched-capacitor operation. Here, the multi-phase nature of a ring-oscillator is fully utilized to apply the multi-phase interleaving technique at the $V_\text{X}$ node to minimize the voltage ripple caused by the switched-capacitor operation \cite{jang2018pll}. The low-$V_\text{TH}$ devices are used to facilitate 0.5 V low-supply operation for the implementation of the switched-capacitor circuit. The \tip{FLL} feedback formed through the $A_{1}$ amplifier with a 27\,dB gain, sub-\tip{OSC}, and switched-capacitor circuit ensures that $V_\text{X}$ equals the reference voltage $V_\text{REF}$ while also ensuring $I_\text{IN}$ equals $I_\text{SC}$. As a result, the output frequency of the \tip{FLL}-based \tip{VCO} is set as in \eqref{eq:fll}.
\begin{align}
f_\text{FLL}&=\frac{V_\text{IN}}{15R_\text{REF}C_\text{REF}V_\text{REF}} \label{eq:fll}\\
K_\text{FLL-VCO}&
=\frac{\partial f_\text{FLL}}{\partial V_\text{IN}}
=\frac{1}{15R_\text{REF}C_\text{REF}V_\text{REF}}
\label{eq:K_fll}
\end{align}
Since $f_\text{FLL}$ is represented by the input voltage $V_\text{IN}$ and reference parameters such as $R_\text{REF}$, $C_\text{REF}$, and $V_\text{REF}$, it leads to a \tip{FLL}-aided linearization of the \tip{VCO} as derived in (\ref{eq:K_fll}), where $K_\text{FLL-VCO}$ corresponds to the voltage-to-frequency tuning gain. This is because the value of passive elements ($R_\text{REF}$, $C_\text{REF}$, and $V_\text{REF}$) has no dependency on the input signal amplitude ($V_\text{IN}$).
The 3\,dB bandwidth of the \tip{VTC} circuit is given as below, which is similar to the equation of resistive-input and current-feedback ring-oscillator-based filter \cite{drost2012ringfilter}
\begin{align}
f_\text{3dB,VTC}=\frac{1}{2\pi}K_\text{FLL-VCO}K_\text{PD}\beta_\text{DAC}\\
\beta_\text{DAC}=\frac{15C_\text{DAC}}{C_\text{IN}+15C_\text{DAC}}V_\text{DD}
\end{align}
where $K_\text{PD}$ is the gain of phase detector and $\beta_\text{DAC}$ is the feedback factor (time-to-voltage). We designed $f_\text{3dB,VTC}$ to be 17\,kHz when the nested \tip{FLL} feedback has a 158\,kHz gain-bandwidth product which contributes as a non-dominant pole to the overall negative feedback loop of the \tip{VTC} circuit. As shown in Fig.~\ref{fig:vtc_sim}, the stability of the \tip{VTC} is verified with a transient simulation.
To allow 0.5\,V low-supply operation for the sub-\tip{OSC}, a varactor-controlled supply-temperature compensator is proposed. This is achieved by sizing the diode-connected transistor $P_{1}$ so that $V_\text{VAR}$ becomes proportional-to-absolute-temperature (PTAT) \cite{seok2012vref}. Therefore, the capacitance of MOS-varactors in the delay cells \cite{Zhao2020VCOADC} adaptively stabilizes the temperature drift of the ring-oscillator frequency. For example, if the temperature increases, then $V_\text{VAR}$ also increases, therefore the MOS-varactors are further turned on. This effect negates the frequency increase of the ring-oscillator with temperature increase. Low-$V_\text{TH}$ MOS capacitors are used to further enhance the varactor effect. In addition, instead of configuring the sub-\tip{OSC} as controlled by the gate voltage of $P_{2}$ only, the $N_{1}-P_{1}$ path is added to reduce $V_\text{DD}$ sensitivity based on the fact that $V_\text{GS,N1}$ is less sensitive to $V_\text{DD}$ than $V_\text{SG,P2}$. The simulation results in Fig. \ref{fig:vtc_sub-osc_sim} show that with the proposed techniques, the supply-temperature variation of the sub-\tip{OSC} is reduced by 19.98$\times$ in the worst case. Note that the baseline sub-\tip{OSC} refers to the \tip{OSC} circuit assuming that the added compensation circuits (marked as red color in Fig. \ref{fig:vtc}) are removed. In this case, $V_\text{CTRL}$ is connected to the gate of the $P_{2}$ transistor and therefore the frequency tuning curve of sub-\tip{OSC} becomes decreasing function as $V_\text{CTRL}$ increases. Fig.~\ref{fig:vtc_sim} shows the post-layout simulation result of the \tip{VTC}. The plotted graph represents the multi-phase \tip{PWM} signal of \tip{VTC} output ($VTC_\text{P}-VTC_\text{N}$). The designed \tip{VTC} converts a voltage-domain input into a time-domain \tip{PWM} output while ensuring $<$-70 dB distortion for dominant harmonics (second and third) even with a \tip{SE} input. The \tip{PWM} tones at higher frequencies are filtered out at the following \tip{BPF} stage.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_sro-integ.jpg}
\caption{\tip{SRO} as an ideal $\phi$-to-$\phi$ integrator.}\label{fig:sro-integ}
\end{center}
\vspace{-5mm}
\end{figure}
\subsection{Time-Domain Band-Pass Filter}\label{sec_bpf}
Fig. \ref{fig:sro-integ} shows a conceptual diagram of using the switched-ring-oscillator (SRO) \cite{elshazly2014sro} as an ideal $\phi$-to-$\phi$ integrator. The \tip{SRO} switches its running frequency between $f_\text{Low}$ and $f_\text{High}$ according to the incoming input \tip{PWM} signal. The averaged value of \tip{SRO} frequency is proportional to the duty-cycle of the input \tip{PWM} signal. If the input \tip{PWM} signal is configured as a multi-phase format, the possible number of running frequencies also increases. When the dual-\tip{SRO} is implemented in a pseudo-differential manner, the output phase difference $\Delta\phi_{OUT}$ becomes an accumulated (or integrated) input phase difference $\Delta\phi_{IN}$ over time. Specifically, this integral procedure flows as following. \textit{Input Phase} $\rightarrow$ \textit{SRO Frequency} $\rightarrow$ \textit{SRO Phase (+Integral)} $\rightarrow$ \textit{Output Phase}. The phase mathematically represents an integral amount of the frequency within an oscillator. This time-domain accumulation process allows an integral of the signal without boundary as long as the \tip{SRO} oscillates, unlike voltage-domain designs that saturate due to headroom. In other words, it shows an infinite DC gain and acts as a true lossless integrator regardless of the intrinsic gain of transistor or supply voltage level \cite{drost2012ringfilter}. As shown in the lowermost description of Fig. \ref{fig:sro-integ}, the $\phi$-to-$\phi$ transfer function is described by $K_\text{SRO}/s$ where $K_\text{SRO}$ is the switching gain of a \tip{SRO}.
Fig. \ref{fig:bpf-concept} shows a conceptual diagram for the implementation of a time-domain $\phi$-to-$\phi$ \tip{BPF}. It adopts the two-integrator-loop Tow-Thomas biquad topology \cite{tow1968biquad,thomas1971biquad} using \tip{SRO} as a lossless integrator ($\omega_{0}/s$). The \tip{PD} extracts the phase difference between two input signals and outputs the phase difference in a \tip{PWM} signal. The output of \tip{PD} is used to close the feedback loops where the inner feedback loop ensures the desired $Q$ factor and the outer feedback loop generates the high-pass shape of the \tip{BPF}. Overall, the time-domain \tip{BPF} receives the \tip{PWM} input and generates the \tip{PWM} output. Note that an external clock $f_\text{REF}$ is fed to the \tip{PD} in Fig. \ref{fig:bpf-concept} since it is represented as a simplified half-circuit diagram. If the two \tips{BPF} are placed in parallel to work as a pseudo-differential configuration as shown in Fig. \ref{fig:sro-integ}, the external clock $f_\text{REF}$ is no longer needed and the \tip{BPF} operates in a fully asynchronous way. The same consideration for a pseudo-differential topology also applies to the \tip{VTC} design as described in Section \ref{sec_vtc}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_bpf-concept.jpg}
\caption{Block diagram of a time-domain second-order \tip{BPF} using \tip{SRO} as a core building block with a half-circuit representation.}\label{fig:bpf-concept}
\end{center}
\vspace{-5mm}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_bpf.jpg}
\caption{0.5 V supply time-domain rectifying \tip{BPF}.}\label{fig:bpf}
\end{center}
\vspace{-5mm}
\end{figure}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{Figures/Fig_fll-bias.jpg}
\caption{Schematic of \tip{FLL}-based bias generator and \tip{SRO}.}\label{fig:fll-bias}
\end{center}
\vspace{-5mm}
\end{figure*}
Fig. \ref{fig:bpf} shows the block diagram of the proposed time-domain \tip{BPF}. It receives the multi-phase \tip{PWM} output of the \tip{VTC} as an input signal and does not require an external clock. It incorporates four \tips{SRO} and two \tips{PFD}. The outputs of the \tip{BPF} are two single-phase \tip{PWM} signals. The two \tips{PFD} implemented in the \tip{BPF} offers an inherent rectification function in time-domain, which will be discussed in Section \ref{sec_rectifier}. A local \tip{FLL}-based bias generator provides the required bias voltage which are shared over the four \tips{SRO}. This bias voltage is different over 16-channel \tip{BPF} bank to set different center frequencies. Note that the outputs of first \tip{PFD} are crossed and connected to the \tips{SRO} with opposite polarities, to realize a subtraction function.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_fwr-pfd.jpg}
\caption{Proposed time-domain \tip{FWR} with operational principle of the \tip{PFD}.}\label{fig:fwr-pfd}
\end{center}
\vspace{-5mm}
\end{figure}
Fig. \ref{fig:fll-bias} shows a schematic of the \tip{FLL}-based bias generator and \tips{SRO} used in our \tip{BPF} design. The \tip{SRO} receives time-domain \tip{PWM} signals as input, such as \tip{VTC} and \tip{PFD}. All the \tip{PWM} signals are summed at the internal node of the \tip{SRO} and these signals drive the buffers which act as an array of current-mode \tips{DAC}. Therefore, the output frequency of \tip{SRO} is proportional to the sum of incoming \tip{PWM} signals. To realize different switching gains of \tip{PWM} inputs, switching transistors are differently sized. The unit current for current-\tip{DAC} operation is provided by a local \tip{FLL} circuit. The \tip{FLL} acts as a bias generator for realization of per-channel center frequency designs in \tips{BPF}, using a replica biasing scheme. As shown in Fig. \ref{fig:fll-bias}, the bias voltage $V_\text{VAR}$ is generated from a diode-connected pFET in the FLL-bias circuit. Therefore, the current-\tip{DAC} in \tip{SRO}\textsubscript{1-2} operates as a current mirror when $V_\text{VAR}$ is shared over the four \tips{SRO} from the \tip{FLL}-bias circuit. This means that the switching gain of each \tip{PWM} input signal is determined by $f_\text{FLL}$ and the sizing ratio of the current-mode \tip{DAC}. For example, the switching gain of \tip{VTC}-port is $K_\text{IN}f_\text{FLL}$ and the switching gain of \tip{PFD}\textsubscript{2}-port fed into the \tip{SRO}\textsubscript{2} is $K_\text{2}f_\text{FLL}$. Note that we adopt the same circuit structure from the sub-\tip{OSC} circuit in Section \ref{sec_vtc} which allows the \tip{BPF} to work at 0.5 V low-supply voltage. As discussed in \eqref{eq:fll}, the locking frequency of the \tip{FLL} circuit is proportional to $1/C_\text{REF}$. To cover the target range of \tip{BPF} center frequencies ranging from 100 Hz to 8 kHz in our design, a coarse-fine approach is used. The output of \tip{SRO} is divided coarsely by $N$ times using \tips{D-FF} and the $C_\text{REF}$ of \tip{FLL} circuit is fine controlled through proper sizing. The complete transfer function $H_\text{BPF}(s)$ of the proposed time-domain \tip{BPF} is given in \eqref{eq:bpf}. Its center frequency $\omega_{0}$ and $Q$-factor are given in \eqref{eq:bpf-w-Q} where the $Q$-factor is designed as 2 for each \tip{BPF} channel by proper sizing of the switching transistors in the \tips{SRO}. The stability of the proposed second-order time-domain \tip{BPF} is verified with a transient simulation.
\begin{equation} \label{eq:bpf}
H_\text{BPF}(s)=\frac{\displaystyle \frac{sK_\text{IN}f_\text{FLL}K_\text{PFD}}{N}}
{\displaystyle s^{2}+s\frac{K_{1}f_\text{FLL}K_\text{PFD}}{N}
+\frac{K_{1}K_{1}f_\text{FLL}^{2}K_\text{PFD}^{2}}{N^{2}}}
\end{equation}
\begin{equation} \label{eq:bpf-w-Q}
\omega_{0}=f_\text{FLL}K_\text{PFD}\sqrt{\frac{K_{1}K_{2}}{N}}\qquad Q=\sqrt{\frac{K_{2}}{K_{1}}}
\end{equation}
\subsection{Time-Domain Rectifier}\label{sec_rectifier}
Fig. \ref{fig:fwr-pfd} shows the schematic of \tip{FWR}. The proposed time-domain \tip{FWR} is based on a simple \tip{PFD} circuit consisting of only two \tips{D-FF} and one NAND gate. Compared to the prior voltage-domain design \cite{yang2019vad} that required several scaling-unfriendly \tips{OTA}, references, and passive elements, this work offers an alternative solution that is fully compatible with standard logic gates. Fig. \ref{fig:fwr-pfd} shows a state diagram and an input-output characteristic of the \tip{PFD}. The \tip{PFD} circuit extracts the input phase difference $\Delta\phi_\text{IN}$, but at the same time asynchronously quantizes the phase difference using a ternary code with UP and down (DN) signals. As shown in the state diagram, there are three states that are activated by the rising edges of incoming \tip{PWM} signals ($\phi_\text{INP}$, $\phi_\text{INN}$). When both UP and DN signals are high, the NAND gate resets two \tips{D-FF} immediately, thereby making itself as a ternary quantizer. Since the state of \tip{PFD} stays the same unless a new rising edge arrives, the UP and DN signal represents a positively and negatively Half-Wave Rectified (HWR) phase difference $\phi_\text{IN}$, respectively. Interestingly, the UP signal has the same form as the \tip{ReLU} activation function widely used in modern \tips{DNN}. In conventional usage of such signals like in \tip{PLL} designs, they are subtracted to derive a linearized phase difference extractor. However, if we add them, a time-domain \tip{FWR} can be implemented. The \tip{PFD}-based \tip{FWR} benefits from its fully time-domain nature, that is, it does not exhibit a headroom-related saturation, assuming that the input signal swing ($\Delta\phi_\text{IN}$) is within $\pm 2\pi$ range. As shown in Fig. \ref{fig:bpf}, the proposed time-domain \tip{FWR} is seamlessly integrated within the time-domain \tip{BPF} circuit and thus the \tip{BPF} provides an inherent rectification function. The rectified \tip{PWM} signals ($BPF_\text{P/N}$) are summed at the subsequent \tip{PFM} stage as described in Fig. \ref{fig:pfm-tdc}.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\textwidth]{Figures/Fig_pfm-tdc.jpg}
\caption{\tip{SRO}-based \tip{PFM} encoder, XOR differentiator, and subsequent post-processing blocks.}\label{fig:pfm-tdc}
\end{center}
\vspace{-5mm}
\end{figure*}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_pfm.jpg}
\caption{Proposed \tip{SRO}-based \tip{PFM} encoder.}\label{fig:pfm}
\end{center}
\vspace{-5mm}
\end{figure}
\subsection{Pulse-Frequency Encoder and Time-to-Digital Converter}\label{sec_tdc}
The analog \tip{FEx} designs presented in \cite{yang2019vad, wang2021kws} used an \tip{IAF} circuit which was originally proposed in \cite{mead1989a-vlsi}. The circuit converts an input current into a rate-encoded spiking \tip{PFM} signal and its spiking frequency is proportional to the input current magnitude. The \tip{IAF} circuit can be interpreted as a \tip{CCO} where the core oscillator topology is equivalent to a relaxation oscillator \cite{abidi1983relaxation}. However, the \tip{IAF} circuit is scaling-unfriendly because of its voltage-domain integral operation and voltage-domain static amplifier as discussed in Section \ref{sec_intro}. In this work, we propose to use the \tip{SRO} as a \tip{PFM} encoder instead of the \tip{IAF} circuit. As shown in Fig. \ref{fig:pfm}, the \tip{SRO} exploits an inherent phase-domain $2\pi$ threshold and its integral operation occurs in the phase-domain which is free from headroom issue. The proposed \tip{SRO}-based design offers a scaling-friendly implementation using only logic gates and bias generator without a static amplifier or passive components.
In previous designs \cite{yang2019vad, wang2021kws}, an asynchronous ripple-carry counter associated with a multi-bit register was used to quantize and sample the input \tip{PFM} signal. Interestingly, given the aforementioned interpretation of \tip{IAF} circuit as an oscillator, the signal flow from an \tip{IAF} to a counter builds an \tip{VCO}/\tip{CCO}-based $\Delta\Sigma$ modulator \cite{iwata1999vcoadc, elshazly2014sro}. However, the design approaches used in \cite{yang2019vad, wang2021kws} had two major problems. First, the ripple-carry counter exhibits metastability-induced data corruption when the sampling occurs at the instant of multibit transition of binary codes. Second, the output digital data from the asynchronous counter, which is $\Delta\Sigma$ modulated, was directly fed to the \tip{DNN} classifier without filtering of high-pass-shaped quantization noise. Our approach uses arrayed 1-bit XOR differentiators \cite{straayer2008vcoadc} to solve the metastability problem and an oversampling associated with a decimation filter to filter out quantization noise.
Fig. \ref{fig:pfm-tdc} shows the overview of implemented 16-channel \tip{SRO}-based \tip{PFM} encoder, XOR differentiator, and subsequent post-processing stages, also with the signal flow domains at the top. The \tip{SRO} receives rectified \tip{PWM} signals from the preceding \tip{BPF} stage and converts it into 15-phase \tip{PFM} signals. The same design of \tip{FLL} circuit discussed in Section~\ref{sec_bpf} is reused for biasing of the \tip{SRO} where the generated bias voltage is shared over 16 channels. As the ring-\tip{OSC} output is represented in thermometer-code, the XOR differentiator ensures the worst-case error to be within 1-\tip{LSB}. In addition, this 1-\tip{LSB} error is noise-shaped \cite{kim2010vcoadc} which can be eliminated through oversampling and decimation filtering. The thermometer-coded output data is aggregated to be represented as binary format and then filtered and decimated through a 1\textsuperscript{st}-order \tip{CIC} filter. We use 2\textsuperscript{10} decimation size, i.e., $f_\text{S,Deci}=f_\text{S,Over}/2^{10}$, and $f_\text{S,Deci}$ is used in the post-processing blocks which incorporates a programmable offset ($\beta$) subtractor to remove the DC offset due to a free-running component of the \tip{SRO}-based \tip{PFM} encoder. A programmable per-channel gain calibrator ($\alpha$) is used to correct inter-channel gain deviations caused by mismatch of SROs in the PFM encoder. A logarithmic compression using a LUT and a programmable input normalizer helps to increase the classification accuracy of the following \tip{GRU}-\tip{FC} neural network. The post-processing stage is clocked at 61\,Hz $f_\text{S,Deci}$ and thus its power dissipation is negligible.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_gru.jpg}
\caption{Architecture of the \tip{GRU}-\tip{FC} classification network (upper) and accelerator (lower).}\label{fig:gru}
\end{center}
\vspace{-5mm}
\end{figure}
\subsection{Recurrent Neural Network Accelerator}\label{sec_rnn}
Fig.~\ref{fig:gru} shows the architectures of the \tip{GRU}-\tip{FC} network and accelerator. The network has two \tip{GRU} layers with 48 units per layer and a final \tip{FC} layer that generates the confidence scores of the 12 classes. The network model size is entirely buffered within the 24\,KB \tip{WMEM}. The accelerator computes the \tip{KWS} classifier network and its input comes from the normalizer shown in Fig. \ref{fig:pfm-tdc}. The \tip{WMEM} block is implemented by on-chip \tip{SRAM} compiled based on the foundry-provided 6-transistor (6T) bit-cell. The classifier weights are loaded into \tip{WMEM} over the SPI interface. The accelerator has 8 Heterogeneous Processing Elements (HPEs) controlled by a \tip{FSM}. Each HPE has a 14-bit multiplier, a 24-bit accumulator, and a LUT-based Sigmoid/Tanh unit. Partial sums of the multiply-accumulate operations and outputs are stored in a shared 1.3\,KB \tip{SRAM} output buffer. Multiplexers before each multiplier operand select inputs from the normalizer, the Sigmoid/Tanh unit, the \tip{WMEM}, and the output buffer to compute element-wise vector multiplication/addition and hyperbolic functions in the \tip{GRU} \tip{RNN}. The high $V_{TH}$ device library is used for logic synthesis to reduce leakage current. Output scores of the classifier are fed to the argmax decoder, which outputs the class with the highest score. The classification result is transmitted over the SPI interface with an interrupt flag to the external host, which is a MiniZed board with a Xilinx Zynq-7007S SoC.
\subsection{Network Training}\label{sec_train}
\paragraph{Dataset preparation}{Our \tip{GSCD} training set is composed of 38,463 samples. The number of samples in the ``Silence" class is 4,044 which are randomly sampled from the background noise tracks in the dataset. The ``Unknown" class also has 4,044 samples which are randomly chosen words outside the target 12 classes. As for the test set, we used the standard \tip{GSCD} test set\footnote{\url{http://download.tensorflow.org/data/speech_commands_test_set_v0.02.tar.gz}} which has roughly equal number of samples (around 400) among the 12 target classes. Thus, the ratio between training and test set is around 8:1.} As shown in the measurement setup of Fig.~\ref{fig:chip_photo}, the samples from our entire training and test set were played from a laptop to $V_\text{IN,VTC}$ through a USB sound card \tip{DAC} (Sound Blaster E1). We normalized the \tip{GSCD} samples with the mean and standard deviation of the entire samples such that the amplitude of $V_\text{IN,VTC}$ is set to $\sim$250\,mV\textsubscript{PP}. The corresponding $\text{FV}_\text{Raw}$ from all samples were recorded. They were then corrected for the DC offset ($\beta$) and the inter-channel gain deviation ($\alpha$). After applying the logarithmic compression, we then normalize the $\text{FV}_\text{Raw}$ with the mean ($\mu$) and standard deviation ($\sigma$) of the recorded feature vectors from the entire \tip{GSCD} training set. This resulting vector called $\text{FV}_\text{Norm}$ (see Fig.~\ref{fig:overall}) are then presented as inputs to the \tip{GRU}-\tip{FC} classifier during training. The same $\mu$ and $\sigma$ are applied to $\text{FV}_\text{Log}$ of the test set to generate the corresponding $\text{FV}_\text{Norm}$ for \tip{KWS} evaluation.
\paragraph{Training schedule} The network is built in the PyTorch 1.8 framework and trained for 200 epochs using the AdamW optimizer \cite{loshchilov2018adamw} with an initial learning rate of 1e-3 and 0.01 weight decay. The ReduceLROnPlateau learning rate scheduler is used with a decay factor of 0.8 and patience of 3 epochs. The lowest learning rate is 5e-4. Using quantization-aware training, the activations and weights are quantized to 14-bits and 8-bits respectively.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_chip_photo.jpg}
\caption{Chip photograph of the prototyped KWS IC with a block-diagram of the measurement setup.}\label{fig:chip_photo}
\end{center}
\vspace{-5mm}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_meas_bpf-spectrum.jpg}
\caption{Measured frequency response of the \tip{FEx} (a) without per-channel correction, (b) with per-channel gain ($\alpha$) correction, and (c) output spectrum of $\text{FV}_\text{Raw}$ (after decimation filter) in channel 8 as shown in Fig.~\ref{fig:pfm-tdc}.}\label{fig:meas_bpf-spectrum}
\end{center}
\vspace{-5mm}
\end{figure}
\section{Measurement Results}\label{sec_measurement}
Fig.~\ref{fig:chip_photo} shows the \tip{KWS} IC, which is fabricated in TSMC 65 nm CMOS LP process with an active area of 2.03mm\textsuperscript{2} for the \tip{KWS} core. The area occupied by the analog and digital circuits is 1.6mm\textsuperscript{2} (79\%) and 0.428mm\textsuperscript{2} (21\%) respectively in the \tip{KWS} core. The \tip{GRU}-\tip{FC} neural network accelerator and the associated peripherals in the digital circuits are synthesized from a standard auto place-and-route (P\&R) flow.
Fig. \ref{fig:meas_bpf-spectrum}(a) and (b) show the measured frequency response of the 16-channel \tip{FEx} with and without per-channel gain calibration. In this case, $V_\text{IN,VTC}$ was connected to a function generator. The center frequencies of the 16 \tip{BPF} channels range from 111\,Hz to 10.4\,kHz. The center frequencies are distributed according to the Mel scale therefore low-frequency ($<$1 kHz) channels are spaced further apart than high-frequency channels. As shown in Fig. \ref{fig:meas_bpf-spectrum}(a), the measured gain curve before the calibration shows the inter-channel gain deviations which are caused by systematic mismatches from the \tip{SRO}-based \tip{PFM} encoder. The main cause of gain deviation is the voltage bias ($V_\text{VAR}$ in Fig. \ref{fig:fll-bias}) which is generated from a single \tip{FLL} circuit and it is shared over the 16-channel \tip{SRO} as depicted as `\tip{FLL} (\tip{SRO})' in the chip photograph. We expect this systematic mismatch due to distribution of the voltage bias can be improved with a better layout floorplan, for example with a centralized placement of the bias circuits while the random mismatch can be improved with larger sizing of the biasing transistors.
Fig. \ref{fig:meas_bpf-spectrum}(c) shows the measured output spectrum of channel 8 for two different input conditions of the \tip{VTC}; the black curve is obtained with a zero input condition while the red curve is obtained with a 2 kHz sinusoidal input of 390 mV\textsubscript{PP}. Here, the amplitude of the input to the \tip{VTC} circuit was assumed to be sufficiently large, and our future work will include an additional ultra-low-power pre-amplifier \cite{Han2013amp} before the \tip{VTC} circuit. The oversampling clock frequency that is fed into the XOR differentiator is 62.5\,kHz. It is clearly seen that the output spectrum has a first-order noise-shaping property with a 20\,dB/dec slope for both input conditions. After the feature data is decimated by 2\textsuperscript{10}, the in-band frequency is limited as 30\,Hz which is translated into a 16\,ms frame shift or 61\,frame/s throughput, and so the 16-channel \tip{FV} is generated every 16\,ms. The integrated in-band noise with zero input is calculated as 248 $\mu$V\textsubscript{RMS}, which is dominated by $1/f$ noise. When the input amplitude is increased to 390 mV\textsubscript{PP}, the in-band noise is dominated by thermal noise. We believe that the noise increase is caused by a higher running frequency of the \tip{SRO}, since the phase noise of ring oscillators increases with operating frequency \cite{abidi2006phasenoise}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_meas_audio.jpg}
\caption{Measured audio response of the \tip{FEx} with an applied sample keyword from \tip{GSCD}.}\label{fig:meas_audio}
\end{center}
\vspace{-5mm}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_meas_accuracy.jpg}
\caption{Measured \tip{KWS} accuracy on the \tip{GSCD} test set. A confusion matrix (left) where the magnitudes are normalized between 0 and 1, and a plot of the true positive rates over 12 different classes (right) are shown.}\label{fig:meas_accuracy}
\end{center}
\vspace{-5mm}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_meas_accuracy_SNR.jpg}
\caption{\tip{KWS} accuracy obtained over different \tip{SNR} levels.}\label{fig:meas_accuracy_SNR}
\end{center}
\vspace{-5mm}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/Fig_meas_powerbreak.jpg}
\caption{Power breakdown of the \tip{KWS} core implemented in the proposed IC (see Fig.~\ref{fig:overall}).}\label{fig:meas_powerbreak}
\end{center}
\vspace{-5mm}
\end{figure}
Fig. \ref{fig:meas_audio} shows the measured audio response of the \tip{FEx}. A 254\,mV\textsubscript{PP} ``Yes'' keyword sample from \tip{GSCD} is selected and applied to the \tip{VTC} while measuring the \tip{FEx} output. The magnitudes of \tip{FV} in this figure are normalized by subtracting DC offset and dividing by the standard deviation of the sample clip, for better visualization. It is clearly seen that the 16-channel \tip{FV} has higher response at low frequencies for the ``Ye" sound, and at higher frequencies for the ``s'' sound.
Fig. \ref{fig:meas_accuracy} shows the measured \tip{KWS} accuracy of the prototype chip obtained using the full 12-class verification flow of the \tip{GSCD}v2 \cite{warden2018gscd}. The 12 classes include ``Silence", ``Unknown", and 10 target keywords. As shown in the measurement setup in Fig.\,\,\ref{fig:chip_photo}, the generated \tips{FV} ($\text{FV}_\text{Raw}$ in Fig.~\ref{fig:overall}) from our time-domain \tip{FEx} are recorded using the \tip{GSCD} training set that is fed to the \tip{VTC} of our chip ($V_\text{IN,VTC}$ in Fig.~\ref{fig:chip_photo}), to train the classifier network. The 16 ms frame window and 16 ms frame shift (stride) are used for recording, so there is no overlap between two consecutive frames. The detected class is the most active output at the end of the \tip{GSCD} sample. The prototype \tip{KWS} IC achieves an overall 86.03\% accuracy with the \tip{GSCD} test set. The measured true positive rates show that ``Silence" is the easiest class with 100\% accuracy and the classifier performed the best on two keywords, ``Stop" and ``Yes," with 93\% accuracy. The most challenging class is ``Unknown" class since it includes 25 non-target keywords such as ``Happy" and ``Dog," which requires the classifier to train more parameters with larger model capacity to improve the accuracy. We expect that the detection accuracy of ``Unknown" and thus the overall accuracy on this \tip{KWS} dataset will improve with a larger network model, but at the expense of additional power consumption and silicon area.
The state-of-the-art accuracy on \tip{GSCD} using \tip{GRU}-\tips{RNN} is 94.2\% \cite{Zhang2017helloedge} with a network which has 499\,KB parameters and running on a Cortex-M7 microcontroller. This network size would require 21$\times$ more on-chip memory leading to higher power consumption and chip area.
\begin{table*}[ht]
\centering
\caption{Performance Comparison Table - Analog \tip{FEx}}\label{comp_table_FEx}
\begin{tabular}{||c||c|c|c|c|c|}
\hline
{Analog FEx} &
\begin{tabular}[c]{@{}c@{}}
{M. Yang}\\{JSSC 2016 \cite{yang2016coch}}
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
{K. Badami}\\{JSSC 2016 \cite{badami2016vad}}
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
{M. Yang}\\{JSSC 2019 \cite{yang2019vad}}
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
{S. Oh}\\{JSSC 2019 \cite{oh2019vad}}
\end{tabular} &
\textbf{This Work}
\\ \hline
{Process (nm)} &
180 &
90 &
180 &
180 &
\textbf{65}
\\ \hline
{Area/Ch. (mm\textsuperscript{2})} &
0.26 &
0.13 &
0.1 &
- &
\textbf{0.1}
\\ \hline
{Architecture} &
g\textsubscript{m}C-\tip{BPF} &
g\textsubscript{m}C-\tip{BPF} &
g\textsubscript{m}C-\tip{BPF} &
Mixer &
\textbf{OSC-BPF}
\\ \hline
{Number of Ch.} &
64$\times$2 &
16 &
16 &
32{\color{Maroon}\textsuperscript{A}} &
\textbf{16}
\\ \hline
{Freq. Range (Hz)} &
8-20k &
75-5k &
100-5k &
75-4k &
\textbf{111-10.4k}
\\ \hline
{Supply (V)} &
0.5 &
- &
0.6 &
1.4 &
\textbf{0.5}
\\ \hline
{Power ($\mu$W)} &
55 &
6 &
0.38 &
0.06 &
\textbf{9.3}
\\ \hline
{Frame Shift (ms)} &
- &
31.25{\color{Maroon}\textsuperscript{B}} &
10 &
512{\color{Maroon}\textsuperscript{A}} &
\textbf{16}
\\ \hline
{Dynamic Range (dB)} &
55 &
45 &
40{\color{Maroon}\textsuperscript{C}} &
47 &
\textbf{54.89}
\\ \hline
{FoM\textsubscript{S,DR} (dB)} &
- &
82.3 &
91.5 &
91.33 &
\textbf{93.11}
\\ \hline
{Target Task} &
General Purpose &
\multicolumn{3}{c|}{
VAD
} &
\textbf{KWS}
\\ \hline
{Building Blocks} &
\begin{tabular}[c]{@{}c@{}}
BPF, ADM\\
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
LNA, BPF\\FWR, LPF
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
LNA, BPF\\FWR, IAF
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
LNA, Mixer\\LPF, DSP
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
\textbf{VTC}\\\textbf{Rec-BPF}\\\textbf{PFM}
\end{tabular}
\\ \hline
{Support SE Mic} &
No &
\checkmark &
No &
\checkmark &
\textbf{\checkmark}
\\ \hline
{Parallel FEx} &
\checkmark &
\checkmark &
\checkmark &
No &
\textbf{\checkmark}
\\ \hline
\multicolumn{6}{|l|}{
{\color{Maroon}\textsuperscript{A}}With 32-32-16-2 FC neural network $\qquad$
{\color{Maroon}\textsuperscript{B}}f\textsubscript{LPF} = 16 Hz
}\\
\multicolumn{6}{|l|}{
{\color{Maroon}\textsuperscript{C}}Measured by the firing rate range of the IAF only, excluding output noise of the feature vector
}\\ \hline
\end{tabular}
\begin{equation}
\label{eq:Norm_Power}
\mathrm{P_{Norm}}=
\frac{P(1-r)}{1-r^{n}}\cdot
\frac{20\mathrm{k}}{f_{H}}
\qquad\qquad
r=\left(\frac{f_{L}}{f_{H}}\right)^{1/(n-1)}
\end{equation}
\begin{equation}
\label{eq:FoM}
\mathrm{FoM_{S,DR}}=
\mathrm{DR}+
10\cdot\mathrm{log}_{10}
\left(\frac{1}{\mathrm{P_{Norm}}\cdot 2\cdot \mathrm{FrameShift}}\right)
\end{equation}
\end{table*}
\begin{table*}[ht]
\centering
\caption{Performance Comparison Table - \tip{KWS}}\label{comp_table_KWS}
\begin{tabular}{||c||c|c|c|c|c|c|c|}
\hline
{KWS} &
\begin{tabular}[c]{@{}c@{}}
{S. Zheng}\\{TCAS-I 2019 \cite{zheng2019kws}}
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
{H. Dbouk}\\{JSSC 2021 \cite{dbouk2021kws}}
\end{tabular} &
\multicolumn{2}{c|}{
\begin{tabular}[c]{@{}c@{}}
{W. Shan}\\{JSSC 2021 \cite{shan2021kws}}
\end{tabular}} &
\begin{tabular}[c]{@{}c@{}}
{J. Giraldo}\\{VLSI 2019 \cite{giraldo2019kws}}
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
{D. Wang}\\{ISSCC 2021 \cite{wang2021kws}}
\end{tabular} &
\textbf{This Work}
\\ \hline
{-} &
\multicolumn{4}{c|}{
Off-Chip ADC
} &
On-Chip ADC &
\multicolumn{2}{c|}{
On-Chip Analog FEx
}
\\ \hline
{Process (nm)} &
28 &
65 &
\multicolumn{2}{c|}{
28
} &
65 &
65 &
\textbf{65}
\\ \hline
{Area (mm\textsuperscript{2})} &
1.29{\color{Maroon}\textsuperscript{A}} &
4.13{\color{Maroon}\textsuperscript{A}} &
\multicolumn{2}{c|}{
0.23{\color{Maroon}\textsuperscript{A}}
} &
1.52 &
2.71{\color{Maroon}\textsuperscript{B}} &
\textbf{2.03}
\\ \hline
{SRAM (KB)} &
52 &
38 &
\multicolumn{2}{c|}{
2
} &
32 &
20 &
\textbf{27}
\\ \hline
{Clock (Hz)} &
2.5M &
1G &
\multicolumn{2}{c|}{
40k
} &
250k &
120k &
\textbf{250k}
\\ \hline
{FEx} &
Digital &
- &
\multicolumn{2}{c|}{
Digital
} &
Digital &
Analog Voltage &
\textbf{Analog Time}
\\ \hline
{Classifier} &
CNN &
RNN &
\multicolumn{2}{c|}{
CNN
} &
RNN &
SNN (MLP) &
\textbf{RNN}
\\ \hline
{KWS Power ($\mu$W)} &
141{\color{Maroon}\textsuperscript{A}} &
11000{\color{Maroon}\textsuperscript{A}} &
\multicolumn{2}{c|}{
0.51{\color{Maroon}\textsuperscript{A}}
} &
16.1 &
0.205-0.570 &
\textbf{23}
\\ \hline
Frame Shift (ms) &
10 &
20 &
\multicolumn{2}{c|}{16} &
16 &
100 &
\textbf{16}
\\ \hline
{Latency (ms)} &
10 &
0.04 &
\multicolumn{2}{c|}{
64
} &
16 &
100 &
\textbf{12.4}
\\ \hline
{Dataset} &
TIDIGITS &
\multicolumn{6}{c|}{
GSCD
}
\\ \hline
\begin{tabular}[c]{@{}c@{}}
{Number of Classes}\\{(Keywords)}
\end{tabular} &
2 &
\begin{tabular}[c]{@{}c@{}}
7{\color{Maroon}\textsuperscript{C}}\\(6)
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
2{\color{Maroon}\textsuperscript{C}}\\(1)
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
5{\color{Maroon}\textsuperscript{C}}\\(4)
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
12{\color{Maroon}\textsuperscript{D}}\\(10)
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
5{\color{Maroon}\textsuperscript{C}}\\(4)
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
\textbf{12{\color{Maroon}\textsuperscript{D}}}\\\textbf{(10)}
\end{tabular}
\\ \hline
\begin{tabular}[c]{@{}c@{}}
{Accuracy (\%)}
\end{tabular} &
96 &
90.38 &
\begin{tabular}[c]{@{}c@{}}
97.3{\color{Maroon}\textsuperscript{E}}
\end{tabular} &
\begin{tabular}[c]{@{}c@{}}
91.7{\color{Maroon}\textsuperscript{E}}
\end{tabular} &
90.87 &
90.2 &
\textbf{86.03}
\\ \hline
{Support SE Mic} &
\multicolumn{4}{c|}{
Off-Chip ADC
} &
No &
No &
\textbf{\checkmark}
\\ \hline
\multicolumn{8}{|l|}{
{\color{Maroon}\textsuperscript{A}}Excluding off-chip ADC $\qquad$
{\color{Maroon}\textsuperscript{B}}Including SNN chip \cite{wang2020snn} $\qquad$
{\color{Maroon}\textsuperscript{C}}Excluding ``Unknown" word detection as a distinct class}\\
\multicolumn{8}{|l|}{
{\color{Maroon}\textsuperscript{D}}2 non-keywords (Silience/Unknown) + 10 keywords $\quad$
{\color{Maroon}\textsuperscript{E}}Accuracy is reported from 16-bit GSCD samples; the design excludes the 16-bit ADC}
\\ \hline
\end{tabular}
\end{table*}
Fig.~\ref{fig:meas_accuracy_SNR} shows the dependence of the \tip{KWS} classification accuracy on added noise levels to the recorded feature vector $\text{FV}_\text{Raw}$ (see Fig.~\ref{fig:overall}). We first computed the average power $P_\text{Avg,GSCD}$ using the recorded $\text{FV}_\text{Raw}$ (see Section~\ref{sec_train}). Then Gaussian noise of different standard deviation values ($\sigma^{2}=P_\text{Avg,Noise}$) were added to create different $\text{SNR}$ values following the equation below.
\begin{align}
\text{SNR}=10\log_{10}
\left( \frac{P_\text{Avg,GSCD}}{P_\text{Avg,Noise}} \right)
\label{eq:SNR}
\end{align}
The noise is randomly generated for each training epoch and test evaluation. For each \tip{SNR} case, our \tip{GRU}-\tip{FC} network is retrained using the noisy training set and the noisy test set is used to evaluate the classification accuracy. The proposed \tip{KWS} IC ensures $<$1\% accuracy drop even for noise levels up to 432\,$\mu$V\textsubscript{RMS} (input-referred to $V_\text{IN,VTC}$), or 40\,dB \tip{SNR}.
Fig.~\ref{fig:meas_powerbreak} shows the power breakdown of the \tip{KWS} IC. As it is stated in Section~\ref{sec_intro}, this paper focuses on the \tip{KWS} core only therefore the power breakdown in Fig.~\ref{fig:meas_powerbreak} does not include energy harvester, low-dropout regulator, and voltage reference circuits. The total power consumption of the \tip{KWS} core is 23\,$\mu$W when it is measured at 25$^{\circ}$C room temperature. The \tip{GRU}-\tip{FC} neural network accelerator accounts for 43\% of the \tip{KWS} core power. When the 16IN-48H-48H-12C \tip{GRU}-\tip{FC} network is updated at 250 kHz clock frequency and 0.75\,V supply voltage while performing continual inference on random \tip{GSCD} samples with 16 ms frame shift, the accelerator consumes 9.96\,$\mu$W. The accelerator power consumption can be further decomposed into dynamic power (75\%) and leakage (or static) power (25\%). The leakage power is dominated by the \tip{SRAM} block (78\%) while both logic (44\%) and \tip{SRAM} (56\%) contributed rather evenly to the dynamic power. We expect that leakage power can be reduced with custom memory cells \cite{shan2021kws}.
Table \ref{comp_table_FEx} compares the performance of our time-domain \tip{FEx} with the state-of-the-art voice processing analog \tip{FEx} \cite{yang2016coch, yang2019vad, badami2016vad, oh2019vad}. The proposed \tip{FEx} circuit is the first that demonstrated ring-oscillator-based \tip{BPF} topology used for the \tip{KWS} task. It supports \tip{SE} microphones, thereby offering a lower system-level power. Unlike sequential \tip{FEx}, our parallel \tip{FEx} does not lose frequency-selective information at any time \cite{oh2019vad}. To allow a fair comparison with previously reported designs with a variety of frame shifts, we derive a Schreier \tip{FoM} \cite{schreier2005dsm} \eqref{eq:FoM}, widely used for \tips{ADC}. The Schreier FoM considers the trade-off between \tip{DR} and bandwidth, also accounting for power consumption. For near DC input \tips{ADC}, the bandwidth is replaced with a reciprocal of conversion time \cite{chae2013dsm}. As a band-pass filtered signal is demodulated into baseband (DC) after the rectifier \cite{yang2019vad, badami2016vad} or mixer \cite{oh2019vad} stage, we consider analog \tip{FEx} as a DC-input \tip{ADC} with a pre-processing stage. The \tip{FoM} equation \eqref{eq:FoM} includes the normalized power consumption $P_\text{Norm}$ (\ref{eq:Norm_Power}) proposed in \cite{yang2016coch}, the \tip{DR}, and the frame shift. The frame shift is part of the denominator of \eqref{eq:FoM} because the \tips{FV} are generated in every frame shift. The amount of integrated in-band noise is reduced with a larger decimation window (i.e., averaged over the longer time interval) in our design and also in \cite{chae2013dsm} where the number of \tip{ADC} cycles was used for decimation window. The proposed \tip{FEx} records the best Schreier FoM among the state-of-the-art designs. In addition, the time-domain processing circuits offer better technology scaling, and will outperform voltage-domain designs \cite{yang2016coch, yang2019vad, badami2016vad, oh2019vad} in terms of power and area when implemented in advanced technology nodes.
Table \ref{comp_table_KWS} compares the performance of our KWS IC with other state-of-the-art \tip{KWS} ICs \cite{zheng2019kws, dbouk2021kws, shan2021kws, giraldo2019kws, wang2021kws}. This work uses an on-chip analog \tip{FEx} while other works needed an off-chip high-resolution (16-bit) \tip{ADC} \cite{zheng2019kws, shan2021kws}. Sometimes even the digital \tip{FEx} and \tip{ADC} were implemented off-chip \cite{dbouk2021kws}. Furthermore, only this work and \cite{giraldo2019kws} support the essential ``Unknown" class to be detected as a distinct class, which is the most challenging class in the \tip{GSCD} test set. As such, it implies that concessions would be made in terms of \tip{KWS} accuracy for \cite{dbouk2021kws, shan2021kws, wang2021kws}, or a larger model size will be required for the classifier to uphold the accuracy, leading to additional power and area costs. In addition, the proposed chip supports \tip{SE} microphone interface and the \tip{KWS} task is verified with a \tip{SE} input condition. This work shows competitive performance and better system-level power efficiency by using a low-power \tip{MEMS} \tip{SE} microphone instead of the differential microphone used in \cite{giraldo2019kws}. Last but not least, our prototype chip is the first silicon-verified analog \tip{FEx}-based voice processing IC that demonstrates 12-class \tip{KWS} task on \tip{GSCD}, using an on-chip classifier.
Our belief is that the 5\% degradation in the classification accuracy of our \tip{KWS} IC (86\%) as compared to the software model accuracy (91\%, Section~\ref{sec_software}) is mainly due to the increased noise floor when the input amplitude is high, as shown in Fig.~\ref{fig:meas_bpf-spectrum}(c). Advanced noise suppression techniques such as chopper stabilization \cite{enz1996chop} and dynamic element matching \cite{ha2019bioz} when applied to the front-end, will help mitigate the accuracy discrepancy. Our time-domain \tip{FEx} still needs to address the per-chip gain calibration requirement due to the mismatch of analog circuits, which is not necessary in a fully-digital approach \cite{giraldo2019kws}. For this, as discussed in the paragraph in Section~\ref{sec_measurement} describing Fig.~\ref{fig:meas_bpf-spectrum}(a), improved layout floorplan and larger device sizes accompanied with mismatch-aware \tip{DNN} training \cite{yang2019vad} will be another opportunity to remove the calibration requirement.
\section{Conclusion}\label{sec_conclusion}
We have presented a low-power \tip{KWS} chip that exploits ring-oscillator-based time-domain processing circuits. Implemented in a 65\,nm CMOS process, it consumes 23\,$\mu$W power dissipation with a power supply of 0.5\,V for analog circuits and 0.75\,V for digital circuits. The nested analog \tip{FLL} enhances the linearity of \tip{VTC}, and thus facilitates the use of single-ended microphones as discussed in Section \ref{sec_vtc}. The usage of \tip{PFD} as a time-domain \tip{FWR} shows significantly reduced implementation cost in comparison with a voltage-domain design. The \tip{PFM} functionality is realized using a \tip{SRO}, instead of the conventional \tip{IAF} circuit to obviate the need for scaling-unfriendly voltage-domain circuits. Table \ref{comp_table_FEx} shows that the proposed time-domain \tip{FEx} achieves the state-of-the-art \tip{DR}-based Schreier \tip{FoM}. The on-chip integrated \tip{GRU}-\tip{FC} digital back-end circuit processes incoming audio \tips{FV} with a 16\,ms frame shift using only $\sim$10\,$\mu$W power, demonstrating $>$86\% classification accuracy with only 12.4\,ms latency on the 12-class \tip{GSCD} \tip{KWS} task. We expect that the proposed time-domain processing techniques can be further expanded in other domains and thus provide various design opportunities for power-efficient circuits, such as the fully time-domain \tip{ReLU} activation unit shown in Fig. \ref{fig:fwr-pfd} for \tips{DNN}. The improvement directions as discussed in Section~\ref{sec_measurement} along with \tip{DR} enhancement techniques such as front-end automatic gain control will enable the time-domain \tip{FEx} to be applied to more challenging real-world audio-inference tasks. A 35-class \tip{KWS} on \tip{GSCD} \cite{rybakov2020kws} or a streaming-mode \tip{KWS} can be such examples.
\section*{Acknowledgment}
The authors would like to thank Frank K. Gürkaynak and Beat Muheim from ETH Zürich, for their technical
support of the digital circuits in this IC technology and Taekwang Jang from ETH Zürich for the valuable discussions on analog frequency-locked loop.
\bibliographystyle{IEEEtran}
|
{
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| 5,130
|
és un ex-jugador de voleibol rus que va competir sota bandera soviètica durant les dècades de 1950 i 1960.
El 1964 va prendre part en els Jocs Olímpics de Tòquio, on guanyà la medalla d'or en la competició de voleibol. En el seu palmarès també destaquen dues medalles d'or (1960 i 1962) i una de bronze (1966) al Campionat del Món de voleibol, una d'or a la Copa del Món de voleibol de 1965 i dues de bronze (1958 i 1963) al Campionat d'Europa.
A nivell de clubs jugà amb l'Energiya Moskva i CSKA Moskvà. Amb el CSKA guanyà sis edicions de la lliga soviètica (1958, 1960, 1961, 1962, 1965 i 1966) i dues copes d'Europa (1960 i 1962). Un cop retirat va exercir d'entrenador en diversos equips.
Referències
Medallistes soviètics als Jocs Olímpics d'estiu de 1964
Jugadors de voleibol soviètics
Jugadors de voleibol russos
Persones de la província de Moscou
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 5,060
|
Destination: Kuwait
Kuwait Highlights
Kuwait, officially the State of Kuwait is a country in Western Asia. It shares borders with Iraq and Saudi Arabia. As of 2016, Kuwait has a population of 4.5 million people: 1.3 million are Kuwaitis and 3.2 million are expatriates. Expatriates account for 70% of the population. Islam is the official religion in Kuwait, and the majority of the citizen population are Muslim. Kuwait City is the capital of Kuwait, and also its political, cultural, and economic center. Kuwait's official language is Arabic.
History and Government
Kuwait City began as a fishing village when its first inhabitants arrived in 1613. It was after the arrival of the BaniUtubs in 1714 that the city transformed into a chief trade center of India and the Far East. It is a suitable location on the trade route stretching from Calcutta to Baghdad and Aleppo enabled the city to become a significant stop by 1750. Kuwait's government is a constitutional monarchy headed by the hereditary leader, the emir.
Most of Kuwait is an entirely flat and dry barren desert of sandy plains. The land (desert) begins to rise slightly in the southwest, along the border with Saudi Arabia. The country's highest point is an unnamed hill that peaks at 1,003 ft. There are no perennial lakes or rivers.
Kuwait has a thriving art scene and features more than 30 art galleries. Laws and customs in Kuwait define and enforce a division of labor by gender. However, women in Kuwait participated in the workforce more than any other country in the Gulf Cooperation Council. Females are often employed in academic, social service or clerical positions, while business leaders, managers, and top-level administrators are male. Marriages are mostly arranged and women usually need their father's permission to marry. Women cannot marry non-Muslim men, while men are allowed this freedom. The prevailing religious laws also allow men to have up to four wives at one time.
Kuwaiti cuisine has been influenced by Arabian, Mediterranean, Persian, and Indian cuisines. Seafood, especially fish, rice, and bread are the staples of Kuwaiti cuisine. For example, the traditional Kuwaiti flatbread, called khubz, is baked in a special oven and topped with sesame seeds and usually served with fish sauce. Other popular dishes of Kuwaiti cuisine include biryani (seasoned rice cooked with lamb or chicken), maglooba (rice cooked with potatoes, eggplant, and meat) and gabout (steamed flour dumplings stuffed with cooked meat or vegetables and served with meat stew). Arabic coffee, karak tea, dried lime tea, and sharbat are examples of some of the popular beverages in Kuwait.
Best Time to Visit Kuwait
The best time to visit any country depends on various factors, namely, your purpose of visit – sightseeing/outdoor activities you are looking for like shopping etc., and also the vacation budget you have set. For sake of convenience and to make prior bookings, remember that the high season to visit Kuwait are the months of November to March while the low seasons are the months of April to September.
Kuwait exudes the charm of Arabian adventure, and therefore, the travelers find plenty of things to do in Kuwait. Apart from enjoying magnificent accommodation and fine dining, there are plenty of things for visitors to do in Kuwait. Located on Arabian Gulf Road in Sharq district, the Kuwait Towers is one of the famous landmarks in Kuwait, it comprises two major towers and a minor tower. It is the symbol of Kuwaiti liberation, the representation of the country's resurgence, second tallest tower in Kuwait, and the fifth tallest telecommunication tower in the world. Doha Village is a place to visit for those interested in gaining some information about the glorious trading history of Kuwait.
Price on call View More
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 7,148
|
Filipe Zau (Lisboa, 2 de novembre de 1950) és un investigador en ciències de l'educació, professor, autor i compositor d'Angola. També és rector de la Universitat Independent d'Angola.
Biografia
Nascut a Lisboa, el seu pare era un mariner originari de la província de Cabinda i la seva mare era capverdiana. Començà tocant en grups com Duo Ouro Negro fins que fou reclutat per la banda Alerta Está, que feia concerts a les casernes, i on es va fer amic del bateria Guilherme Inês.
L'u d'abril de 1975 tornà a Luanda i contactà amb músics portuguesos com Fausto, Zeca Afonso, Vitorino Salomé o Sérgio Godinho. En 1990 fou agregat cultural a l'ambaixada angolesa a Lisboa. En 1996 va gravar l'opereta O Canto da Sereia: o Encanto en homenatge al seu pare juntament amb el cantant Filipe Mukenga. El mateix any treu Luanda Lua e Mulher i amb Mukenga componen cançons per a Celina Pereira.
Filipe Zau i Filipe Mukenga van rebre el premi Common Ground Music Award 2008 atorgat per l'associació Search for Common Ground, en maig de 2008, pel disc en col·laboració Angola solta a tua voz. Ambdós també foren autors de la cançó himne de la Copa d'Àfrica de Nacions 2010, Angola, país de futuro, acompanyats de la banda Maravilha.
Obra editada
Llibres
Encanto de um Mar que eu Canto (poesia, Universitária, 1996)
Meu canto à razão e à quimera das circunstâncias (poesia, Universitária, 2005)
Marítimos africanos e um clube com história (2007)
Notas fora da pauta (prosa, Chá de Caxinde/Prefácio, 2007)
Angola – Trilhos para o Desenvolvimento (tesi; Universidade Aberta)
Educação em Angola. Novos Trilhos de Desenvolvimento (tesi, Movilivros, 2009)
Discografia
Congresso/Tania (Single, CDA, 1974)
Luanda Lua e Mulher (CD, Strauss, 1996)
Referències
Enllaços externs
Filipe Zau a Discogs
Músics angolesos
Músics lisboetes
Escriptors lisboetes
Escriptors angolesos
Músics africans
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 5,206
|
Q: Extend Polka Request object using Typescript First of all, I know this topic looks really similar to this other topic talking about extending Express Request object using Typescript
Basically, I try to do the same thing but with Polka this time
Has anyone managed to do this?
The path I followed is similar to this
At project root lever I created this folder structure :
app/
├─ src/
│ ├─ @types/
│ │ ├─ polka/
│ │ │ ├─ index.d.ts
And I add this in index.d.ts
import * as polka from "polka";
declare global {
namespace polka {
interface Request {
foo: string;
}
}
}
I also updated my tsconfig.json by adding this :
"typeRoots": [ "@types" ]
The middlware where I assign a value to the request looks like this
import type { Middleware } from "polka";
export const dummyMiddleware: Middleware = (req, res, next) => {
req.foo = "hello";
next();
};
By doing this I have this Typescript error :
Property 'foo' does not exist on type 'Request<ParamsDictionary, any, any, ParsedQs,
When I look at the Polka Middleware definition I see this type in generic.
I tried to do something like this
import type { Middleware } from "polka";
export const dummyMiddleware: Middleware<{foo : string}> = (req, res, next) => {
req.foo = "hello";
next();
};
But the error message only turns into this
Property 'foo' does not exist on type 'Request<{ foo: string; }, any, any, ParsedQs, Record<string, any>>'.ts(2339)
So, question, is declaration merging the best way to achieve this ? If yes, do you have a proper way to achieve this ?
A bit of context, this middle ware will be use for seeding Sapper session data.
Versions :
*
*"typescript": "^4.0.3"
*"polka": "next"
*"@types/polka": "^0.5.2",
Full TypeScript config :
{
"extends": "@tsconfig/svelte/tsconfig.json",
"compilerOptions": {
"module": "esnext",
"lib": ["DOM", "ES2017", "WebWorker", "ESNext"],
"strict": true
},
"include": ["src/**/*", "src/node_modules/**/*"],
"exclude": ["node_modules/*", "__sapper__/*", "static/*"],
"typeRoots": [ "@types" ]
}
Disclaimer :
I am a novice in polka and typescript so it is not impossible that I missed something obvious
A: I settled with first casting IncomingMessage to Request.
import { IncomingMessage, ServerResponse } from 'node:http';
import polka, { Next, Request } from 'polka';
function one(req:IncomingMessage, res:ServerResponse, next:Next) {
(req as Request).foo = 'world';
next();
}
function two(req:IncomingMessage, res:ServerResponse, next:Next) {
//req.foo = '...needs better demo ';
next();
}
const app = polka();
app.use(one, two);
app.get('/users/:id', (req:IncomingMessage, res) => {
console.log(`~> Hello, ${(req as Request).foo}`);
res.end(`User: ${(req as Request).params.id}`);
});
app.listen(3000, (err:any) => {
if (err) throw err;
console.log(`> Running on localhost:3000`);
});
Then in index.d.ts try this:
declare global{
module 'polka' {
interface Request {
foo: string;
}
}
}
Not great, but something to work with.
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 1,915
|
{"url":"https:\/\/libraryguides.centennialcollege.ca\/c.php?g=645085&p=5148023","text":"It looks like you're using Internet Explorer 11 or older. This website works best with modern browsers such as the latest versions of Chrome, Firefox, Safari, and Edge. If you continue with this browser, you may see unexpected results.\n\n# Math help from the Learning Centre\n\nThis guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.\n\n## Measuring Electricity\n\nElectricity is measured in units of power called Watts, named to honor James Watt, the inventor of the steam engine. A Watt is the unit of electrical power equal to one ampere under the pressure of one volt.\n\nOne Watt is a small amount of power. Some devices require only a few Watts to operate, and other devices require larger amounts. The power consumption of small devices is usually measured in Watts, and the power consumption of larger devices is measured in kilowatts (kW), or 1,000 Watts.\n\nElectricity generation capacity is often measured in multiples of kilowatts, such as megawatts (MW) and gigawatts (GW). One MW is 1,000 kW (or 1,000,000 Watts), and one GW is 1,000 MW (or 1,000,000,000 Watts).\n\nExamples:\n\nConvert $$0.0058$$ mW into kW.\n\n$0.0058 mW \\times \\frac{1\\, kW}{1,000,000\\, mW} = 0.0000000058\\, kW$\n\n## Measuring Resistance, Current, and Voltage\n\nThe\u00a0ohm\u00a0($$\\Omega$$) is a unit of electrical resistance, name after German physicist George Ohm. It is correlated to voltage ($$V$$) or the force of electricity, and the electric current, measured in amperes ($$A$$).\n\nExamples:\n\n1. Convert $$12.85\\, k\\Omega$$ to $$\\Omega$$\n\n$12.85\\, k\\Omega\\times \\frac{1000\\, \\Omega}{1\\, k\\Omega}= 12,850\\, \\Omega$\n\n2. $$874\\, \\mu A$$ to $$A$$\n\n$874\\, \\mu A \\times \\frac{1\\, A}{1,000,000\\, \\mu A}= 0.000874\\, A$\n\n3. $$82$$ MV\u00a0to V\n\n$82\\, MV \\times \\frac{1,000,000\\, V}{1\\, MV}= 82,000,000\\, V$","date":"2022-12-01 03:57:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5004717707633972, \"perplexity\": 2062.4830156316207}, \"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-2022-49\/segments\/1669446710789.95\/warc\/CC-MAIN-20221201021257-20221201051257-00483.warc.gz\"}"}
| null | null |
I like to think I've got skills. I've got this. I don't like to think about how I fall short. Who does? We live our lives to be capable and independent and successful. And while I know we shouldn't play the comparison game, I admit I do sometimes. I suspect you do too!
So when it comes to living life well, I often think I'm doing just fine. Ever feel that way?
But then I am hit by the reality that I don't measure up. Sometimes (many times) to others, but definitely not to the powerful perfect standard of God. I am confronted with my brokenness or my insufficiency or insecurity or pride or or or.
And I realize I have nothing.
Nothing to bring to God that will make a difference in the decision of my eternal outcome. I realize that all my effort or goodness or skills aren't enough. And you know what? That's just fine with Him. He loves us and accepts us as we are, knowing we are in desperate need of Him. The great reality is that He is EVERYTHING and He has made a way for us.
We've been invited to a great feast! And it's not a potluck. So BRING YOUR NOTHING and HE WILL BE YOUR EVERYTHING.
Why would He do that? Simply, because He loves us in our weakness and wants for us to be restored to Him.
I am fine, but I am tired. I am on autopilot. That's not entirely bad if the path is a good trajectory. But I am not taking as much joy in this part of the journey as I would like. I find frustration and grumpiness to be expressed too often. I am feeling more envy than I wish of those who travel and have leisure. I am confident in my call and choices, but it does have times when it tires one out. I am not going to crash and burn so don't worry for me, but I welcome prayers.
Today marks the anniversary of my last marathon, the marathon that did me in. You can read about it in an earlier post. The result of this marathon has been a year of the most incredible pain and huge measure of discouragement. Not a day has gone by without some of the most intense pain. Pain in my feet mostly, but my legs as well. The doctors could not find any obvious problem with my feet. X-rays revealed nothing. But clearly there was some soft tissue damage in the foot bed, the arches, the heels, and more. Most days I could barely walk, let alone run. I would shuffle about, not having any ability to raise the heel and get into a fluid pace. Cycling was virtually impossible as well as the pressure on the pedal also caused pain. To this day the achilles pain has moments of intensity but the at least the feet seemed to have finally healed. That endured until about October.
But without exercise or movement and the heaviness of discouragement, my weight has increased. The added weight has now caused further pain, some sort of joint reaction in my elbows and arms. I can barely grip things and shaking hands gives me intense pain. I find myself caught in this vicious cycle that a lot of overweight people find themselves in. One is heavy and needs to exercise but is too heavy to do so. I know to many (most) this sounds like excuses, but it just isn't.
When I was nearing my goal weight and doing the running, I felt a lightness and freedom in movement. It motivated me to get out and move more. But when one is heavy you feel anchored and every move is an exercise in frustration – not the type of exercise that motivates. Much of the battle is mental and emotional, particularly when pain is in the mix, but there is a physical component as well.
As a result this has been a very discouraging year. Constant pain, extreme weight gain, and loss of fitness have led me to wrestle with a lot of frustration and perhaps even a bit of depression.
As today approached I have been asking myself, "Do I regret running that last marathon?". Of course there is no practical reason to second guess – nothing can be done about that. OF COURSE I would prefer to not have all the pain and weight gain this past year. I think not having run the race would have made the year very different. But I also completed a major goal. I learned some incredible lessons during and from the run that have been valuable. I understand empathy and the value of truthfully acknowledging one's pain so much more. So no, I do not "regret" doing the run. I can not live by the "If only…" trap. I have to stand by my choices, whatever comes. But the choice to run, even with all the lessons learned, did have consequences. As all choices do.
Ups and downs in my weight will always be my battle. Fitness will be harder for me than most. It is the genetic cards I was dealt. The emotional and mental, and even physical, battle will face me everyday. I hope I can find my way back. I hope I receive the empathy I have learned to offer. And as depressed and depressing as this last paragraph sounds, I do have hope. I really do.
I recently had the honor to take a team from my church to serve at Door of Faith Orphanage (dofo.org) in the town of La Mision, B.C. Mexico. It is located between Rosarito and Ensenada. It is a most surprisingly charming place. If a child has to be in an orphanage, this is the place to be! As the Director, DJ, gave us his welcome and shared the values of the orphanage that he has led for over 20 years, one of the three values were Healing through Service. He began to share that the children and staff of DOFO are encouraged to serve, and not just chores. But real ministry outreach service. The kids raise funds, go on trips and make real differences in the places they go. This leads to a healing in their own lives as most are social orphans, having been removed from broken or abusive homes. Teaching them to serve others brings a lasting and deep healing in their lives.
Javier is a man we met when we went to make and serve breakfast with the Baja Family Outreach located the Tijuana dump area. A church has been built in the heart of the community of thousands who live on reclaimed dump land. Most are uneducated and find their income by recycling garbage. Their homes are built with items found in the dump. We saw more than one roof made from the vinyl banners of highway billboards. As we made the breakfast a family of 8 came in, Javier being the dad. His children and wife sat down to eat the food we had prepared, but Javier stood at the entrance to greet people and watch over his family. I asked him why he wasn't eating and he shared he normally volunteers. By our team being there, we had taken his job. Of course he was blessed by our presence and asked we pray for his family and to please return. But I saw something powerful in him. He had no money to care for his family as I am sure he would like, but by volunteering he felt he was part of providing. Serving was healing for him. Serving was empowerment for him. Serving made him feel like he was being a provider for his family.
No better example of healing through service came in the story of Martita, a lady we met serving at another organization, Life in the Canyon (lifeinthecanyon.vpweb.com). This group is led by Dave Hessler and he found basically the poorest community of Tijuana and began serving a number of years back and was an outgrowth of his time with the Baja Family Outreach. He landed in the former TJ dump that has created massive hills of garbage covered in soil and is now being reclaimed by squatters and the poor who have built pallet shacks and small homes. A little over two years ago Dave came across Martita as he walked through a cemetery. She was high on crystal meth and was digging through burn piles for scraps of metal to sell to support her habit. He offered, through his ministry to serve the community, some help. At first, if I understood her story, she didn't accept, but time after time he continued to reach out to her and eventually she came to the community center. There she saw loving people serving their community and something drew her.
She asked if she could serve and they found a place for her. The desire to quit drugs grew and at a point she quietly decided to stop. After a week went by being clean, she came to Dave and shared her good news. He told her to go one day at a time and every day to come and give him a number; the number '1' for one more day of sober. He wrote that number on a white board and eventually it grew from 7, to 25, to 50. As of our meeting her it has been over 2 years! Her love for God, her purpose found in serving her community, and the love shown her by Dave and others gave her hope and strength to go on. Today she is trusted with keys to everything, is one of Dave's trusted assistants, and has a restored relationship with her daughter and grandchildren. She is no longer homeless but has a tiny micro-home built with money from a 16 year old boy who did a fund raiser in his church! She can't read or write but has learned to post on Facebook in order to be connected. She gave us a tour of her community, the dump, and you could see the love she had for it's residents. She found healing through serving!
One of the things that I found most exciting with DOFO is that they don't have teams come just to stay focused on them as an orphanage. DJ said many times to me, almost to the point of sounding a humorously insulting, "We don't want you to stay here. Our kids have seen every drama and are pretty saturated with Americans!" But as he clarified you knew exactly what he was saying. Come and use DOFO as a hub to serve the broader area of Baja. Go to the dumps, build a house for a resident, serve at the rehab homes, do an outreach to drunk Americans at the beaches and bars, give out groceries and pray with residents of the dump. I believe DOFO is as blessed and beautiful as it is because it too has become whole by serving others and not manipulating donors and guests to selfishly stay committed to them. They have a mature faith, a generous faith.
Over the years I've heard from many who don't understand why so many in the Church appear to have a "woe is me" mindset. I can see their confusion, because if anybody should be joyful, it should be the Christ follower! And yet there is a paradox that exists in our faith.
In fact there are many! We're already saved yet are working out our salvation. We are at the same time both in eternity and yet also bound by time. We are righteous but know full well we are being sanctified. This last one leads to the what I think is the most beautiful paradox of all. It's the place where mourning and joy exist in their fullest at once. It is in that state where grace is realized in such beauty!
The Sermon on the Mount is a powerful and core teaching by Jesus that can be found in Matthew 5-7. The first section has become known as the beatitudes and has challenged and transformed me as I have wrestled with what seems like an impossible way to live. We know Jesus came to set us free from the "law" and yet at first glance this sermon calls us to a life even more severe. And so one is compelled to look deeper to reconcile the apparent contradiction.
The first section of the beatitudes is where the tone is set.
I used to think of each of the Blesseds like individual traits to attain. "I like the peace maker, but mourning, not so much." Sort of like a buffet. But I've come to realize it's more like a 6 course meal. It's not a buffet, not individual attributes that we pick and choose from, but a layering, a building to become the person Jesus calls us to. And the first is necessary to move on to the next.
When you are poor in spirit having come to the end of yourself, acknowledging you have nothing to bring to God, it's then you mourn deeply for the reality of sin and it's destruction in your life. It's then that the power of meekness, the control of yourself is able to set in. The choosing of God's control rather than sin's. And at that point one is spiritually bankrupt, desperately hungering, desperately thirsting for God's righteousness! The promise? We are filled. God imputes, or puts in us HIS righteousness! Then and only the does the rest of the sermon makes sense. Then and only then can any of us begin to live the sermon out!
For example, take the passages on murder and adultery (Mt 5:21-30). It's easy in our righteousness to live a life of fidelity and never murdering someone. Most succeed in that! But by age three we've all called someone a name in anger and by 12 we've all lusted. And so in my abilities I am a failure. And if we haven't come to the end of ourselves and realized we have nothing, we are nothing apart from Christ, then we will never live the life God has made available to us through Jesus.
Do I have a purpose and potential? Of course! Do I have gifts I bring to the table? Absolutely! Are they of any value? For this life and the common good? Sure. But for eternity, no! My righteousness is nothing and His is everything! And in His hands those gifts and potential will become something wholly different, better.
So the beautiful paradox, the sweet spot…is being both mournful and filled with joy at the same time. Not dwelling in ashes, but acknowledging my capacity and propensity to sin. Not "woe is me" but most definitely full on mourning. "Woe is me" is a self-focused declaration. The mourning Jesus calls us to is recognition of our sinful condition.
And the joy? It can and should be full-on crazy celebration! As I mentioned earlier, the sermon can seem like an even more severe life than the law. And here is what most fail to see and even when we see it, we find hard to experience. Jesus knows we can't. We can't live free of anger and lust and judgement and unforgiveness! That's why He offers us His righteousness. That's why mercy is given. That. Is. Grace!
And until I come to the end of me and bring my nothing to God, there is no room for grace to be experienced. And so the dance of the paradox begins. Living in spiritual poverty and the riches of Christ together is the challenge.
If you're like me, and you are, you begin to take credit for spiritual maturity at times. You take for granted the all-encompassing nature grace must have in our lives. The longer I live in Christ the more I realize how much I need Him. The dos and don'ts are easy, grace is not. It requires dying and mourning and spiritual bankruptcy. But then and only then do we experience freedom and joy.
By the way, this why we desperately need each other! But that's another post.
It's a journey and one in which I hope you find joyful mourning.
For the past six weeks our church has gone through a campaign using Rick Warren's, What On Earth Am I Here For? (formerly, The Purpose Driven Life) book and materials. It has been a wonderful experience as all our LifeGroups went through a study together and our weekend services addressed the same topic each week. It also coincided with each person reading the daily readings from the book. And of course we threw in extra special events and activities along the way creating a really fun and unifying season in the church.
Each week I sent out special update emails with various bits of information about what groups were doing, the reading schedule and more. It also included a "blog post" of that week's purpose, adding some of my insight into the topic. Each week got a bit longer as I grew into doing this (I had not originally intended to include this). Even though some of post is specifically "to" our church, I wanted to capture them in and repost them here. Just in case you are not aware, Rick Warren's premise is that God has designed his followers to live lives that demonstrate 5 purposes: worship, fellowship, discipleship, ministry, and evangelism. Here are the repostings of my insights for each of those.
As you in your reading and in your LifeGroups look at the purpose of worship, remember that worship is not characterized as an act or ritual like bowing down. Instead it is an attitude of the heart and will – recognizing the value of someone or something and giving it the proper attention in our lives. So when we hear something like, "He worships money or sports more than God" it makes more sense. Of course nobody is bowing to money or sports or hobbies, etc. But we certainly can give them greater value and importance in our lives than they deserve.
That is why Mariners believes so strongly in the role of LifeGroups. Of course thats not the only way to express and live in fellowship in our lives, but it sure is a great one! If your group is a temporary one started for this series, please begin to consider what it might look like to stay formed and work together to see Christ formed in each other. If you are not in a group yet, please seek one out.
Take the "risk", it's worth it!
Maturing in Christ, having Him formed in us, is our calling. To live like and become like Jesus! Somewhere along the way, the idea of disciple became synonymous with bible study and prayer times and daily devotions. And yes, disciples do those things! We do develop habits or "disciplines" of faith in order to understand and grow in faith. But the habits are not the mark of a disciple. Christ-like character is! Do we live humbly, exhibit the "fruit of the Spirit", consider others…Love one another?
Over the past few months a number of our messages have included themes of sufferings and difficulties in life, and one might get the impression that to follow Christ is all down-and-out hardships. We don't share those things to be depressing or because we have a cup-half-empty worldview. We do that to help each of us have a clear perspective that, despite the modern western view that we should have a life of ease and happiness, we live in a broken world and following God doesn't exempt us from experiencing it's hardships.
What makes our message different, however, is that the sufferings of life are not wasted in God's plan. A verse in the message this past weekend was, "Trouble produces patience, and patience produces character, and character produces hope." Romans 5:3-4 The process "in God's plan" leads to hope. Outside of God, there is no hope.
So do we as followers of Christ have times of happiness and joy and ease? Yes, of course. Can we experience God and grow to be like Him in those times? Yes, of course! But let's not be naive or caught off guard or run from the difficult times in life for as we face them with the view that God is working in us, then we are led to a hope, a Christ formed in us hope!
Ministry is an interesting concept There are so many aspects and layers to it.
As I brought out in my message this weekend it is "anything" you do to bless others and honor God. And yet, I also brought out that we are called to find a role in our church family to fulfill our corporate calling to reach our community. So there is this tension between "where" and "how" I serve that seems to be casting a shadow on all of this. That is until we remember, ministry is less of what we DO and more of who we ARE.
We are ministers! The creation of the career pastor has kind of messed us all up a bit in our idea of ministry. God has shaped and called all of us to be ministers. And as we understand our new identity as Christ-followers we can begin to tune our "ears" to hearing the Spirit of God lead us to bless others at any given time.
There is another tension that casts a shadow, and that is the "need" versus "gift/skill" area. I'm going to speak to this in context of finding a role in our church. At the Connection Sunday this past weekend you saw serving teams with sign-up sheets expressing they need people to be on their teams. And as you walk from table to table you may think, "well, I don't know where I fit in! I love to cook, or I love organize things, or I love to help people …(fill in the blank)." And you just don't know how your gift or skill fits in with so many of the teams.
We had a discussion about this as a staff this week and so I want to pass this on to you – we understand the potential disconnect for many of us! And we are committed to figure out ways to make opportunities to serve be more relatable to your gift mix rather than our team/department name. Our next Connection Sunday will be based more on gifts and skills required instead of ministry teams! And to start off the opportunity to serve… if you are a person who likes to organize things and help communicate systems, I would love to have you help me create an even more relatable Connection Sunday Ministry Fair!
Until then, here is what you can do to find a place to serve now. If you know what your gifts and interests are, I and many others on our leadership would LOVE to talk to you about custom areas for you to jump in. There is a place for EVERY gift here and there are more opportunities than you can imagine to be part of the great calling YOUR church has to make difference in Half Moon Bay. If you really hunger to take those next steps to serve, then please reach out and we will help you find a role that can be very fulfilling for you and a powerful blessing for those you serve.
It starts with a couple of "reframing" actions. First, begin to embrace your calling as a full-time minister rather than an occasional volunteer. Second, become self-aware of what you bring to the table. You ARE gifted in some way by God and likely more than you give yourself, or frankly God, credit for! Third, recognize that those gifts can be used for more than your career or hobby and instead or in addition can lead people to experience God!
So much can be drawn from that passage and the verses surrounding it. But what struck me that week and again today is the "serve the Lord enthusiastically" phrase. I want for us all to love serving and give our all to this high calling, and to just go for it! As you move forward in your adventure, the adventure to see others blessed by God working through you, I pray you experience the incredible joy that God has in store for you!
Evangelism; The Highest Form of Worship!
This is our final update email and therefore my last blog post on our purposes. I have been privileged to be part of this event with our church and I hope you have been moved to take steps closer to Christ in trust and faith.
Evangelism is really an interesting call or purpose for our lives. Certainly, if any of these are controversial, it's this one. Just the word…evangelism…has developed an awkwardness these days. For many it conjures up images of TV preachers or street corner harassment or door-to-door campaigns. Our fears of rejection or turning people off or being labeled judgmental or just not knowing what to say gets the best of us.
And yet it's the sharing of the "good news" that is the last thing Jesus told us to do when he left this earth.
I don't know about you, but I get pretty excited about my latest tech gadget or running shoe, and want to tell people. And practically every survey I have ever taken about a product or customer support asks how likely I am to tell somebody. So if it's expected that I share about the insignificant, then how much more should I about the eternal? I don't ask that to guilt any of us! I understand our fears.
But let me present this in just a bit of a different way for a moment. As we have learned of the 5 purposes, they have been presented as individual attributes or callings that we should aspire to develop in our lives. Sort of a measuring stick to see how healthy we are in our faith development. I want us to do just that, and yet instead of seeing them as separate, we should also see them as integrated and intertwined and all of them as "worship".
An old statement from church history says, "The chief end of man is to glorify God and enjoy Him forever." John Piper takes it a step further and changes the last half to "by enjoying Him forever". So if worship is our goal, then how does evangelism fit in as the highest form of worship as my title suggests?
Our God is worthy enough for all of mankind, and more personally, our friends and family, to give their lives. Our God is worthy enough for us to "shout from the mountain tops" of what He has done in our lives. By sharing the good news, we in fact worship! And if one more person gives their life to Him, one more person is a worshipper.
I don't know if that helps any of us overcome our internal obstacles to sharing more. But perhaps by seeing it's importance, we might make the opportunity to work through our obstacles more intentionally.
We have the good news! Let's pass it on.
In recent months a number of thoughts have been bouncing around in my mind. But even deeper, my heart. Life has been going on with all it's busyness. Demands for work projects, recovery pain from latest marathon, family activities – just like everyone, I am busy. But like an underground spring, there have been thoughts of what could be. What I hope for. What I sense is missing or coming up short in my life.
I'm not depressed at all. Rather quite hopeful and encouraged!
These thoughts have been spurred on by a few passages of the bible that have come up in some of my readings or small group discussions.
They devoted themselves to the apostles' teaching and to fellowship, to the breaking of bread and to prayer. Everyone was filled with awe at the many wonders and signs performed by the apostles. All the believers were together and had everything in common. They sold property and possessions to give to anyone who had need. Every day they continued to meet together in the temple courts. They broke bread in their homes and ate together with glad and sincere hearts, praising God and enjoying the favor of all the people. And the Lord added to their number daily those who were being saved. Acts 2:42-47 (NIV).
Each of these passages have burrowed deep into my heart to create this longing.
As pastors, church leaders, and workers in our churches (paid or not) we are part of a powerful experience, opportunity and responsibility to be part of presenting "everyone fully mature in Christ"! I know life can be busy and overwhelming. I know that we can lose some of the wonder and joy and passion for what we do in ministry and see it as another obligation or task. And at seasons in my life it has been reduced to that.
I want the awe! I want the amazement!
I don't long for some 1st century re-creation of the church. It can never be exactly like that. I'm not expecting signs and wonders. I'll take them, sure. I don't envision a series of revival meetings with ecstatic behavior either.
What I see, what I hunger for is a revival of community. The principals of the Acts community can be developed. And there is no greater miracle to be seen than "Christ formed in us". I want to see the Lord adding to our numbers daily.
I want these things for my generation! My time!
I want to see a church strenuously contending with all the energy Christ so powerfully works in us! I'm seeing signs of that in my particular local church right now, but my hope is for the Church, not just my church.
I don't want to dismiss what I know He is already doing now, or imply we are not "strenuously contending" now. I see the power of God at work all the time and am moved to praise and gratefulness.
I just have this sense there is something…more.
I want to encourage us today to make space in our hearts for the wonder to grow – the wonder that God would use each of us to transform lives.
Let's not get caught up in the distractions of politics…God is bigger than America!
Let's stop whining about the marginalization of the church in America and be the church America…and the world needs. Our authority is not given to us by congress or the zeitgeist of culture, but by Christ!
Let's not try to impose our faith on culture, but compel our culture to adopt our God through the irresistible grace He offers!
Let's devote ourselves to the teaching and the fellowship and the prayer. Let's fill our worship centers with praise and celebration and let's fill our homes with each other – admonishing and teaching and encouraging and proclaiming.
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\section{Introduction}
Exploiting historical data to iteratively improve the performance of Model Predictive Controllers (MPC) has been an active theme of research over the past few decades \cite{rosolia2018data, hewing2019learning, aswani2013provably,kocijan2004gaussian,koller2018learning,berkenkamp2016safe,hewing2018cautious,hewing2019cautious,terzi2018learning,bacic2003general,ostafew2014learning,ostafew2016robust,brunner2013stabilizing,rosolia2019learning,lee1997model,lee2000convergence,lee2000model,lu2019robust,lorenzen2019robust,lorenzen2017adaptive,tanaskovic2013adaptive,tanaskovic2014adaptive,bujarCDC18,bujarbaruahAdapFIR,bujarArxivAdap}. The key idea is to use the stored state, input, and cost data to compute at least one of the following control design elements: $\emph{i})$ a \textit{model} to predict the system trajectory for a given initial state and input sequence, $\emph{ii})$ a \textit{safe set} of states from which the control task can be completed using a known safe policy and $\emph{iii})$ a \textit{value function}, which for a given safe policy, maps each state of the safe set to the closed-loop cost of completing the task.
Policy evaluation strategies used to estimate value functions from historical data are studied in Approximate Dynamic Programming (ADP) and Reinforcement Learning (RL)~\cite{bertsekas1996neuro,bertsekas2019feature,recht2018tour}. For instance, direct strategies compute the estimate value function which best fits the realized closed-loop cost over the stored states. On the other hand, in indirect strategies the estimate value function is computed by iteratively minimizing the temporal difference~\cite{sutton1988learning,bradtke1996linear}. A survey on policy evaluation strategies goes beyond the scope of this paper, we refer the reader to~\cite{bertsekas2019feature,bertsekas1996neuro} for a comprehensive review on this topic.
The integration of MPC with system identification strategies used to estimate the prediction model has been extensively studied in literature~\cite{aswani2013provably,kocijan2004gaussian,berkenkamp2016safe,koller2018learning,hewing2018cautious,hewing2019cautious,rosolia2019learning, terzi2018learning,bacic2003general,brunner2013stabilizing,ostafew2014learning,ostafew2016robust}. In adaptive MPC strategies~\cite{lu2019robust,lorenzen2019robust,lorenzen2017adaptive,tanaskovic2013adaptive,tanaskovic2014adaptive,bujarCDC18,bujarbaruahAdapFIR,bujarArxivAdap}, set-membership approaches are used to identify the set of possible parameters and/or the domain of the uncertainty which characterize the system's model. Afterwards, robust MPC strategies for additive~\cite{bemporad1999robust} or parametric~\cite{fleming2014robust,evans2012robust} uncertainty are used to guarantee recursive constraint satisfaction. Another strategy to identify the system dynamics is to fit a Gaussian Process (GP) to experimental data~\cite{koller2018learning,hewing2018cautious,kocijan2004gaussian,berkenkamp2016safe, hewing2019cautious}. GP can be used to identify a nominal model and confidence bounds, which may be used to tighten the constraint set over the planning horizon. This strategy provides high-probability safety guarantees~\cite{koller2018learning, hewing2019cautious}. The effectiveness of GP-based strategies on experimental platforms has been shown in~\cite{hewing2018cautious, berkenkamp2016safe}, where an MPC is used to race a 1/43-scale vehicle and to safely fly a drone. Regression strategies may also be used to identify the system model~\cite{rosolia2019learning, terzi2018learning}. For instance, the authors in~\cite{terzi2018learning} used a linear regression strategy to identify both a nominal model and the disturbance domain used for robust MPC design. In~\cite{rosolia2019learning}, we used local linear regression to identify the system model used by the controller, which was able to drive a 1/10-scale race car at the limit of handling.
Model-based and data-based approaches for computing safe sets have also been proposed in literature \cite{fisac2018general,kaynama2011continual,lygeros1999controllers, liniger2017real, wabersich2018linear, bacic2003general,brunner2013stabilizing,blanchini2005relatively, LMPClinear, LMPC}. In reachability-based strategies safe sets are computed solving a two players game between the controller and the disturbance \cite{fisac2018general,kaynama2011continual,lygeros1999controllers}.
Furthermore, these strategies provide a control policy, which can be used to guarantee safety by robustly constraining the evolution of the system within the safe set \cite{fisac2018general}.
Also viability theory may be used to compute safe sets \cite{liniger2017real}. The authors in \cite{liniger2017real} showed how to compute an inner approximation of the viability kernel and demonstrated the effectiveness on a RC-car set-up. In \cite{wabersich2018linear} the authors showed how to compute safe sets for uncertain systems exploiting data from a robust controller, afterwards they used the safe set in a linear model predictive safety certification framework. In~\cite{bacic2003general,brunner2013stabilizing} the authors computed safe sets and the associated control policy combining stored trajectories with polyhedral and ellipsoidal invariant sets. Another approach is proposed in~\cite{blanchini2005relatively} where the stored trajectories are mirrored to construct invariant sets. Finally, in \cite{LMPClinear, LMPC} we have shown how data from a deterministic system can be trivially used to compute safe sets. However, these strategies cannot be used to compute safe sets for uncertain systems.
In this paper we present an iterative \emph{robust} Learning Model Predictive Control (LMPC) design procedure for uncertain systems. We consider control tasks where the goal is to minimize a given cost function while satisfying state and input constraints. An iteration is a finite time execution of a control task from a given initial condition. We assume that at each iteration the constraint set and cost function are unchanged, but the initial condition and duration can be selected by the control designer.
At each iteration we exploit historical data and the LMPC policy from previous iterations to construct robust safe sets and approximations to the value function. Furthermore, we show how to compute a safe control policy which is defined over the robust safe set and it is used in the LMPC design. In particular, the controller plans the system trajectory using either a safe policy or a disturbance feedback policy.
We show that the proposed strategy guarantees that: \emph{i}) state and input constraints are recursively satisfied, \emph{ii}) the closed-loop system is Input-to-State Stable (ISS) and \emph{iii}) the performance of the certainty equivalent system is bounded by a function $Q^j(\cdot)$ which is non-increasing with the iteration index (i.e. $Q^{j+1}(\cdot) \leq Q^j(\cdot)$).
Finally, we show that the proposed iterative design procedure can be used to collect data on progressively larger regions of the state space. Thus, in applications where a conservative robust controller is available, our iterative algorithm may be used to improve the closed-loop performance and to enlarge the region of attraction.
Compared with standard robust MPC strategies~\cite{bemporad1999robust, mayne2005robust, chisci2001systems, bemporad2003min, chen1998robust}, where the terminal cost and constraint set used in the design are fixed, the proposed strategy iteratively updates these components used to synthesize the control policy. This strategy, which is tailored to iterative tasks, allows us to iteratively enlarge the region of attraction of the controller and to improve the performance of the certainty equivalent closed-loop system.
Compared to the robust certainty equivalent LMPC in~\cite{rosolia2017robust}, where we constructed a control invariant set and a control Lyapunov function for the certainty equivalent system, in this work we compute a \textit{robust} control invariant set and a control Lyapunov function for the uncertain system. We underline that extending the nominal and certainty equivalent LMPC approaches~\cite{LMPC,LMPClinear, rosolia2017robust} to uncertain systems is not straightforward.
In fact, as it will be clear later in this paper,
standard shifting MPC arguments for proving recursive robust constraint satisfaction do not apply.
In particular, the use of different feedback policies along the prediction horizon is one of the key elements of the proposed control design which is necessary for providing robustness guarantees.
This paper is organized as follows: in Section~\ref{sec:background} we recall some definitions from set theory and the definition of Input-to-State Stability (ISS). Section~\ref{sec:invSetFromData} describes the challenges of learning safe sets from stored data of uncertain systems. Section~\ref{sec:ProbForm} describes the problem formulation and design requirements. The strategy proposed to compute safe sets and the $Q$-function is described in Section~\ref{sec:LMPCpre}. Section~\ref{sec:LMPCpolicy} describes the control design strategy and Section~\ref{sec:properties} illustrates the controller properties. A strategy to approximate the region of attraction of the controller is presented in Section~\ref{sec:regAtt}. Finally, in Section~\ref{sec:results} we show the effectiveness of the proposed controller an a double integrator system subject to bounded additive uncertainty.
\section{Technical Background}\label{sec:background}
In this section we recall some definitions from set theory \cite[Chapter 10]{BorrelliBemporadMorari_book}, which will be used later in this work.
\begin{definition}[Positive Invariant Set]\label{def:Inv} A set $\mathcal{O} \subseteq \mathcal{X}$ is said to be a positive invariant set for the autonomous system $x_{t+1} = A x_t$, if
\begin{equation*}
x \in \mathcal{O} \rightarrow Ax \in \mathcal{O}.
\end{equation*}
\end{definition}
\begin{definition}[Robust Positive Invariant Set]\label{def:RobustInv} A set $\mathcal{O} \subseteq \mathcal{X}$ is said to be a robust positive invariant set for the uncertain autonomous system $x_{t+1} = A x_t + w_t$, with $w_t \in \mathcal{W}$
if
\begin{equation*}
x \in \mathcal{O} \rightarrow Ax+w\in \mathcal{O},~ \forall w \in \mathcal{W}.
\end{equation*}
\end{definition}
\begin{definition}[Robust Control Positive Invariant Set]\label{def:RobustInv} A set $\mathcal{C} \subseteq \mathcal{X}$ is said to be a robust control positive invariant set for the uncertain system $x_{t+1} = A x_t + Bu_t+ w_t$, with $w_t \in \mathcal{W}$ and $u_t \in \mathcal{U}$, if
\begin{equation*}
x \in \mathcal{C} \rightarrow \exists u \in \mathcal{U}: A x + B u + w \in \mathcal{C},~ \forall w \in \mathcal{W}.
\end{equation*}
\end{definition}
\begin{definition}[Robust Successor Set]\label{def:RobustSuc}
Given a control policy $\pi(\cdot)$ and the closed-loop system $x_{t+1} = A x_t + B\pi(x_t) + w_t$, we denote the robust successor set from the set $\mathcal{S}$ as
\begin{equation*}
\begin{aligned}
\Succ(\mathcal{S}, \mathcal{W}, {\pi}) = \{ & x_{t+1} \in \mathbb{R}^n : \exists x_t \in \mathcal{S}, \exists w_t \in \mathcal{W} \\
& \text{ such that } x_{t+1} = A x_t + B \pi(x_t) + w_t \}.
\end{aligned}
\end{equation*}
\end{definition}
Given the initial state $x_t$, the robust successor set $\Succ( x_t , \mathcal{W}, {\pi})$ collects the states that the uncertain autonomous system may reach in one time step.
\begin{definition}[N-Step Robust Reachable Set]\label{def:RobustSuc} Given a \\ control policy $\pi(\cdot)$ and the closed-loop system $x_{t+1} = A x_t + B\pi(x_t) + w_t$ with $w_t \in \mathcal{W}$ for all $t\geq 0$, for $k=\{0, \ldots, N-1 \}$ we recursively define the $N$-step robust reachable set from the set $\mathcal{S}$ as
\begin{equation*}
\begin{aligned}
\mathcal{R}_{t\rightarrow t+k+1}(\mathcal{S}, {\pi}) = \Succ(\mathcal{R}_{t\rightarrow t+k}(\mathcal{S}, {\pi}), \mathcal{W}, {\pi})
\end{aligned}
\end{equation*}
where $\mathcal{R}_{t\rightarrow t}(\mathcal{S}, {\pi})=\mathcal{S}$. Robust reachable sets are also referred to as forwards reachable sets.
\end{definition}
Given a linear time-invariant system, the $N$-Step robust reachable set $\mathcal{R}_{t\rightarrow t+N}(\mathcal{S}, {\pi})$ collects the states which can be reached from the set $\mathcal{S}$ in $N$-steps.
Finally, we recall the definition of Input-to-State Stability (ISS) of a robust invariant set \cite{lin1995various}, which extends the more widely known notion of ISS of an equilibrium point \cite{jiang2001input, goulart2006optimization,khalil2002nonlinear,grune2014iss}. We use the standard function classes $\mathcal{K}$, $\mathcal{K}_\infty$, and $\mathcal{KL}$ notation (see \cite{kellett2014compendium}) and we define the distance from a point $x\in\mathbb{R}^n$ to a set $\mathcal{O}\subseteq \mathbb{R}^n$ as
\begin{equation*}
|x|_\mathcal{O} \overset{\Delta}{=} \inf_{d\in\mathcal{O}} ||x-d||_2.
\end{equation*}
\begin{definition}[Input to State Stability (ISS)~\cite{lin1995various}]\label{def:iss}
Let $\mathcal{O}$ be a robust positive invariant set for the autonomous system~$x_{t+1} = A x_t + B \pi (x_t) + w_t$ with $w_t \in \mathcal{W}$. We say that the closed-loop system is ISS with respect to $\mathcal{O}$ if for all $w_t \in \mathcal{W}$, $t \geq 0$, and $x_0 \in \mathcal{X}$ we have:
\begin{equation*}
|x_{t+1}|_\mathcal{O} \leq \beta(|x_0|_{\mathcal{O}}, t+1) + \gamma \big( \textstyle \text{sup}_{k \in \{0, \ldots, t\}} ||w_k|| \big),
\end{equation*}
where $\beta(\cdot, \cdot)$ is a class-$\mathcal{KL}$ function and $\gamma(\cdot)$ is a class-$\mathcal{K}$ function.
\end{definition}
\section{Computing Invariant Sets From Data}\label{sec:invSetFromData}
In order to motivate our work, this section highlights the challenges associated with the computation of invariant sets from data. First, we recall from~\cite{LMPC, LMPClinear} how historical data can be used to compute invariant sets for deterministic systems. Consider the discrete time linear system
\begin{equation*}
\bar x_{t+1}^j = A \bar x_t^j + B \pi^j(\bar x_t^j)
\end{equation*}
where $\pi^j(\cdot)$ is a feedback policy known only along the $j$th stored trajectory $ \bar {\bf{x}}^j = [\bar x_0^j, \ldots, \bar x_t^j, \ldots, \bar x_{T^j}^j]$.
Assume that $\pi^j(\cdot)$ is able to execute the desired task safely, meaning that $\bar x_{T^j} \in \mathcal{O}$.
At any iteration $i>j$ and time $k\geq 0$, if the system state $x_k^i$ equals a state $x_t^j$ which has been visited at the previous $j$th iteration, then the feedback policy $\pi^j(\cdot)$ will drive the system along the $j$th trajectory. This obvious fact is a consequence of the system being deterministic. More importantly, as the policy $\pi^j(\cdot)$ brings the system to the invariant set $\mathcal{O}$, the convex hull of visited states and $\mathcal{O}$ is a control invariant set.
Therefore, invariant sets for deterministic systems can be easily built from data.
In contrast, when dealing with \emph{uncertain systems}, the set of visited states is not an invariant set. In fact, consider the discrete time uncertain system
\begin{equation*}
x^j_{t+1} = A x^j_t + B \pi^j(x^j_t) + w_t^j,
\end{equation*}
where the random disturbance $w_t^j$ belongs to the set $\mathcal{W}$ and the $j$th stored trajectory is ${\bf{x}}^j = [x_0^j, \ldots, x_t^j, \ldots, x_{T^j}^j]$.
Assume that $\pi^j(\cdot)$ is able to execute the desired task safely at iteration~$j$.
Notice that the stored trajectory ${\bf{x}}^j$ is associated with a specific disturbance realization $[w_0^j,\ldots,w_t^j,\ldots]$. For this reason at any iteration $i>j$ and time $k\geq 0$, if the system state $x_k^{i}$ equals a state $x_t^j$ that has been visited, applying the feedback policy $\pi^j(\cdot)$ may drive the system to a state neither stored nor safe, due to a potentially different disturbance realization $[w_0^{i},\ldots,w_t^{i},\ldots]$. In conclusion, the set of visited states cannot be naively exploited to compute invariant sets.
Furthermore, we underline that even if a control invariant set for uncertain systems can be computed from data, its use for MPC design is not straightforward. This issue will become clear later in this paper. In the following, we first present a strategy to construct robust control invariant sets using stored data from a linear uncertain system. Afterwards, we leverage these sets to iteratively synthesize robust LMPC policies.
\section{Problem Formulation}\label{sec:ProbForm}
We consider a discrete time uncertain system of the form:
\begin{equation}\label{eq:system0}
x_{t+1} = A x_t + B u_t + w_t,
\end{equation}
where $x_t \in \mathbb{R}^n$ and $u_t \in \mathbb{R}^d$ are the state and the input at time $t$ and the matrices $A$ and $B$ are known. The disturbances $w_t^j$ are zero mean independent and identically distributed (\textit{i.i.d.}) with bounded support $\mathcal{W}\subset\mathbb{R}^n$.
\begin{assumption}\label{ass:first}
The disturbance's support $\mathcal{W}$ is a compact polytope described by $l$ vertices $\{v^1_w, \ldots, v^l_w\}$ and it contains the origin.
\end{assumption}
Furthermore, the system is subject to the following convex constraints on states and inputs:
\begin{equation}\label{eq:constraintSet}
x_t \in \mathcal{X} \text{ and } u_t \in \mathcal{U}, ~ \forall t \geq 0,
\end{equation}
where the sets $\mathcal{X}$ and $\mathcal{U}$ contain the origin and are assumed to be compact.
In this paper, we consider control tasks where we would like to steer system~\eqref{eq:system0} towards the set $\mathcal{O}$, while minimizing the summation of the stage cost $h:\mathbb{R}^n \times \mathbb{R}^d \rightarrow \mathbb{R}$ and satisfying constraints~\eqref{eq:constraintSet}. Throughout the paper we make the following assumptions.
\begin{assumption}\label{ass:O_inf}
The set $\mathcal{O} \subset \mathcal{X} \subset \mathbb{R}^n$ is a robust positive invariant set for the autonomous system $x_{t+1} = (A + BK) x_t+w_t$ with $w_t \in \mathcal{W}$. Furthermore, $\mathcal{O}$ is a polyhedron defined through its vertices $\{v_o^1, \ldots, v^{m}_o \}$ and
\begin{equation*}
K \mathcal{O} = \{ u \in \mathbb{R}^d: \exists x \in \mathcal{O}, u = Kx \}.
\end{equation*}
\end{assumption}
\begin{assumption}\label{ass:cost}
The stage cost $h(\cdot,\cdot)$ is continuous and jointly convex in its arguments. Furthermore, we assume that for all $x \in \mathbb{R}^n$, and for all $ u \in \mathbb{R}^d$ the stage cost satisfies:
\begin{equation*}
\begin{aligned}
\alpha_x^l(|x|_\mathcal{O}) \leq h(x,0) &\leq \alpha_x^u(|x|_\mathcal{O})\\
&\text{and } \alpha_u^l ( |u|_{K\mathcal{O}} ) \leq h(0,u) \leq \alpha_x^u(|u|_{K\mathcal{O}}),
\end{aligned}
\end{equation*}
where $\alpha_{x}^u,\alpha_{x}^l, \alpha_{u}^u$, and $\alpha_{u}^l \in \mathcal{K}_\infty$.
\end{assumption}
\vspace{0.2cm} \begin{remark}
In Assumption~\ref{ass:O_inf} a robust invariant $\mathcal{O}$ is required. In the proposed approach $\mathcal{O}$ can be a very small neighborhood of the origin. In fact, the iterative nature of the control design will allow us enlarge the closed-loop region of attraction at each iteration.
\end{remark}
\subsection{Iteration and Control Design Objectives}\label{sec:Obj}
Iteration $j$ refers to a finite time execution of the control task from the initial conditions $x_0^j$. At each iteration, the initial condition may be different, however the system model~\eqref{eq:system0}, the constraints~\eqref{eq:constraintSet}, the set $\mathcal{O}$, and the cost function $h(\cdot,\cdot)$ are identical. We denote $x_t^j$ and $u_t^j$ as the state and input of the system at time $t$ of iteration $j$, i.e.,
\begin{equation}\label{eq:sys}
x_{t+1}^j = Ax_t^j + B u^j_t + w_t^j.
\end{equation}
Furthermore, we define $T^j$ as the finite time duration of iteration $j$. As we will discuss in Section~\ref{sec:LMPCpre}, the time $T^j$ is selected by the control designer.
At each iteration $j$, our objective is to design a state feedback policy for the uncertain system~\eqref{eq:sys}
\begin{equation}\label{eq:feedbackPolicy}
\pi^j(\cdot): \mathcal{C}^j \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^d,
\end{equation}
and the associated region of attraction $\mathcal{C}^j$ such that at each $j$th iteration and for all $x_0^j \in \mathcal{C}^j \subseteq \mathbb{R}^n$ we have that:
\begin{enumerate}
\item The certainty equivalent system
\begin{equation}\label{eq:disturbanceFreeSys}
\bar x_{t+1}^j = A \bar x^j_t + B \bar u^j_t
\end{equation}
with $\bar u^j_t = \pi^j(\bar x^j_t)$ converges asymptotically to the set $\mathcal{O}$, i.e. $\lim_{t \rightarrow \infty} \bar x_t^j \in \mathcal{O}$.
\item The closed-loop system $x_{t+1}^j = A x_t^j + B \pi^j(x_t^j) + w_t^j$ is Input-to-State Stable (ISS) with respect to the set $\mathcal{O}$ (see Section~\ref{sec:background} for the definition of ISS).
\item The closed-loop state and input constraints are robustly satisfied, namely
\begin{equation*}
x_t^j \in \mathcal{X} \text{ and } \pi^j(x_t^j) \in \mathcal{U }, \forall w_t^j \in \mathcal{W},~\forall t \geq 0.
\end{equation*}
\item The domain $\mathcal{C}^j$ of policy $\pi^j(\cdot)$ does not shrink with the iteration index, i.e., $\mathcal{C}^j \subseteq \mathcal{C}^{j+1}$.
\item The iteration cost of the certainty equivalent system~\eqref{eq:disturbanceFreeSys}, defined as
\begin{equation*}
J^{j}_{0\rightarrow T^j}(\bar x_0^{j}) = \sum_{k=0}^{T^j} h(\bar x_t^j,\pi^j(\bar x_t^j)),
\end{equation*}
is an upper-bound to the function $Q^{j-1}(\cdot)$ (i.e. $J^{j}_{0\rightarrow T^j}( x_0^{j}) \leq Q^{j-1}( x_0^{j})$), which is non-increasing at each iteration
\begin{equation*}
Q^{j}(\bar x) \leq Q^{k}(\bar x), \forall j \geq k.
\end{equation*}
\end{enumerate}
Property 5) implies that, as more data is collected, the upper-bound on the performance of the certainty equivalent closed-loop system is non-increasing.
\vspace{0.2cm} \begin{remark}
Note that convergence and stability properties 1) and 2) hold for $t\rightarrow \infty$, although we study iterations with finite time duration. In particular, an iteration is terminated after $T^j$ time steps and, therefore, at the end of the $j$th iteration the closed-loop system may not reach the set $\mathcal{O}$. As we will discuss later on, the duration of an iteration $T^j$ may affect the closed-loop performance of the controller.
\end{remark}
\section{LMPC Preliminaries}\label{sec:LMPCpre}
In this section, we assume that a robust policy at iteration $j$ is given and we describe how to compute the terminal cost and constraints used to synthesize the control policy at the next iteration $j+1$.
In particular, we show how historical data and the $j$th robust MPC policy can be used to build a \textit{robust safe set} of states from which the iteration $j+1$ can be executed.
The iterative synthesis procedure used to update the control policy is described in Section~\ref{sec:LMPCpolicy}. This iterative procedure is based on the robust safe set, $Q$-function, and safe policy, which are defined in this section. Their initialization at iteration $j=0$ is also discussed.
\subsection{Robust Safe Set}\label{sec:robSS}
This section shows how to iteratively construct robust control invariant sets. In particular, we run the closed-loop system at iteration $j$ and we exploit the closed-loop trajectory to construct a robust safe set.
We initialize the \textit{robust convex safe set} at iteration $j=0$ as
\begin{equation}\label{eq:SS_init}
\mathcal{CS}^0=\mathcal{O}.
\end{equation}
Afterwards, we design the robust $N$-steps policy
\begin{equation}\label{eq:NstepPolicy}
\boldsymbol{\pi}^{1,*}_t(\cdot)= [\pi^{1,*}_{t|t}(\cdot), \ldots, \pi^{1,*}_{t+N\text{-}1|t}(\cdot)],
\end{equation}
which steers the following closed-loop system
\begin{equation}\label{eq:NstepPredicted}
x^1_{t+1}= A x^1_t +B \pi^{1,*}_{t|t}(x_t) + w_t
\end{equation}
to the robust convex safe set $\mathcal{CS}^0$. Next, we show that the control policy~\eqref{eq:NstepPolicy} and the convex safe set~\eqref{eq:SS_init} can be used to construct the convex safe set at iteration $1$.
\begin{assumption}\label{ass:policyAssumption}
For all $t \in \{0, \ldots, T^1\}$, the $N$-steps policy $\boldsymbol{\pi}^1_t(\cdot)$ in~\eqref{eq:NstepPolicy} robustly steers the predicted closed-loop system
\begin{equation*}
\begin{aligned}
x_{k+1|t}^1 = A x_{k|t}^1 + B \pi_{k|t}^{1,*}( x_{k|t}^1 & ) + w^1_{k|t}, \\
&\forall k = t, \ldots, t+N-1
\end{aligned}
\end{equation*}
from the state $x_t^1{\in \mathcal{C}^1}$ to the robust convex safe set $\mathcal{CS}^0=\mathcal{O}$ in $N$-steps, while robustly satisfying state and input constraints~\eqref{eq:constraintSet}.
\end{assumption}
\vspace{0.2cm} \begin{remark}
The control policy $\boldsymbol{\pi}^{1,*}_t(\cdot)$, which satisfies Assumption~\ref{ass:policyAssumption}, can be computed using the iterative procedure that we will be describing in Section~\ref{sec:LMPCpolicy}.
\end{remark}
\vspace{0.2cm} \begin{remark}
We underline that $T^j$ represents the duration of the $j$th closed-loop simulation. On the other hand, $N$ is the length of the prediction horizon associated with the control policy $\boldsymbol{\pi}^{1,*}_t(\cdot)$. Notice that $T^j$ and $N$ are not related but in general $N<T^j$. Indeed, the horizon length $N$ is usually chosen much smaller than the task duration $T^j$ to reduce the computational burden associated with the control policy. Finally, we underline that the duration of the control task $T^j$ is chosen by the control designer.
\end{remark}
Let the vectors
\begin{equation}\label{eq:recordedUncertainTrajectory}
\begin{aligned}
[x_0^{1}, \ldots, x_{T^1}^{1}] \\
[u_0^{1}, \ldots, u_{T^1}^{1}]
\end{aligned}
\end{equation}
collect states and inputs associated with a simulation of the closed-loop system~\eqref{eq:NstepPredicted}. As we will discuss later on, by linearity of the system, any state in the convex-hull of the closed-loop trajectory in~\eqref{eq:recordedUncertainTrajectory} can be robustly steered to~$\mathcal{O}$.
However, the convex hull of the stored states in~\eqref{eq:recordedUncertainTrajectory} and $\mathcal{O}$, denoted as $\text{Conv}\{\{\cup_{t=0}^{T^j}x_t^1\}\cup\mathcal{O} \}$, is not invariant. In fact, the set $\text{Conv}\{\{\cup_{t=0}^{T^j}x_t^1\}\cup\mathcal{O} \}$ does not necessarily contain the $k$-steps robust reachable sets $\mathcal{R}_{t \rightarrow t + k}( x_0^1, { \boldsymbol{\pi}^{1,*}_t})$ from the starting state $x_0^1$, as shown in Figure~\ref{fig:stateNominalEvolution}.
\begin{figure}[h!]
\centering
\includegraphics[width= 0.9\columnwidth]{figures/cvxRollOut.png}
\caption{The figure shows the robust invariant set $\mathcal{O}$ (solid black), the robust reachable sets $\mathcal{R}_{t\rightarrow t+k}(x_0^j, \boldsymbol{\pi}^{1,*}_t)$ (dashed blue line) and the convex hull of the states in~\eqref{eq:recordedUncertainTrajectory} and $\mathcal{O}$, denoted as $\text{Conv}\{\{\cup_{t=0}^{T^j}x_t^1\}\cup\mathcal{O} \}$ (dashed red line). We notice that the set $\text{Conv}\{\{\cup_{t=0}^{T^j}x_t^1\}\cup\mathcal{O} \}$ (dashed red line) does not contain the robust reachable sets $\mathcal{R}_{t\rightarrow t+k}(x_0^j, \boldsymbol{\pi}^{1,*}_t)$ (dashed blue line) and therefore it is not a robust invariant for the closed-loop system~\eqref{eq:NstepPredicted}.}
\label{fig:stateNominalEvolution}
\end{figure}
Now, we notice that robust control invariant sets can be computed using $k$-steps robust reachable sets $\mathcal{R}_{t\rightarrow t+k}( x_0^1, { \boldsymbol{\pi}^{1,*}_t})$ from the stored states in~\eqref{eq:recordedUncertainTrajectory}. In particular, we notice that the union of the $k$-steps robust reachable sets $\mathcal{R}_{t\rightarrow t+k}( x_0^1, { \boldsymbol{\pi}^{1,*}_t})$ for $k=0, \ldots, N$ and the robust convex safe set $\mathcal{CS}^0=\mathcal{O}$ is a robust control invariant. Therefore, we define the \textit{robust convex safe set} at iteration $j=1$ as the convex hull of the robust reachable sets $ \mathcal{R}_{t\rightarrow t+k}(x_t^1, { \boldsymbol{\pi}^{1,*}_t})$ and the robust convex safe set $\mathcal{CS}^0$ at iteration~$0$:
\begin{equation}
\mathcal{CS}^1 = \text{Conv}\Bigg( \bigg\{ \bigcup_{t=0}^{T^1} \bigcup_{k=0}^N \mathcal{R}_{t\rightarrow t+k}(x_t^1, { \boldsymbol{\pi}^{1,*}_t}) \bigg\}\bigcup \mathcal{CS}^0\Bigg).
\end{equation}
The above robust convex safe set at iteration $j=1$ is constructed for the closed-loop system~\eqref{eq:NstepPredicted} and it is shown in Figure~\ref{fig:cs}.
\begin{figure}[h!]
\centering
\includegraphics[width= 0.9\columnwidth]{figures/cs.png}
\caption{The figure shows the robust invariant set $\mathcal{O}$ (solid black line), the convex safe set $\mathcal{CS}^1 = \text{Conv}( \{ \bigcup_{t=0}^{T^1} \bigcup_{k=0}^N \mathcal{R}_{t\rightarrow t+k}(x_t^1, { \boldsymbol{\pi}^{1,*}_t}) \} \bigcup \mathcal{CS}^0)$ (dashed green line) and the robust reachable sets $\mathcal{R}_{t\rightarrow t+k}(x_0^1, { \boldsymbol{\pi}^{1,*}_t})$ (dashed blue line).}
\label{fig:cs}
\end{figure}
The above process is repeated at each iteration $j\geq 1$. Clearly, Assumption~\ref{ass:policyAssumption} must hold when $\mathcal{CS}^{0}$ is replaced with $\mathcal{CS}^{j-1}$ and iteration 1 with $j$.
More formally, given the $N$-steps policy
\begin{equation}\label{eq:policy_j_iteration}
\boldsymbol{\pi}^{j,*}_t(\cdot) = [\pi_{t|t}^{j,*}(\cdot), \ldots, \pi_{t+N\text{-}1|t}^{j,*}(\cdot)]
\end{equation}
and the closed-loop system
\begin{equation}\label{eq:cl_sys}
x_{t+1}^j = Ax_{t}^j + B \pi_{t|t}^{j,*}(x_{t}^j) + w_t^j
\end{equation}
we assume that the following holds.
\begin{assumption}\label{ass:policyAss}
For all $t \in \{0, \ldots, T^j\}$, the $N$-steps policy $\boldsymbol{\pi}^{j,*}_t$ from~\eqref{eq:policy_j_iteration} robustly steers the predicted closed-loop system
\begin{equation*}
\begin{aligned}
x_{k+1|t}^j = A x_{k|t}^j + B \pi_{k|t}^{j,*}( x_{k|t}^j & ) + w^j_{k|t}, \\
&\forall k = t, \ldots, t+N-1
\end{aligned}
\end{equation*}
from the state $x_t^j {\in \mathcal{C}^j}$ to the robust convex safe set $\mathcal{CS}^{j-1}$ in $N$-steps, while robustly satisfying state and input constraints~\eqref{eq:constraintSet}.
\end{assumption}
Later in Section~\ref{sec:LMPCpolicy}, we will show how to synthesize a control polity $\boldsymbol{\pi}^{j,*}_t$ which satisfies Assumption~\ref{ass:policyAss}.
At iteration $j$, we iteratively define the \textit{robust convex safe set}:
\begin{equation}\label{eq:CS}
\mathcal{CS}^j = \text{Conv}\Bigg( \bigg\{ \bigcup_{t=0}^{T^j} \bigcup_{k=0}^N \mathcal{R}_{t\rightarrow t+k}(x_t^j, { \boldsymbol{\pi}^{j,*}_t}) \bigg\}\bigcup \mathcal{CS}^{j-1}\Bigg).
\end{equation}
Details on the computation and storage of the convex safe set are provided next.
\subsection{Robust Convex Safe Set: Vertex Representation}
Recall from Assumption~\ref{ass:O_inf} that $l$ denoted the number of vertices of the disturbance support. Now, we define the $l^k$ vertices of the $k$-step robust reachable set $\mathcal{R}_{t\rightarrow t+k} (x^j_t, { \boldsymbol{\pi}^{j,*}_t})$ from $x^j_t$ as
\begin{equation}\label{eq:verticesReachSet}
[v_{t+k|t}^{j,1}, \ldots, v_{t+k|t}^{j,l^k}].
\end{equation}
The vertices of the robust reachable sets $\mathcal{R}_{t\rightarrow t+k} (x^i_t, { \boldsymbol{\pi}^{i,*}_t})$ for all $k \in \{0,\ldots, N-1\}$, $i\in\{0,\ldots,j\}$ and $t \in \{0, \ldots, T^j\}$ are collected by the following matrix
\begin{equation}\label{eq:xMatrix}
\begin{aligned}
{\bf{X}}^j = [{\bf{X}}^{j-1}, &~v_{0|0}^{j,1},\ldots, v_{N-1|0}^{j,l^{N-1}}, \ldots\\
&~v_{t|t}^{j,1}, \ldots, v_{t+N-1|t}^{j,l^{N-1}}, \ldots\\
&~v_{T^j|T^j}^{j,1}, \ldots, v_{T^j + N-1|T^j}^{j,l^{N-1}}],
\end{aligned}
\end{equation}
where at the $j$th iteration $v_{t+k|t}^{j,i}$ represents the $i$th vertex of the robust reachable set $\mathcal{R}_{t\rightarrow t+k} (x^j_t, { \boldsymbol{\pi}^{j,*}_t})$.
In the above recursive definition, we set ${\bf{X}}^{0}=[v_o^1, \ldots, v_{o}^m]$, where $v_o^i$ for $i\in\{1,\ldots,m\}$ are the vertices of $\mathcal{O}$ from Assumption~\ref{ass:O_inf}. The matrix ${\bf{X}}^j \in \mathbb{R}^{n \times \text{col}({\bf{X}}^j)}$ , where the number of columns $\text{col}({\bf{X}}^j) = (T^j +1)\sum_{k=0}^{N-1}l^k + \text{col}({\bf{X}}^{j-1})$.
Finally, as the columns of the matrix ${\bf{X}}^j$ in~\eqref{eq:xMatrix} collect all vertices of the robust reachable sets $\mathcal{R}_{t\rightarrow t+k}(x_t^j, { \boldsymbol{\pi}^{j,*}_t})$, the robust convex safe set $\mathcal{CS}^j$ from \eqref{eq:CS} can be written as
\begin{equation}\label{eq:RS_vertex}
\mathcal{CS}^j = \Big\{ x \in \mathbb{R}^n : \exists {\bm{\lambda}}^j \geq 0, {\bf{X}}^j {\bm{\lambda}}^j = x \text{ and } \mathds{1}^\top{\bm{\lambda}}^j = 1 \Big\},
\end{equation}
where $\mathds{1} \in \mathbb{R}^{\text{col}({\bf{X}}^j)}$ is a vector of ones.
\vspace{0.2cm} \begin{remark}
Notice that the strategy presented in this paper is based on a commonly used ``vertex enumeration approach"~\cite{scokaert1998min, BorrelliBemporadMorari_book}. Its worst case complexity is exponential in the horizon $N$ of the feedback policy~\eqref{eq:policy_j_iteration}, although independent on the length of the task duration $T^j$. We underline that this paper focuses on the fundamental properties of the controller design. Computational tractability can be obtained as in any MPC scheme by using a different disturbance model or feedback parametrization.
\end{remark}
\subsection{Robust Q-Function}\label{sec:itCost}
The robust $Q$-function approximates the cost-to-go over the robust convex safe set and it is constructed iteratively as explained next.
At iteration $j$, we assume that we are given the robust $Q$-function $Q^{j-1}(\cdot)$ which maps each state $x \in \mathcal{CS}^{j-1}$ to the closed-loop cost, and we show how to construct a robust $Q$-function at the next iteration $j$.
This recursion is initialized at iteration $0$ setting the robust $Q$-function
\begin{equation}\label{eq:Qfun_init}
Q^0(x) = 0, ~\forall x \in \mathcal{CS}^0 = \mathcal{O}.
\end{equation}
While in the nominal case from~\cite{LMPClinear} the vertices of the convex safe set are a subset of the stored trajectory, the convex safe set from~\eqref{eq:CS} may introduce additional vertices representing the worst case predicted realizations. For this reason, a cost-to-go associated with such predicted worst case realizations should be defined. In the following we define the cost-to-go $J^{j}_{t|t}$ associated with the stored states $x_{t|t}^j = x_t^j$ and the predicted cost-to-go $J^{j}_{k|t}$ associated with the predicted state $x^{j}_{k|t}$ at time $k$. In particular, after completion of the iteration $j$ for $t\in \{0, \ldots, T^j\}$, $k \in \{0, \ldots, N-1\}$ and $i \in \{1,\ldots,l^{k}\}$, we compute the cost-to-go for the vertices $v_{k|t}^{j,i}$ of $\mathcal{CS}^j$ from ${\bf{X}}^j$ in~\eqref{eq:xMatrix} as
\begin{equation}\label{eq:J_definition1}
\begin{aligned}
J^{j}_{k|t} \big( v_{k|t}^{j,i}\big)\! = \! \min_{\boldsymbol{\gamma}_k \geq 0}~~ & h\big( v_{k|t}^{j,i}, \pi_{k|t}^{j,*} (v_{k|t}^{j,i} ) \big) \! + \! \sum_{r=1}^{l^{k+1}} \gamma^r J^{j}_{k+1|t} ( v_{k+1|t}^{j,r} ),\\
\text{s.t.}~~~~ & \sum_{r=1}^{l^{k+1}} \gamma^r v_{k+1|t}^{j,r} = A v_{k|t}^{j,i} + B \pi_{k|t}^{j,*} (v_{k|t}^{j,i})\\
& \sum_{r=1}^{l^{k+1}} \gamma^r = 1
\end{aligned}
\end{equation}
where $\boldsymbol{\gamma}_k = [\gamma^1,\ldots, \gamma^{l^{k+1}}]$ and ${\pi}^{j,*}_{k|t}(\cdot)$ is the control policy from~\eqref{eq:policy_j_iteration}. In the above recursion we set
\begin{equation}\label{eq:J_definition2}
J^{j}_{t+N |t }\big( v_{t+N|t}^{j,i} \big) = Q^{j-1}\big( v_{t+N|t}^{j,i} \big),~\forall i \in \{1,\ldots, l^{N}\}.
\end{equation}
Basically, the cost-to-go $J^{j}_{k|t} ( v_{k|t}^{j,i})$ at time $k$ is computed summing up the running cost and the interpolated cost-to-go at the next predicted time $k+1$.
Given $Q^{j-1}(\cdot)$, the cost-to-go $J^{i}_{k|t} ( \cdot )$ is computed for all $i\in \{0, \ldots, j\}$, $t \in \{0,\ldots, T^j\}$ and $k\in \{0, \ldots, N\text{-}1\}$. Then, these cost values are collected in the following vector
\begin{equation*}
\begin{aligned}
{\bf{J}}^{j} = [ {\bf{J}}^{j-1}, &~J_{0|0}^j\big(v_{0|0}^{j,1} \big),\ldots, J_{N-1|0}^j\big(v_{N-1|0}^{j,l^{N-1}} \big), \ldots\\
&~J_{t|t}^j\big(v_{t|t}^{j,1}\big), \ldots, J_{t+N-1|t}^j\big(v_{t+N-1|t}^{j,l^{N-1}} \big), \ldots\\
&~J_{T^j|T^j}^j\big(v_{T^j|T^j}^{j,1} \big), \ldots, J_{T^j+N-1|T^j}^j\big(v_{T^j+N-1|T^j}^{j, l^{N-1}}\big)],
\end{aligned}
\end{equation*}
where ${\bf{J}}^{0} = [0, \ldots, 0]$ represents the cost-to-go associated with the vertices of $\mathcal{O}$.
Finally, we define the $Q$-function at iteration $j$ which interpolates the cost-to-go over the robust safe set,
\begin{equation}\label{eq:Qfun}
\begin{aligned}
Q^j(x) = \min_{ {\bm \lambda}^j \in \Lambda^{j}(x)} & \quad {\bf{J}}^{j} {\bm \lambda}^j, \\
\end{aligned}
\end{equation}
where for the matrix ${\bf{X}}^j$ the set
\begin{equation}\label{eq:LambdaSet}
\begin{aligned}
\Lambda^{j}(x) = \Big\{{\bm{\lambda}}^j \in \mathbb{R}^{\text{col}({\bf{X}}^j)} : {\bm{\lambda}}^j \geq 0, ~&{\bf{X}}^j{\bm{\lambda}}^j = x \\&\text{ and } \mathds{1}^\top{\bm{\lambda}}^j = 1 \Big\}
\end{aligned}
\end{equation}
collects the vectors ${\bm{\lambda}}^j$ which can be used to express $x$ as a convex combination of the columns of ${\bf{X}}^j$.
\vspace{0.2cm} \begin{remark}
Notice that a longer task duration $T^j$ results in more data points used to compute the cost vector ${\bf{J}}^{j}$. Consequently, the $Q$-function from~\eqref{eq:Qfun} can only decrease when the duration of the control task $T^j$ is increased. As we will discuss later on, the $Q$-function from~\eqref{eq:Qfun} is an upper-bound of the closed-loop performance of the certainty equivalent system, therefore the upper-bound on the cost can only decrease as the duration of the control task $T^j$ increases.
\end{remark}
\subsection{Set of Safe Policies}\label{sec:dataBasedPolicy}
At this point we have shown how to compute a robust control invariant set and a $Q$-function based on data collected at the $j$th iteration. The last missing element needed for an MPC design is the feedback controller associated with the terminal set.
Here we show how to construct a set of safe policies $\mathcal{SP}^j$, which may be used to robustly constrain the evolution of system~\eqref{eq:sys} within $\mathcal{CS}^j$, while satisfying state and input constraints~\eqref{eq:constraintSet}.
We begin by presenting an implicit parametrization of the set of policies $\mathcal{SP}^j$ which is amenable for optimization and it can be used to design a predictive controller that guarantees recursive constraint satisfaction. Afterwards, we define a safe policy $\kappa^{j,*}(\cdot) \in \mathcal{SP}^j$, which is able to complete the task from any state within the robust convex safe set $\mathcal{CS}^j$.
First, we define the matrix ${\bf{U}}^j$ collecting the inputs associated with the data stored in~\eqref{eq:xMatrix},
\begin{equation}\label{eq:uMatrix}
\begin{aligned}
{\bf{U}}^j \! = \! [{\bf{U}}^{j-1}, & \pi^{j,*}_{0|0} \big( v_{0|0}^{j,1}\big),\ldots, \pi^{j,*}_{N-1|0} \big( v_{N-1|0}^{j,l^{N{-}1}}\big), \ldots\\
& \pi^{j,*}_{t|t} \big( v_{t|t}^{j,1}\big), \ldots, \pi^{j,*}_{t+N-1|t} \big( v_{t+N-1|t}^{j,l^{N-1}}\big), \ldots\\
& \pi^{j,*}_{T^j|T^j} \big( v_{T^j|T^j}^{j,1} \big), \ldots, \pi^{j,*}_{T^j+N-1|T^j} \big( v_{T^j+N-1|T^j}^{j,l^{N-1}}\big)]
\end{aligned}
\end{equation}
where the policies $ \pi^{j,*}_{k|t}$ are defined in~\eqref{eq:policy_j_iteration}.
In the above recursive definition ${\bf{U}}^{0}= [Kv_o^1, \ldots, Kv_{o}^m]$, where the feedback gain $K$ and the vertices $v_o^i$ of $\mathcal{O}$ are defined in Assumption~\ref{ass:O_inf}.
Now, we notice that by linearity of system \eqref{eq:sys}, if a state $x \in \mathcal{CS}^j$ is expressed as a convex combination of the stored states $x = {\bf{X}}^j {\bm{\lambda}}^j$, then the input $u = {\bf{U}}^j {\bm{\lambda}}^j \in \mathcal{U}$ will keep the evolution of the system within $\mathcal{CS}^j$ for all disturbance realizations. More formally, given the set $\Lambda^j(\cdot)$ defined in~\eqref{eq:LambdaSet}, we have that
$\forall x \in \mathcal{CS}^j \subseteq \mathcal{X}, \forall {\bm{\lambda}}^j \in \Lambda^j(x)$
\begin{equation*}
{\bf{U}}^j {\bm{\lambda}}^j \in \mathcal{U} \text{ and } Ax+B {\bf{U}}^j {\bm{\lambda}}^j +w \in \mathcal{CS}^j, \forall w \in \mathcal{W}.
\end{equation*}
Therefore, the set of feedback policies $\kappa^j(\cdot): \mathbb{R}^n \rightarrow \mathbb{R}^d$
\begin{equation}\label{eq:feasPolicy}
\begin{aligned}
\mathcal{SP}^j = \big\{ \kappa^j(\cdot) : \forall x \in \mathcal{CS}^j, \exists & {\bm{\lambda}}^j \in \Lambda^j(x), \\ &\text{ such that }\kappa^j(x) ={\bf{U}}^j {\bm{\lambda}}^j \big\},
\end{aligned}
\end{equation}
guarantees that $\forall \kappa^j(\cdot) \in \mathcal{SP}^j$ the robust convex safe set $\mathcal{CS}^j$ is a robust positive invariant set for the closed-loop system $x_{t+1} = Ax_t+B\kappa^j(x_t)+w_t$. This statement is formalized by the following Proposition~\ref{prop:invariance}.
\begin{proposition}\label{prop:invariance}
Let Assumptions~\ref{ass:first}--\ref{ass:policyAss} hold. Then, for all control policies $\kappa^j(\cdot) \in \mathcal{SP}^j$ and $\forall x \in \mathcal{CS}^j$ we have that
\begin{equation*}
Ax+B\kappa^j(x)+w \in \mathcal{CS}^j\subseteq \mathcal{X}~ \forall w \in \mathcal{W}
\end{equation*}
and $\kappa^j(x) \in \mathcal{U}$.
\end{proposition}
\begin{proof}
The proof can be found in the Appendix.
\end{proof}
Finally, we define the safe policy
\begin{equation}\label{eq:theSafePolicy}
\kappa^{j,*}(x) = {\bf{U}}^j {\bm{\lambda}}^{j,*}(x),
\end{equation}
where ${\bm{\lambda}}^{j,*}(x)$ is the minimizer in~\eqref{eq:Qfun}. Basically, the above safe policy evaluated at $x$ is given by the convex combination of stored inputs, for the multipliers ${\bm{\lambda}}^{j,*}(x)$ which define the robust $Q$-function at $x$. In the following propositions, we show that the $Q$-function is a Lyapunov function for the certainty equivalent closed-loop system~\eqref{eq:disturbanceFreeSys} and \eqref{eq:theSafePolicy}. Furthermore, we show that the policy~\eqref{eq:theSafePolicy} in closed-loop with system~\eqref{eq:sys} guarantees Input-to-State Stability (ISS).
\begin{proposition}\label{prop:QfunDecrease}
Let Assumptions~\ref{ass:first}--\ref{ass:policyAss} hold. Consider the $Q$-function $Q^j(\cdot)$ in~\eqref{eq:Qfun}, we have that for all $x \in \mathcal{CS}^j$
\begin{equation}
Q^j(x) \geq h(x, \kappa^{j,*}(x)) + Q^j(Ax+B \kappa^{j,*}(x)),
\end{equation}
where $\kappa^{j,*}(\cdot)$ is the safe policy defined in~\eqref{eq:theSafePolicy}.
\end{proposition}
\begin{proof}
The proof can be found in the Appendix.
\end{proof}
\begin{proposition}
\label{prop:safePolicyISS}
Consider system~\eqref{eq:sys} in closed-loop with the safe policy~\eqref{eq:theSafePolicy}. Let Assumptions~\ref{ass:first}--\ref{ass:policyAss} hold and assume that $x_0 \in \mathcal{CS}^j$, then the closed-loop system~\eqref{eq:sys} and \eqref{eq:theSafePolicy} is input to state stable for the robust positive invariant set $\mathcal{O}$.
\end{proposition}
\begin{proof}
The proof can be found in the Appendix.
\end{proof}
\section{Learning Model Predictive Control}\label{sec:LMPCpolicy}
This section introduces the iterative control design procedure.
At the end of iteration $j-1$, we collect a data set of costs, inputs, and states, which are used to construct the robust convex safe set and robust $Q$-function at iteration $j-1$, as described in Section~\ref{sec:LMPCpre}. Finally, we leverage these quantities to design a robust Learning Model Predictive Controller (LMPC) for the $j$th iteration. The LMPC policy is able to safely execute the control task at the $j$th iteration and it can be used to collect new closed-loop data to design the controller at the next iteration $j+1$.
\subsection{Policy Synthesis}
In this section, we introduce the LMPC policy.
For more details on the control design choices and the differences with standard MPC design, we refer to the discussion in Section~\ref{sec:discussion} and to the properties description in Section~\ref{sec:properties}.
We define the following optimal control problem for the state $x_t^j \in \mathbb{R}^n$ and the parameter $N_t^j \in \{0,\ldots, N\}$:
\begin{subequations}\label{eq:FTOCP}
\begin{align}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!C_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t^j, N_t^j) = \notag\\
\min_{\substack{\boldsymbol{M}_t^j,\\ {\bm{\lambda}}^j_{t}, \boldsymbol{g}_t^j } } ~ & \sum_{k=t}^{t+N-1} h(\bar x^j_{k|t}, \pi^j_{k|t}(\bar x^j_{k|t})) + Q^{j-1}(\bar x^j_{t+N|t}) \label{eq:FTOPC_Cost} \\
\textrm{s.t. }~\!~
& x^j_{t|t}= \bar x^j_{t|t}=x_t^j,\label{eq:FTOPC_Init} \\
& \bar x^j_{k+1|t}=A \bar x^j_{k|t}+B \pi^j_{k|t}(\bar x^j_{k|t}), \label{eq:FTOCP_dynNom}\\
& x^j_{k+1|t}=A x^j_{k|t}+B \pi^j_{k|t}(x^j_{k|t}) + w^j_{k|t}, \label{eq:FTOCP_dyn}\\
& x^j_{k|t} \in \mathcal{X},~~\pi^j_{k|t}(x^j_{k|t}) \in \mathcal{U}, \label{eq:FTOCP_InpCons}\\
& x^j_{t+N|t} \in ~\mathcal{CS}^{j-1}, \label{eq:FTOCP_TermCons}\\
& \pi^{\text{d}}_{k|t}(x_{k|t}) = \textstyle\sum_{s=0}^{k-t-1} M^j_{ks|t}w^j_{s|t} + g^j_{k|t}, \label{eq:FTOCP_DFpolicy} \\
& \kappa_{k|t}^{j-1}(x^j_{k|t}) = {\bf{U}}^{j-1} {\bm{\lambda}}^j_{k|t}, \label{eq:FTOCP_DataBasedpolicy1}\\
& {\bm{\lambda}}^j_{k|t} \in \Lambda^{j-1}(x_{k|t}^j), \label{eq:FTOCP_DataBasedpolicy2} \\
& \pi^j_{i|t}(x^j_{i|t}) = \pi^{\text{d}}_{i|t}(x_{i|t}),\forall i\in \{t,...,t+N_t^j-1 \}, \label{eq:FTOCP_policy1} \\
& \pi^j_{i|t}(x^j_{i|t}) = \kappa_{i|t}^{j-1}(x^j_{i|t}), \forall i\in \{t+N_t^j, ..., t\!+\!N\!-\!1\},\label{eq:FTOCP_policy2} \\
& \forall w^j_{k|t} \in \mathcal{W}, ~ \forall k = \{t, ..., t+N-1\}, \notag
\end{align}
\end{subequations}
where the optimization variables are
\begin{equation*}
\boldsymbol{M}_t^j = \begin{bmatrix} 0 & \ldots \\
M_{21|t}& 0 & \ldots \\
M_{31|t}& M_{32|t} & \ddots \\
\vdots
\end{bmatrix},~\boldsymbol{g}_t^j = \begin{bmatrix} g^j_{t|t}\\
\vdots \\
g^j_{t+N-1|t},
\end{bmatrix}
\end{equation*}
${\bm{\lambda}}^j_{t} = [{\bm{\lambda}}^j_{t|t},\ldots, {\bm{\lambda}}^j_{t+N-1|t}]$.
Equation~\eqref{eq:FTOCP_policy2} and the parameter $N_t^j$ describe the control policy which defines the evolution of the predicted nominal and uncertain trajectories in \eqref{eq:FTOCP_dynNom}--\eqref{eq:FTOCP_dyn}. In particular, for the first $N_t^j$ predicted time steps the control policy $\pi_{k|t}^j(\cdot)$ equals the disturbance feedback policy~\eqref{eq:FTOCP_DFpolicy}, and for the last $N-N_t^j$ predicted steps $\pi_{k|t}^j(\cdot)$ equals the safe feedback policy~\eqref{eq:FTOCP_DataBasedpolicy1}--\eqref{eq:FTOCP_DataBasedpolicy2}.
Equations \eqref{eq:FTOPC_Init}--\eqref{eq:FTOCP_InpCons} represent input and state constraints which must be satisfied robustly for all disturbance realizations.
For $k \in \{t, \ldots, N_t^j-1\}$, the robust state and input constraints are reformulated using the disturbance feedback policy~\eqref{eq:FTOCP_DFpolicy} \cite{goulart2006optimization}.
On the other hand, from Proposition~\ref{prop:invariance}, we have that for time $k\in\{ N_t^j, \ldots, N \}$ the safe policy~\eqref{eq:FTOCP_DataBasedpolicy1} guarantees robust constraints satisfaction by construction.
Finally, the terminal constraint \eqref{eq:FTOCP_TermCons} robustly enforces $x_{t+N|t}$ within the robust control invariant set $\mathcal{CS}^{j-1}$.
The finite time optimal control problem~\eqref{eq:FTOCP} is used to define the LMPC policy from Algorithm~\ref{LMPCpolicy}. Given the measured state $x_t^j$, Algorithm~\ref{LMPCpolicy} solves $N+1$ instances of Problem~\eqref{eq:FTOCP} and it returns the optimal robust $N$-steps policy at time $t$
\begin{equation}\label{eq:optNstepPolicy}
{\boldsymbol{\pi}_t^{j,*}(\cdot)=[\pi^{j,*}_{t|t}(\cdot), \ldots, \pi^{j,*}_{t+N-1|t}(\cdot)]}
\end{equation}
and the LMPC cost $J_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t^j)$. Then, we apply to system~\eqref{eq:sys}
\begin{equation}\label{eq:MPCpolicy}
u_t^j = \pi^{j,*}_{t|t}(x_t).
\end{equation}
Algorithm~\ref{LMPCpolicy} is resolved at time $t+1$, based on the new state $x_{t+1|t+1} = x_{t+1}^j$, yielding a \textit{moving} or \textit{receding horizon} control strategy. Finally, the control policy~\eqref{eq:optNstepPolicy} is used to compute the robust convex safe set $\mathcal{CS}^j$ as discussed in Section~\ref{sec:LMPCpre}.
\begin{algorithm}[h!]
\SetAlgoLined
\textbf{Input:} System's state $x_t^j$\\
Set $N_t^{j,*} = \textstyle \text{argmin}_{N_t^j \in \{0, \ldots, N\} } C_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t^j, N_t^j) $ \\
Let $\boldsymbol{\pi}_t^{j,*}(\cdot)=[\pi^{j,*}_{t|t}(\cdot), \ldots, \pi^{j,*}_{t+N-1|t}(\cdot)]$ be the optimal solution to problem $C_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t^j,N_t^{j,*})$\\
Set $J_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t) = \textstyle \text{min}_{N_t^j \in \{0, \ldots, N\} } C_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t^j, N_t^j)$\\
\textbf{Output:} $\boldsymbol{\pi}_t^{j,*}(\cdot)$ and $J_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t^j)$
\caption{LMPC Algorithm}
\label{LMPCpolicy}
\end{algorithm}
The above algorithm solves different instances of Problem~\eqref{eq:FTOCP} to synthesize the control policy at iteration $j$. In particular, in line~2 we first solve Problem~\eqref{eq:FTOCP} for all $N_t^j \in \{0,\ldots, N\}$ and then set $N_t^{j,*}$ equal to the parameter which led to the minimum open-loop cost $C_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t^j, N_t^{j,*})$. Afterwards, in lines $3$-$4$ we set the optimal control policy $\boldsymbol{\pi}_t^{j,*}(\cdot)$ and the open-loop cost $J_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t)$ equal to the optimal policy and the open-loop cost of problem $C_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t^j, N_t^{j,*})$, respectively.
\vspace{0.2cm} \begin{remark}
The optimal control problem~\eqref{eq:FTOCP} computes a sequence of policies ${\boldsymbol{\pi}_t^{j,*}(\cdot)=[\pi^{j,*}_{t|t}(\cdot), \ldots, \pi^{j,*}_{t+N-1|t}(\cdot)]}$, which robustly steer system~\eqref{eq:sys} from $x_t^j$ to the robust convex safe set $\mathcal{CS}^{j-1}$.
Notice that if at iteration $j-1$ more data is collected, then the robust convex safe set $\mathcal{CS}^{j-1}$ from~\eqref{eq:CS} may be enlarged. As a result, when the tasks are defined over longer time periods $T^j$, the proposed iterative control design procure may synthesize control policies with larger regions of attraction.
\end{remark}
\subsection{Design Choices}\label{sec:discussion}
In standard robust MPC at each time step we solve an optimal control problem over a fixed space of feedback policies. On the other hand, in Problem~\eqref{eq:FTOCP} the space of feedback policies changes as a function of the predicted time step $k$. In particular, the predicted trajectory is computed using a disturbance feedback policy for $k \leq N_t^j$ and a safe feedback policy~\eqref{eq:FTOCP_DataBasedpolicy1}--\eqref{eq:FTOCP_DataBasedpolicy2} for $k>N_t^j$. In the following we discuss why this strategy allows us to guarantee recursive constraint satisfaction.
Recall that in predictive control recursive constraint satisfaction is ensured using a terminal constraint set. In particular, the terminal constraint set should be (robust) control invariant, for a feedback policy that can be used by the (robust) MPC to forecast the evolution of the system~\cite{BorrelliBemporadMorari_book}.
Notice that a disturbance feedback policy (or equivalently an affine state feedback policy~\cite{goulart2006optimization}) may not be able to robustly constrain the evolution of the system within the terminal constraint set $\mathcal{CS}^j$.
For this reason, in Problem~\eqref{eq:FTOCP} we used a time-varying feedback policy, which is defined by the parameter $N_t^j$, and in Algorithm~\ref{LMPCpolicy} we solved Problem~\eqref{eq:FTOCP} for different values of $N_t^j$. This strategy guarantees that the safe policy can be used to robustly constrain the evolution of the predicted system within the robust safe set $\mathcal{CS}^j$, and it is used in Theorem~\ref{th:recFeas} to show that the LMPC~\eqref{eq:FTOCP} and~\eqref{eq:MPCpolicy} guarantees recursive feasibility.
Furthermore, we comment on the computational tractability of the proposed strategy. As already mentioned, Algorithm~\ref{LMPCpolicy} solves $N+1$ instances of Problem~\eqref{eq:FTOCP} to forecast the evolution of the system using either the disturbance feedback policy or the safe policy from Section~\ref{sec:dataBasedPolicy}. We underline that these $N+1$ optimal control problems are independent and can be solved in parallel. Therefore, when parallel computing is available, the online computational complexity of the proposed strategy is independent on the controller horizon.
Finally, we emphasize the differences between the linear control policy defined by the feedback gain $K$ from Assumption~\ref{ass:O_inf}, the safe control policy $\kappa^{j,*}(\cdot)$ in~\eqref{eq:theSafePolicy}, and the LMPC policy~\eqref{eq:MPCpolicy}. We notice that the linear feedback policy $\pi^{\mathrm{linear}}(x) = K x$ may be used to robustly constrain the evolution of the system within the robust positive invariant set $\mathcal{O}$, which may be a small neighborhood of the origin. On the other hand, the safe control policy $\kappa^{j,*}(\cdot)$ is defined on the robust convex safe set $\mathcal{CS}^j \subseteq \mathcal{O}$ and it may be used to steer the system from any state $x \in \mathcal{CS}^j$ to the neighborhood of the origin $\mathcal{O}$, as shown in the result section. Finally, the LMPC policy~\eqref{eq:MPCpolicy} is defined on a larger domain $\mathcal{C}^j \subseteq \mathcal{CS}^j$ and it may be used to enlarge the robust convex safe set by collecting new data, when the initial condition $x_0^j$ is selected on the boundary of $\mathcal{C}^j$. A strategy to select such initial condition is described in Section~\ref{sec:regAtt} and its efficacy is demonstrated in the result Section~\ref{sec:resIterativeEnlargement}.
\section{Properties}\label{sec:properties}
This section shows that the LMPC policy~\eqref{eq:MPCpolicy} satisfies our design requirements from Section~\ref{sec:Obj}.
\subsection{Recursive Feasibility}
We show that if Problem~\eqref{eq:FTOCP} is feasible at time $t=0$ for some $N_0^j \in \{0, \ldots, N\}$, then the LMPC policy~\eqref{eq:MPCpolicy} guarantees that state and input constraints are recursively satisfied.
First, we define the set
\begin{equation}\label{eq:feasRegion}
\mathcal{C}^j = \{ x \in \mathbb{R}^n : \exists N_0^j \in \{0, \ldots, N\}, C_{0\rightarrow N}^{\scalebox{0.5}{LMPC},j}(x, N_0^j) < \infty \}
\end{equation}
collecting the states from which Problem~\eqref{eq:FTOCP} is feasible for some $N_0^j \in \{0, \ldots, N\}$. Next, we show that if $x_0^j \in \mathcal{C}^j$, then Problem~\eqref{eq:FTOCP} is feasible for all time $t\geq 0$ and for some $N_t^j \in \{0, \ldots, N\}$.
\begin{theorem}\label{th:recFeas}
Consider the closed-loop system~\eqref{eq:sys} and \eqref{eq:MPCpolicy}. Let Assumptions~\ref{ass:first}--\ref{ass:cost} hold and $x_0^j \in \mathcal{C}^j$. Then, for all time $t\geq0$ Problem~\eqref{eq:FTOCP} is feasible for some $N_t^j \in \{0, \ldots, N\}$, and the closed-loop system~\eqref{eq:sys} and \eqref{eq:MPCpolicy} satisfies state and input constraints.
\end{theorem}
\begin{proof}
Assume that at time $t$ Problem~\eqref{eq:FTOCP} is feasible for some $N_t^j \in \{0, \ldots, N\}$. At the next time $t+1$, by Proposition~\ref{prop:invariance} we know that, for $\kappa^{j-1,*}(\cdot) \in \mathcal{SP}^{j-1}$, the following candidate policy
\begin{equation}\label{eq:candidateSolution}
[\pi^{j,*}_{t+1|t}(\cdot), \ldots, \pi^{j,*}_{t+N-1|t}(\cdot), \kappa^{j-1,*}(\cdot)]
\end{equation}
is feasible for the Problem~\eqref{eq:FTOCP} for some $N_{t+1}^j \in \{0, \ldots, N\}$. \\
By assumption $x_0^j \in \mathcal{C}^j$, thus we have that Problem~\eqref{eq:FTOCP} is feasible at time $t=0$ for some $N_0^j \in \{0, \ldots, N\}$. Concluding, we have shown that if Problem~\eqref{eq:FTOCP} is feasible for some $N_t^j \in \{0, \ldots, N\}$ at time $t$, then Problem~\eqref{eq:FTOCP} is feasible for some $N_{t+1}^j \in \{0, \ldots, N\}$ at $t+1$. Therefore by induction we have that for all time $t\geq0$ Problem~\eqref{eq:FTOCP} is feasible for some $N_t^j \in \{0, \ldots, N\}$ and the closed-loop system~\eqref{eq:sys} and~\eqref{eq:MPCpolicy} satisfies state and input constraints~\eqref{eq:constraintSet}.
\end{proof}
\subsection{Input to State Stability (ISS)}
In this section, we show that the closed-loop system~\eqref{eq:sys} and~\eqref{eq:MPCpolicy} is ISS with respect to $\mathcal{O}$.
We recall that in standard MPC strategies the finite time optimal control problem can be reformulated as a parametric Quadratic Program (QP). This fact is used in~\cite{goulart2006optimization} to show continuity of the open-loop cost and then to prove ISS of the origin.
In the proposed approach, the open-loop cost from Algorithm~\ref{LMPCpolicy}
\begin{equation*}
J_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t) = \textstyle \text{min}_{N_t^j \in \{0, \ldots, N\} } C_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t^j, N_t^j)
\end{equation*}
is not given by the solution to a parametric QP.
Therefore, continuity cannot be guaranteed, and the standard technique from~\cite{goulart2006optimization} cannot be used to prove ISS.
Instead, we introduce the standard definition of dissipative-form ISS-Lyapunov function for the robust invariant set $\mathcal{O}$ \cite{lin1995various,grune2014iss} and we show that the cost of the LMPC $J_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t)$ is an ISS-Lyapunov function.
\begin{definition}
A dissipative-form ISS-Lyapunov function for the closed-loop system \eqref{eq:sys} and~\eqref{eq:MPCpolicy} and the invariant set $\mathcal{O}$ is a function $V:\mathbb{R}^n \rightarrow \mathbb{R}_{\geq0}$ such that there exists $\alpha_1, \alpha_2, \alpha \in \mathcal{K}_\infty$, and $\sigma \in \mathcal{K}$ so that for all $x_t \in \mathbb{R}^n$ and $w_t \in \mathbb{R}^m$,
\begin{subequations}
\begin{align}
&\alpha_1(|x_t|_\mathcal{O}) \leq V(x_t) \leq \alpha_2(|x_t|_\mathcal{O}) \label{eq:zeroProperty}\\
&V(Ax_t+B\pi^{j,*}_{t|t}(x_t)+w)-V(x_t) \leq -\alpha(|x_t|_\mathcal{O}) + \sigma(||w_t||).
\end{align}
\end{subequations}
\end{definition}
Notice that, as in \cite{grune2014iss}, no assumptions on the continuity of $V(\cdot)$ are required. However \eqref{eq:zeroProperty} implies that $V(\cdot)$ is continuous on the boundary of $\mathcal{O}$. The above definition can be used to show that the closed-loop system \eqref{eq:sys} and~\eqref{eq:MPCpolicy} is ISS with respect to the invariant set $\mathcal{O}$, as described by the following proposition.
\begin{proposition}\label{prop:ISS}
The following statements are equivalent:
\begin{itemize}
\item The closed-loop system \eqref{eq:sys} and~\eqref{eq:MPCpolicy} is ISS with respect to the robust invariant set $\mathcal{O}$.
\item There exists a dissipative-form ISS-Lyapunov function $V(\cdot)$.
\end{itemize}
\end{proposition}
\begin{proof}
The proof follows from \cite[Theorem~2.3]{grune2014iss} substituting $|x|$ with $|x|_\mathcal{O}$. Note that we can replace $|x|$ with $|x|_\mathcal{O}$ as by \eqref{eq:zeroProperty} we have that $V(x)=0$ iff $|x|_\mathcal{O}=0$.
\end{proof}
\begin{proposition}\label{prop:costBound}
Let Assumptions~\ref{ass:first}--\ref{ass:cost} hold, $x_t^j \in \mathcal{C}^j$ and define the closed-loop system dynamics
\begin{equation*}
f^j_t(x_t^j,w_t^j)=Ax_t+B \pi^{j,*}_{t|t}(x_t^j)+w_t^j,
\end{equation*}
where $\pi^{j,*}_{t|t}(\cdot)$ is the optimal policy~\eqref{eq:optNstepPolicy}. Then there exists a constant $L > 0$ such that
\begin{equation*}
\begin{aligned}
J^{\scalebox{0.5}{LMPC},j}_{t+1 \rightarrow t+1+N}(f_t^j(x_t^j,w_t^j) ) - & J^{\scalebox{0.5}{LMPC},j}_{t \rightarrow t + N}(x_t^j )\\
& \leq -\alpha(|x_t^j|_\mathcal{O}) + L ||w_t^j||,
\end{aligned}
\end{equation*}
$\forall t \geq 0$, $\forall w_t^j \in \mathcal{W}$ and $\alpha \in \mathcal{K}_\infty$.
\end{proposition}
\begin{proof}
The proof can be found in the Appendix.
\end{proof}
The above propositions allow us to prove that the closed-loop system~\eqref{eq:sys} and~\eqref{eq:MPCpolicy} is ISS with respect to $\mathcal{O}$.\\
\begin{theorem}\label{th:conv}
Consider the closed-loop system~\eqref{eq:sys} and~\eqref{eq:MPCpolicy}. Let Assumptions~\ref{ass:first}--\ref{ass:cost} hold and assume that $x_0^j \in \mathcal{C}^j$, then the closed-loop system~\eqref{eq:sys} and~\eqref{eq:MPCpolicy} is Input-to-State Stable (ISS) for the robust positive invariant set $\mathcal{O}$.
\end{theorem}
\begin{proof} First we show that the set $\mathcal{O}$ is robust positive invariant for the closed-loop system~\eqref{eq:sys} and~\eqref{eq:MPCpolicy}. Assume that at time $t$ of iteration $j$ the state $x_t^j \in \mathcal{O}$ and recall that the disturbance feedback policy \eqref{eq:FTOCP_DFpolicy} is equivalent to state a feedback policy \cite{goulart2006optimization}, then we have that the candidate policy
\begin{equation*}
[\pi^{j,*}_{t|t}(x)=Kx,\ldots,\pi^{j,*}_{t+N-1|t}(x)=Kx]
\end{equation*}
is feasible at time $t$ of the $j$th iteration for $N_t^j = N$. Now we notice that the cost associated with the above feasible policy is zero. Therefore, we have that $u_t^j=\pi^{j,*}_{t|t}(x_{t}^{j})=Kx_t^j$, which together with Assumption~\ref{ass:O_inf} implies that the closed-loop system $x_{t+1}^j = Ax_t^j + B\pi^{j,*}_{t|t}(x_{t}^{j}) +w_t^j \in \mathcal{O}, \forall w_t^j \in \mathcal{W}$ and that $\mathcal{O}$ is robust positive invariant for the closed-loop system~\eqref{eq:sys} and~\eqref{eq:MPCpolicy}. Next, notice that for a fixed $N_t^j$ the function $C_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(\cdot, N_t^j):\mathbb{R}^n\rightarrow\mathbb{R}$ is continuous at the boundary of the set $\mathcal{O}$.
Therefore, the function $J^{\scalebox{0.5}{LMPC},j}_{t \rightarrow t+ N}( \cdot )$, which is defined as the point-wise minimum of the functions $C_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(\cdot, N_t^j)$ for $N_t^j \in \{0,\ldots,N\}$, is continuous at the boundary of $\mathcal{O}$.
Continuity at the boundary of $\mathcal{O}$, Assumption~\ref{ass:cost} and \eqref{eq:FTOPC_Cost} imply the existence of $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ such that $\forall x_t \in \mathcal{C}^j$
\begin{equation}
\alpha_1(|x_t|_\mathcal{O}) \leq h(x_t, 0) \leq J^{\scalebox{0.5}{LMPC},j}_{t \rightarrow t+N}( x_t ) \leq \alpha_2(|x_t|_\mathcal{O}).
\end{equation}
In the above equation, we used compactness of the sets $\mathcal{X}$ and $\mathcal{U}$.
Finally, from Proposition~\ref{prop:costBound}, we have that $\forall x_t \in \mathcal{C}^j$
\begin{equation*}
\begin{aligned}
J^{\scalebox{0.5}{LMPC},j}_{t+1 \rightarrow t+1+N}( Ax_t+B\pi^{j,*}_{t|t}(x_t)&+w_t )-J^{\scalebox{0.5}{LMPC},j}_{t \rightarrow t+N}( x_t ) \\
&~\leq - \alpha(|x_t|_\mathcal{O})+\sigma(||w_t||_2)
\end{aligned}
\end{equation*}
and therefore $J^{\scalebox{0.5}{LMPC},j}_{t \rightarrow t+N}( \cdot)$ is a ISS-Lyapunov function and the closed-loop system~\eqref{eq:sys} and~\eqref{eq:MPCpolicy} is Input-to-State Stable for the robust positive invariant set $\mathcal{O}$.
\end{proof}
\vspace{0.1cm}
\vspace{0.2cm} \begin{remark}
We underline that the ISS property does not imply that the closed-loop system converges in finite time to the set $\mathcal{O}$. However, finite time convergence may be obtained following the approach presented in~\cite{langson2004robust}, if at time $T^j$ the system has not reach $\mathcal{O}$. In particular, the optimal policy ${\boldsymbol{\pi}_{T^j}^{j,*}(\cdot)=[\pi^{j,*}_{{T^j}|t}(\cdot), \ldots, \pi^{j,*}_{{T^j}+N-1|t}(\cdot)]}$ computed by the finite time optimal control problem~\eqref{eq:FTOCP} can be used to robustly steer the system to the terminal set~$\mathcal{CS}^{j-1}$. Then, from a state $x_{{T^j}+N-1}^j\in\mathcal{CS}^j$ it would be possible to solve the finite time optimal control problem~\eqref{eq:FTOCP} for $j-1$ and leverage the time-varying optimal policy ${\boldsymbol{\pi}_{T^j+N}^{j-1,*}(\cdot)}$ to steer the system in finite time to $\mathcal{CS}^{j-2}$. This procedure may be iterated to steer the system in finite time to $\mathcal{CS}^{0} = \mathcal{O}$. Finally, we point out that robust exponential stability of the closed-loop system may be obtained decoupling the nominal and error dynamics as shown in~\cite{mayne2005robust}. However, this strategy cannot be applied when using a disturbance feedback policy as in~\eqref{eq:FTOCP}.
\end{remark}
\subsection{Performance Bound}
Finally, we show that whenever $x_0^j\in \mathcal{CS}^{j-1}$ the robust $Q$-function at iteration $j-1$ is an upper-bound to the closed-loop cost of the certainty equivalent system at the next $j$th iteration.
\begin{theorem}\label{th:cost}
Consider the certainty equivalent system \eqref{eq:disturbanceFreeSys} in closed-loop with the LMPC \eqref{eq:FTOCP} and~\eqref{eq:MPCpolicy}. Let Assumptions~\ref{ass:first}--\ref{ass:cost} hold and $x_0^j = \bar x_0^j \in \mathcal{CS}^{j-1} \subseteq \mathcal{C}^j$. Then, we have that the iteration cost of the certainty equivalent closed-loop system \eqref{eq:disturbanceFreeSys} and~\eqref{eq:MPCpolicy} is an upper-bound to the $Q$-function constructed at the previous iteration, i.e.,
\begin{equation}
J^{j}_{0\rightarrow T^j}(\bar x_0^j) = \sum_{t=0}^{T^j} h\big(\bar x_t^j, \pi_{t|t}^{j,*}(\bar x_t^j)\big) \leq Q^{j-1}(\bar x_0^j)
\end{equation}
where $\pi_{t|t}^{j,*}(\cdot)$ is the optimal control policy in~\eqref{eq:optNstepPolicy}.
\end{theorem}
\begin{proof}
By Proposition~\ref{prop:QfunDecrease} we have that
\begin{equation}\label{eq:boundQfun}
\begin{aligned}
Q^{j-1}(\bar x_0^j) & \geq h(\bar x_0^j, \kappa^{j-1,*}(\bar x_0^j) ) + Q^{j-1}(\bar x_1^j) \\
& \geq h(\bar x_0^j, \kappa^{j-1,*}(\bar x_0^j) ) + h(\bar x_1^j, \kappa^{j-1,*}(\bar x_1^j) ) \\
& \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + Q^{j-1}(\bar x_2^j) \\
& \geq \sum_{k=0}^{N-1} h(\bar x_k^j, \kappa^{j-1,*}(\bar x_k^j) ) + Q^{j-1}(\bar x_{N}^j) \\
& \geq J^{\scalebox{0.5}{LMPC},j}_{0\rightarrow N}(\bar x_0^j),
\end{aligned}
\end{equation}
where the last inequality holds by the feasibility of the safe policy from Section~\ref{sec:dataBasedPolicy} for $x_0^j \in \mathcal{CS}^{j-1} \subseteq \mathcal{C}^j$.\\
Now consider the LMPC cost at time $t$, by Proposition~\ref{prop:QfunDecrease} we have that
\begin{equation*}
\begin{aligned}
&J^{\scalebox{0.5}{LMPC},j}_{t \rightarrow t + N}(\bar x_t^j) = \sum_{k = t}^{t+N-1} h(\bar x_{k|t}^{j,*}, \pi_{k|t}^{j,*}(\bar x_{k|t}^{j,*})) + Q^{j-1}( \bar x_{t+N|t}^{j,*} ) \\
&~~\geq \sum_{k = t}^{t+N-1} h(\bar x_{k|t}^{j,*}, \pi_{k|t}^{j,*}(\bar x_{k|t}^{j,*})) + h\Big(\bar x_{t+N|t}^{j,*} , \kappa^{j-1,*}\big( \bar x_{t+N|t}^{j,*} \big)\Big) \\
&~~~~~~~~~~~~~~~+ Q^{j-1}\Big(A\bar x_{t+N|t}^{j,*} + B \kappa^{j-1,*}\big( \bar x_{t+N|t}^{j,*} \big)\Big) \\
& ~~~= h(\bar x_{t|t}^{j,*}, \pi_{t|t}^{j,*}(\bar x_{t|t}^{j,*})) + \sum_{k = t+1}^{t+N-1} h(\bar x_{k|t}^{j,*}, \pi_{k|t}^{j,*}(\bar x_{k|t}^{j,*})) \\
& ~~~ + h\Big(\bar x_{t+N|t}^{j,*} , \kappa^{j-1,*}\big( \bar x_{t+N|t}^{j,*} \big)\Big) \\
&~~~~~~~~~~~~~~~~~~+ Q^{j-1}\Big(A\bar x_{t+N|t}^{j,*} + B \kappa^{j-1,*}\big( \bar x_{t+N|t}^{j,*} \big)\Big) \\
& ~~~ \geq h(\bar x_{t}^{j}, \bar u_{t}^{j}) + J^{\scalebox{0.5}{LMPC},j}_{t+1 \rightarrow t+1 + N}( A \bar x_t^{j} + B \bar u_{t}^{j} ),
\end{aligned}
\end{equation*}
where $\bar u_{t}^{j} = \pi_{t|t}^{j,*}(\bar x_{t}^{j})$.
The above equation implies that the LMPC cost is non-increasing and for the closed-loop trajectory of the certainty equivalent we have
\begin{equation*}\label{eq:costDecrease}
J^{\scalebox{0.5}{LMPC},j}_{t+1 \rightarrow t+1 + N}( \bar x_{t+1}^{j} ) - J^{\scalebox{0.5}{LMPC},j}_{t \rightarrow t + N}(\bar x_t^j) \leq -h(\bar x_{t}^{j}, \bar u_{t}^{j}),
\end{equation*}
which in turns implies that
\begin{equation*}
\begin{aligned}
J^{\scalebox{0.5}{LMPC},j}_{0 \rightarrow N}(\bar x_0^j) & \geq \sum_{k=0}^{T^j} h(\bar x_{k}^{j}, \bar u_{k}^{j}) + J^{\scalebox{0.5}{LMPC},j}_{T^j+1 \rightarrow T^j+1 + N}( \bar x_{T^j+1}^{j} ) \\
&\geq \sum_{k=0}^{T^j} h(\bar x_{k}^{j}, \bar u_{k}^{j})=J^{j}_{0\rightarrow T^j}(\bar x_0^j).
\end{aligned}
\end{equation*}
Finally, from the above equation and~\eqref{eq:boundQfun} we conclude that
\begin{equation*}
Q^{j-1}(\bar x_0^j) \geq J^{\scalebox{0.5}{LMPC},j}_{0 \rightarrow N}(\bar x_0^j) \geq J^{j}_{0\rightarrow T^j}(\bar x_0^j).
\end{equation*}
\end{proof}
\section{Region of Attraction Approximation}\label{sec:regAtt}
In this section, we present an algorithm which may be used to select the initial condition $x_0^j$ or to approximate the region of attraction of the LMPC policy $\mathcal{C}^j$ in \eqref{eq:feasRegion}. We underline that the region of attraction $\mathcal{C}^j$ may be computed solving problem~\eqref{eq:FTOCP} as a set of parametric optimization problems with parameter $x_t^j$~\cite{BorrelliBemporadMorari_book}. However, this computation may be prohibitive and therefore we propose a strategy to compute an inner approximation to the region of attraction $\mathcal{C}^j$.
Given a vector $d\in \mathbb{R}^n$, we define the following optimization problem:
\begin{equation}\label{eq:regAttApp}
\begin{aligned}
P(N_t^j, d) =
\min_{\substack{x_0, \boldsymbol{M}_t^j,\\ {\bm{\lambda}}^j_{t}, \boldsymbol{g}_t^j } } & ~~ d^\top x_0 \\
\textrm{s.t. }& ~~ (d^\perp )^\top x_0 =0 \\
& ~~ x^j_{t|t}= \bar x^j_{t|t}=x_0\\
& ~~ \eqref{eq:FTOCP_dynNom}-\eqref{eq:FTOCP_policy2}\\
& ~~ \forall w^j_{k|t} \in \mathcal{W}, \forall k = \{t, ..., t\!+\!N\!\!-\!\!1\},
\end{aligned}
\end{equation}
where $d^\perp \in \mathbb{R}^n$ is a vector perpendicular to $d \in \mathbb{R}^n$. Basically, the above optimization problem finds the optimal initial state $x_0^*$, which is farthest from the origin along the direction $d$ and guarantees that problem $C_{t\rightarrow t+N}^{\scalebox{0.5}{LMPC},j}(x_0, N_t^j)$ from~\eqref{eq:FTOCP} is feasible. Therefore, given a user-defined set of vectors $\mathcal{D}=\{d^1, \ldots, d^k\}$, problem~\eqref{eq:regAttApp} can be solved repeatedly to approximate the region of attraction~${\mathcal{C}}^j$. In particular, for each vector $d^i \in \mathcal{D}$ and parameter $N_t^j \in \{1,\ldots,N\}$ we solve problem $P(N_t^j, d^i)$. Afterwards, we define the approximated region of attraction $\mathcal{\tilde{C}}^j$ as the convex hull of the optimal initial states $x_0^*$ associated with different $N_t^j$ and vector $d^i$. Algorithm~\ref{RegAttrAppr} summaries this procedure.
\begin{algorithm}[h!]
\SetAlgoLined
\textbf{Input:} Set of vectors $\mathcal{D}=\{d^1, \ldots, d^k\}$ and horizon $N$\\
Initialize $\mathcal{\tilde{C}}^j = \varnothing$ \\
\For{$d^i \in \mathcal{D}$}{
\For{$N_t^j \in \{1,\ldots,N\}$}{
Solve $P(N_t^j, d^i)$ from~\eqref{eq:regAttApp}\\
Let $x_0^*$ be the optimal initial state from $P(N_t^j, d^i)$\\
Set $\mathcal{\tilde{C}}^j = \text{Conv}\{\mathcal{\tilde{C}}^j\cup \{x_0^*\}\}$
}}
\textbf{Output:} Approximate region of attraction $\mathcal{\tilde{C}}^j$
\caption{Region of Attraction Approximation}
\label{RegAttrAppr}
\end{algorithm}
Finally, we underline that the optimization problem~\eqref{eq:regAttApp} may be used to select the initial condition at iteration $j$. In particular given a direction $d^j$, the initial condition $x_0^j$ may be chosen as the optimal initial state $x_0^*$ for problem $P(N, d^j)$, as shown in the result section.
\section{Simulation Results}\label{sec:results}
We test the proposed controller on a system subject to bounded additive uncertainty. First, we show that the proposed strategy is able to improve the performance of a system executing an iterative task. Afterwards, we show that the proposed LMPC can be used to iteratively construct a robust convex safe set $\mathcal{CS}^j$, which is defined over progressively larger regions of the state space. Finally, we show that data collected by the LMPC can be exploited to construct the safe policy~\eqref{eq:theSafePolicy}, which robustly steers the uncertain system from any state within the robust safe set $\mathcal{CS}^j$ to the set $\mathcal{O}$.
We consider the following double integrator system
\begin{equation*}
x_{t+1} = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} x_t + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u_t + w_t,
\end{equation*}
where $w_t \in \{w \in \mathbb{R}^2: ||w||_\infty \leq 0.1 \}$, subject to the following constraints $x_t \in \mathcal{X} = \{x\in \mathbb{R}^2 : ||x||_\infty \leq 10 \}$
and $u_t \in \mathcal{U} = \{ u \in \mathbb{R}: ||u||_\infty \leq 1 \}$ for all time instant $t \geq 0$. Furthermore, we define the running cost $h(x,u) = 10|x|_\mathcal{O} + |u|_{K\mathcal{O}}$, which satisfies Assumption~\ref{ass:cost}.
\subsection{Iterative Task}
We use the LMPC~\eqref{eq:FTOCP} and~\eqref{eq:MPCpolicy} to iteratively steer the system from $x_0 = [5.656; 0]$ to the robust invariant set $\mathcal{O}$. We designed a sub-optimal robust MPC to perform the $0$th iteration and to construct the robust safe set $\mathcal{CS}^0$ and the robust $Q$-function $Q^0(\cdot)$, which are used to initialize the LMPC with $N=3$. We underline that the closed-loop performance of the robust MPC can be improved tuning its parameters. The goal of this section is to show that, given a safe sub-optimal policy, the proposed strategy may be used to iteratively improve the closed-loop performance. In the next section, we will show that the proposed strategy allows us also to enlarge the region of attraction associated with the safe sub-optimal policy used to initialize the algorithm.
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{figures/iterativeQfunEvolution.eps}
\caption{Comparison between the robust safe set and $Q$-function at the first and last iteration.}
\label{fig:iterativeQfunEvolution}
\end{figure}
We perform $4$ iterations of the control task for the certainty equivalent system. At each $j$th iteration, we store the LMPC predicted policy and the closed-loop data in order to construct the robust safe set $\mathcal{CS}^j$ and the robust $Q$-function $Q^j(\cdot)$. Table~\ref{table:closedLoopCost} shows that the closed-loop cost of the certainty equivalent system decreases, until it converges to a steady state value after $4$ iterations.
\begin{table}[h!]
\caption{Closed-loop cost $J^j_{0\rightarrow T^j}(x_0)$ for iteration $i\in \{0,\ldots,4\}$.}
\label{table:closedLoopCost}
\centering
\begin{tabular}{cccccc} \midrule
$i=0$& $i=1$& $i=2$&$i=3$&$i=4$ \\
$863.4245$ & $827.9588$ & $827.9380$ & $827.9371$ & $827.9371$\\ \midrule
\end{tabular}
\end{table}
Finally, in Figure~\ref{fig:iterativeQfunEvolution} we compare the robust safe set and the robust $Q$-function at the first and last iteration. First, we notice that the robust safe set, which represents the domain of the $Q$-function, is enlarged. Furthermore, we confirm that $Q^j(\cdot)$ is non-increasing (i.e. $Q^0(x) \leq Q^{4}(x), \forall x \in \mathcal{CS}^{4}$) and therefore it guarantees better bounds on the performance of the certainty equivalent closed-loop system~\eqref{eq:disturbanceFreeSys} and~\eqref{eq:MPCpolicy}, as shown in Theorem~\ref{th:cost}.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{figures/safeSetEvolution.eps}
\caption{Evolution of the robust safe set at each iteration.}
\label{fig:evolutionRS}
\end{figure}
\begin{figure}[b]
\centering
\includegraphics[width=\columnwidth]{figures/closedLoop.eps}
\caption{Closed-loop trajectories for different disturbance realizations.}
\label{fig:closedloop}
\end{figure}
\subsection{Domain Enlargement}\label{sec:resIterativeEnlargement}
We show that the domain of the LMPC policy may be iteratively enlarged.
At each iteration, we simulate the uncertain closed-loop system~\eqref{eq:sys} and~\eqref{eq:MPCpolicy} and we store both the closed-loop data and the predicted LMPC policy~\eqref{eq:MPCpolicy}. The stored data are used to construct the robust safe set $\mathcal{CS}^j$ and robust $Q$-function as described in Section~\ref{sec:LMPCpre}.
\begin{figure*}[h!]
\includegraphics[width=1.0\textwidth]{figures/evolutionValueFunction_new.eps}
\caption{Evolution of the robust $Q$-function $Q^j$ at each iteration. Notice that $Q^j(\cdot)$ (in blue) is a lower-bound to $Q^{j+1}(\cdot)$ (in red) for all $i \in \{3,5,7,11\}$, until convergence is reached and $Q^{11}(\cdot) =Q^{12}(\cdot)$.}
\label{fig:evolutionQ}
\end{figure*}
Furthermore, we use the strategy proposed in Section~\ref{sec:regAtt} to iteratively select the initial condition. At each iteration $j$, we define the vector $d^j =[(-1)^j,0]$ and we pick $x_0^j$ as the optimal initial state $x_0^*$ from problem $P(N, d^j)$ in~\eqref{eq:regAttApp}.
Notice that by definition~\eqref{eq:CS} we have that $\mathcal{CS}^j \subseteq \mathcal{CS}^{j+1}$, therefore the set of states which can be steered to $\mathcal{O}$ by the LMPC~\eqref{eq:FTOCP} and~\eqref{eq:MPCpolicy} does not shrink, i.e., $\mathcal{C}^j \subseteq \mathcal{C}^{j+1}$.
Figure~\ref{fig:evolutionRS} shows that the robust convex safe set $\mathcal{CS}^j$ grows at each iteration, until it converges to a set which saturates the state constraints. We underline that the closed-loop data used to enlarge $\mathcal{CS}^j$ are generated by the LMPC, which steers the system to regions of the state space associated with low cost values. In other words, the growth of the robust safe set is cost-driven. More importantly, the iterative enlargement of the $\mathcal{CS}^j$ is performed safely. Indeed, the LMPC guarantees robust state and input constraints satisfaction at each iteration.
Figure~\ref{fig:closedloop} shows $100$ Monte Carlo simulations of the closed-loop system for the $12$th iteration. We notice that the closed-loop trajectories satisfy state constraints and converge to the set $\mathcal{O}$, regardless of the disturbance realization.
\begin{table}[b!]
\centering
\caption{Performance of the LMPC policy~\eqref{eq:MPCpolicy} and the safe policy~\eqref{eq:theSafePolicy} in closed-loop with the uncertain system~\eqref{eq:sys}.}\label{tab:comparison}
\begin{tabular}{c|cc}
& Average solver time & Average closed-loop cost \\\midrule
LMPC & $4.6$s & $77.1$ \\
Safe Policy & $5$ms & $80$ \\\midrule
\end{tabular}
\end{table}
Finally, Figure~\ref{fig:evolutionQ} shows the growth of the $Q$-function $Q^j(\cdot)$, which is non-increasing at each iteration. It is important to underline that $Q^j(\cdot)$ is piece-wise affine as it is the solution to a parametric LP \cite{BorrelliBemporadMorari_book}. Furthermore, we notice that $Q^j(\cdot)$, which is an upper-bound of the closed-loop cost of the disturbance-fee system, resembles a quadratic function. This result is expected as the optimal value function for this problem is piece-wise quadratic \cite{BorrelliBemporadMorari_book}.
\subsection{Exploiting the safe policy}
In this section, we use the stored data from Section~\ref{sec:dataBasedPolicy} to construct the safe policy~\eqref{eq:theSafePolicy}. We tested this policy for $1000$ Monte Carlo simulations, where we randomly sampled the initial condition $x_0$ from the robust convex safe set $\mathcal{CS}^{12}$. We confirm that, for all initial conditions $x_0 \in \mathcal{CS}^{12}$ and disturbance realizations, the safe policy~\eqref{eq:theSafePolicy} steered the system to the set $\mathcal{O}$, as shown in Figure~\ref{fig:safePolicy}.
\begin{figure}[h!]
\centering
\includegraphics[width=\columnwidth]{figures/dataBasedPolicy.eps}
\caption{Closed-loop trajectories for different disturbance realizations and initial conditions.}
\label{fig:safePolicy}
\end{figure}
We compare the performance of the uncertain system \eqref{eq:sys} in closed-loop with the safe policy~\eqref{eq:theSafePolicy} and the LMPC~\eqref{eq:FTOCP} and~\eqref{eq:MPCpolicy}. In particular, we simulated both the closed-loop system \eqref{eq:sys} and \eqref{eq:theSafePolicy} and the closed-loop system \eqref{eq:sys} and~\eqref{eq:MPCpolicy} for the same random initial condition $x_0 \in \mathcal{CS}^{12}$ and disturbance realization. As reported in Table~\ref{tab:comparison}, on average it takes $\sim 5$ms to evaluate the safe policy~\eqref{eq:theSafePolicy} and $\sim 4.6$s to evaluate the LMPC policy~\eqref{eq:MPCpolicy}. This result is expected as the safe policy~\eqref{eq:theSafePolicy} is evaluated solving a LP and the LMPC policy~\eqref{eq:MPCpolicy} solving $N+1=4$ QPs. On the other hand, it is interesting to notice that the closed-loop cost associated with the safe policy~\eqref{eq:theSafePolicy} is on average $\sim 3\%$ higher than the cost associated with the LMPC policy~\eqref{eq:MPCpolicy}, as shown in Table~\ref{tab:comparison}. This result suggests that, in applications where the computational power is not always available, one can first use the proposed LMPC to iteratively construct a large robust convex safe set $\mathcal{CS}^j$ and robust $Q$-function $Q^j$. Afterwards, these quantities can be leveraged to synthesize a safe control policy, which at the cost of slickly worse performance is able to reduce the computational burden.
\section{Conclusions}\label{sec:conclusions}
In this work we proposed a robust Learning Model Predictive Controller (LMPC) for linear systems subject to bounded additive uncertainty.
At each execution of the control task, we store both the closed-loop data and the optimal predicted policy of the LMPC. First, we show how the stored data can be combined with the optimal policy from the LMPC to construct a safe set and an approximation to the value function. Afterwards, we design a safe policy which may be used to complete the task from any state in the safe set. Finally, the safe set, the value function approximation, and the safe policy are used in the control design, which guarantees input to state stability, robust constraint satisfaction, and performance bounds for the certainty equivalent closed-loop system. The effectiveness of the proposed LMPC is tested on a numerical example.
\section{Acknowledgment}
The authors would like to thank Monimoy Bujarbaruah and Siddharth Nair for helpful discussions and
reviews. Some of the research described in this review was funded by the Hyundai Center of Excellence
at the University of California, Berkeley. This work was also sponsored by the Office of Naval
Research. The views and conclusions contained herein are those of the authors and should not be
interpreted as necessarily representing the official policies or endorsements, either expressed or
implied, of the Office of Naval Research or the US government.
\section{Appendix}\label{sec:appendixProof}
\subsection{Proof of Proposition~\ref{prop:invariance}}
\begin{proof}
The proof follows from linearity of system~\eqref{eq:sys} and convexity of the constraint set~\eqref{eq:constraintSet}. \\
By Assumption~\ref{ass:policyAss}, for all vertices $v_{k|t}^{j,i}$ and associated control action $\pi^{j,*}_{k|t}( v_{k|t}^{j,i} )$ collected in the columns of the matrices ${\bf{X}}^j$ in~\eqref{eq:xMatrix} and ${\bf{U}}^j$ in~\eqref{eq:uMatrix}, we have that
\begin{equation*}
Av_{k|t}^{j,i} + B\pi^{j,*}_{k|t}( v_{k|t}^{j,i} ) + w \in \mathcal{CS}^j \subseteq \mathcal{X}, ~\forall w \in \mathcal{W}.
\end{equation*}
The above equation implies that $\forall x \in \mathcal{CS}^j$ and $\forall {\bm{\lambda}}^j \in \Lambda(x)$
\begin{equation*}
A {\bf{X}}^j {\bm{\lambda}}^j + B {\bf{U}}^j {\bm{\lambda}}^j+w \in \mathcal{CS}^j, \forall w \in \mathcal{W}.
\end{equation*}
By definition $\forall \kappa^j(\cdot) \in \mathcal{SP}^j$ there exists ${\bm{\lambda}}^j \in \Lambda(x)$ such that $x = {\bf{X}}^j{\bm{\lambda}}^j$ and $\kappa^j(x) = {\bf{U}}^j {\bm{\lambda}}^j$. Consequently, from the above equation we have that $\forall x \in \mathcal{CS}^j$ and $\forall \kappa^j(\cdot) \in \mathcal{SP}^j$
\begin{equation*}
A x + B \kappa^j(x) + w \in \mathcal{CS}^j, \forall w \in \mathcal{W}.
\end{equation*}
\end{proof}
\subsection{Proof of Proposition~\ref{prop:QfunDecrease}}
\begin{proof}
Recall that we initialized $\mathcal{CS}^{0}=\mathcal{O}$ and $Q^{0}(x)=0~\forall x \in \mathcal{CS}^{0}$, then we trivially have that
\begin{equation*}
Q^{0}(x) \geq h(x, \kappa^{0,*}(x)) + Q^{0}(Ax+B \kappa^{0,*}(x)), \forall x \in \mathcal{CS}^{0}.
\end{equation*}
Next, we show that $\forall j \geq 1$ and $\forall x \in \mathcal{CS}^{j}$
\begin{equation*}
Q^{j}(x) \geq h(x, \kappa^{j,*}(x)) + Q^{j}(Ax+B \kappa^{j,*}(x)).
\end{equation*}
Given $ x \in \mathcal{CS}^j$, we have that
\begin{equation}\label{eq:defQProof}
Q^j(x) = ({\bf{J}}^{j})^\top {\bm \lambda}^{*,j},
\end{equation}
Now notice that by definitions~\eqref{eq:J_definition1}--\eqref{eq:J_definition2} each element of ${\bf{J}}^{j}$ can be written as
\begin{equation}\label{eq:Qwritten_1}
\begin{aligned}
J_{k|t}^{j}\big(v_{k|t}^{j,i} \big) &= h\big(v_{k|t}^{j,i} , \pi_{k|t}^{j,*} (v_{k|t}^{j,i} ) \big) \\
& \quad \quad \quad \quad \quad \quad +\sum_{r= 1}^{l^{k+1}} \gamma^{r,*} J^j_{k+1|t}\big(v_{k+1|t}^{j,r}\big) \\
&\geq h\big(v_{k|t}^{j,i} , \pi_{k|t}^{j,*} (v_{k|t}^{j,i} ) \big) \\
& \quad \quad \quad \quad + Q^j \big(A v_{k|t}^{j,i} + B \pi_{k|t}^{j,*} (v_{k|t}^{j,i}) \big).
\end{aligned}
\end{equation}
where the above inequality holds as
\begin{equation*}
\sum_{r=1}^{l^{k+1}} \gamma^{r,*} v_{k+1|t}^{j,r} = A v_{k|t}^{j,i} + B \pi_{k|t}^{j,*} (v_{k|t}^{j,i})
\end{equation*}
and for $x^+ = A v_{k|t}^{j,i} + B \pi_{k|t}^{j,*} (v_{k|t}^{j,i})$
\begin{equation*}
Q^j \big( x^+ \big) = \min_{\boldsymbol{\lambda}^j \in \Lambda(x^+)} ({\bf{J}}^{j})^\top {\bm \lambda}^{j}.
\end{equation*}
Convexity of $h(\cdot, \cdot), Q^j(\cdot)$ and Equation~\eqref{eq:Qwritten_1} imply that
\begin{equation}
\begin{aligned}
& Q^j(x) = {\bf{J}}^j {\bm \lambda}^{*,j} \\
& \geq h\Bigg(\begin{bmatrix} v_{0|0}^{0,1} \\
\vdots \\
v_{k|t}^{j,i} \\
\vdots \\
v_{t+T^j\text{-}1|T^j}^{l^{N\text{-}1},j} \end{bmatrix} {{ { \bm \lambda}}}^{j,*} ,
\begin{bmatrix} \pi_{0|0}^{0,*}(v_{0|0}^{0,1}) \\
\vdots \\
\pi_{k|t}^{j,*}(v_{k|t}^{j,i}) \\
\vdots \\
\pi_{t+T^j\text{-}1|T^j}^{j,*}(v_{t+T^j\text{-}1|T^j}^{j,l^{N\text{-}1}}) \end{bmatrix} {{ { \bm \lambda}}}^{j,*}\Bigg) \\
& \quad \quad + \begin{bmatrix} Q^j\big(A v_{0|0}^{0,1} + B \pi_{0|0}^{0,*} (v_{0|0}^{0,1}) \big) \\
\vdots \\
Q^j(A v_{k|t}^{j,i} + B \pi_{k|t}^{j,*} (v_{k|t}^{j,i}) \big) \\
\vdots \\
Q^j\big(A v_{t+T^j\text{-}1|T^j}^{j,l^{N\text{-}1}} + B \pi_{t+T^j\text{-}1|T^j}^{j,*} ( v_{t+T^j\text{-}1|T^j}^{j,l^{N\text{-}1}} ) \big) \end{bmatrix} {\bm \lambda}^{*,j} \\
& ~~ \geq h\big({\bf{X}}^j {{ { \bm \lambda}}}^{j,*} , {\bf{U}}^j {{ { \bm \lambda}}}^{j,*}\big) + Q^j ( {\bf{X}}^j \tilde {\bm \lambda}^{j,*}),
\end{aligned}
\end{equation}
for some $\tilde {\bm \lambda}^{j,*}$ such that
\begin{equation*}
{\bf{X}}^j \tilde {\bm \lambda}^{j,*} = \begin{bmatrix} A v_{0|0}^{0,1} + B \pi_{0|0}^{0,*} \big(v_{0|0}^{0,1} \big) \\ \vdots \\ A v_{k|t}^{j,i} + B \pi_{k|t}^{j,*} \big(v_{k|t}^{j,i} \big) \\ \vdots \\ A v_{t+T^j\text{-}1|T^j}^{j,l^{N\text{-}1}} + B \pi_{t+T^j\text{-}1|T^j}^{j,*} \big(v_{t+T^j\text{-}1|T^j}^{j,l^{N\text{-}1}} \big) \end{bmatrix} {\bm \lambda}^{j,*}.
\end{equation*}
The above equation implies that
\begin{equation*}
\begin{aligned}
Q^j(x) & \geq h\big({\bf{X}}^j {{\bm \lambda}}^{j,*} , {\bf{U}}^j {{\bm \lambda}}^{j,*}\big) + Q^j ( {\bf{X}}^j \tilde {\bm \lambda}^{j,*}) \\
& \geq h({\bf{X}}^j {\bm \lambda}^{j,*} , {\bf{U}}^j {\bm \lambda}^{j,*} ) + Q^j( A{\bf{X}}^j {\bm \lambda}^{j,*} + B {\bf{U}}^j {\bm \lambda}^{j,*} )
\end{aligned}
\end{equation*}
Finally, we notice that by definition~\eqref{eq:theSafePolicy} $\kappa^{j,*}( x )={\bf{U}}^j {\bm \lambda}^{j,*}$, therefore the above equation can be rewritten as
\begin{equation*}
Q^j(x) \geq h(x , \kappa^{j,*}(x) ) + Q^{j}(Ax + B \kappa^{j,*}(x)).
\end{equation*}
\end{proof}
\subsection{Proof of Proposition~\ref{prop:safePolicyISS}}
\begin{proof} First we show that the set $\mathcal{O}$ is robust positive invariant for the closed-loop system~\eqref{eq:sys} and \eqref{eq:theSafePolicy}. Assume that at time $t$ of iteration $j$ the state $x_t^j \in \mathcal{O}$, by definitions~\eqref{eq:xMatrix}, \eqref{eq:J_definition1}, \eqref{eq:Qfun}, and \eqref{eq:theSafePolicy} we have that $Q^j(x_t^j)=0$ and $\kappa^{j,*}(x_{t}^{j})=Kx_t^j$. Therefore, by Assumption~\ref{ass:O_inf} we have that closed-loop system $x_{t+1}^j = Ax_t^j + B\kappa^{j,*}(x_{t}^{j}) +w_t^j \in \mathcal{O}, \forall w_t^j \in \mathcal{W}$ and that $\mathcal{O}$ is robust positive invariant for the closed-loop system~\eqref{eq:sys} and \eqref{eq:theSafePolicy}. \\
Continuity of $Q^j( \cdot )$, Assumption~\ref{ass:cost} and \eqref{eq:FTOPC_Cost} imply the existence of $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ such that $\forall x_t \in \mathcal{CS}^j$
\begin{equation}
\alpha_1(|x_t|_\mathcal{O}) \leq h(x_t, 0) \leq Q^j( x_t ) \leq \alpha_2(|x_t|_\mathcal{O}).
\end{equation}
We recall that $Q^j( \cdot )$ is Lipschitz continuous as it is the solution to a parametric LP \cite{BorrelliBemporadMorari_book}.
Therefore, from Proposition~\ref{prop:QfunDecrease} and Lipschitz continuity of $Q^j( \cdot )$ for a Lipschitz constant $L$, we have that $\forall x_t \in \mathcal{CS}^j$
\begin{equation*}
\begin{aligned}
Q^j&( Ax_t+B\kappa^{j,*}(x_t)+w_t )-Q^j( x_t ) \\
&=Q^j( Ax_t+B\kappa^{j,*}(x_t)+w_t )-Q^j( Ax_t+B\kappa^{j,*}(x_t))\\&+Q^j( Ax_t+B\kappa^{j,*}(x_t))-Q^j( x_t )\\
&~\leq L||w_t||_2+Q^j( Ax_t+B\kappa^{j,*}(x_t))-Q^j( x_t )\\
&\leq L||w_t||_2 - h(x_t, \kappa^{j,*}(x_t)).
\end{aligned}
\end{equation*}
Therefore $Q^j(\cdot)$ is a ISS-Lyapunov function according with Definition~\ref{def:iss} and by Proposition~\ref{prop:ISS} the closed-loop system~\eqref{eq:sys} and~\eqref{eq:theSafePolicy} is ISS for the robust positive invariant set $\mathcal{O}$.
\end{proof}
\subsection{Proof of Proposition~\ref{prop:costBound}}
\begin{proof} By assumption $x_t^j \in \mathcal{C}^j$, therefore by Theorem~\ref{th:recFeas} Problem~\eqref{eq:FTOCP} is feasible at all times $t\geq0$ for some $N_t^j \in \{0,\ldots, N\}$. Let $\boldsymbol{\pi}_t^{j,*}(\cdot)$ be the optimal $N$-steps policy~\eqref{eq:optNstepPolicy} and $J_{t\rightarrow t+N}^{\scalebox{0.4}{LMPC},j}(x_t^j)$ the optimal cost. At time $t$, let
\begin{equation*}
[\bar x_{t|t}^{j,*}, \ldots, \bar x_{t+N|t}^{j,*} ]
\end{equation*}
be the optimal trajectory of the certainty equivalent system associated with the optimal policy $\boldsymbol{\pi}_t^{j,*}(\cdot)$. Then, the LMPC cost at time $t$ can be written as
\begin{equation}\label{eq:cost_t}
\begin{aligned}
J^{\scalebox{0.5}{LMPC},j}_{t \rightarrow t+N}( x_t^j ) &= \sum_{k=t}^{t+N-1} h(\bar x_{k|t}^{j,*}, \pi_{k|t}^{j,*}(\bar x_{k|t}^{j,*})) + Q^{j-1}(\bar x_{t+N|t}^{j,*})\\
&= h(\bar x_{t|t}^{j,*}, \pi_{t|t}^{j,*}(\bar x_{k|t}^{j,*})) + p(\bar x_{t+1|t}^{j,*})
\end{aligned}
\end{equation}
where the function $p(\bar x_{t+1|t}^{j,*})$, which represents the total cost from time $t+1$ to time $t+N$ for the optimal policy $\boldsymbol{\pi}_t^{j,*}(\cdot)$, is Lipschitz as it is composed of summation and composition of Lipschitz functions. Now we notice that by feasibility of \eqref{eq:candidateSolution} the cost of the LMPC at time $t+1$ satisfies the following inequality
\begin{equation}\label{eq:cost_t+1}
\begin{aligned}
&J^{\scalebox{0.5}{LMPC},j}_{t+1 \rightarrow t+1+N}(f(x_t,w_t)) \leq \sum_{k=t+1}^{t+N-1} h\big(\bar x_{k|t+1}^j , \pi_{k|t}^{*,j}(\bar x_{k|t+1}^j) \big) \\
&~~+ h\big(\bar x_{t+N|t+1}^j, \kappa^{j-1,*} (\bar x_{t+N|t+1}^j )\big)\\
&~~+ Q^{j-1}\big(A \bar x_{t+N|t+1}^j+B \kappa^{j-1,*}(\bar x_{t+N|t+1}^j)\big)\\
&~~\leq \sum_{k=t+1}^{t+N-1} h\big(\bar x_{k|t+1}^j , \pi_{k|t}^{*,j}(\bar x_{k|t+1}^j) \big)+ Q^{j-1}\big( \bar x_{t+N|t+1}^j \big)\\
&~~= p(\bar x_{t+1|t+1}^j),
\end{aligned}
\end{equation}
where the last inequality follows from Proposition~\ref{prop:QfunDecrease}, the feasible nominal trajectory is given by
\begin{equation*}
\begin{aligned}
\bar x_{k|t+1}^j = A^{k-t-1}(Ax_t+&B\pi_{t|t}^{*,j}(x_t) + w_t) \\
&+ \sum_{i=t+1}^{k-1}A^{k-1-i}B\pi_{i|t}^{*,j}(\bar x_{i|t+1}^j)
\end{aligned}
\end{equation*}
for $k=\{t+2, \ldots, t+N \}$, and the initial state
\begin{equation}\label{eq:equalityProp}
\bar x_{t+1|t+1}^j = Ax_t^j+B\pi_{t|t}^{*,j}(x_t^j) + w_t^j = \bar x_{t+1|t}^{j,*} + w_t^j,
\end{equation}
$\forall w_t^j \in \mathcal{W}$. Therefore, from the $L$-Lipschitz continuity of $p(\cdot)$, \eqref{eq:cost_t}, \eqref{eq:cost_t+1} and \eqref{eq:equalityProp} we have that
\begin{equation}
\begin{aligned}
& J^{\scalebox{0.5}{LMPC},j}_{t+1 \rightarrow t+1+N}(f(x_t^j,w_t^j)) - J^{\scalebox{0.5}{LMPC},j}_{t \rightarrow t+N}(x_t^j)\\
&\quad \quad \quad = p(\bar x_{t+1|t}^{j,*} + w_t)-p(\bar x_{t+1|t}^{j,*}) -h(\bar x_{t|t}^{*,j},\pi_{t|t}^{*,j}( x_t^j))\\
&\quad \quad \quad \quad \quad \leq -h(\bar x_{t|t}^{*,j},\pi_{t|t}^{*,j}(x_t^j)) + L ||w_t^j||\\
&\quad \quad \quad \quad \quad \quad \leq -h(\bar x_{t|t}^{*,j},0) + L ||w_t^j||\\
&\quad \quad \quad \quad \quad \quad \quad \leq -\alpha_x^l( x_{t}^{j}) + L ||w_t^j||,
\end{aligned}
\end{equation}
for all $w_t^j \in \mathcal{W}$, where the last inequality holds by Assumption~\ref{ass:cost}.
\end{proof}
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Opening a Tab only once I have my 4 static HTML pages created as below.
*
*Home
*About
*Services
*Contact
On each page, there is a link, which opens up another static page called myVideo.html which plays video about me on a new tab.
I have kept that link on each page so that the user can open it up for anywhere.
The problem is, when it opens up from home page, it can again open from about page, so that makes the video autoplay two times which creates a bad impact.
What I want is, when the user has started the video from any page and when they again execute the same from another page, they should get an Alert mentioning that "The video is already being played on a new tab", so that my video plays only once. They can again play the video only on close of the tab which is playing that video.
My question is, is this possible ??
A: Use a target="" in the link. The target is meant for frames, but when you specify the same target name in every link the browser will open a new tab or window when no window with the name exists, and will reuse the existing window when it exists.
<a href="/link/to/video/" target="video">link</a>
A: Nearly everything is possible.
But this functionnality need some server-side script in my opinion.
The real problem is the impossibility of storing data in a file with JavaScript.
You can detect video events (exemple here: Detect when an HTML5 video finishes) but you're not able to make this information passing through pages.
A JavaScript-only solution will automatically fail when a user will refresh the page, eso...
A: You can handle this in jquery.
<a href="" title="" target="_blank" onclick="myFunction()" >content</a>
<script>
function myFunction() {
window.open("http://stackoverflow.com","stackoverflow");
}
</script>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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{"url":"http:\/\/dictionnaire.sensagent.leparisien.fr\/Completely%20regular%20space\/en-en\/","text":"Publicit\u00e9 \u25bc\n\n## d\u00e9finition - Completely regular space\n\nvoir la d\u00e9finition de Wikipedia\n\nWikipedia - voir aussi\n\nPublicit\u00e9 \u25bc\n\nWikipedia\n\n# Tychonoff space\n\n(Redirected from Completely-regular space)\nKolmogorov (T0) version Topological spaces in separation axiom T0 | T1 | T2 | T2\u00bd | completely T2 T3 | T3\u00bd | T4 | T5 | T6\n\nIn topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms.\n\nTychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (\u0422\u0438\u0445\u043e\u043d\u043e\u0432) is variously rendered as \"Tychonov\", \"Tikhonov\", \"Tihonov\", \"Tichonov\" etc.\n\n## Definitions\n\nSuppose that X is a topological space.\n\nX is a completely regular space if and only if, given any closed set F and any point x that does not belong to F, there is a continuous function f from X to the real line R such that f(x) is 0 and f(y) is 1 for every y in F.In fancier terms, this condition says that x and F can be separated by a function.\n\nX is a Tychonoff space, or T space, or T\u03c0 space, or completely T3 space if and only if it is both completely regular and Hausdorff.\n\nNote that some mathematical literature uses different definitions for the term \"completely regular\" and the terms involving \"T\".The definitions that we have given here are the ones usually used today; however, some authors switch the meanings of the two kinds of terms, or use all terms synonymously for only one condition.In Wikipedia, we will use the terms \"completely regular\" and \"Tychonoff\" freely, but we'll avoid the less clear \"T\" terms.In other literature, you should take care to find out which definitions the author is using.(The phrase \"completely regular Hausdorff\", however, is unambiguous, and always means a Tychonoff space.)For more on this issue, see History of the separation axioms.\n\nCompletely regular spaces and Tychonoff spaces are related through the notion of Kolmogorov equivalence.A topological space is Tychonoff if and only if it's both completely regular and T0.On the other hand, a space is completely regular if and only if its Kolmogorov quotient is Tychonoff.\n\n## Examples and counterexamples\n\nAlmost every topological space studied in mathematical analysis is Tychonoff, or at least completely regular.For example, the real line is Tychonoff under the standard Euclidean topology.Other examples include:\n\n## Properties\n\n### Preservation\n\nCompletely regularity and the Tychonoff property are well-behaved with respect to initial topologies. Specifically, complete regularity is preserved by taking arbitrary initial topologies and the Tychonoff property is preserved by taking point-separating initial topologies. It follows that:\n\n\u2022 Every subspace of a completely regular or Tychonoff space has the same property.\n\u2022 A nonempty product space is completely regular (resp. Tychonoff) if and only if each factor space is completely regular (resp. Tychonoff).\n\nLike all separation axioms, completely regularity is not preserved by taking final topologies. In particular, quotients of completely regular spaces need not be regular. Quotients of Tychonoff spaces need not even be Hausdorff. There are closed quotients of the Moore plane which provide counterexamples.\n\n### Real-valued continuous functions\n\nFor any topological space X, let C(X) denote the family of real-valued continuous functions on X and let C*(X) be the subset of bounded real-valued continuous functions.\n\nCompletely regular spaces can be characterized by the fact that their topology is completely determined by C(X) or C*(X). In particular:\n\n\u2022 A space X is completely regular if and only if it has the initial topology induced by C(X) or C*(X).\n\u2022 A space X is completely regular if and only if every closed set can be written as the intersection of a family of zero sets in X (i.e. the zero sets form a basis for the closed sets of X).\n\u2022 A space X is completely regular if and only if the cozero sets of X form a basis for the topology of X.\n\nGiven an arbitrary topological space (X, \u03c4) there is a universal way of associating a completely regular space with (X, \u03c4). Let \u03c1 be the initial topology on X induced by C\u03c4(X) or, equivalently, the topology generated by the basis of cozero sets in (X, \u03c4). Then \u03c1 will be the finest completely regular topology on X which is coarser than \u03c4. This construction is universal in the sense that any continuous function\n\n$f:(X,\\tau)\\to Y$\n\nto a completely regular space Y will be continuous on (X, \u03c1). In the language of category theory, the functor which sends (X, \u03c4) to (X, \u03c1) is left adjoint to the inclusion functor CRegTop. Thus the category of completely regular spaces CReg is a reflective subcategory of Top, the category of topological spaces. By taking Kolmogorov quotients, one sees that the subcategory of Tychonoff spaces is also reflective.\n\nOne can show that C\u03c4(X) = C\u03c1(X) in the above construction so that the rings C(X) and C*(X) are typically only studied for completely regular spaces X.\n\n### Embeddings\n\nTychonoff spaces are precisely those spaces which can beembedded in compact Hausdorff spaces. More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K such that X is homeomorphic to a subspace of K.\n\nIn fact, one can always choose K to be a cube (i.e. a possibly infinite product of unit intervals). Every cube is compact Hausdorff as a consequence of Tychonoff's theorem. Since every subspace of a compact Hausdorff space is Tychonoff one has:\n\nA topological space is Tychonoff if and only if it can be embedded in a cube.\n\n### Compactifications\n\nOf particular interest are those embeddings where the image of X is dense in K; these are called Hausdorff compactifications of X. Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.\n\nAmong those Hausdorff compactifications, there is a unique \"most general\" one, the Stone\u2013\u010cech compactification \u03b2X.It is characterised by the universal property that, given a continuous map f from X to any other compact Hausdorff space Y, there is a unique continuous map g from \u03b2X to Y that extends f in the sense that f is the composition of g and j.\n\n### Uniform structures\n\nComplete regularity is exactly the condition necessary for the existence of uniform structures on a topological space. In other words, every uniform space has a completely regular topology and every completely regular space X is uniformizable. A topological space admits a separated uniform structure if and only if it is Tychonoff.\n\nGiven a completely regular space X there is usually more than one uniformity on X that is compatible with the topology of X. However, there will always be a finest compatible uniformity, called the fine uniformity on X. If X is Tychonoff, then the uniform structure can be chosen so that \u03b2X becomes the completion of the uniform space X.\n\n## References\n\nContenu de sensagent\n\n\u2022 d\u00e9finitions\n\u2022 synonymes\n\u2022 antonymes\n\u2022 encyclop\u00e9die\n\n\u2022 definition\n\u2022 synonym\n\nPublicit\u00e9 \u25bc\n\ndictionnaire et traducteur pour sites web\n\nAlexandria\n\nUne fen\u00eatre (pop-into) d'information (contenu principal de Sensagent) est invoqu\u00e9e un double-clic sur n'importe quel mot de votre page web. LA fen\u00eatre fournit des explications et des traductions contextuelles, c'est-\u00e0-dire sans obliger votre visiteur \u00e0 quitter votre page web !\n\nEssayer ici,\u00a0t\u00e9l\u00e9charger le code;\n\nSolution commerce \u00e9lectronique\n\nAugmenter le contenu de votre site\n\nAjouter de nouveaux contenus Add \u00e0 votre site depuis Sensagent par XML.\n\nParcourir les produits et les annonces\n\nObtenir des informations en XML pour filtrer le meilleur contenu.\n\nIndexer des images et d\u00e9finir des m\u00e9ta-donn\u00e9es\n\nFixer la signification de chaque m\u00e9ta-donn\u00e9e (multilingue).\n\nRenseignements suite \u00e0 un email de description de votre projet.\n\nJeux de lettres\n\nLes jeux de lettre fran\u00e7ais sont :\n\u25cb\u00a0\u00a0\u00a0Anagrammes\n\u25cb\u00a0\u00a0\u00a0jokers, mots-crois\u00e9s\n\u25cb\u00a0\u00a0\u00a0Lettris\n\u25cb\u00a0\u00a0\u00a0Boggle.\n\nLettris\n\nLettris est un jeu de lettres gravitationnelles proche de Tetris. Chaque lettre qui appara\u00eet descend ; il faut placer les lettres de telle mani\u00e8re que des mots se forment (gauche, droit, haut et bas) et que de la place soit lib\u00e9r\u00e9e.\n\nboggle\n\nIl s'agit en 3 minutes de trouver le plus grand nombre de mots possibles de trois lettres et plus dans une grille de 16 lettres. Il est aussi possible de jouer avec la grille de 25 cases. Les lettres doivent \u00eatre adjacentes et les mots les plus longs sont les meilleurs. Participer au concours et enregistrer votre nom dans la liste de meilleurs joueurs ! Jouer\n\nDictionnaire de la langue fran\u00e7aise\nPrincipales R\u00e9f\u00e9rences\n\nLa plupart des d\u00e9finitions du fran\u00e7ais sont propos\u00e9es par SenseGates et comportent un approfondissement avec Littr\u00e9 et plusieurs auteurs techniques sp\u00e9cialis\u00e9s.\nLe dictionnaire des synonymes est surtout d\u00e9riv\u00e9 du dictionnaire int\u00e9gral (TID).\nL'encyclop\u00e9die fran\u00e7aise b\u00e9n\u00e9ficie de la licence Wikipedia (GNU).\n\nChanger la langue cible pour obtenir des traductions.\nAstuce: parcourir les champs s\u00e9mantiques du dictionnaire analogique en plusieurs langues pour mieux apprendre avec sensagent.\n\n8006 visiteurs en ligne\n\ncalcul\u00e9 en 0,062s\n\nJe voudrais signaler :\nsection :\nune faute d'orthographe ou de grammaire\nun contenu abusif (raciste, pornographique, diffamatoire)","date":"2021-01-16 19:11:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 1, \"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\": 0.8021560311317444, \"perplexity\": 2060.543733302142}, \"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\/1610703506832.21\/warc\/CC-MAIN-20210116165621-20210116195621-00404.warc.gz\"}"}
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El Globus d'Or a la millor actriu secundària (Best Performance by an Actor in a Supporting Role in a Motion Picture) és un premi de cinema atorgat anualment des de 1944 per l'Associació de la Premsa Estrangera de Hollywood.
Llista de les premiades
1944: Katína Paxinoú a For Whom the Bell Tolls
1945: Agnes Moorehead a Mrs. Parkington
1946: Angela Lansbury a The Picture of Dorian Gray
1947: Anne Baxter a El tall de la navalla (The Razor's Edge)
1948: Celeste Holm a Gentleman's Agreement
1949: Ellen Corby a Mai l'oblidaré (I Remember Mama)
1950: Mercedes McCambridge a All the King's Men
1951: Josephine Hull a Harvey
1952: Kim Hunter a A Streetcar Named Desire
1953: Katy Jurado a Sol davant el perill (High Noon)
1954: Grace Kelly a Mogambo
1955: Jan Sterling a The High and the Mighty
1956: Marisa Pavan a The Rose Tattoo
1957: Eileen Heckart a The Bad Seed
1958: Elsa Lanchester a Testimoni de càrrec (Witness For the Prosecution)
1959: Hermione Gingold a Gigí
1960: Susan Kohner a Imitació de la vida
1961: Janet Leigh a Psicosi (Psycho)
1962: Rita Moreno a West Side Story
1963: Angela Lansbury a El missatger de la por (The Manchurian Candidate)
1964: Margaret Rutherford a The V.I.P.s
1965: Agnes Moorehead a Hush… Hush, Sweet Charlotte
1966: Ruth Gordon a La rebel (Inside Daisy Clover)
1967: Jocelyne LaGarde a Hawai
1968: Carol Channing a Millie, una noia moderna (Thoroughly Modern Millie)
1969: Ruth Gordon a La llavor del diable (Rosemary's Baby)
1970: Goldie Hawn a Flor de cactus (Cactus Flower)
1971: ex aequo Karen Black a Five Easy Pieces i Maureen Stapleton a Airport
1972: Ann-Margret a Carnal Knowledge
1973: Shelley Winters a L'aventura del Posidó (The Poseidon Adventure)
1974: Linda Blair a L'exorcista (The Exorcist)
1975: Karen Black a El gran Gatsby (The Great Gatsby)
1976: Brenda Vaccaro a Jacqueline Susann's Once Is Not Enough
1977: Katharine Ross a El viatge dels maleïts (Voyage of the Damned)
1978: Vanessa Redgrave a Julia
1979: Dyan Cannon a El cel pot esperar (Heaven Can Wait)
1980: Meryl Streep a Kramer contra Kramer (Kramer vs. Kramer)
1981: Mary Steenburgen a Melvin and Howard
1982: Joan Hackett a Només quan ric (Only When I Laugh)
1983: Jessica Lange a Tootsie
1984: Cher a Silkwood
1985: Peggy Ashcroft a Passatge a l'Índia (A Passage To India)
1986: Meg Tilly a Agnès de Déu (Agnes of God)
1987: Maggie Smith a Una habitació amb vista (A Room With a View)
1988: Olympia Dukakis a Encís de lluna (Moonstruck)
1989: Sigourney Weaver a Working Girl
1990: Julia Roberts a Magnòlies d'acer (Steel Magnolias)
1991: Whoopi Goldberg a Ghost
1992: Mercedes Ruehl a El rei pescador (The Fisher King)
1993: Joan Plowright a Un abril màgic (Enchanted April)
1994: Winona Ryder a L'edat de la innocència (The Age of Innocence)
1995: Dianne Wiest a Bales sobre Broadway (Bullets Over Broadway)
1996: Mira Sorvino a Poderosa Afrodita (Mighty Aphrodite)
1997: Lauren Bacall a L'amor té dues cares
1998: Kim Basinger a L.A. Confidential
1999: Lynn Redgrave a Gods and Monsters
2000: Angelina Jolie a Innocència interrompuda (Girl, Interrupted)
2001: Kate Hudson a Gairebé famosos (Almost Famous)
2002: Jennifer Connelly a Una ment meravellosa
2003: Meryl Streep a Adaptation: el lladre d'orquídies (Adaptation)
2004: Renée Zellweger a Cold Mountain
2005: Natalie Portman a Closer
2006: Rachel Weisz a The Constant Gardener
2007: Jennifer Hudson a Dreamgirls
2008: Cate Blanchett a I'm Not There
2009: Kate Winslet a El lector
2010: Mo'Nique a Precious
2011: Melissa Leo a The Fighter
2012: Octavia Spencer a The Help
2013: Anne Hathaway a Les Misérables
2014: Jennifer Lawrence a American Hustle
2015: Patricia Arquette a Boyhood
2016: Kate Winslet a Steve Jobs
Enllaços externs
Lloc web oficial
Actriu secundària
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| 7,874
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Vlaho Bukovac (* 4. Juli 1855 in Cavtat bei Dubrovnik, Kroatien, als Biagio Faggioni; † 23. April 1922 in Prag) war ein aus Dalmatien stammender Maler, der lange Zeit in Prag gelebt hat. Er gilt als herausragender Vertreter des kroatischen Jugendstils.
Leben
Nachdem er 1866 mit einem Onkel nach New York ausgewandert war, lebte er in den 1870er Jahren in Peru und San Francisco. Dort lernte er autodidaktisch das Malen. 1877 reiste er mit Unterstützung von Medo Pucić und Josip Juraj Strossmayer nach Paris, wo er vom Alexandre Cabanel ausgebildet wurde, zu dieser Zeit unterschrieb er zum ersten Mal ein Werk (Türkin im Harem) mit dem slawisierten Namen Bukovac, unter dem er dann berühmt wurde. In den 1880er Jahren arbeitete er unter anderem am Hof des serbischen Königs in Belgrad und in Dalmatien. 1893 ließ er sich in Zagreb nieder, bis er nach Streitigkeiten mit anderen Künstlern 1902 nach Wien ging. 1903 erhielt er eine außerordentliche (ab 1910 ordentliche) Professur an der Akademie der Bildenden Künste in Prag. Er bildete zahlreiche bekannte Maler wie Emil Filla, Bohumil Kubišta und Bedřich Feigl aus und unternahm einige Reisen nach England, wo er Porträts anfertigte.
1918 wurde er zum Mitglied der tschechischen Delegation auf der Konferenz von Versailles ernannt. Noch kurz vor seinem Tod fertigte er 1922 ein Porträt von Alexander I. (Jugoslawien) an.
Ab 1905 war er Mitglied der Serbischen Königlichen Akademie. 1919 wurde er Ehrenmitglied der Jugoslawischen Akademie der Wissenschaften und Künste.
Werke
Bukovac malte vornehmlich Porträts und Landschaften. Ursprünglich zählt er zu Vertretern des Realismus, seine späteren Werke werden zum Impressionismus und Pointillismus gezählt. Seine Bilder signierte er teilweise mit seinem Namen, aber auch mit Boukovatz oder unter dem Pseudonym Andrez, in Belgrad verwendete er den Namen Bukovac in kyrillischer Schrift.
Galerie
Literatur
Mara von Berks: Vlaho Bukovac. In: Österreichs Illustrierte Zeitung, Jahrgang 12 (1902/03), Heft 21 (22. Februar 1903), S. 400–401 (Online bei ANNO)
Hrvatska Enciklopedija, Band 2, 2000
Weblinks
Walker Art Gallery
Porträtmaler
Landschaftsmaler
Maler des Realismus
Maler des Impressionismus
Person (kroatische Geschichte)
Mitglied der Kroatischen Akademie der Wissenschaften und Künste
Hochschullehrer (Akademie der Bildenden Künste, Prag)
Mitglied der Serbischen Akademie der Wissenschaften und Künste
Person (Österreich-Ungarn)
Historische Person (Südosteuropa)
Geboren 1855
Gestorben 1922
Mann
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Aistulf (ca. 700-756) was koning van de Longobarden van 749 tot 756.
Veroveringen
Hij kwam aan de macht in 749 toen zijn broer Ratchis besloot monnik te worden. Hij keerde terug naar een agressievere veroveringspolitiek. In 751 veroverde hij, naar het voorzichtige plan van zijn broer, het exarchaat Ravenna, tot ontsteltenis van paus Zacharias. Aistulf kwam nooit onder de indruk van pauselijke interventies, ook nu niet (in tegenstelling tot zijn afgetreden broer Ratchis). In 753 reisde de nieuwe paus Stephanus II (III) - op verzoek van Constantijn V van Byzantium - naar Pavia om Aistulf tot teruggave van Ravenna te bewegen. Deze ging daar echter niet op in. Aistulf viel in 754 de Kerkelijke Staat binnen, waarop de paus de hulp van Pepijn de Korte inriep.
Franken in Italië
Stefanus III reisde over de Alpen om de hulp van Pepijn af te smeken. Dit manoeuvre van de paus was ongezien en gaf aan wie de echte machtshebber werd in het Westen. De Frankische bisschop Chrodegang begeleidde de paus doorheen zijn reis in het Frankische Rijk. In 755 werd Aistulf verslagen door het Frankische leger van Pepijn onder leiding van diezelfde Chrodegang. De Franken namen Pavia in en kregen Aistulf zo ver kreeg dat hij de teruggave van Ravenna beloofde in het verdrag van Pavia (756). Aistulf loste zijn belofte nooit in, want Pepijn ging kort na zijn overwinning terug naar Francia met buit en gijzelaars, alsook met de titel patricius romanus als dank omdat hij het veroverde land geschonken had aan de paus (donatio pepini).
Nalatenschap van Aistulf
Aistulf stierf door een val van zijn paard op de jacht (756). Aistulf had geen nakomelingen. Zijn broer-monnik Ratchis volgde hem kort op (756-757). De nieuwe koning Desiderius zou de laatste koning der Longobarden worden.
De veroveringspolitiek van Aistulf was afgeblokt door een nieuwe machtige entiteit in Midden-Italië: de Pauselijke Staat, gecreëerd met Frankische hulp. De macht van de Longobarden en van de Byzantijnen was over haar hoogtepunt heen in Italië.
Koning van de Longobarden
Persoon in de 8e eeuw
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\section{Introduction} \label{sec:introduction}
\input{sec_introduction}
\section{Spatial Consistency}\label{sec:spatial_cons}
\input{sec_spatial_consistency}
\section{Measurements}\label{sec:measurements}
\input{sec_measurements}
\section{Numerical Results} \label{sec:numerical_results}
\input{sec_numerical_results}
\section{Conclusion} \label{sec:conclusion}
In this paper, we evaluate the spatial consistency feature based on \ac{SIMO} measurements with angular information of the \acp{MPC}. By studying the behavior of \(d_{\mathrm{CMD}}\) over distance we categorize each track into one of the ``\ac{LoS} radial'', ``\ac{LoS} tangential'', ``Far away'' and ``Uncorrelated'' classes. In general, the similarity decreases over distance, however, the speed of decreasing varies form class to class due to individual geometry. Moreover, low similarity is observed in the \ac{NLoS} scenarios, proving it difficult to cluster users in such scenarios. These findings could be beneficial for future \ac{3GPP} proposal in channel model.
This work is a first step to motivate a further and deeper analysis of the measurement data with respect to spatial consistency. As a next step, we will analyze the dependency of the spatial consistency from angular spread and K-factor to better understand in which radio environments massive \ac{MIMO} schemes that utilize similarity in covariance matrices can be applied. This will also include a direct comparison with the \ac{3GPP} spatial consistency feature.
\section*{Acknowledgment}\label{sec:acknowledgment}
A part of this work has been performed in the framework of the Horizon 2020 project ONE5G (ICT-760809), funded by the European Union. The authors would like to acknowledge the contributions of their colleagues in the project, although the views expressed in this contribution are those of the authors and do not necessarily represent the project.
Moreover, the authors thank their Fraunhofer \ac{HHI} colleagues for conducting the measurements, which is done by Fabian Undi, Leszek Raschkowski and Stephan Jaeckel. Stephan Jaeckel also did the data post-processing.
We also thank Boonsarn Pitakdumrongkija\footnote{While Boonsarn was with NEC at the time of the measurements, in the meantime he left NEC.}, Xiao Peng and Masayuki Ariyoshi from NEC Japan for enabling these measurements in the first place.
\bibliographystyle{IEEEtran}
\subsection{Track Classification}\label{sec:track_class}
After a thorough examination of each track with its serving \ac{BS} and corresponding surroundings shown in \cref{fig:meas_map}, we sort each measurement into one of the following classes:
\begin{itemize}
\item \textbf{\acs{LoS} radial}: The track is approximately radial/perpendicular to the \ac{BS} and \ac{LoS} is available throughout the whole track.
\item \textbf{\acs{LoS} tangential}: The track is approximately tangential \ac{w.r.t.} the position of the \ac{BS} with a \ac{LoS} condition over the complete track.
\item \textbf{Far away}: The track is located far away (at least \SI{100}{\m}) from the \ac{BS} \ac{w.r.t.} the track distance of \(\approx \SI{40}{\meter}\). Only available in the open street scenario.
\item \textbf{Uncorrelated}: Tracks with low correlation, with either \ac{NLoS} condition or obstructed \ac{LoS} condition\footnote{Obstructed \ac{LoS} means objects with a size similar to the wavelength between transmitter and receiver, e.g. three branches.}.
\item \textbf{Other}: The curve shape of the spatial consistency does not follow the other classes.
\end{itemize}
Table~\ref{tab:track_classes} maps the measurement tracks from \cref{fig:meas_map} to the classes given above. Note that due to the highly dynamic transmission environment, e.g. transition from \ac{LoS} to \ac{NLoS} or vice versa, and obstacles that are not shown in the measurement map, e.g. parking cars, the above-mentioned classes can not cover all measurements. Therefore, we leave out evaluation of the measurements in the ``Other'' class for future work. We can see from the table that except for the ``Far away'' class, each class contains tracks from both ``Campus'' and ``Open Street'' scenario.
In order to study the behavior of the \ac{CMD} similarity \ac{w.r.t.} distance, we set the starting \ac{LTTS} of each measurement as user 1, while treating each of the remaining \acp{LTTS} as user 2 moving away from user 1. Therefore, the \(d_{\mathrm{CMD}}\) between two users can be calculated according to \cref{eq:cmd} for each track. \cref{fig:LoS_radial} shows the \ac{CMD} similarity of class ``\ac{LoS} radial'' over distance. For visualizing purpose, the area between the minimum and maximum \(d_{\mathrm{CMD}}\) over all tracks in the class at each \ac{LTTS} is shown as a gray cover plot. Otherwise, the many lines within the same figure would be hard to distinguish. Instead, two representative \(d_{\mathrm{CMD}}\) curves are given to show the dynamic range of the measurement tracks. We can see in \cref{fig:LoS_radial} that in this class the average \(d_{\mathrm{CMD}}\) drops steadily over distance but overall remains high, e.g. mostly above 0.5 up to \SI{20}{\m}. This high correlation can be explained by the \ac{LoS} condition and that most of the power is received by the \ac{LoS} path.
Similarly, \cref{fig:LoS_tangential} shows results for the ``\ac{LoS} tangential'' class. Here \(d_{\mathrm{CMD}}\) decreases almost proportionally with the distance down to 0.1. In comparison to the ``\ac{LoS} radial'' class, the ``\ac{LoS} tangential'' class decreases faster. This can be explained by the change in angles. In the ``\ac{LoS} radial'' class, where the tracks are perpendicular to the \ac{BS}, moving along tracks only changes the elevation angle of the \acp{MPC}, whereas in class ``\ac{LoS} tangential'', where the tracks are tangential to the \ac{BS}, moving along tracks changes both azimuth and elevation angles of the \acp{MPC}. The additional change in the azimuth angle of the \acp{MPC} causes the steeper decrease in \(d_{\mathrm{CMD}}\).
Next, the ``Far away'' class is shown in \cref{fig:Far_Away}. It is of interest to note that the \ac{CMD} similarity fluctuates over distance and no significant decrease is observed for tracks in this class. This is due to the fact that distance between users has a smaller impact on the channel for users located far away from the \ac{BS} than the ones which are closer to the \ac{BS}. To further elaborate, the change in channel coefficients is dependent on the ratio of the relative distance between users to the total distance from the \ac{BS} to the track. The smaller the ratio, the smaller the changes in \(d_{\mathrm{CMD}}\).
At last, \cref{fig:Uncorrelated} shows the ``Uncorrelated'' class. In this case, low correlation is observed in the measurements, since \(d_{\mathrm{CMD}}\) drops below 0.4 after the first few \acp{LTTS}. A cross-check with \cref{fig:meas_map} shows that most of these measurements are taken on tracks without direct \ac{LoS} to the serving \ac{BS}. This indicates that the angles of the received \acp{MPC} at the \ac{BS} change rapidly even by moving the transmitter for \SI{1}{\meter}. One explanation for this could be a fast change of scattering objects that results in non-continuous phase and amplitude jumps of \acp{MPC}. This is also called ``death'' and ``birth'', or lifetime, of scatterers. In our measurements this relates to scatterers observed by the \ac{BS} which was a \ac{UCA} deployed on a pole and therefore able to receive \acp{MPC} from all azimuth directions.
\subsection{Result Analysis}\label{sec:result_analysis}
These 4 classifications show that the spatial consistency depends less on the \acp{LSP} of a given scenario and more on the individual geometry, as both the ``Campus'' and ``Open Street'' scenarios appear in the same classes.
In general, the similarity of covariance matrices decreases over distance. However, depending on each classification, the similarity can have large variation locally, e.g. the similarity remains high over \SI{40}{\m} distances in the ``Far-Away'' category, whereas in the ``Uncorrelated'' category, the similarity is barely seen.
Furthermore, the similarity of the covariance matrices is decreasing within a very short distance in many of the \ac{NLoS} scenarios. This indicates that clustering of users based on covariance matrix similarity can be difficult to achieve in \ac{NLoS} scenarios.
The above described effects are not captured by the current \ac{3GPP} proposal of the spatial consistency feature where only a single dependency on the ``correlation distance'' and the distance between users is captured, see \cite{KDJT18}. Therefore, an extension of the spatial consistency feature in \cite{3GPP17-38901} is required.
\begin{table}
\renewcommand{\tabcolsep}{1pt}
\centering
\begin{tabular}{C{1.3cm}|C{0.3cm}|C{4.8cm}}
\hline
Track Class & \acs{BS} ids & Track ids \\ \hline \hline
\multirow{4}{\linewidth}{\acs{LoS} radial} & 1 & 2 (\SI{6}{\meter}) \\ \cline{2-3}
& 2 & 4 (\SI{6}{\meter}) \\ \cline{2-3}
& 5 & 13, 25 \\ \cline{2-3}
& 6 & 24 (\SI{6}{\meter})\\ \hline
\multirow{3}{\linewidth}{\acs{LoS} tangential} & 1 & 4 (\SI{3}{\meter})\\ \cline{2-3}
& 2 & 3 \\ \cline{2-3}
& 3 & 4 (\SI{6}{\meter})\\ \cline{2-3}
& 5 & 23, 24 \\ \cline{2-3}
& 6 & 14, 15 (\SI{3}{\meter}), 22\\ \cline{2-3}
& 7 & 17 (\SI{6}{\meter}), 18 (\SI{3}{\meter}), 19 (\SI{3}{\meter}), 20 (\SI{3}{\meter}) \\ \hline
\multirow{3}{\linewidth}{Far away} & 5 & 15 (\SI{3}{\meter}), 16, 18 (\SI{3}{\meter}), 19, 20 (\SI{3}{\meter}), 21, 22 \\ \cline{2-3}
& 6 & 17 (\SI{6}{\meter}), 18 (\SI{6}{\meter}), 19 (\SI{6}{\meter}) \\ \cline{2-3}
& 7 & 13 (\SI{6}{\meter}), 15 (\SI{3}{\meter}),23 (\SI{6}{\meter}), 25 (\SI{3}{\meter}) \\ \hline
\multirow{4}{\linewidth}{Un-correlated Tracks}
& 1 & 5 (\SI{6}{\meter}), 6, 7, 9, 10, 11, 12 \\ \cline{2-3}
& 2 & 1 (\SI{6}{\meter}), 2 (\SI{6}{\meter}), 5, 6, 7, 8, 10 (\SI{6}{\meter}) \\ \cline{2-3}
& 3 & 2, 5 (\SI{6}{\meter}), 9, 10 (\SI{3}{\meter}), 11, 12 \\ \cline{2-3}
& 5 & 26 (\SI{3}{\meter}) \\ \cline{2-3}
& 6 & 21 (\SI{6}{\meter}) \\ \hline
\end{tabular}
\caption{Mapping of measurement tracks to track-classes}
\label{tab:track_classes}
\vspace{-0.35cm}
\end{table}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Paper_LoS_radial}
\caption{\ac{CMD} similarity over distance for the tracks in class ``\acs{LoS} radial'' according to \cref{tab:track_classes}. }
\label{fig:LoS_radial}
\vspace{-0.35cm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Paper_LoS_tangential}
\caption{\ac{CMD} similarity over distance for the tracks in class ``\acs{LoS} tangential'' according to \cref{tab:track_classes}. }
\label{fig:LoS_tangential}
\vspace{-0.35cm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Paper_Far_Away}
\caption{\ac{CMD} similarity over distance for the tracks in class ``Far away'' according to \cref{tab:track_classes}. }
\label{fig:Far_Away}
\vspace{-0.35cm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Paper_Uncorrelated}
\caption{\ac{CMD} similarity over distance for the tracks in class ``Uncorrelated'' according to \cref{tab:track_classes}. }
\label{fig:Uncorrelated}
\vspace{-0.35cm}
\end{figure}
\subsection{Covariance Matrix}\label{sec:cov_mtrx}
The covariance matrix is obtained by averaging the channel coefficients over the time duration $\tau$ and the number of \acl{OFDM} subcarriers $N$. Channel coefficients between the transmitter and the receiver at time $t$ on the $n$-th subcarrier are denoted as \(\mathbf{H}_{t,n} \in \mathbb{C}^{n_r \times n_t}\), where \(n_r\) and \(n_t\) are the number of antennas at the receiver and the transmitter, respectively.
The covariance matrix at the receiver side $\mathbf{R}\in\mathbb{C}^{n_r\times n_r}$ is often defined as
\begin{equation}
\mathbf{R}^{(\mathrm{Lit})} = \mathbb{E}\left[\mathbf{H}_{t,n}\mathbf{H}^\mathrm{H}_{t,n}\right],
\label{eq:cov_literature}
\end{equation}
without further explanation on how exactly the covariance matrix can be obtained \cite{ANAC13,MHA+17}, because the covariance matrix is directly generated in the simulations and Gaussian noise is added for individual channel realizations. In \cref{eq:cov_literature}, $\mathbb{E}[\cdot]$ denotes the expectation value. However, in our case the covariance matrix has to be obtained from discrete measurements and is calculated as
\begin{equation}
\mathbf{R}^{(\mathrm{Meas})}=\frac{1}{\tau N}\sum_{t=1}^{\tau}\sum_{n=1}^N\mathbf{H}_{t,n}\mathbf{H}^\mathrm{H}_{t,n}.
\label{eq:cov_measurements}
\end{equation}
It can be seen from \cref{eq:cov_measurements} that the covariance matrix depends on the selection of the averaging time \(\tau\) and the averaging bandwidth represented by \(N\).
Based on the covariance matrix according to \cref{eq:cov_measurements}, details on the \ac{CMD} to study the spatial correlation between two user is given next.
\subsection{Correlation Matrix Distance}\label{sec:cmd}
The \ac{CMD} was proposed in \cite{GVL12} and served as a novel measure to track the changes in spatial structure of non-stationary \ac{MIMO} channels.
Results in \cite{KDJT18} have shown a strong correlation between the physical distance and the \ac{CMD}.
Given the covariance matrices of two users ($\mathbf{R}_1$,$\mathbf{R}_2$), the similarity measure based on \ac{CMD}, according to \cite{GVL12}, can be obtained by
\begin{equation}
\label{eq:cmd}
d_\mathrm{CMD}\left(\mathbf{R}_1,\mathbf{R}_2\right) =\frac{\mathrm{Tr}(\mathbf{R}_1^\mathrm{H}\mathbf{R}_2)}{\|\mathbf{R}_1\|_\mathrm{F}\cdot\|\mathbf{R}_2\|_\mathrm{F}} \\
\end{equation}
where \(\mathrm{Tr}(\cdot)\) denotes the ``trace'' operator.
The \ac{CMD} based similarity measure is a normalized metric which is upper-bounded by $1$ in the case of $\mathbf{R}_1$ and $\mathbf{R}_2$ being collinear, and lower-bounded by $0$ in the case of $\mathbf{R}_1$ and $\mathbf{R}_2$ being orthogonal.
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Stanisław Franciszek Sosabowski (ur. 8 maja 1892 w Stanisławowie, zm. 25 września 1967 w Hillingdon) – polski dowódca wojskowy, działacz niepodległościowy, członek ruchu strzeleckiego i skautingu, uczestnik I wojny światowej w szeregach armii austro-węgierskiej, pułkownik dyplomowany piechoty Wojska Polskiego, 15 czerwca 1944 roku mianowany generałem brygady, organizator i dowódca 1 Samodzielnej Brygady Spadochronowej, z którą walczył w bitwie o Arnhem. Kawaler Krzyża Srebrnego Orderu Virtuti Militari, pośmiertnie odznaczony Orderem Orła Białego.
Dzieciństwo i wykształcenie
Urodził się i wychował w Stanisławowie, jako najstarszy z trojga rodzeństwa. Syn Władysława i Franciszki z domu Garbarskiej. Ojciec był kolejarzem. W wieku jedenastu lat stracił ojca. Brat Julian Andrzej, późniejszy podpułkownik WP, miał wówczas siedem lat, a siostra Janina – cztery. W latach 1903–1910 uczył się w siedmioklasowej cesarskiej i królewskiej Wyższej Szkole Realnej w Stanisławowie przy ulicy Sapieżyńskiej. Pomagał matce w utrzymaniu rodziny udzielając korepetycji. Jak sam pisał w swych wspomnieniach "nasze skromne bytowanie na granicy głodu polepszyło się nieco od chwili, gdy zarabiałem lekcjami". W gimnazjum brał udział w tajnych kółkach samokształceniowych. Będąc uczniem klasy piątej został przewodniczącym wszystkich kółek samokształceniowych funkcjonujących w szkole oraz członkiem tzw. "piątki" grupującej przedstawicieli pięciu stanisławowskich szkół średnich. 30 czerwca 1910 złożył maturę z odznaczeniem.
Działalność niepodległościowa
Od 1909 był członkiem Organizacji Młodzieży Niepodległościowej "Zarzewie" i Armii Polskiej. W 1910 kontynuował naukę w Akademii Handlowej w Krakowie, która prowadziła kursy o profilu ekonomicznym dla absolwentów szkół średnich ogólnokształcących. Na początku 1912 przerwał naukę i wrócił do Stanisławowa. W styczniu tego roku, po przemianowaniu Armii Polskiej w Polskie Drużyny Strzeleckie, został mianowany dowódcą 24 Polskiej Drużyny Strzeleckiej w Stanisławowie i awansowany na podchorążego. Był to najwyższy stopień w organizacji, w czasie pokoju. W dowodzonej przez niego drużynie służył między innymi Stanisław Lityński. Równolegle, od 11 listopada 1911 organizował skauting. Do 1913 pełnił funkcję komendanta hufca w rodzinnym mieście. Z komendy hufca zrezygnował w następstwie konfliktu z miejscowymi władzami "Sokoła", pod auspicjami którego organizowano skauting.
Służba w Armii Austro-Węgier
W 1913 roku został powołany do odbycia obowiązkowej służby w armii austro-węgierskiej. W sierpniu 1914, w czasie mobilizacji był kapralem w austriackim 58 pułku piechoty w Stanisławowie. Pułk w czasie pokoju wchodził w skład 30 Dywizji Piechoty XI Korpusu, a po przeprowadzeniu mobilizacji został włączony do XXII Brygady 11 Dywizji Piechoty. Na początku października, w rejonie Twierdzy Przemyśl przeszedł swój chrzest bojowy. W listopadzie jego pułk od Podgórza do rejonu Zakliczyna toczył walki odwrotowe. W styczniu 1915 pomaszerował na Górne Węgry (obecnie Słowację), gdzie w ataku na bagnety zdobył Czarną Górę. Od wiosny tego roku, w ofensywie zapoczątkowanej bitwą pod Gorlicami, maszerował wzdłuż Sanu, następnie przez Lubelszczyznę w rejon Brześcia, gdzie przeprawił się przez Bug. 15 czerwca 1915 nad rzeką Leśną został ranny w kolano z porażeniem nerwu. "Unieruchomiło to moją prawą nogę na szereg lat, zanim odzyskałem pełną władzę". W czasie walk na froncie awansował do stopnia starszego sierżanta. "Pierś moją pokryły wszystkie dostępne podoficerom medale za waleczność łącznie z dodatkami do żołdu (...) Z moich 250 współtowarzyszy broni, którzy wyruszyli razem ze mną w pole, pozostało zaledwie trzech". Ewakuowany do szpitala w Ołomuńcu. W trakcie leczenia zawarł związek małżeński z Marią Tokarską oraz został awansowany do stopnia podporucznika. Po rehabilitacji w zakładach ortopedycznych na Morawach otrzymał przydział do Urzędu Cenzury w Złoczowie. W 1916 został przydzielony do Dowództwa XI Korpusu stacjonującego w Morawskiej Ostrawie. Tam ukończył kurs wyszkolenia archiwalnego. W styczniu 1917 urodził mu się syn Stanisław. W lutym tego roku został przeniesiony do Bozen w Dolomitach. W miejscowym dowództwie zajmował się archiwum. Na początku 1918, na własną prośbę, przeniesiony został do Lublina i przydzielony do Generalnego Gubernatorstwa. W tym okresie awansował do stopnia porucznika i nawiązał kontakt z komendantem lubelskiego okręgu Polskiej Organizacji Wojskowej, Stanisławem Burhardt-Bukackim.
Służba w Wojsku Polskim w latach 1918–1939
1 listopada 1918 został kierownikiem Komisji Likwidacyjnej byłego austriackiego Generalnego Gubernatorstwa w Lublinie. 15 listopada awansował do stopnia kapitana. 1 stycznia 1919 przeniesiony do Warszawy i przydzielony do Głównego Urzędu Likwidacyjnego w Ministerstwie Spraw Wojskowych. Z powodu kontuzji kolana nie brał bezpośredniego udziału w wojnie polsko-bolszewickiej i wojnie polsko-ukraińskiej. Od 1 marca 1920 do 15 marca 1921 pełnił służbę w Oddziale IV Zaopatrzenia i Komunikacji Sztabu Ministerstwa Spraw Wojskowych, kierowanym przez płk. Jana Niesiołowskiego.
3 maja 1922 roku został zweryfikowany w stopniu majora ze starszeństwem z 1 czerwca 1919 roku i 21. lokatą w korpusie oficerów intendentów. Później został przeniesiony do korpusu oficerów piechoty. W latach 1922–1923 był słuchaczem Kursu Doszkolenia Wyższej Szkoły Wojennej w Warszawie. Z dniem 15 października 1923, po ukończeniu kursu i uzyskaniu dyplomu naukowego oficera Sztabu Generalnego, przydzielony został do Oddziału IV Sztabu Generalnego WP. 12 kwietnia 1927 awansował na podpułkownika ze starszeństwem z dniem 1 stycznia 1927 i 37. lokatą w korpusie oficerów piechoty. 23 maja 1927 został przeniesiony do 75 pułku piechoty w Królewskiej Hucie na stanowisko dowódcy II batalionu. 31 października 1927 został przesunięty na stanowisko dowódcy I batalionu 75 pp w Rybniku. 26 kwietnia 1928 został wyznaczony na stanowisko zastępcy dowódcy 3 pułku strzelców podhalańskich w Bielsku. W grudniu 1929 został przeniesiony do Wyższej Szkoły Wojennej w Warszawie. Początkowo zajmował stanowisko wykładowcy, a od roku szkolnego 1930/1931 kierownika Katedry Operacyjnej Służby Sztabów. W 1937 objął dowództwo 9 pułku piechoty Legionów w Zamościu. Pod jego dowództwem 9 pułk piechoty Legionów zdobył nagrodę dywizyjną w strzelaniu oraz w okresie zimowym w jeździe na nartach.
Latem 1938 starszy syn pułkownika, Stanisław Janusz, wówczas podchorąży III rocznika Szkoły Podchorążych Sanitarnych uległ poważnemu wypadkowi. 8 października 1938 szesnastoletni Jacek Sosabowski postrzelił się śmiertelnie z pistoletu ojca. Tragedia rodzinna wstrząsnęła pułkownikiem. Został zwolniony z dowodzenia pułkiem, urlopowany i skierowany na leczenie w Sanatorium Wojskowym w Zakopanem.
21 lutego 1939 został wyznaczony na stanowisko dowódcy 21 pułku piechoty "Dzieci Warszawy" w Cytadeli Warszawskiej, a 19 marca 1939 awansował na pułkownika.
Kampania wrześniowa 1939
W kampanii wrześniowej dowodził 21 pułkiem piechoty "Dzieci Warszawy".
29 września dowódca Armii "Warszawa", generał dywizji Juliusz Rómmel wydał "Rozkaz pożegnalny do 8 Dywizji Piechoty", w którym stwierdził: "ze szczególnym uznaniem muszę podkreślić bohaterstwo, upór i ofiarność żołnierzy 21 pp pod dowództwem wyśmienitego dowódcy pułku płk dypl. Stanisława Sosabowskiego, który w najcięższych chwilach kryzysu bitwy w dniach 2.IX. do 6.IX. utrzymał swój pułk w ręku i razem z dyonem artylerii 8 pal osłonił odwrót całej dywizji, odchodząc krok za krokiem; skutecznie zatrzymał nieprzyjaciela i tym dał możność innym oddziałom swojej dywizji zebrania się i zajęcia stanowisk wyjściowych do dalszej walki. Działanie to razem z akcją płk Sosabowskiego na wschodnim brzegu Wisły oraz udział chlubny w obronie Warszawy jeszcze raz stwierdza, że 21 pp zwany pułkiem «Dzieci Warszawy», nie zawiódł i jest godzien tej zaszczytnej nazwy". Tym samym rozkazem generał Rómmel nadał pułkownikowi dyplomowanemu Stanisławowi Sosabowskiemu Krzyż Srebrny Orderu Virtuti Militari "w uznaniu zasług i wykazane męstwo i ofiarność w czasie walk z Niemcami".
Zgodnie z postanowieniami aktu kapitulacyjnego 30 września 1939, razem z synem, wyjechał samochodem z Warszawy do Żyrardowa, pierwszego etapu niewoli. W Żyrardowie, symulując chorobę, zbiegł z niewoli. W pierwszej dekadzie października 1939 powrócił do Warszawy i wstąpił do Służby Zwycięstwu Polski.
Na obczyźnie
Francja
Tam otrzymał zadanie przedarcia się przez Węgry do Francji, by przekazać meldunki o sytuacji w kraju rządowi na wychodźstwie. 21 grudnia 1939, pod przybranym nazwiskiem "Emil Hełm", przybył do Paryża. Po przyjeździe do stolicy Francji spotkał się z Naczelnym Wodzem, generałem dywizji Władysławem Sikorskim, który przydzielił go do dyspozycji generała broni Kazimierza Sosnkowskiego. W lutym 1940 został wyznaczony na stanowisko dowódcy piechoty dywizyjnej 1 Dywizji Piechoty, która 3 maja 1940 została przemianowana na 1 Dywizję Grenadierów. Następnie był słuchaczem dwumiesięcznego kursu taktyki artylerii w Mailly-le-Camp. W kwietniu 1940 został wyznaczony na stanowisko dowódcy piechoty dywizyjnej 4 Dywizji Piechoty.
W kwietniu 1940 dywizja przeniosła się do obozu w Parthenay. Tam zastał ją początek działań na froncie zachodnim. Mimo nalegań dowódców jednostki i polskiego rządu, Francja zwlekała z dostarczeniem niezbędnego zaopatrzenia. Gdy w końcu sprzęt dotarł, nie było już czasu na wyszkolenie rekrutów. Do wybuchu działań wojennych jedynie ok. 3150 żołnierzy (spośród 11 000) otrzymało broń. Wobec tego dowódca dywizji gen. bryg. Rudolf Dreszer podjął 16 czerwca decyzję o przebijaniu się jednostki w kierunku portów atlantyckich. 19 czerwca Sosabowskiemu udało się dotrzeć do portu w La Pallice, skąd wraz z 6000 żołnierzy dywizji został ewakuowany do Wielkiej Brytanii.
Wielka Brytania
Natychmiast po dotarciu do Wielkiej Brytanii Sosabowski zgłosił się do polskiego sztabu, gdzie otrzymał przydział na dowódcę formującej się właśnie 4 Brygady Kadrowej Strzelców. Początkowo planowano jej przeniesienie na jakiś czas do Kanady, gdzie miała zostać uzupełniona ochotnikami spośród tamtejszej Polonii. Jednak wkrótce okazało się, że wyjazd skadrowanej brygady za ocean jest nie tylko trudny z powodów logistycznych i transportowych, ale też bezcelowy: nie było tam wystarczającej liczby Polaków.
Sosabowski postanowił ze swej brygady utworzyć pierwszą w historii Wojska Polskiego jednostkę spadochronową. Powstał ośrodek szkoleniowy w Largo House zwany "małpim gajem", w którym prowadzono szkolenia, natomiast skoki spadochronowe odbywały się na lotnisku Ringway: "Gdy przyjdzie chwila, jak orły zwycięskie spadniecie na wroga – mówił na ćwiczeniach w Szkocji 23 września 1941 gen. Sikorski – i przyczynicie się pierwsi do wyzwolenia naszej ojczyzny. Jesteście odtąd Pierwszą Brygadą Spadochronową...". Tym samym gen. Sikorski nadał brygadzie oficjalną nazwę i pozostawił ją do swej wyłącznej dyspozycji.
Hasłem brygady było "Najkrótszą drogą!", co znaczy, że jako pierwsi powrócą do Polski tą właśnie drogą – na spadochronach. Dowódcę zaś – jako znakomitego fachowca – wysoko cenili także Brytyjczycy, zwykle niechętni obcokrajowcom. We wrześniu 1943 gen. Frederick Browning, wysoki oficer brytyjskich wojsk powietrznodesantowych, złożył płk. Sosabowskiemu wręcz niezwykłą propozycję: zaproponował mu objęcie dowództwa brytyjsko-polskiej dywizji spadochronowej. Brygada liczyła wówczas ok. tysiąca żołnierzy. Resztę – 11 tysięcy – mieli stanowić Brytyjczycy, a płk Sosabowski miał otrzymać automatycznie awans na generała. Sosabowski odmówił. 15 czerwca 1944 płk Sosabowski awansował do stopnia generała brygady.
W pierwszych dniach sierpnia przyszła wiadomość o wybuchu powstania w Warszawie. W brygadzie wszyscy byli gotowi do lotu nad Warszawę, jednak rozkazu do startu ze strony Brytyjczyków nie było, zarówno z przyczyn politycznych, wojskowych, jak i technicznych. Wraz z upływającymi kolejnymi dniami powstania narastała w Brygadzie atmosfera buntu, którego zarzewie w kilku kompaniach generał musiał gasić swoim autorytetem. Nie wiedział, że jego syn Stanisław Janusz Sosabowski – lekarz, porucznik AK i dowódca plutonu "Stasinek" utracił w powstańczych walkach wzrok. Brytyjczycy zagrozili rozbrojeniem brygady. Nowy Wódz Naczelny gen. Sosnkowski pod naciskiem aliantów podjął decyzję o przekazaniu 1.SBS do dyspozycji naczelnego dowództwa sprzymierzonych.
Operacja Market-Garden
Ostatecznie Brygada wzięła udział w największej w II wojnie światowej operacji powietrznodesantowej sprzymierzonych, noszącej kryptonim Market Garden. Miała razem z brytyjską 1. Dywizją Powietrznodesantową zająć most na Dolnym Renie. Część Brygady dotarła pod Arnhem w zrzutach 18 i 19 września. Zrzut większości Brygady, na czele z Sosabowskim, miał nastąpić 20 września, jednak przełożono go na następny dzień ze względu na złe warunki pogodowe. Zmieniono również strefę zrzutu: spadochroniarze wylądowali w Driel na południowym brzegu Renu, gdzie znajdowało się mniej niemieckich jednostek. Tymczasem Brytyjczycy pod dowództwem gen. Roberta Urquharta, wyparci z Arnhem przez niemieckie oddziały pancerne, bronili się na zachód od miasta, w Oosterbeek na północnym brzegu - zamierzali ponownie zaatakować Arnhem po dotarciu brytyjskiego XXX Korpusu Pancernego. Urquhart poprosił Sosabowskiego o przybycie z pomocą, jednak w nocy z 21 na 22 września okazało się to niemożliwe (nie znaleziono promu, który miał się znajdować w okolicach Driel). 22 września do brygady Sosabowskiego dotarły oddziały zwiadowcze brytyjskiego XXX Korpusu Pancernego. Niemiecki atak na pozycje zajmowane przez Polaków tego samego dnia został odparty. Następnej nocy podjęto próbę sforsowania Renu na dmuchanych gumowych łódkach dostarczonych przez Brytyjczyków z Oosterbeek, jednak wszystkie one zostały przez Niemców zniszczone i jedynie kilkudziesięciu żołnierzy zdołało się przeprawić.
23 września do Driel przybyła część brytyjskiej 43. Dywizji Piechoty z nowymi łodziami. W nocy z 23 na 24 przeprawiono nimi pod niemieckim ostrzałem około 150 kolejnych żołnierzy. Gdy z 16 łodzi pozostały 3, zaniechano dalszej przeprawy. 24 września do Driel przybył dowódca brytyjskiego XXX Korpusu, gen. Brian Horrocks. Ocenił sytuację operacji jako bardzo trudną - Niemcy stawiali zaciekły opór XXX Korpusowi na południe od Arnhem i zablokowali drogę do innego mostu na Renie, zlokalizowanego kilkanaście kilometrów na zachód od Driel. Brytyjczycy gen. Urquharta w Oosterbeek, dysponując resztkami amunicji, zdziesiątkowani ostrzałem artyleryjskim, odpierali kolejne niemieckie ataki.
Podczas spotkania Horrocksa, Sosabowskiego, gen. Fredericka Browninga i dowódcy brytyjskiej 43. Dywizji Piechoty, Ivora Thomasa, w pobliskim Valburgu zdecydowano, że kolejnej nocy przeprawieni zostaną kolejni żołnierze brytyjscy i polscy. Sosabowski ostro zaprotestował, obawiając się kolejnych strat wśród swoich ludzi. Ostatecznie przeprawiło się jedynie 350 Brytyjczyków z 43. Dywizji, a Polacy pozostali w okolicach Driel, gdyż łodzie dotarły zbyt późno. Kolejnej nocy, z 25 na 26 września, brytyjscy i kanadyjscy inżynierowie zorganizowali ewakuację resztek 1. Dywizji Powietrznodesantowej na południowy brzeg Renu. Do rana przeprawiono większość żołnierzy, w tym 160 Polaków, którzy razem z Brytyjczykami z 43. Dywizji osłaniali przeprawę. Jednakże nie wszyscy osłaniający przeprawę zdążyli się ewakuować przed świtem: po kilkudziesięciu żołnierzy polskiej Brygady i 43. Dywizji dostało się do niewoli.
Sosabowski razem ze swoją brygadą pozostali w Holandii do października, kiedy przeniesiono ich z powrotem do Wielkiej Brytanii. Za bitwę pod Arnhem generał został odznaczony Krzyżem Walecznych. W listopadzie gen. Browning, pozostający w konflikcie z Sosabowskim już od czasu przygotowań do operacji Market Garden, zażądał odebrania mu dowództwa Brygady. Rozkazem z 27 grudnia 1944 1. SBS została odebrana Sosabowskiemu, którego mianowano inspektorem Jednostek Etapowych i Wartowniczych.
Lata powojenne
W lipcu 1948 gen. Stanisław Sosabowski został zdemobilizowany. Miał już wtedy przy sobie ociemniałego syna wraz z żoną, których udało się ściągnąć z Polski. Pozostał na emigracji w Wielkiej Brytanii i pracował jako robotnik magazynowy w fabryce silników elektrycznych, później telewizorów. Jak sam napisał: "przez 17 lat pracowałem w fabryce jako oficjalnie nie znany, prowadząc żywot podwójny: zwykłego robotnika przez 5 dni w tygodniu, "jako szeregowiec fabryczny" – "Stan", oraz dostojny żywot polskiego generała, poniekąd "ojca" polskich spadochroniarzy, znanego wśród swoich i Brytyjczyków, Amerykanów i Holendrów". Przystąpił do Koła Lwowian w Londynie. Wśród swoich żołnierzy nosił wciąż przydomek "Sosab".
Wraz z żołnierzami swojej Brygady był zrzeszony w Związku Polskich Spadochroniarzy i odbywał z nimi coroczne spotkania, organizowane w Instytucie im. gen. Sikorskiego w Londynie w rocznicę sformowania jednostki przy jej sztandarze. Z uwagi na chorobę serca nie przybył na spotkanie stowarzyszenia w dniu 24 września 1967. Nazajutrz zmarł na zawał serca 25 września 1967 w szpitalu w Hillingdon w wieku 75 lat. Pogrzeb odbył się 30 września 1967 w kościele św. Andrzeja Boboli w Londynie.
W 1969 wciąż wierni swemu dowódcy spadochroniarze generała przywieźli jego prochy do Polski, gdzie spoczęły – zgodnie z jego wolą – na Cmentarzu Wojskowym na Powązkach w Warszawie (kwatera A19-7-10/11). W tym miejscu zostali pochowani także jego żona Maria (zm. 1958) i syn Stanisław Janusz (zm. 2000).
Upamiętnienie
9 czerwca 1982 roku – powstała w Łodzi 27 Spadochronowa Drużyna Harcerzy i Harcerek im gen. Stanisława Franciszka Sosabowskiego
1987 – nadanie sztandaru 27 Spadochronowej Drużynie Harcerskiej w Lodzi, oraz odsłonięcie 2 w Polsce Izby pamięci poświęconej 1 Samodzielnej Brygadzie Spadochronowej
25 czerwca 1983 – nadanie imienia gen. Stanisława Franciszka Sosabowskiego 7 Harcerskiej Drużynie Czerwonych Beretów z Lęborka.
Ciche nadanie nazwy Sosabowskiego ulicy w Warszawie na Gocławiu, w obrębie bronionego w 1939 r. odcinka Grochów.
1995 – Szkoła Podstawowa nr 69 im. gen. Stanisława Sosabowskiego w Gdańsku.
31 maja 2006 – Beatrix, królowa Holandii przyznała generałowi pośmiertnie Medal Brązowego Lwa.
Chorągiew 1. Samodzielnej Brygady Spadochronowej została udekorowana Wojskowym Orderem Wilhelma.
15 września 2006 – w Oosterbeek (w miejscu przeprawy żołnierzy gen. Sosabowskiego z 1 SBS na północnym brzegu Renu) została odsłonięta tablica z nazwą "Generaal Sosabowskiwaard" (Polder Generała Sosabowskiego), nadana przez Gminę Renkum.
16 września 2006 – w ramach 62 rocznicy obchodów bitwy pod Arnhem odbyło się uroczyste odsłonięcie przez brytyjskich weteranów pomnika generała Sosabowskiego w Driel.
5 lipca 2007 – Jednostka Strzelecka 2031 Związku Strzeleckiego "Strzelec" OSW w Rybniku otrzymuje imię gen. bryg. Stanisława Sosabowskiego.
7 października 2007 – nadanie sztandaru 7 Harcerskiej Drużynie Czerwonych Beretów im. gen. Stanisława Franciszka Sosabowskiego z Lęborka.
18 czerwca 2008 – premiera filmu dokumentalnego "Honor generała", reż. Joanna Pieciukiewicz w TVP Polonia.
18 października 2008 – w Szczecinie odsłonięto tablicę poświęconą generałowi i nadano jednej z ulic jego imię.
W 2008 roku gen. Sosabowski został ogłoszony patronem Gimnazjum nr 27 w Szczecinie.
26 września 2009 – odsłonięcie tablicy na Centrum Kultury Praga Południe – ul. Podskarbińska, w pobliżu ul. Grochowskiej, na linii obrony odcinka Grochów (nazwa pochodzi od okręgu Grochów zbliżonego powierzchnią do dzisiejszej dzielnicy Praga Pd.); propozycja wystawienia pomnika w dzielnicy.
29 września 2009 roku Decyzją MON Nr PF-42/Org./P1 6.Brygada Desantowo-Szturmowa otrzymuje nazwę 6.Brygada Powietrznodesantowa im. gen. Stanisława F. Sosabowskiego
11 września 2011 Binowo – nadanie sztandaru 64 Obronnej Drużynie Starszoharcerskiej im. gen. bryg. Stanisława Franciszka Sosabowskiego ze Szczecina w ramach obchodów 30-lecia drużyny
9 maja 2012 – Senat RP podjął uchwałę w sprawie uczczenia 120. rocznicy urodzin Generała Stanisława Franciszka Sosabowskiego i ustanowienia września 2012 roku miesiącem Jego pamięci
18 maja 2012 – nadanie sztandaru Gimnazjum nr 27 im. gen. bryg. Stanisława Franciszka Sosabowskiego w Szczecinie
17 września 2012 – odsłonięcie przez harcerzy z 7 Harcerskiej Drużyny Czerwonych Beretów tabliczek z nazwą ul. gen. Sosabowskiego na Osiedlu Harcerzy w Lęborku
1 września 2013 – odsłonięcie popiersia generała w Parku im. Henryka Jordana w Krakowie
23 września 2017 – odsłonięcie Pomnika gen. Stanisława Sosabowskiego w Warszawie przy al. Wojska Polskiego.
7 października 2017 – odsłonięcie Pomnika gen. Stanisława Sosabowskiego w Lęborku przy ul. Armii Krajowej 16
13 grudnia 2017 – na mocy Zarządzenia zastępczego Wojewody Wielkopolskiego Zbigniewa Hoffmanna dawna aleja Braterstwa Broni na poznańskiej Cytadeli otrzymała nazwę alei gen. Stanisława Sosabowskiego
15 lutego 2020 – nadanie sztandaru i imienia Szkole Podstawowej nr 2 w Proszowicach im. 6 Brygady Powietrznodesantowej gen. bryg. Stanisława Sosabowskiego i odsłonięcie tablicy pamiątkowej upamiętniającej to wydarzenie
Ordery i odznaczenia
Order Orła Białego (pośmiertnie 6 listopada 2018)
Krzyż Srebrny Orderu Wojennego Virtuti Militari nr 12236
Krzyż Komandorski z Gwiazdą Orderu Odrodzenia Polski (pośmiertnie, 1988)
Krzyż Niepodległości (9 października 1933)
Krzyż Kawalerski Orderu Odrodzenia Polski (2 maja 1923)
Krzyż Walecznych (dwukrotnie)
Złoty Krzyż Zasługi z Mieczami
Złoty Krzyż Zasługi (19 marca 1937)
Medal Pamiątkowy za Wojnę 1918–1921
Srebrny Wawrzyn Akademicki (7 listopada 1936)
Medal Dziesięciolecia Odzyskanej Niepodległości
Srebrny Medal za Długoletnią Służbę
Brązowy Medal za Długoletnią Służbę
Złoty Medal Waleczności (Austria)
Srebrny Medal Waleczności (Austria)
Krzyż Oficerski Orderu Korony Rumunii (Rumunia)
Krzyż Oficerski Orderu Orła Białego (Królestwo SHS, 1926)
Krzyż Oficerski Orderu Świętego Sawy (Królestwo SHS, 1926)
Komandor Orderu Imperium Brytyjskiego honorowy, kategoria wojskowa (Wielka Brytania, 1943)
Medal Obrony 1939–1945 (Wielka Brytania)
Medal Wojny 1939–1945 (Wielka Brytania)
Gwiazda Francji i Niemiec (Wielka Brytania)
Medal Brązowego Lwa (Holandia, pośmiertnie 2006)
Dorobek pisarski
Wspomnienia:
Opracowania:
Wychowanie żołnierza-obywatela, Warszawa 1931.
Kwatermistrzostwo w polu, 1935.
Artykuły:
Armia i przemysł, "Polska Zbrojna" Nr 79 z 30 grudnia 1921 r., s. 4–5.
Odpowiedzialność w administracji państwowej, "Polska Zbrojna" Nr 147 z 3 czerwca 1922 r., s. 2.
Powrót, "Polska Zbrojna" Nr 166 z 23 czerwca 1922 r., s. 1–2.
Depresja w sferach oficerskich, "Polska Zbrojna" Nr 263 z 26 września 1923 r., s. 3.
Odniesienia w kulturze masowej
W 1974 amerykański dziennikarz i pisarz Cornelius Ryan wydał książkę pod tytułem A Bridge Too Far (polski tytuł: O jeden most za daleko, wyd. 1 – 1979), opisującą operację Market-Garden. W ekranizacji książki w postać Sosabowskiego wcielił się Gene Hackman.
Zobacz też
generałowie Wojska Polskiego
Cora Baltussen
Geertjan Lassche
Przypisy
Bibliografia
Dzienniki Personalne Ministra Spraw Wojskowych.
Roczniki oficerskie 1923, 1924, 1928 i 1932.
Spis oficerów służących czynnie w dniu 1.6.1921 r. Dodatek do Dziennika Personalnego M.S.Wojsk. Nr 37 z 24 września 1921 r.
Sprawozdania Dyrekcyi C. K. Wyższej Szkoły Realnej w Stanisławowie (...) 1904–1910.
Zbigniew Mierzwiński, Generałowie II Rzeczypospolitej, Wydawnictwo Polonia, Warszawa 1990, .
Aneta Chmielewska-Szymańska, Generał Brygady Stanisław Sosabowski 1892–1967.
Englert Juliusz L., Barbarski Krzysztof, Generał Stanisław Sosabowski: dowódca 1 Samodzielnej Brygady Spadochronowej – Commander 1st Polish Independent Parachute Brigade, Wyd. Instytut Polski i Muzeum im. gen. Sikorskiego; Polska Fundacja Kulturalna, Londyn, 1996.
Wojciech Markert, Arnhem: udział 1 Samodzielnej Brygady Spadochronowej w operacji Market–Garden, Wydawnictwo "Ajaks", Pruszków 1999, , .
George F. Cholewczynski "Rozdarty naród. Polska brygada spadochronowa w bitwie pod Arnhem", Wydawnictwo: Andrzej Findeisen – A.M.F. Plus Group, 2006; .
Wojciech Markert "Arnhem 1944", Wydawnictwo: Almapress, 2006 (Seria: Wielkie bitwy historii), .
Wojciech Markert "Na drodze do Arnhem", Wydawnictwo: Ajaks, 2000, .
Wojciech Markert "Najcichsi bohaterowie spod Arnhem", Wydawnictwo: Ajaks, 2003, .
Piotr Zarzycki, Plan mobilizacyjny "W", Wykaz oddziałów mobilizowanych na wypadek wojny, Oficyna Wydawnicza "Ajaks" i Zarząd XII Sztabu Generalnego Wojska Polskiego, Pruszków 1995, .
Linki zewnętrzne
Sosabowski Family Website
Stanisław Sosabowski i polski skauting
Biografie kanonu polskiej Wikipedii
Członkowie Organizacji Młodzieży Niepodległościowej "Zarzewie"
Członkowie Polskich Drużyn Strzeleckich
Członkowie Służby Zwycięstwu Polski
Dowódcy 9 Pułku Piechoty Legionów
Dowódcy 21 Pułku Piechoty (II RP)
Generałowie brygady Polskich Sił Zbrojnych
Harcerze
Ludzie urodzeni w Stanisławowie
Ludzie związani z Zamościem
Odznaczeni Krzyżem Kawalerskim Orderu Odrodzenia Polski (II Rzeczpospolita)
Odznaczeni Krzyżem Komandorskim z Gwiazdą Orderu Odrodzenia Polski (władze RP na uchodźstwie)
Odznaczeni Krzyżem Niepodległości
Odznaczeni Krzyżem Srebrnym Orderu Virtuti Militari (II Rzeczpospolita)
Odznaczeni Krzyżem Walecznych
Odznaczeni Medalem Dziesięciolecia Odzyskanej Niepodległości
Odznaczeni Medalem Pamiątkowym za Wojnę 1918–1921
Odznaczeni Medalem za Długoletnią Służbę (II Rzeczpospolita)
Odznaczeni Orderem Orła Białego (III Rzeczpospolita)
Odznaczeni Srebrnym Wawrzynem Akademickim
Odznaczeni Złotym Krzyżem Zasługi (II Rzeczpospolita)
Odznaczeni Złotym Krzyżem Zasługi z Mieczami
Oficerowie 75 Pułku Piechoty (II RP)
Oficerowie dowództwa 1 Samodzielnej Brygady Spadochronowej
Oficerowie dowództwa 4 Dywizji Piechoty (WP we Francji)
Oficerowie dyplomowani II Rzeczypospolitej
Pochowani na Powązkach-Cmentarzu Wojskowym w Warszawie
Polacy – uczestnicy kampanii francuskiej 1940
Polacy – żołnierze Cesarskiej i Królewskiej Armii w I wojnie światowej
Polacy odznaczeni Gwiazdą Francji i Niemiec
Polacy odznaczeni Medalem Brązowego Lwa
Polacy odznaczeni Medalem Obrony
Polacy odznaczeni Medalem Waleczności
Polacy odznaczeni Medalem Wojny 1939–1945
Polacy odznaczeni Orderem Imperium Brytyjskiego
Polacy odznaczeni Orderem Korony Rumunii
Polacy odznaczeni Orderem Orła Białego (Serbia)
Polacy odznaczeni Orderem św. Sawy
Pułkownicy piechoty II Rzeczypospolitej
Uczestnicy bitwy o Arnhem 1944 (strona polska)
Uczestnicy bitwy pod Mławą (1939)
Urodzeni w 1892
Zastępcy dowódcy 3 Pułku Strzelców Podhalańskich
Zmarli w 1967
Dowódcy brygad Polskich Sił Zbrojnych
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Q: Cron jobs output on console I have written a shell script (myscript.sh):
#!/bin/sh
ls
pwd
I want to schedule this job for every minute and it should display on the console. In order to do that I have done crontab -e:
*/1 * * * * /root/myscript.sh
Here, it's display the output in the file /var/mail/root rather than printing on the console.
What changes I have to make inorder to print the output on console?
A: The easiest way I can think of is to log the output to disk and have a console window constantly checking to see if the logfile has been altered and printing the changes.
crontab:
*/1 * * * * /root/myscript.sh | tee -a /path/to/logfile.log
in console:
tail -F /path/to/logfile.log
The problem with this is that you will get an ever growing log file which will need to be periodically deleted.
To avoid this you would have to do something more complicated whereby you identify the console pid that you wish to write to and store this in a predefined place.
console script:
#!/usr/bin/env bash
# register.sh script
# prints parent pid to special file
echo $PPID > /path/to/predfined_location.txt
wrapper script for crontab
#!/usr/bin/env bash
cmd=$1
remote_pid_location=$2
# Read the contents of the file into $remote_pid.
# Hopefully the contents will be the pid of the process that wants the output
# of the command to be run.
read remote_pid < $remote_pid_location
# if the process still exists and has an open stdin file descriptor
if stat /proc/$remote_pid/fd/0 &>/dev/null
then
# then run the command echoing it's output to stdout and to the
# stdin of the remote process
$cmd | tee /proc/$remote_pid/fd/0
else
# otherwise just run the command as normal
$cmd
fi
crontab usage:
*/1 * * * * /root/wrapper_script.sh /root/myscript.sh /path/to/predefined_location.txt
Now all you have to do is run register.sh in the console you want the program to print to.
A: I tried to achieve the output of a cron job to a gnome terminal , and managed it by this
*/1 * * * * /root/myscript.sh > /dev/pts/0
I suppose if you do not have a GUI and you only have CLI you might use
*/1 * * * * /root/myscript.sh > /dev/tty1
to achieve the output of a crontab job to be redirected to your console. But first be sure to find the name of your terminal. if it is /dev/tty1 or something else. I am not sure how this can be done for all cases, probably with something like
env | grep -i tty
A: One option would be to use "wall" in your script to display a message to all terminals. E.G.
wall -n Display this message to all terminals.
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KUALA LUMPUR: The Health Ministry (KKM) has to amend the existing law before the proposal to make two vaccinations compulsory for children is implemented.
KUALA LUMPUR, March 9 ― The Health Ministry (KKM) has to amend the existing law before the proposal to make two vaccinations compulsory for children is implemented.
LETTER | The housemen placement woes due to a glut of medical graduates in Malaysia could be a paradox.
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{"url":"https:\/\/www.zenodo.org\/record\/2576892\/export\/xd","text":"Journal article Open Access\n\n# Evaluation of Mycotoxin Screening Tests in a Verification Study Involving First Time Users\n\nVeronica M. T. Lattanzio, Christoph von Holst, Vincenzo Lippolis, Annalisa De Girolamo, Antonio F. Logrieco, Hans G. J. Mol, Michelangelo Pascale\n\n### Dublin Core Export\n\n<?xml version='1.0' encoding='utf-8'?>\n<oai_dc:dc xmlns:dc=\"http:\/\/purl.org\/dc\/elements\/1.1\/\" xmlns:oai_dc=\"http:\/\/www.openarchives.org\/OAI\/2.0\/oai_dc\/\" xmlns:xsi=\"http:\/\/www.w3.org\/2001\/XMLSchema-instance\" xsi:schemaLocation=\"http:\/\/www.openarchives.org\/OAI\/2.0\/oai_dc\/ http:\/\/www.openarchives.org\/OAI\/2.0\/oai_dc.xsd\">\n<dc:creator>Veronica M. T. Lattanzio, Christoph von Holst, Vincenzo Lippolis, Annalisa De Girolamo, Antonio F. Logrieco, Hans G. J. Mol, Michelangelo Pascale<\/dc:creator>\n<dc:date>2019-02-25<\/dc:date>\n<dc:description>Rapid screening methods are currently recognized as a strategic tool for mycotoxin issues management. Specific guidelines for validation and verification of mycotoxin screening methods are set in the Commission Regulation (EU) No 2014\/519. This regulation establishes that the \u201caim of the validation is to demonstrate the fitness-for-purpose of the screening method\u201d and focuses the entire validation procedure on determining specific cut-off values ensuring a maximum rate of false negative results of 5%. In addition, the assessment of the rate of false suspect results is addressed. With regard to rapid test-kits, \u2018fitness-for-purpose\u2019 includes not only the criteria more commonly considered when discussing laboratory-based methods (specificity, accuracy, and precision), but also more \u201cpractical\u201d parameters such as speed and ease of implementation in a new operational environment. The latter means demonstrating under local conditions that performance parameters, as established during the validation, can be achieved by first time users. This goal can be achieved through \u201cmethod verification\u201d. The aim of the present study was to verify the fitness-for-purpose of mycotoxin screening methods when applied by first time users. This was achieved in one laboratory facility via results of a training course with multiple technicians attending. The verification study was organized similarly to a collaborative exercise and involved two groups comprising of 10 technicians each that used the methods for the first time. Different screening methods were applied for deoxynivalenol (DON) in wheat, which was mainly Enzyme Linked Immunosorbent Assay (ELISA), lateral flow device (LFD), fluorescence polarization immunoassay (FPIA), and liquid chromatography-high resolution mass spectrometry (LC-HRMS). An additional verification was done for aflatoxin B1 (AFB1) in maize and wheat using LFD and LC-HRMS, respectively. The results of analyses were used to calculate intermediate precision (RSDip, covering the inter-analyst variability in preparing the analytical samples and the precision under repeatability conditions) cut-off values and false suspect rates. RSDip ranged from 6.5% to 30% for DON, and from 16% to 33% for AFB1. The highest obtained variances were associated with the AFB1 analyses due to working with much lower mass fractions. The rate of false suspect results were lower than 0.1% for all tested methods. All methods showed a fit-for-purpose method performance profile, which allowed a clear distinction of samples containing the analytes at the screening target concentration (STC) from negative control samples. Moreover, the first time users obtained method performances similar to those obtained for validation studies previously performed on the screening methods included in the training course.<\/dc:description>\n<dc:identifier>https:\/\/zenodo.org\/record\/2576892<\/dc:identifier>\n<dc:identifier>10.3390\/toxins11020129<\/dc:identifier>\n<dc:identifier>oai:zenodo.org:2576892<\/dc:identifier>\n<dc:relation>info:eu-repo\/grantAgreement\/EC\/H2020\/678781\/<\/dc:relation>\n<dc:rights>info:eu-repo\/semantics\/openAccess<\/dc:rights>\n<dc:subject>mycotoxins; screening; validation; immunoassay; mass spectrometry; cereals<\/dc:subject>\n<dc:title>Evaluation of Mycotoxin Screening Tests in a Verification Study Involving First Time Users<\/dc:title>\n<dc:type>info:eu-repo\/semantics\/article<\/dc:type>\n<dc:type>publication-article<\/dc:type>\n<\/oai_dc:dc>\n\n65\n45\nviews","date":"2021-06-19 10:18:07","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\": 0.32911229133605957, \"perplexity\": 6859.432237187218}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487647232.60\/warc\/CC-MAIN-20210619081502-20210619111502-00260.warc.gz\"}"}
| null | null |
Can Facebook Predict Dallas's Next Mayor? A Social Media "Psychic" Works His Magic.
Leslie Minora
| Politics |
Leslie Minora | May 13, 2011 | 3:00pm
Not surprisingly, all four of Dallas's mayoral candidates have Facebook fan pages, some (David Kunkle) more active than others (Edward Okpa). But what, if anything, do Facebook numbers mean? Are they indicators of electoral momentum or of quick-thumbed interns?
A Chicago-based social media guru says Facebook numbers, when viewed in the context of overall page activity, are very telling. In fact, he says, he predicted Chicago's mayoral election based on semi-scientific analysis of candidates' pages. And he happily analyzed our local would-bes' pages to make a prediction about tomorrow's election.
Johnny Campbell, The Transition Man (who has nearly 500 Facebook fans and a social media training business in Chicago), tells Unfair Park that a candidate's Facebook fan numbers are an important indicator election success, along with what he calls "engagement" -- the amount of social media interaction between a candidate and his fan base.
"I'm looking at their number of fans, the amount of engagement, the amount of responses on their walls by other people than themselves, as well as events and discussions," Campbell says. "I'm just using my own gut instinct to a degree."
The night before the Chicago mayoral election, he correctly predicted that Rahm Emanuel would win and accurately predicted the standings of the candidates behind him. We know: Emanuel killed it in Chicago, raking in well over the majority of the vote he needed, so that prediction doesn't mean much.
So: Is his "science" legit, just fun or somewhere in between? There's only one way to find out.
Johnny Campbell
Let's put Johnny Facebook to work. And remember, if Saturday's outcome turns out as he says, you heard it here first.
Campbell's predictions:
1. David Kunkle
2. Mike Rawlings
3. Ron Natinsky
4. Edward Okpa
... and no runoff.
The analysis:
"It seemed like Kunkle had a little bit more of engagement than Rawlings did on his fan page," Campbell says, adding that it was a close margin. "Him being police chief created more of a conversation." Campbell notes the "links floating around on Kunkle's about Angela Hunt," calling her an "influencer," as far as he could tell on Facebook. "And those make a difference when you're running for office."
Hedging his bets, Campbell adds that campaign finances are not clear on Facebook, but can heavily influence an election's outcome. With that major caveat, Rawlings could be the man to unhinge the Facebook activity/election day correlation. Tomorrow will tell.
When Unfair Park asked Campbell for a professional title to sum up his emerging side-career of making Facebook predictions, he laughed at the suggestion of "social media psychic" and instead suggested "social media strategist." Either way, he's confident things will play out as he says they will.
"I'm probably 90 percent sure," he says.
|
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| 9,142
|
HITLER'S
FORGOTTEN
CHILDREN
_This book is dedicated to all the victims of Nazi Germany –
men, women and, above all, children – and to those throughout
the world today who suffer from the persisting evil which
teaches that one race, creed or colour is superior to another._
## CONTENTS | _Preface_
---|---
ONE | August 1942
TWO | Year Zero
THREE | Escape
FOUR | Home
FIVE | Identity
SIX | Walls
SEVEN | Source of Life
EIGHT | Bad Arolsen
NINE | The Order
TEN | Hope
ELEVEN | Traces
TWELVE | Nuremberg
THIRTEEN | Rogaška Slatina
FOURTEEN | Blood
FIFTEEN | Pure
SIXTEEN | Taken
SEVENTEEN | Searching
EIGHTEEN | Peace
|
_Afterword_
|
_Acknowledgements_
|
_Index_
## PREFACE
Blood runs through this story. The blood of young men spilled on the battlefields of war; the blood of civilians that ran through the gutters of cities, towns and villages across Europe; the blood of millions destroyed in the pogroms and death camps of the Holocaust.
But blood, too, as an idea: the Nazi belief – absurd as this seems today – in 'good blood', precious Ichor to be sought out, preserved and expanded. And with it, the inevitable counterpart: 'bad blood', to be ruthlessly eradicated.
I was born in 1941 in the depths of the Second World War. I grew up in its wake, and under the shadow of its brutal progeny, the Cold War.
My history is the history of millions of ordinary German men and women like me. We are the victims of Hitler's obsession with blood, as well as the beneficiaries of the post-war economic miracle that transformed our devastated and pariah nation into the powerhouse of modern Europe. Our story is that of a generation raised in the shadow of infamy, but which found a way to struggle towards honesty and decency.
But my own story is also that of a much more secret past, still cloaked in silence and shrouded in shame.
I am a child of Lebensborn _._
Lebensborn is an ancient German word meaning 'fountain of life', twisted and distorted by the word-smelters of National Socialism. What did it mean in the madness of Nazism? What does it mean today? My search for the answers – to uncover my own story – has taken me on a long and painful road: a physical journey that has led me across the map of modern Europe. It has been an historical expedition, too: an often uncomfortable return to the Germany of more than seventy years ago, and into the troubled stories of those countries overrun by Hitler's armies.
The journey has also forced me to make a psychological voyage into everything I have known and grown up with: a fundamental questioning of who I am, and what it means to be German. I will not pretend that this is a simple story: it will not always be easy to read. But neither has it been easy to live.
I am not, by nature, overtly emotional. The expression of emotion, such a commonplace thing in twenty-first-century society, does not come easily to me. I have spent my life attempting to suppress my inner self, to subordinate my feelings to the circumstances in which I have grown up, as well as to the needs of others.
But this is a story which, I believe, needs to be heard. More, much more, it needs to be understood. It is not unique, in that there are others who have endured much of what has shaped my life. But while I share a common thread with thousands of others who passed through the vile experiment of Lebensborn, to the best of my knowledge no one else shares the particular twists of fate, history and geography that have defined my seventy-four years on earth.
Lebensborn. The word runs through my life like the blood coursing through my body. To see it, to understand it, demands much more than a superficial examination. The search for the roots of this story requires a deep and intrusive investigation of the most hidden places.
We must start in a town and a country that no longer exist.
## ONE | AUGUST 1942
_'Men... must be shot, the women locked up and transported to concentration camps, and the children must be torn from their motherland and instead accommodated in the territories of the old Reich.'_
REICHSFÜHRER-SS HEINRICH HIMMLER, 25 JUNE 1942
### Cilli, German-occupied Yugoslavia, 3–7 August 1942
The schoolyard was crowded. Hundreds of women – young and old – clutched the hands of their children and found what space they could in the packed courtyard. Nearby, Wehrmacht soldiers, rifles slung over their shoulders, looked on as the families slowly drifted in from towns and villages across the area.
These women had been summoned by their new German masters, ordered to bring their children to the school for 'medical tests'. Upon arrival they were arrested and told to wait. Otto Lurker, commander of the police and security services for the region, watched relaxed and impassive – his hands resting comfortably in his pockets – as the yard filled with families. Once, Lurker had been Hitler's gaoler: now he was the Führer's leading henchman in Lower Styria. He held the rank of SS-Standartenführer – the paramilitary equivalent of a full colonel in the army – but that summer's morning he was casually dressed in a two-piece civilian suit.
Yugoslavia had been under Nazi rule for sixteen months. In March 1941, with the surrounding countries of Hungary, Romania and Bulgaria having recently joined the Reich's alliance of Balkan nations, Hitler put pressure on the kingdom's ruler, the Regent Prince Paul, to fall into line. He and his cabinet bowed to the inevitable, formally tying Yugoslavia to the axis powers, but the Serb-dominated army launched a coup d'état, replacing Paul with his seventeen-year-old second cousin, Prince Peter.
News of the revolt reached Berlin on 27 March. Hitler took the coup as a personal insult and issued Directive 25, formally designating the country an enemy of the Reich. The Führer ordered his armies 'to destroy Yugoslavia militarily and as a state'. A week later, the Luftwaffe began a devastating bombing campaign while divisions of Wehrmacht infantry and tanks of the Panzer Corps swept through towns and villages. The Royal Yugoslavian Army was no match for Germany's Blitzkrieg troops: on 17 April the country surrendered.
The occupying troops immediately set about fulfilling Hitler's instruction to dismantle all vestiges of the state. Some 65,000 people – primarily intellectuals and nationalists – were exiled, imprisoned or murdered, their homes and property handed over to their new German masters. The Slovene language was prohibited.
But for the rest of 1941 and throughout the first half of 1942, partisan groups, led by the communist Josip Broz Tito, fought a determined campaign of resistance. Germany retaliated with a brutal crackdown: the Gestapo swooped on fighters and civilians alike, deporting thousands to concentration camps across the Reich. Others were executed as a warning against resistance. In the nine months following September 1941, 374 men and women were lined up against the walls of the prison yard at Cilli and summarily shot. Photographers recorded the murders for the purposes of both posterity and propaganda.
On 25 June 1942, Heinrich Himmler – the second most powerful and feared man in Nazi Germany – issued orders to his secret police and SS officers for the elimination of partisan resistance.
This campaign possesses every required element to make harmless the population which has supported the bandits and provided them with human resources, weapons and shelter. Men from such families, and often even their relatives, must be shot, the women locked up and transported to concentration camps, and the children must be torn from their motherland and instead accommodated in the territories of the old Reich. I expect to be provided with a special report on the number of children and their racial values.
Against this bloody backdrop, 1,262 people – many the surviving relatives of those executed as an example to the rest of the population – assembled in the schoolyard that August morning to await their fate.
Among them was a family from the nearby village of Sauerbrunn. Johann Matko came from a family of known partisans: his brother, Ignaz, had been one of those lined up and shot against the wall of Cilli prison in July. Johann had been dragged off to Mauthausen concentration camp. After seven months in the camp he was allowed to return home to his wife, Helena, and their three children: eight-year-old Tanja, her brother Ludvig – then six – and nine-month-old baby Erika.
When all the families were accounted for, an order was given to separate them into three groups – one each for the children, the women, and the men. Under Lurker's direction the soldiers moved in and pulled children from the grasp of their mothers; a local photographer, Josip Pelikan, recorded the harrowing scene for the Reich's obsessive archivists. His rolls of film captured the fear and alarm of women and children alike: his shots included scores of toddlers held in low pens of straw inside the school buildings.
As the mothers waited outside, Nazi officials began a cursory examination of the children. Working with charts and clipboards, they painstakingly noted each child's facial and physical characteristics.
These, though, were not 'medical tests' as any doctor would know them: instead they were crude assessments of 'racial value' which assigned each youngster to one of four categories. Those who met Himmler's strict criteria for what a child of true German blood should look like were placed in Category 1 or 2: this formally registered them as potentially useful additions to the Reich population. By contrast, any hint or trace of Slavic features – and certainly any sign of 'Jewish heritage'– consigned a child to the lowest racial status of Categories 3 and 4. Thus branded as _Untermensch_ , their value was no more than future slave labour for the Nazi state.
By the following day this rudimentary sifting had finished. Those children deemed racially worthless were handed back to their families. But 430 other youngsters, from young babies to twelve-year-old boys and girls, were taken away by their captors. Marshalled by nurses from the German Red Cross, they were packed into trains and transported across the Yugoslavian border to an _Umsiedlungslager_ – or transit camp – at Frohnleiten, near the Austrian town of Graz.
They did not stay long in this holding centre. By September 1942, a further selection had been made – this time by trained 'race assessors' from one of the myriad organisations established by Himmler to preserve and strengthen the pool of 'good blood'.
Noses were measured and compared to the official ideal length and shape; lips, teeth, hips and genitals were likewise prodded, poked and photographed to sort the genetically precious human wheat from the less-valuable chaff. This finer, more rigorous sieving re-assigned the captives within the four racial categories.
Older children newly listed in Categories 3 or 4 were shipped off to re-education camps across Bavaria in the heartland of Nazi Germany. The best of the younger ones in the top two categories would – in time – be handed over to a secretive project run by the Reichsführer himself. Its name was Lebensborn and among the infants assigned to its care was nine-month-old Erika Matko.
## TWO | YEAR ZERO
_'It is our will that this state shall endure for a thousand years. We are happy to know that the future is ours entirely!''_
ADOLF HITLER: _T RIUMPH OF THE WILL_, 1935
At 2.40 a.m. on Monday, 7 May 1945, in a small red-brick schoolhouse in the French city of Reims, Generaloberst Alfred Jodl, Chief of the German Armed Forces High Command, signed the unconditional surrender of the Thousand Year Reich. The five terse paragraphs of this act of capitulation handed over Germany and all its inhabitants to the mercy of the four victorious Allied powers – Britain, America, France and Russia – from 11.01 p.m. the following night.
A week earlier, Hitler and most of his inner circle had committed suicide in the bowels of the Berlin Führerbunker. Heinrich Himmler – Hitler's chief henchman and the man in charge of the entire Nazi apparatus of terror – was on the run, disguised in the coarse grey serge of an enlisted soldier and equipped with forged papers proclaiming him to be a humble sergeant.
It was over: six years of 'total war' in which my country had murdered and plundered its way across Europe. Now we had to live with the peace.
Who were we then, on that May morning? What was Germany – once the begetter of Bach and Beethoven, Goethe and Schiller – in the aftermath of the brutality of the Blitzkrieg, let alone the Final Solution? What would peace look like to the victors and to the vanquished?
A new term was coined to describe our situation in 1945: _Die Stunde Null_. Literally translated, this means 'zero hour' but for the smouldering remains of Germany – a country of ruins, shame and starvation – it was more accurately 'Year Zero': both an end and a beginning.
What did it mean to be a German from 11.01 p.m. on Tuesday, 8 May 1945? To the Allies – the new owners of every metre of turf and of every individual life from the Mass in the west to the Memel in the east – it meant subjugation, suspicion and suppression. Never again, said the four occupying powers, would the poisonous twin rivers of German nationalism and militarism be allowed to rise up and flood the continent. Within hours there would be mechanisms and procedures in place to enforce this ideal – systems that, though I was too young to know then, would direct the course of my life.
To Germans, this question of identity meant something different. Something much less philosophical, something that could perhaps be categorised as the three Ps: physical, political and psychological. Of these, the greatest – the most pressing – was undoubtedly the physical.
Germany in May 1945 was a wasteland of blown-up bridges, damaged roads, burned-out tanks. In the dying weeks and months of his Reich, consumed by madness and impotent rage, Hitler had issued orders to create 'fortress cities'. The Fatherland was to be defended to the last drop of pure German blood and the last brick of German building. There was to be no surrender but, instead, a _Götterdämmerung_ of flame and sacrifice to mark the final days of his self-proclaimed Master Race.
The result was less a noble funeral pyre than a thousand-mile-wide bonfire of his vanity. Forced to fight for every inch of territory – and bludgeoned by Allied carpet-bombing – Germany was reduced to a post-apocalyptic desert. Piles of rubble lay where buildings had once stood: in Berlin alone there were seventy-five million tons of it piled up along and across almost every street. Other German cities suffered equally, obliterated by bombing and house-to-house fighting that damaged or left derelict seventy per cent of their buildings. And everywhere, now hollow and haggard, a once-proud people who had subjugated those they believed to be inferior.
Newsreels and photos (Allied ones, since the German press had been shut down from the moment of surrender) captured previously unimaginable scenes. Clustered around half-destroyed buildings, blown apart so that the remnants of a once normal life were exposed for all to see – a fireplace, shreds of wallpaper, the remains of a toilet – were the living ghosts of women and children. Orphans, refugees, the aged and the wounded: everywhere a dystopian tableau of anonymous bodies lying dead in the street, watched – or more often avoided – by skeletal figures who might well soon join them.
All of Germany, at least in the cities, was picking through debris, creating makeshift shelters, scrounging for food and either hiding from or fearfully fraternising with the victorious occupying armies. Not from choice, but from necessity.
In the last weeks of the war, the country's economy – so long directed by and for the benefit of the Nazi Party – had collapsed as badly as its buildings. Ironically, there was plenty of money, but coins and paper bills were useless: as every available resource was diverted away from the people to the needs of the army, and as explosions ripped up the railway network, preventing what food was harvested from being distributed, there was little or nothing to buy with the now-useless marks.
Nor did Germany's new masters appear to have a coherent idea of what to do with it. Between July and August 1945, the Allied leaders – Churchill (and, later, Attlee), Truman and Stalin – met at Potsdam to plan the future. Unlike the end of the First World War, when Germany was defeated and subjected to severe punishment and reparations but not wiped from the geographical and political map, the decision was taken that the country would cease to exist once the war ended. In its place would be four separate 'Occupation Zones', each owned and ruled by one of the war's victors, according to its own principles and plans.
Yet beyond that there had been little concerted thinking about what, practically, would be done with the former German state once Hitler had been defeated. France had favoured breaking the Reich into a series of small independent states while America had considered returning Germany to a pre-industrialised nation focused and dependent on farming. Washington would come to relent, to accept that requiring tens of millions of Germans to live as medieval peasants was unworkable as well as undesirable. But the Allies failed to contemplate how their separate occupations would function, or to address the monumental problem of feeding both a conquered people – a population swelled by more than ten million refugees from the east – and the massive armies imposing the peace.
There was simply not enough food – and without a functioning transport system, what little there was couldn't be moved to the places where it was most needed. Worse, there was a widespread feeling among the occupying armies that the Germans were long overdue a taste of their own medicine: had the Nazi rampage across Europe not deliberately starved villages, cities, entire nations to the point of death?
This, then, was Hitler's true legacy: a nation starving to death; a population reduced to a desperate struggle for survival, subsisting at best on half the calories needed to sustain life. A country not simply beaten and half-destroyed but wiped completely out of existence.
I was three and a half when peace came. A small, quiet and archetypally blonde German child, I lived in Bandekow, a tiny hamlet in the rural heart of the Mecklenburg region, with my mother, grandmother and younger brother Dietmar. Our home was a big farmhouse, half-timbered and characteristic of the region, set in acres of forest. We were, I think, typical both of a particular class of pre-war Germans and, by contrast, of the post-war country at large. On both sides our family was old, well established and, notwithstanding the wrecked economy, well off.
My mother, Gisela, was the daughter of a shipping line magnate from Hamburg. The Andersens belonged to the old Hanseatic class – the patrician and prestigious ruling elite which had made its money and its name from trade since Hamburg was declared a free city by the 1815 Congress of Vienna.
Our house in Bandekow had been in my mother's family for generations: it belonged to my great uncle, but had almost certainly been used as a country retreat in the years before 1945. Certainly, the Andersens kept their main residence in Hamburg itself and my grandfather remained there, with my grandmother dividing her time between the two homes.
Gisela was one of four Andersen children. Her brother had been killed, serving in the Wehrmacht in the last days of the war; her eldest sister was estranged – the result of some unspoken act of dishonesty that tarnished the otherwise respectable family name – but her remaining sibling, my Aunt Ingrid (known universally as Erika, or 'Eka'), was a constant companion in my childhood. At the end of the war, Gisela was thirty-one. She was young, bright – in the brittle and privileged way of her class – and pretty. She was also married, though not, as it turned out, happily.
Hermann von Oelhafen was a career soldier. He had served with honour in the First World War: he was seriously injured in 1914, again in 1915, and, after a final wound in 1917, was awarded the Iron Cross for his pains. Like Gisela, he came from an aristocratic background: both his father and mother could boast the tell-tale 'von' – the mark of the upper class – in their family names.
But where Gisela was young and lively, Hermann was the complete opposite. He was thirty years older than Gisela and suffered from severe epileptic seizures. Whether these were the cause of his peevish, mean-spirited nature I do not know: what I am certain of is that their marriage – which took place in 1935, during the first confident years of Hitler's reign – was, by 1945, effectively over. As I grew from a toddler to a young child, I rarely saw my father: we lived in the farmhouse at Bandekow, while Hermann lived 1,000 kilometres away in the Bavarian town of Ansbach.
Perhaps outwardly there was nothing very strange in a married woman living alone with her children and mother. In this our family was typical of the now-dissolved German nation in the immediate months after the war: most adult men – even the very young and the elderly – had been drafted into military service and were now either dead, missing or held in prisoner of war camps across Europe. Germany was a country – more accurately, a former country – of women and children.
But though it played its part, the war was not the prime reason for the separation of my parents. There was an unbridgeable gulf between them; an emotional fracture even less tractable or open to resolution than the divisions imposed upon their nation. I was too young to know it at the time, but it would render my childhood as bleak as the deteriorating political situation in which we found ourselves.
Politics. The second 'P' which defined life at the end of the war. Not politics as modern generations have come to know and disregard it; not the jockeying for position and power between rival parties in a settled democracy: politics in 1945 was truly red in tooth and claw.
The last days of the war had seen the Allied forces smashing their way through Germany from all points of the compass. American tanks and troops rolled eastward from France, Belgium and Holland; the British fought their way northwards, up through the country from Italy and Austria; and the vast armies of the Soviet Union raced westwards from what had, before the war, been Poland. For each there was an overriding imperative to conquer and control as much German territory as possible: whatever they held when the war finally ended would, under the Potsdam Agreement, become their property with little prospect of subsequent redistribution. In those last weeks of spring 1945, the borders of post-war Europe were being claimed and, at the same time, the seeds of the Cold War were being planted.
When the fighting was over, it turned out that my father's home was in the American zone: henceforth his fate would depend on the way Washington saw its duties and rights over the territory it now owned. Bandekow, however, was in the Soviet occupation zone, and Moscow had very different ideas about how to dismantle the infrastructure of Nazi Germany – as well as what it wanted to do with its share of the former Reich.
Initially, at least, there was agreement between the Allies on the need to bring Hitler's surviving henchmen to justice. A four-power war crimes tribunal was established to put the National Socialist machine on trial; Göring, Jodl, Hess, von Ribbentrop and twenty other leaders of the National Socialist state were locked up in cells beneath the Palace of Justice in Nuremberg to await trial for crimes of war and crimes against humanity. Other than Hitler and Goebbels, the most notable absentee from this roll call of infamy was Himmler, creator of the SS and mastermind of the Nazi's apparatus of terror: after being captured he had committed suicide before he could be transported to Nuremberg.
The eventual trial and conviction of almost all these men was undoubtedly a triumph for justice, but it also marked the high point of cooperation between the occupying powers. After Nuremberg, America, Britain, France and the Soviet Union would each take a radically different approach to the land and populations they controlled: the individual fates of tens of millions of former Germans depended on which zone they happened to have been in when the war ended. Very soon these great political divides would change the lives of our little family for ever.
The contrast between the four occupying powers was played out first in the way they viewed Nazi Party members. Denazification was a phrase coined in Washington during the last years of war: President Franklin Roosevelt and his successor, Harry Truman, recognised that the party's tendrils had wound themselves throughout every aspect of German life, from the political to the judicial, the public to the personal. In May 1945 there were more than eight million members of the Nazi Party – around 10 per cent of the total population. What was to be done about this entwining of the mechanics of fascism with the warp and weft of everyday life?
The search for an answer was not confined to America, of course. Each Allied power faced the problem of how to pull out the roots of National Socialism while ensuring that its own zone of occupation kept functioning. The first step was to outlaw the party. On 20 September 1945, Control Council Proclamation No. 2 announced that 'The National Socialist German Workers Party (NSDAP) is completely and finally abolished and declared to be illegal' throughout the former Reich.
But the party itself was only the most visible of a byzantine tangle of Nazi organisations. Beneath it were more than sixty other official associations, ranging from internationally notorious bodies like the SS, Gestapo and Hitler Youth to more obscure societies (even within Germany) such as the Reich Committee for the Protection of German Blood and the Deutsche Frauenschaft, the National Socialist Women's Movement. All were duly made illegal: more importantly, previous association with any one of them would be enough to mark someone as a possible Nazi sympathiser.
Neither Hermann nor Gisela were – to the best of my knowledge – Nazi Party members. I never heard them express fascist opinions or support for Hitler. But their personal histories (my father as a career soldier, who had been a desk officer in the Wehrmacht for much of the war; my mother as a former member of Deutsche Frauenschaft) must have led to some investigation by the denazification officials of their respective Occupation Zones.
The Americans were initially fiercely committed to denazification, but quickly became the most pragmatic of the occupying armies. Washington's military government realised that, however desirable, widespread purges of suspected Nazis would mean that the entire responsibility for organising day-to-day life fell exclusively on its shoulders – a burden that, for a war-weary nation anxious to bring its troops home, was simply too onerous.
And so while my father, like every adult living in the American zone, was required to fill out a questionnaire (termed variously a _Fragebogen_ or a _Meldebogen_ ) in which he affirmed that he had never been a member of any Nazi organisation, there was little follow-up or detailed examination of these self-declarations. With little or no oversight, most applicants were issued with official documents pronouncing them to be 'good Germans', free of the stain of fascism. They quickly became known as _Persilschein_ – pieces of paper that were able to wash the past as clean as any soap powder.
The Soviet approach was very different. Perhaps because it had suffered greater losses and devastation than any of the four Allied powers – or, more likely, because Stalin had clear plans for the future of the Soviet zone – Moscow adopted a much less relaxed approach.
The Soviet Military Administration in Germany – known by its acronym, SMAD – controlled a vast swathe of territory from the Oder river in the east to the Elbe in the west. On April 18, 1945, Lavrenti Beria, Stalin's much-feared head of secret police, issued order Number 00315: it mandated the immediate internment of active Nazis and senior members of Party organisations. No investigations were required prior to these arrests. Ultimately, 123,000 Germans were rounded up and incarcerated in ten special camps set up across the Soviet zone.
The existence of these prisons – run by the NKVD, Stalin's equivalent of the Gestapo, and frequently on the site of former Nazi concentration camps – was in itself a secret. No contact was allowed between prisoners and the outside world, but inevitably word did leak out: the often random nature of arrests and internment (by February 1946, genuine Nazi Party members formed less than half of the total number of prisoners), and fear of being dragged off to the network of _Schweigelager_ (literally, 'Silence Camps') weighed heavily on an already fearful German population under Soviet military rule.
Almost anything – anonymous denunciation, previous membership of an obscure Nazi society or contact with anyone in the other three Occupation Zones – was enough to earn a knock on the door and transport to a _Schweigelager_. All too often this proved to be a one-way ticket: almost 43,000 men and women would die behind the barbed wire of these post-war concentration camps.
Did my mother worry about the risk that her involvement with Deutsche Frauenschaft posed to our household in Bandekow? I do not know: the von Oelhafens were a close-lipped family, rarely given to discussion of emotions, much less those of the past. It would be many years before I discovered the secret at the heart of my childhood, a secret that tied Gisela, Hermann and me to a sinister Nazi organisation, one which would certainly have spelled trouble for us if SMAD came to hear of it.
Was this an added worry, clouding my mother's mind? Again, I do not know. What I do know is that as summer turned to winter, Gisela was terrified of something else: rape.
Throughout 1945, as the Soviet Army fought its way into Germany, its troops mastered one phrase above all: _Komm, Frau_. It was an order that brooked no disobedience and led to the same inevitable conclusion. Tens of thousands – perhaps ten times that number – of German women paid, with their bodies, the price for Hitler's brutal treatment of Russian cities and populations. Rape was so commonplace in the Soviet sector that the question for many women, of all ages, was not _whether_ they had been violated but _how many_ times.
It was also quasi-officially sanctioned. Although SMAD commanders in some parts of the Occupied Zone paid lip service to stamping out the violation of German women, in reality others paid a heavy price for doing so. One young Red Army captain, Lev Kopelev, intervened to stop the gang rape of a group of girls and was sentenced for his troubles to ten years in a labour camp: a tribunal convicted him of the crime of 'bourgeois humanism'.
It was, of course, true that neither the internment camps nor rape were confined to the Soviet sector. The Americans imprisoned thousands of suspected Nazis, often in appalling conditions for years, and French troops frequently ravaged German women in cities under their control. But in the final months of the war, Hitler and Goebbels had fanned the flames of national fear by issuing a constant stream of propaganda about the brutality of the Red Army – and from the moment they fought their way onto German soil, the Soviet occupiers fulfilled the worst of these predictions.
Our family was as vulnerable as any, if not more so. My mother and my Aunt Eka were young and pretty and we came from the hated bourgeoisie: our home was large, comfortable and well-stocked with food from the farm, but it was also isolated, and my brother was the only man in the household. The fear of rape hung over us as the winter wore on. My mother would later remember – one of only a sparse handful of personal feelings she ever shared with me – hiding under the bed whenever she heard rumours that Red Army soldiers were in the area.
But however debilitating the fear, in truth we were better off than most of the population in the Soviet Zone. We had a roof over our heads, unlike the vast majority of people in the bombed-out cities. The winter of 1946–7 was one of the harshest in living memory: temperatures plummeted to -30° and for the millions struggling to exist in the bombed-out basements of their former homes there was no protection from the biting cold. And since what remained of the rail network after the final, disastrous months of fighting was rapidly dismantled by the Soviet Army and taken back east as war reparation, there was little coal to be had: thousands of people simply froze to death.
But it was food – or rather, the lack of it – that soon became the overriding preoccupation. German ration cards were no longer valid: whatever limited provisions had previously been available were now being claimed by SMAD to feed the Red Army. In cities across the country, hunger joined fear as the measure of existence.
In the areas under Moscow's control, new rationing measures were introduced. The Russians created a new five-tier system: the highest level was reserved, bizarrely, for intellectuals and artists; the next level down was assigned to the women – _Trümmerfrauen,_ as they were called _–_ who worked in chain gangs, tearing down and clearing semi-derelict buildings, often with nothing more than their bare hands. This was much more valuable than the official wages of 12 Reichsmarks they received for cleaning up every thousand bricks. Hard physical labour was the only way to survive and, in the ruins of the nation, Germany's women dug for the salvation of their families.
The levels of rationing below this fell incrementally and dramatically. The lowest card, nicknamed the _Friedhofskarte_ (literally meaning 'cemetery ticket'), was issued to those who performed no useful function in the eyes of our Soviet masters: housewives who did no work and the elderly.
Two new words joined the lexicon of post-war lives that winter. The first was _Fringsen_ : it emerged after the Catholic cardinal of Cologne, Josef Frings, gave formal blessing to what many of his flock were already doing – stealing in order to survive. Crime rose dramatically: in addition to the uncountable tally of thefts and rapes by Red Army soldiers, Germans under Soviet occupation began preying on each other. Berlin alone averaged 240 robberies and five murders every day. Urban crime may not have been a pressing concern for the von Oelhafens, living in the relative security of rural Mecklenburg, but the second new word had a very real meaning. _Hamstern_ meant, quite literally, 'to hamster': in practice, it was a constant procession of city dwellers to and from the countryside, desperate to trade their few remaining possessions for the food we had in relative abundance.
This was the reality of _Stunde Null_ : an existence defined by three constant companions: fear – especially of the Red Army and of its determination to exact revenge on German civilians for Hitler's war – hunger and cold. This was Germany, my country and my life on my fourth birthday. This was the legacy of the glorious Reich. And there was worse in store. Throughout 1946, as relations between the occupying powers worsened, Moscow's intentions towards those under its rule in SMAD grew starker. As well as stripping the zone of wealth and food, it began the process of removing the one, flickering hope we had enjoyed when the war ended: freedom.
The boundaries between the four zones were becoming ever less passable. An 'Inner German Border', as SMAD termed it, had been established around the Soviet-held territory in July 1945, but since then it had been only sporadically policed. Although anyone wanting to move between the Soviet sector and the other Allied Occupation Zones officially needed an _Interzonenpass,_ at least one and a half million Germans had managed to flee into the American or British zones. Now that began to change.
In the summer of 1947, preparations were underway for the eventual transformation of SMAD into the new communist-ruled German Democratic Republic. New contingents of Soviet soldiers were assigned to the official border checkpoints. Unofficial crossing places would soon be blocked by newly dug ditches and barbed wire barricades. The Cold War was beginning and we were living on the wrong side of the coming Iron Curtain. In the summer of 1947 my parents – separated both physically and emotionally – made a remarkable joint decision. It was time to escape.
## THREE | ESCAPE
_'Ingrid is very brave and overcomes the strenuous walk without complaining.'_
GISELA VON OELHAFEN'S DIARY, JUNE 1947
My mother kept a journal. Unknown to me – she never told me about it, even when I was an adult – she jotted down the barest of details of my early years across a sparse handful of pages. This slim, black leather-bound notebook contains all I know about my first eight years of life.
It begins with a small black and white photo of me, three years old, barefoot and wearing shorts, captioned: 'Bandekow – Ingrid, Summer 1944'. Over the page is an envelope dated June 4, 1944 and containing – according to my mother's note – a few strands of my hair. If that seems fairly conventional, the sort of journal any loving mother might keep as a record of her daughter's childhood, the rest of the content fails to match that impression. There are very few entries – no more than four or five for each of the five years my mother wrote in it. And the nature of the inscriptions themselves are curious: they are all in the third person. Gisela refers to herself as ' _Mutti_ ' (the German colloquial word for Mummy) throughout, never 'I'.
This sort of third-person journal was, I have since gathered, not uncommon in Germany during the later war years – perhaps she did it to make it easier for her children to read later. But given that she never told us about it, that benign explanation seems unlikely. Her curiously detached writing style seems rather to emphasise the difficulty my mother found in conforming to a maternal stereotype, and the distance I always felt between us.
Still, the journal does give me some idea of what sort of child I was. The entry for my birthday on 11 November 1944 reads:
Today Ingrid is three years old. At her age she is not tall, but she thrives and prospers wonderfully and has a healthy constitution. She has a strong will and a disposition to violent temper. Her character is calm and persistent. She does not pay attention to people she does not know: at that point her little ego takes centre stage and claims a little too much of her small world.
The next item, dated a month afterwards, hints at a desire to win my mother's affections. For reasons undisclosed, I was left alone with Dietmar at lunchtime; when my mother returned she noted that:
Ingrid – with a serious face – was very busy feeding her brother just as Mummy always does.
If the diary is any guide, I appear not to have succeeded. Throughout the twelve months of 1945, my mother only managed to put pen to paper on five occasions: two to record the effect of measles on me, one noting the happy news that I was no longer afraid of our family dog, and two more in which my slowness to speak ('She doesn't make complete sentences, her maximum is three or four words') is observed and preserved. There was, as my mother must have known perfectly well, a very good reason why I might be slow to get my childish tongue around German words. But there is no mention of this in the pages of the notebook.
Nor does she dwell on what must have been a traumatic period in my life the following year. In the summer, my mother laconically recorded that I (and presumably Dietmar) had been sent away to a children's home, more than 250 kilometres away at Lobetal, near Berlin. How did we get there? I do not know: her diary is silent on this, as on so much else. All that she wrote was that:
Mummy is ill – meanwhile Ingrid lives from 5 August to 1 November in a children's home in Lobetal. There she suffers from mumps – but not too badly.
My mother's illness, I learned decades later, was in fact a nervous breakdown. Perhaps it was the collapse of her marriage and the burden of looking after two small children. Perhaps it was the strain of living under Soviet occupation; the constant fear of arrest or – worse – rape by the Red Army. The first entry in the notebook for 1947 shows that she had made up her mind to escape – and that she had enlisted my father, though they were still estranged, into her dangerous plans.
1 May 1947. Daddy takes both children to the children's home in Lobetal. Mummy wants to cross the border illegally.
I cannot pretend that I was ever close to my mother, nor can I claim to have ever really felt from her the sort of love that a child should take for granted from a parent. Gisela also plainly knew this; another terse diary entry in my mother's handwriting noted that I was always much fonder of my grandmother. 'Granny is loved over all others, often more than Mummy. She gets on with the children very well.' But even so, I have to acknowledge that the decision to make a bid for freedom was immensely brave.
The border between what would, less than two years later, become the German Democratic Republic and the British zone of occupied Germany was both political and physical. It was, of course, forbidden to leave the Soviet zone without a special permit, and these were far from easy to obtain. Even writing the idea of crossing illegally in her diary could – had it been discovered – have led to interrogation, imprisonment in the Silence Camps, or worse.
In addition to this, the journey to the border was as arduous and complicated as it was dangerous. Bandekow might have been less than fifteen kilometres as the crow flies from the Elbe river, which marked out much of the boundary with the British zone, but there was no way to cross it. My mother had already made a secret trial run and must have discovered that the nearest bridges at Lauenburg and Dömitz had been blown up by the retreating German army in 1945. The nearest bridge left intact was 150 kilometres further south at Magdeburg.
With the country's railways still in chaos and with private cars (let alone the petrol to run them) a rarity, getting to Magdeburg would have been a challenge for a healthy adult, travelling alone. My mother was far from well – and she would have two very young children to drag with her every step of the way: it must have been a daunting prospect. Encumbered by Dietmar and me, she could not carry anything with her: the three of us would make the trek in whatever clothes we had and – if successful – would arrive in the safety of British territory with nothing more than the clothes we stood up in.
The near-impossibility of getting simply from one place to another in 1947 is clear from the convoluted escape route my mother carefully wrote down in the diary. Tracing it now on a map, the first part of the journey took us east, not west: ever deeper into the Soviet zone and away from the sanctuary we sought. We began on 30 June, travelling – I think – by horse and cart, twenty-five kilometres to the little town of Lübtheen. Here, my mother found a hotel to put us up for the night and in which we could await the arrival of her co-conspirator the next morning.
I have no idea how my father managed to get papers to cross from the American zone into Soviet-controlled territory, nor how he obtained the car into which the four of us crowded that morning. All I know is that the thirty-kilometre journey, eastwards to the city of Ludwigslust, was the last time our family was together.
The reason for heading so far east was waiting for us in the station at Ludwigslust. Both the platform and the train which would take us back westwards to Magdeburg must have been very crowded. That summer more than ten million refugees and released prisoners of war were on the move: like us, many were desperately trying to find a route out of the Soviet zone. Somehow – I was never told how – we had precious train tickets: my mother's diary records only that the train was so crowded my father had to push the two of us into her arms through the train window. She makes no mention of the fact that he did not make the journey with us, staying behind on the platform to (so I like to imagine) wave farewell to his wife and children.
Magdeburg was more than 150 kilometres south, and the train journey took all day. When we finally arrived, it was evening and we must have been both tired and hungry. Finding food could not have been an easy task: Magdeburg had been heavily bombed in 1945 and by the time we arrived it was still a city of ruins and ruination. And although it was in the Soviet zone, our Soviet-issued ration cards were not valid there. Alone with two small children in a strange and devastated city, my mother took the only available option: she found a black-market trader and handed over sixty marks for a few pieces of bread.
There is no information in my mother's diary about where we stayed that night. In the chaos that was Magdeburg, it seems unlikely that we would have found a hotel: it says simply that we stayed in the city all the next day, changing accommodation in the evening to be closer to the next stage of our path to freedom.
We had, first, to get a train out of Magdeburg, heading north to the village of Gehrendorf. Here the little river Aller was the boundary between east and west. On the other side was the hamlet of Bahrdorf, safely inside the British zone. All that lay between us and sanctuary were the slow-running waters of the Aller. But there were no boats and no bridges: the only way across was to wade. So that was what we did.
It cannot have taken too long, for the Aller is small at the best of times, and in the height of summer would have been reasonably shallow. Nonetheless it must have been challenging for a fraught young woman with two young children in the heat of a midsummer's day. She must have been scared, hoping not to be seen by Red Army border guards and praying that neither Dietmar nor I would cry out and give our position away. The only record I have is what my mother later set down in her notebook:
The temperature is very high. Ingrid is very brave and overcomes the strenuous walk without complaining.
Finally we reached the sanctuary of the other side. We crawled up the bank and after a long trek through No Man's Land, we reached the British zone. We were free.
My mother could not have known it – though, given the urgency and determination inherent in her succinct account of our flight from east to west, she must have sensed the Iron Curtain beginning to fall – but we had made our getaway just in time. By September 1947, the borders between the Soviet Zone and those of its former western allies were closely guarded by a new influx of NKVD troops; it was not long before orders were given to shoot would-be escapers on sight.
What did freedom look like to Gisela von Oelhafen that summer's evening? What did it mean to her to have reached safety after two years under Soviet occupation, and to have escaped with her children from Moscow's iron rule? I wish I could ask her now.
A day's hard travelling later, we arrived in Wunstorf, a little town just west of Hanover and the penultimate stop on my mother's journey to her family home in Hamburg. I say my mother's journey quite deliberately because she would make the last leg of her trek alone. Her diary entry – terse as ever – recorded the very different fate allocated to Dietmar and me: '4 July: To Loccum, children's home.'
She had taken us out of the Soviet zone and into the less dangerous territory of the British sector. But that is as far as her maternal protection extended. No sooner had she spirited us away to safety, than she sent us away. My second night of freedom ended in the surroundings of a home for unwanted children.
I would spend the next six years, lonely and isolated, in the care of the church. In fact, my new life began exactly as the old life had ended: cold and frightened.
## FOUR | HOME
_'Dear Mummy, please take me home for ever.
I'm longing for you and Granny and Aunt Eka.'_
LETTER TO MY MOTHER
FROM THE CHILDREN'S HOME
My first real memory is an orange. I have snatches of other, possibly earlier, recollections – lying, cold, under a blanket on the floor of a train; a line of camp beds in a long room and a rat running over my feet – but the first actual memory I know to be true is of the orange. I am at a long wooden dining table in a big room. There are a lot of people, grown-ups and children. I know that many of the adults are homeless men and women who have been invited here for the day; the children, though, live in this building. Each of us is given a plate with fruit on it, including a single orange as a special treat.
I know where and when this memory comes from. The year was 1947 and I was almost six years old. The room with the long table was in the children's home to which Dietmar and I were dispatched. It was Christmas Day.
The home was run by the Protestant church and was called Nothelfer, which, literally translated, means 'help us in affliction'. There were sixty-five boys and girls living there, all under the age of ten. Some were displaced persons – children who had lost their parents during the war or in the chaotic mass migrations of the immediate post-war months. Dietmar and I were different: we had two living parents who knew where we were but who, for reasons best known to themselves, had sent us to be cared for by others.
We were physically as well as psychologically isolated. Nothelfer was on Langeoog, a small island in the North Sea ten kilometres from the coast of mainland Germany and 200 kilometres from Hamburg. To be fair to my parents, I don't think they had intended to send us so far away: when we first arrived in July, it had been situated near Hanover. But at some point in the subsequent five months those premises were closed and we were moved up to Langeoog.
Given its location, it was hardly surprising that Nothelfer was cold. I can still feel the wind whipping up sand from the island's long beach, seemingly stripping the skin from my legs and arms.
The home was staffed by sisters from the religious order and at times the regime could be harsh: physical punishment was part of our daily routine. If we were disobedient, if we wet the bed, if we broke the cardinal rule forbidding us to slide down the sand dunes, we were spanked. One by one we had to line up and pull our pants down and one of the sisters would beat our bare bottoms with a stick.
We would stay here for four years. From time to time our parents made the journey out to the island to visit us. Their visits were rare and they never came together, always alone. My father had moved from the American Occupied Zone into the British sector, and was building himself a new house in the Westphalian spa resort of Bad Salzuflen. Although he and my mother were separated, they had not divorced (and would never do so). Occasionally they spent a little time together – generally when Dietmar and I were allowed out to visit one of them – but my mother had begun to make a new life for herself in Hamburg.
Immediately after sending us to Nothelfer, she had returned to her family's home in the city, a large three-storey building in an exclusive neighbourhood. Number 39 Blumenstrasse had three floors and a basement, and large gardens that ran down to Rondeelteich – one of the big lakes in the heart of Hamburg where people went boating, sailing or swimming.
There she lived with her mother, my Aunt Eka, and a housekeeper-cum-cook. Initially, they also shared the house with British Army officers who had been billeted there at the end of the war. Their presence was, ostensibly, the reason why Dietmar and I had been bundled off to the children's home: according to my mother there was not enough room for us all.
Whether or not this was actually true (since the position didn't change when the soldiers moved out), it did enable her to start a new life, unencumbered by either husband or two small children. She enrolled at college and began training to become a physiotherapist: once qualified, she turned one of the ground-floor rooms into a surgery where she worked with a growing clientele of patients.
She also took advantage of her essentially single status to find a boyfriend. Neither Dietmar nor I would ever meet this man, but within two years she gave birth to a baby boy, whom she named Hubertus. He was not, I am certain, my father's child, but he was formally registered as a von Oelhafen.
I cannot, in all honesty, say that my father's visits to Langeoog meant a great deal to us. Whether this was because of his age or his stiff and disciplinarian nature I do not know: what I do recall with awful clarity is the heartache of being apart from my mother, and the loss I felt after her occasional trips to see us. I missed her terribly.
Among her effects, years later, I found a note from one of the kinder sisters who ran the home.
Highly honored madam,
I'd like to add a few sentences to the letter Ingrid has written to you. For the past few weeks I have been worrying about Ingrid. She yearns intensely for her 'Mummy'. Every day she talks about 'Mummy' or asks: do you think I'm allowed once to go to Mummy? I'd like so much to see her. Aunt Emi, do you think that I would be allowed to leave the island and stay for a short time with Mummy?
Ingrid is eating very little and feels miserable. In my opinion, the reason for this calamity is her longing for Mummy.
In school Ingrid is one of the best. She is hard-working. In general she is a nice child.
I feel committed to inform you about this.
Best wishes from Schwester [Sister] Emi
Did she ever reply to this letter, or to any of the letters I wrote to her? I don't recall: and yet she kept at least some of them. Along with Sister Emi's letter, I found this undated note, scrawled in my childish hand after an all-too-brief visit to see her.
Dear Mummy,
Thanks a lot for the parcel. I am only writing a little today. Dear Mummy, please take me home for ever. I'm longing for you and Grandmother and Aunt Eka. I always cry if someone talks about you, or if I think of you. On the trip [back to Langeoog] I couldn't eat. I still have the chocolate and the two Deutschmarks.
Please write as fast as possible to the authorities and say that I'm allowed to leave the island soon. Christa [another girl in the home] will leave the island this month or next. But I want to leave this month!
Dietmar told me he has got a lot of fruits and sweets. Dietmar teases me a lot and asks: why didn't you stay in Hamburg? I want so much to leave the island. Dear Mummy, please arrange that I can leave. Christa told me, she had cried too when she had to go away from her parents.
Many greetings and kisses from Ingrid. Don't write to Daddy about me having written to you. Dear, dear Mummy, please come and pick me up.
She never did. And even the sporadic, terse entries in the little journal she had – apparently – kept for me ceased abruptly in the summer of 1949.
I was ten years old when I finally left Langeoog for good in 1952. I had passed the exams to go to _Mittelschule_ , the intermediary level of education between elementary and senior schools, and had high hopes that at last I would be allowed to live with my mother in Hamburg. Instead my father sent for my brother and me: we were to share his new home in Bad Salzuflen.
By this point Hermann von Oelhafen was sixty-eight years old, bitter at the loss of his wife, suffering from poor health and utterly ill-equipped to look after young children he barely knew. Of the ten years of my life and nine years of Dietmar's, he had lived with us for a matter of months at most. It was surely a little too late to start being a father.
I believe I know now why Hermann ordered us to Bad Salzuflen. I think my father still loved Gisela and he hoped our presence there might somehow draw her back to him; that despite her evident love affair with another man – and their child – Dietmar and I would be the glue that mended their broken marriage.
In this, as in much else, he was to be disappointed. My mother came occasionally to visit us (a spare bedroom in the smart, if far from grand, house on Akazienstrasse was maintained for her exclusive use), but there was never any question of reconciliation.
From the moment we arrived, life in Bad Salzuflen was horrible. Even as a young child, Dietmar was spirited and difficult, though not particularly naughty. Today I think he might have been diagnosed with ADHD: certainly he and Hermann fought a battle of wills. He was routinely late coming home from school – even though he never seemed to want to learn there – and Hermann, who was hot-tempered at the best of times, had no understanding or patience for this irritating little boy. Very soon he began to beat Dietmar.
It was frightening to watch: on one occasion he physically threw him across the room. And yet somehow Dietmar wasn't frightened of him. I, by contrast, was terrified: even though my father never hit me, I lived in fear of his temper. I began to rely on Dietmar to ask for Hermann's permission to do even the smallest things. One day we wanted to go swimming but I didn't dare ask. Dietmar went straight away in my place, and permission was granted, but it didn't change anything for me. I was still too scared to speak to my father. And then Dietmar was taken away.
Someone – presumably the children's welfare department of the local government – decided that since Hermann did not live with his wife and that there was therefore no mother in our house, Dietmar could no longer live with us.
There were a number of odd aspects to this decision. For a start it only applied to Dietmar: although I was less than a year older than him, the authorities did not appear to believe I had to be removed from my father's care. Officially the reason for this discrepancy was that I was to be looked after by the middle-aged couple who lived with us as Hermann's cook-cum-housekeeper and general help (the Hartes). But no one explained why the paid-for care of Emmi and Karl Harte was good enough for me yet insufficient for Dietmar.
More puzzling still was the revelation that Dietmar had family – a completely different family from ours – living in Munich. Had I been older I would, of course, have realised that the nine months between his birthday and mine meant that we were unlikely to have had the same mother. But even if I'd understood that, I would never have anticipated the truth. It was a complete shock to discover – aged ten – that the boy I had always known as my brother had in fact been fostered by my parents.
And so it transpired that Dietmar had an uncle, an aunt and a sister – blood relatives all – who had, presumably, been looking for him. I don't remember anyone explaining how he had originally come to live with us instead of them: one day he was simply taken away from Bad Salzuflen. As it turned out, he never rejoined his long-lost family: instead – for reasons I have never understood – they placed him in the care of another children's home.
I missed Dietmar very badly. And I was now on my own (save for the Hartes) in my father's house and, if my letters to Gisela are anything to go by, absolutely terrified.
22th June 1952
Dear Mummy!
Please send me some envelopes and stamps. Dear Mummy, please pick me up this week and bring me to Hamburg, I'm not able to stay longer with Daddy. I must tell you that my fear of him is greater than before. He told me off once because I've cried about you. Now I cry every day.
Dear Mummy, please pick me up immediately, I cannot stand it here with Daddy any longer. Or come and stay here for ever. But 'Uncle Harte' says you are afraid of Daddy too. Dear Mummy, perhaps you can send the stamps to 'Uncle Harte', but don't tell Daddy that I've written to you.
Mummy, we can organise it this way: You come and pick me up forever and explain to Daddy you had written to Munich [to the child welfare department] to ask whether they would allow me to stay with you. You [explain that you] didn't bring the letter with you, but promise to send it when we get back to Hamburg.
And in Hamburg we'll write a letter on typewriter and send it to Daddy and pretend it has come from Munich. I'd like so much to come to you just this week, please pick me up quickly. Today I cried again because I have been thinking of you. I'm not in the mood to play because you are not here. Please pick me up at the 25th of June. Greetings and kisses from Ingrid. Please pick me up the 25th. Please, please dear Mummy.
These pleas went unheeded. Although my mother continued to come for short visits, she never took me home with her. Whether this was because my father forbade it or because she didn't want me to live with her, the outcome was the same: I was effectively imprisoned in the house in Bad Salzuflen with an increasingly bitter and parsimonious old man.
Even as an eleven-year-old, I was aware that my father was less well off than my mother. He received a state pension from the new West German government; reward for his years serving both the Kaiser and the Reich as an officer in the army _._ But this did not, for example, seem to be sufficient to pay the price of a daily newspaper to be delivered. Instead he would make the journey into town where he would stand outside the newspaper offices and read the morning's edition, which was always pasted up in the window. Occasionally I was permitted to go with him.
Hermann's health was deteriorating. He had suffered from epilepsy for many years (a condition he had apparently concealed from Gisela when asking her to marry him). Now it became increasingly severe. Although I never saw him have a full-blown grand mal fit, when a seizure took him he became 'absent' – completely lost within himself. There was no way to communicate with him, and his behaviour was strange and frightening. Often he picked up a knife and waved it around wildly. Once he was hospitalised and while he was away, Frau Harte was more generous than he was with the marmalade for my breakfast. When Hermann came home again he saw how much had been used from the jar and became angry. I was punished for my evident greed by being denied marmalade for a week.
School became my refuge. I had made friends with other children and was fortunate that their parents, perhaps seeing how unhappy I was at home, were kind and loving towards me. I loved spending time with them, seeing in their lives the thing that I valued – and missed – most: a real family. And then, when I was eleven, I discovered that I was not who I thought I was.
I woke up one morning and found that I couldn't open my eyes. My father took me to the doctor's surgery.
We sat in the reception area, waiting for my turn. When the doctor called out the name 'Erika Matko', my father stood up and led me into the consulting room. He handed over my health insurance card and I saw that it too had the name 'Erika Matko' printed on it.
I had no idea why I was being called by a different name. But I didn't dare say anything to the doctor or my father; I was still too frightened of him to question anything. At the end of the consultation I was prescribed a course of sun lamps – a common enough treatment in those days for vitamin deficiency (most likely a problem dating back to my years in the children's home at Langeoog) – and we went back home. Nothing was said about that different name, but I had not forgotten.
Shortly after that I had a conversation with Frau Harte. Every Friday it was our routine to clean the house together, and I could talk freely to her about whatever I had on my mind. It was the closest thing I had to a normal relationship with an adult. As we were polishing, I asked her if she knew why my name was written down as Erika Matko.
Emmi told me that Hermann and Gisela were not my biological parents. She said that when I was a baby they had fostered me, just like Dietmar, and that my original name was Erika Matko. Emmi wasn't embarrassed to tell me that I had been fostered. The war had fractured so many families and left so many children without parents that our situation was far from unusual.
I don't recall being upset at discovering the truth about myself. I was not close to Hermann and I think that I processed the information by deciding that it explained his coldness towards me, and why I was not allowed to live with Gisela.
But of course I wondered where I had come from. I assumed that my real parents were German – it never occurred to me to think otherwise – and I speculated about what had happened to them. Perhaps they had been in prison; maybe they had died in the war. Emmi said she had wondered whether I was originally Jewish because of my prominent nose. But although my father had told her that I was a foster child, she didn't know any more than that. Everything else was just speculation.
Of course I never said anything to Hermann. Nor, the next time Gisela came to visit, did I ask her about it. But Hermann must have told her about the visit to the doctor's surgery and I assume she felt she had to say something. She started to tell me that I was fostered and how she had fetched me from a children's home but I quickly cut her off, saying 'I know.' I don't really know why I stopped her: perhaps it was my childish way of showing her that it was all too late, that I was hurt that she had kept the truth from me. The subject was never mentioned again.
The one person I would have liked to talk to was Dietmar. We had been close in the children's home and in the few months we had spent together in Hermann's house. But by then he had been taken away and I had no way to contact him: I didn't even have an address that I could write to.
Life carried on as before. Every morning I went to school – where I was registered and known as Ingrid von Oelhafen – and returned in the afternoon to the house in Bad Salzuflen and the man I now knew was not my biological father, of whom I was still very afraid.
Over the next two years, Hermann's heath continued to deteriorate and he was often still in bed when I left for school. I would go into his room to wish him good morning, but in truth this was no more than living up to my duties as a good daughter.
Then, one morning in April 1954, towards the end of the spring term and with the long summer holidays approaching, I said goodbye to him as usual. I noticed that he seemed a little disorientated when I left, but I didn't say anything to the Hartes because I assumed it was just another symptom of his illness. When I came back from school he was in a very bad way: it was clear he had had a stroke. My father – or rather my foster father, as I now knew him to be – was taken to hospital and died two weeks later.
I have to admit that I was not sad. I felt happy to be free of him and his harsh, unforgiving ways. And I assumed that at long last I would be allowed to live with Gisela in Hamburg. What did hurt me was Emmi and Karl's reaction: they criticised me severely for not telling them about Hermann's condition that morning.
My high hopes for a new life with my mother – I still thought of her as 'Mummy' then, even though I knew I wasn't her 'real' child – were not to be: or not immediately, at least. Gisela was too busy with her thriving physiotherapy practice and her five-year-old son, Hubertus.
For six long months I carried on living in Hermann's home with the Hartes looking after me. It was not until nearly October 1954 that I was finally sent to Hamburg. And by then, the strange story of Erika Matko and my true identity seemed to have been forgotten.
## FIVE | IDENTITY
_'The lost identity of individual children is the social problem of the day on the continent of Europe.'_
INTERNATIONAL REFUGEE ORGANISATION INTERNAL MEMORANDUM, MAY 1949
When I was fifteen years old I saw my face on a poster in the street. A decade after the end of the war, and seven years after the formation of our new Federal Republic, Germany was still a nation of displaced and unclaimed children. United Nations agencies had spent those years searching across Europe for close to two million missing boys and girls, separated from their parents by bombings, military service, evacuation, deportation, forced labour, ethnic cleansing, or murder. By 1956, it had traced just 343,000 of them.
The Red Cross had decided that one way to discover the origins of children who may have been brought to Germany during the war was to post photos of the children as they were then in newspaper advertising columns. Underneath these lists of faces and names ran the headline: _'Who knows our parents and our origins?'_ They also pasted up large posters on columns and lamp posts on streets across West Germany. It was from one of these, in the centre of Hamburg, that my younger face peered back at me.
It was, to say the least, a shock. I had no idea that anyone was looking for me, nor how they would have obtained my photograph. I had to presume that Gisela had given it to the authorities, but no one had said anything about it to me.
By that point I had been living in my mother's house on Blumenstrasse in Hamburg for two years. Two years during which my dreams of a happy family life had proved to be no more than an unrealistic and childish fantasy. I had spent half of my young life longing to be with my mother, aching to feel loved and looked after. By the time I saw my photograph on the poster, reality had set in – and set me in my ways.
I knew, of course, that Gisela was not my real mother, but I still had no idea when – much less how or why – she and Hermann had taken me in, and I had pushed the whole business to the back of my mind. I wanted so much to cling to the belief that I belonged to Gisela and her family.
What I couldn't hide from, though, was the way Gisela treated me. She was not cruel; I could never call her that. But she was noticeably cold – emotionally and physically – towards me. This was in stark contrast to her other relationships. Professionally, she was an extremely successful physiotherapist: her patients clearly loved her, and she returned their affection.
With her own relatives, too, she was warm: to her mother and her sister (Aunt Eka, to whom I increasingly turned for love and understanding), and to her son. Hubertus was eight years younger than me; a very handsome boy, who – unlike me at his age – could speak well and fluently. It might have been easy not to like him: he was, after all, Gisela's natural child, and had been living in the house in Hamburg before I was allowed to go there. But although I resented the fact that Gisela seemed able to show love to almost anyone but me, I had come to care very much for Hubertus and we had a strong bond between us.
But this was a rare glimpse of light. Teenage years are always difficult, especially, I think, for girls. Those crucial years between thirteen and fifteen are generally a time of uncertainty and insecurity, and a time when it is all too easy to be critical of adults. But in Germany in 1956, that biological confusion was exacerbated by national crisis.
The Nazis and the war had broken the previously close bonds of German family life just as surely as the bombs and tanks had destroyed the country's houses, bridges and railways. In addition to creating a huge population of orphans, Hitler's desperate last-ditch battles had blurred the lines between childhood and adult life by throwing young boys into the doomed fighting.
In the immediate post-war years, an army of international psychologists and social workers was drafted in to address the problems for Germany's next generation. The men and women of the United Nations Refugee Relief Organisation (UNRRA) and its successor, the International Refugee Organisation (IRO), recognised that many teenagers in the late 1940s and early 1950s were growing up without the emotional security they needed – both individually and as part of an emerging new nation. An internal IRO memorandum in May 1949 highlighted the crisis in stark terms: _'_ The lost identity of individual children is _the_ social problem of the day...'
And so, while the American Secretary of State, George Marshall, put in place a vast economic aid plan to rebuild Germany's shattered infrastructure and economy (and the rest of Europe), UNRRA and IRO set to work on what they termed a 'psychological Marshall Plan' for its children.
First they had to identify us. Along with the posters, radio announcements instructed those fostering children from other countries to report to their local youth administration office.
How did this affect us? I could not have told you then what Gisela did: it would be decades before I learned that she met with the investigators without telling me. But when I came face to face with my photo on the poster, I had conflicting emotions.
Of course, I wondered who my real parents were. Perhaps my father had been – like Hermann – an officer in the Wehrmacht _,_ who went away to war, leaving me with a mother who either didn't want me, or could not cope alone with a baby. Those were my rational thoughts. But behind them were the sharp pangs of fear and hope. Hope that my biological mother would see the posters and suddenly turn up to say that she now wanted me. Fear, because if she ever did I was worried what sort of person this woman would turn out to be. Perhaps she would be worse than Gisela; maybe she wouldn't even like me?
But these were only flickering emotions and in the end I found it was easier to snuff them out than to dwell on them. Even though I wasn't happy and I knew that the von Oelhafens and the Andersens were not my blood relatives, I clung to the belief that in some way I belonged to them.
Does it sound odd that the mystery of who I was and where I had come from was never discussed? Perhaps. At the time it was simple: I did not have enough of a relationship with Gisela to ask her difficult questions. It would be a long time before I understood that she might have had good reasons for wanting to leave the past alone.
Whatever the reason, the subject was never broached: to all intents and purposes I was Ingrid von Oelhafen, the name under which I was registered at school. I did not have one of the new identity cards, issued by the Federal government from 1951 onwards, but since I was a child no one thought I would need one until I reached the legal age of majority: twenty-one in those days.
As it turned out, the problem of my identity surfaced rather sooner. I wasn't doing well at school: academic work – particularly maths – was not my strong point. I had decided that I wanted a career as either a children's nurse or a vet, but Gisela had other ideas. Although she sent me for tests that showed that I had sufficient potential to take the German equivalent of A Levels, Gisela wanted me to earn money as soon as possible. And so it was arranged that I would leave school at the age of sixteen.
I was unhappy about this decision and felt convinced that behind everything lay the fact that I was not Gisela's biological child. But I didn't ask her to change her mind. I made a point of never asking her for anything because I was afraid she would refuse. Looking back I think this was a form of self-protection stemming from the time when I had pleaded in vain with her to take me away from Hermann's house.
Gisela's plan was for me to train as a physiotherapist, with a view to at some point coming to work in her practice. But as it turned out, I couldn't begin the college course in physiotherapy for another two years. I still have no idea why I was pulled out of school so early but as a stop-gap, I was sent off to live with the son of a friend of Gisela's mother, who owned a farm near Lake Constance on the border between Germany, Switzerland and Austria. Here I was supposed to learn household management. The farm was in a village called Heiligenholz: it was remote and tiny, with only three or four houses nearby. For the first four weeks I cried every night because I was so homesick. Gradually, though, I settled in: the farmer had six daughters and the youngest two, aged twelve and fourteen, became good friends. The farmer's wife was kind and warm, just as I supposed a mother should be. I stayed with them for eleven months and though I didn't really learn any household or cooking skills – my duties were mostly washing up and helping in the fields – they were very good to me, and inadvertently forced Gisela to do something about my lack of official documentation.
At some point during my stay, the family wanted to go on holiday to Switzerland. But I had no papers – no ID card, no passport, not even so much as a birth certificate – which would be needed to cross the border. The only documentation that anyone seemed to have for me was my state health insurance certificate, and that was in the name of the mysterious Erika Matko.
In 1957, children could be included on their parents' documents. Faced with the prospect of leaving me behind (or abandoning his holiday plans altogether), the farmer passed me off as one of his own daughters. We crossed and re-crossed the border without incident. But it prompted him to write to Gisela, urging her to sort out my identity documents – if for no other reason than I was shortly to be dispatched to somewhere where border controls were likely to be less relaxed.
There was still almost a year before I was to begin my physiotherapy training. Rather than spending it back in Hamburg, it was arranged that I was to be sent to England to work as an au pair. I would need a passport.
To this day I have no idea how Gisela arranged it. I never saw a passport in my name, and given what was to follow I'm as sure as can be that I was never issued one. Some form of documentation must have been procured, however, as I was able to make the long journey – alone, again – to a small village in Hertfordshire, 30 miles north of London.
The family I was to live with were evidently wealthy. The father was a banker who travelled into London every day. His wife was much younger than him and spent most of her time with the family's horses. Of their four children, two were away at Gordonstoun – the famous boarding school where Prince Charles was a pupil. The third child, an eight-year-old boy, joined them not long after I arrived, leaving me with only the couple's five-year-old daughter to look after. I spent six months in their grand house and thoroughly enjoyed the experience. The couple treated me very well; I had a lovely bedroom with a private bathroom, and they made me feel like part of their family.
Looking back, I doubt I recognised then the irony of finding in the land of my country's former enemy the emotional warmth I had longed for at home. I was only seventeen and not as aware of history as I have since become. I returned to Hamburg with happy memories.
Completely unknown to me, while I had been away the problem of my identity had once again surfaced. The start date for my physiotherapy course was approaching and the university needed my birth certificate to register me as a student. I assume that Gisela was somehow involved in dealing with this. (It would be many years before I discovered the flurry of correspondence between various local government offices about me – and in those letters the first hints about my origins.) But whatever she told the officials was not, I think, wholly truthful.
My new birth certificate – dated 1959, the first time my existence was formally registered – was in the name of Erika Matko. It was issued by Standesamt 1 in Berlin, the Federal government registry which had been specifically created to issue papers for people who had come (or had been forcibly brought) into Germany mainly from the east and who had no other documentation. And yet, oddly, it recorded my birthplace as St Sauerbrunn in Austria. It was a record that would, many years later, hamper the search for my true identity.
At the time, regardless of my birth certificate, I continued to insist that I was Ingrid von Oelhafen. That was the name I answered to and the one by which my friends at university came to know me.
To the university authorities, however, I was someone else: they had registered me in the name of Erika Matko, and when I graduated three years later, aged twenty-one, that was the name on my degree certificate. When I asked the university to change this to Ingrid von Oelhafen, my request was refused: without any official paperwork to prove that I was Ingrid, the administration insisted that I was Erika.
It was 1962 and I was now an adult, about to enter the world of work (and to pay my taxes and social security contributions) for the first time. My first job was in an institute in the Black Forest. I was quite used to being away from home by now: I had not really lived with Gisela and her family in Blumenstrasse for any length of time since I left school. And I found myself thoroughly enjoying my new life away from her and without the complications that had dogged my existence back in Hamburg. It was not to last.
In the last year of my training, shortly before my twenty-first birthday, Gisela had a serious accident. She had fallen down the stairs and lapsed into a coma, which lasted six months. Even when she eventually woke, she was so severely ill that she remained in hospital for another year. My grandmother and Aunt Eka had taken charge of her affairs while she was in hospital but when it was time for her to be discharged, I was needed. With much regret, I left my job in the Black Forest and returned to Hamburg.
Gisela was then forty-nine years old: still relatively young and the mother of a young boy, but now severely disabled. The fall had left her with brain damage and she was quite unable to walk. There was no prospect of her picking up the reins of her physiotherapy practice again: instead it was decided that I would have to do so.
It was the last thing I wanted to do. I felt uncomfortable about taking over Gisela's business and it meant I had to abandon plans to go to America, where I had wanted to study a new technique. I was conscious, also, that the tangled relationships in the von Oelhafen family would not be easy to manage.
I moved back into my old room in the house in Blumenstrasse. It was a difficult time: my grandmother, Aunt Eka, Hubertus and I all had to adjust to our new circumstances – and to Gisela's condition. It was particularly hard for Hubertus to see his mother so disabled, but as she learned to walk in the garden and even to share occasional laughter, his unhappiness eased.
The big problem was that Hubertus and Eka didn't understand one another, and this often led to rows. I was caught between them, which I hated, because each sought my support against the other. Nor were things easy with Gisela. To some extent, she was like a small child and my aunt tended to speak to her as a strict teacher would address a recalcitrant pupil: understandably my mother resented this and became stubborn.
Hubertus and I found it easier to accept her as she now was, though I often found myself feeling aggrieved: I was still made to feel very much like an outsider, with few rights but enormous responsibility. It didn't seem right or fair, but I knew that I had no choice but to get on with the job and make the best of it.
After a while, for the first time in as long as I could remember, my relationship with Gisela improved. We didn't talk much, but she lost some of the coldness towards me that had marred my younger years. I realised, of course, why this was: her disability had softened her and as she was now increasingly dependent on me, she let me see that I was needed.
In other circumstances or other families, it might have opened the door to an open and frank discussion about my past, and how I came to be fostered by her and Hermann. But that never happened. We never spoke about Hermann: I think that after his death she, like me (although for very different reasons), felt free of him and of the burden of their failed marriage. Perhaps because they had never divorced, she had felt tied to him and haunted by the need to justify her refusal to live with him.
Or so I suppose now. Gisela never discussed her marriage with me, just as she never talked to me about my origins.
The question of who I really was had not, of course, gone away. In the mid-1960s I decided to take matters into my own hands. Although I called myself Ingrid von Oelhafen, I was still officially Erika Matko. I felt that the time had come formally to change my name by the equivalent of deed poll.
But the process turned out to be more difficult than I had imagined. I discovered that German law required me to seek the permission of the von Oelhafen family. Even if Gisela had been well enough to do this, the regulations did not recognise her as a von Oelhafen. She had married into the name: the law recognised only those who had been born into it as the true owners of its heritage. Once again, the old German belief in the sanctity of blood resurfaced.
Ironically, Hubertus was registered as a von Oelhafen and so, in theory, allowed to grant permission. But he was legally a child and too young to sign any official documents. I don't know why but Hubertus had an official guardian, a lawyer, and so I had to write to this guardian to plead my case. Eventually he agreed, with one qualification: I was not permitted to call myself Ingrid von Oelhafen, since I wasn't part of the family by blood. Instead I could style myself 'Ingrid Matko-von-Oelhafen' – a signal to the outside world that I was, in some way, a lesser member of the clan. It was hurtful, but there was nothing to be done: I signed the papers and obtained my new name. The certificate cost 100 marks.
I needed to apply for a passport around the same time and was alarmed to discover when I did so that the authorities wanted to classify me as 'stateless'. Apparently the unresolved question of where I had been born, and to whom, was still an obstacle to being recognised as a genuine German citizen. I was stunned: the ruling made me feel worthless, as though I were a nothing, a nobody. Nor could I understand why it should be – after all, I had been paying taxes for more than three years. Worse still, the classification could have meant that I was not allowed to vote in elections and that I would not be able to travel freely abroad.
It took many months, and the assistance of a lawyer friend of Eka's, before the government relented and I was issued with a passport showing me to be a real German. I didn't know it then, but had Gisela been less disabled (or had she been honest with me years earlier) she could have given me a document that she had kept hidden away for more than twenty years which would have cut though all the red tape and bureaucracy. But she wasn't, and she hadn't been – and it would be another three decades before I found it in a cache of other vital paperwork.
Of those decades there is little to tell that is truly relevant to this story. I spent six years in Gisela's house in Hamburg, running her physiotherapy practice. I wasn't particularly happy with this arrangement: Gisela's clientele was largely made up of the elderly, and my interests lay elsewhere. On one occasion a three-year-old girl who couldn't walk came to the surgery: much as I wanted to help her, I wasn't qualified to do so. From that moment I knew that I wanted to work with children.
I found out about a course at Innsbruck, in the Austrian Tyrol, to learn a new technique for helping disabled youngsters. It would take me away from Hamburg for ten weeks, and a locum would be needed to look after Gisela's practice (as I still thought of it) while I was away. Aunt Eka was not happy about me going, but I was determined.
Towards the end of the course, I was offered a job on the staff at Innsbruck University clinic: I worried about what my aunt would say – and what would happen to Gisela – but in the end I accepted and spent a happy year doing what I loved in a place where I felt comfortable.
It was during my time in Innsbruck that I fell in love. I met a young man who came from Osnabrück – close to Bad Salzuflen, where I had lived with Hermann, and even closer to the house in Hamburg. We began a life together in Osnabrück, albeit in separate apartments.
Our relationship didn't last. For whatever reason, I did not find – have never found – it easy to maintain adult relationships with men. Whether this has something to do with my childhood I cannot say: I know only that whilst I like the idea of falling in love, those to whom I have often been attracted don't feel the same way about me, and I never seemed to like those men who did see something in me that appealed to them.
I do not say any of this to elicit your sympathy: I am not, by nature, comfortable with that, and if I have never known the intimacy of married life or had my own children, I have been fortunate always to find and keep good friendships with other women which have sustained me. And I have known, too, the joy of helping many, many children: in the early 1970s, after years of working in hospitals, I established my own physiotherapy practice dedicated to working with disabled youngsters. From then on I worked six days a week, twelve hours a day, consumed by the need to help them. Every year I travelled across Europe, England and the United States, attending specialist courses that extended my understanding and developed my skills. It has been a lifelong and rewarding career that has brought me immense happiness.
But what of Gisela, the von Oelhafens and the Andersens? What of the life I stepped away from in Hamburg, the strange mystery of my birth and the circumstances by which I came to be fostered? Although I remained in touch with the family throughout my adult life – and stayed close to Eka, in particular – I did not return to live with them, nor work in Gisela's business.
By the time I did – briefly – go back, the walls had come down across Germany and the east.
## SIX | WALLS
_'Because she is a child of German stock, on the orders of the Reichsführer she is to be brought up in a German family'_
STURMBANFÜHRER GÜNTHER TESCH
At 10.45 p.m. on Thursday, 9 November 1989, the Berlin Wall – that most visible and entrenched symbol of the Iron Curtain – began to crumble. I was forty-eight years old: for almost half a century my life, and the lives of my compatriots, had been shaped by the division of our country into East and West.
It had been a bitter partition: beyond the wall East Germany had re-enforced its borders, imprisoning its population inside a rigid ideological police state. Those who sought to flee, as Gisela had done with Dietmar and me, regularly found their way blocked by barbed wire and checkpoints – and by troops under orders to shoot anyone trying to make their way to freedom. More than a thousand men, women and children had been killed trying to escape the iron grip of communism.
And then it was over. After a day of confusion and rumour, the East German commander of the key border checkpoint opened the gate and ordered his guards to allow people through. Hundreds of _Ossis_ – as those from the East were known – swarmed through to be greeted by West Berliners waiting with flowers and champagne.
Before long, a crowd of _Wessis_ climbed on top of the wall, where they were joined by East German youngsters. They danced together, joyously celebrating their new freedom. Within hours, television cameras captured images of people using hammers and chisels to chip lumps off the wall: soon these _Mauerspechte_ (literally 'wall woodpeckers') demolished entire lengths of it, creating several unofficial border crossings.
The speed of events took the governments of both East and West Germany by surprise. Yet, in truth, change had been in the air for months. It began in August when Hungary – physically and politically one of the outliers of the Eastern bloc Moscow had created – effectively dismantled its physical border with Austria. Within weeks more than 13,000 _Ossis_ had travelled to Hungary and then on into Austria. When the government in Budapest tried to stop the flow, East Germans simply marched into the West German embassy and refused to return home. It was an unprecedented act of civil disobedience from a nation which had, for fifty years, grown used to obeying the orders of its communist masters, and there was more to come.
Throughout that early autumn, mass demonstrations broke out across East Germany. Protestors took to the streets chanting _'Wir wollen raus!'_ ('We want out!') and _'Wir sind das Volk!'_ ('We are the people!'). Newspapers and television stations began proclaiming the dawn of a peaceful revolution.
By the time Erich Honecker resigned as General Secretary of the ruling Socialist Unity Party in October, the movement was plainly unstoppable. Honecker was not merely the head of state: as the man who had been in control of East Germany since the early 1970s, he was seen as the embodiment of the communist state itself.
Despite the warning signs, the collapse of the physical borders was chaotic and unplanned. In the early afternoon of 9 November, a televised press conference in East Berlin first hinted that a limited exodus might be permitted. But after hearing the broadcast, people began gathering at the six checkpoints between East and West Berlin, demanding that border guards open the gates immediately. The soldiers were taken by surprise and overwhelmed by the sheer number of those seeking to cross into the West: they made panicky phone calls demanding instructions.
It soon became clear that no one in the disintegrating East German government would take personal responsibility for issuing shoot-to-kill orders: as a result the border guards simply stepped aside and allowed the huge crowds to pass peacefully into the West. At a little before 11 p.m., West German television pronounced the last rites of the German Democratic Republic.
This is a historic day. East Germany has announced that, starting immediately, its borders are open to everyone. The GDR is opening its borders ... the gates in the Berlin Wall stand open.
The opening – and the determined dismantling – of the Berlin Wall was followed inevitably by the abandoning of all checkpoints between East and West Germany. By 1 July 1990, the day the deutschmark was adopted throughout Germany, all border controls officially ceased to exist. Three months later, East Germany was dissolved and absorbed into a new unified Republic.
What did all this mean to me? Although I was born in the early years of the war, I was really a child of the 1950s and 60s – decades in which West Germany had sought to hide the crimes of the past amid the divisions and troubles of the present. I cannot pretend that the reunification of my country meant any more to me than it did to most people of my generation: we were thankful to have grown up on the 'right' side of the Iron Curtain and vaguely reassured that the tide of history had somehow restored the proper and natural order. To be sure, there were economic concerns: no one seemed quite certain what the cost of our new country might be, though there were regular and dire predictions that the German economic miracle, for so long the envy of Europe, would be threatened by the need to support our less developed, and bankrupt, former neighbour. But such fears were primarily for politicians, less alarming for a physiotherapist with her own successful practice living in the security of Lower Saxony. When reunification happened I was fifty years old. I had never married and my life was comfortable: I was financially secure, had a lovely home and I was working harder than ever. There were, though, clouds building. And, inevitably, they centred around Gisela.
My foster mother's health had worsened as the years had passed and she was now severely disabled. Tragedy had also struck the family. Hubertus, the handsome little boy I had known as a child, had grown up into an attractive gay man: in the mid-1980s he was among the first German men to be diagnosed with the terrifying – and then always fatal – new disease of AIDS. In 1988 it claimed him.
The decision to hire a full-time carer to look after Gisela seemed the best way of securing her future. Gisela was well off. Her practice had provided well for her, and both the von Oelhafens and Andersens had money. But the woman we hired saw an opportunity. Not long after Hubertus's death, she took advantage of Gisela's grief and enfeebled mind and persuaded her to move to Gran Canaria, where, she said, they would benefit from a warmer climate. And so the two of them set up home together, more than 3,000 miles away from any of Gisela's relations. Worse, her companion worked hard to cut us all off from Gisela and to isolate her: none of us was able or allowed to contact her.
Only when Gisela developed dementia was I permitted to visit her. What I found in Gran Canaria disturbed me greatly: it was apparent that Gisela was entirely dependent on a woman whose chief concern was to extract as much money as possible from her before she died. Something had to be done.
Together with Aunt Eka I petitioned the German Guardianship Court to order that we be allowed to intervene in Gisela's life. At first the court refused to hear our plea, saying I had no claim because I was only Gisela's foster daughter, not her biological child. But unusually for me (I am not, by nature, forceful), I dug my heels in. I said to the judges: 'I will sit here until you listen to me. I will stay here in this court until you listen to what I have to say.'
Eventually they agreed to hear me. I told the court that Gisela was being controlled by her carer and that she had manipulated the relationship to such an extent that she had been named as one of the main beneficiaries of Gisela's will. I begged them to safeguard her interests. But listening to me was as far as the judges were prepared to go. Ultimately, the court declined to intervene.
It was left to Aunt Eka to work out a private compromise settlement, which provided some measure of protection for Gisela. But the damage had been done: Gisela lived on until 2002, but never again would we be a family.
Her exile in Gran Canaria did have one positive outcome. When Aunt Eka and I finally realised that Gisela would never come back to Hamburg, we set about clearing out her rooms. Which is how I came to find the diary she had kept of my earliest years.
I will remember for ever the moment I laid my hands on it, and the emotion I felt reading its few handwritten pages. I was so very thankful: I had found something about me and my early life – it was the first time I could reach out and touch my past. But alongside the joy there was pain too.
I think, perhaps, I hadn't realised the extent to which I had for nearly forty years blocked off my feelings about the mystery of my childhood. Holding the little volume, the sense of loss and uncertainty was overwhelming. Why had she not given me this diary but instead kept it hidden? How could she not have realised what it would mean to me?
What made this all the more painful was the knowledge that I had only discovered the book because Gisela had – to all intents and purposes – abandoned me once again. That she was in no state to understand this, and that her carer was deliberately exploiting her frailty, did not change the fact that I could not contact her to ask all the questions which the diary prompted.
Perhaps my overpowering sense of loss and hurt explains why I did not look more closely at the other documents I found in Gisela's room. I glanced at them and saw that they seemed to be legal papers about the process by which Gisela and Hermann had fostered me. But rather than pay them the attention I should have, I put them away and devoted myself to my work. It wasn't until the end of the twentieth century that I was reminded of their existence.
One day in the autumn of 1999, I was at my practice as usual when the phone rang. I assumed the caller was a patient or perhaps a referral for a new client. But the lady on the phone that morning was neither of these things. She first asked whether I was Ingrid von Oelhafen and then explained that she was from the German Red Cross. I was initially puzzled: why would the Red Cross be ringing me? I had no professional connection with the organisation: certainly none of my patients had ever come from there.
Then she asked a question that took me completely by surprise: would I be interested in looking for my birth parents?
I find it hard to describe the feelings that ran through me in that moment. For so long I had put the questions regarding who I was and where I had come from to the back of my mind, telling myself that working with disabled children was more important; in truth, though, I think I was really avoiding the issue, perhaps for fear of what I might find. And so I was surprised to find that my overriding emotion was one of real excitement: at long last I had the chance to find out about my origins. Perhaps I was now finally ready to face the truth.
I have thought about this a great deal and come to the conclusion that it was age that made the difference. I was fifty-eight when I received that phone call, and looking back now I can see that the older I became, the more I wondered about my personal history. I am not alone in this: it is part of the human condition to revisit the past as the years slip away. There were practical considerations, too. Whenever I had cause to go to the doctor – something that becomes more frequent with advancing years – I was asked about my family medical history, and of course I had to say that I had no idea.
I didn't ask how the Red Cross knew where to find me or how they knew that I had a family mystery to solve. I simply said yes, and hoped for the best. They couldn't give me any concrete information about my past. Instead, the woman told me to contact an academic historian at the university in Mainz.
I owe an immense debt to Georg Lilienthal. When I sat down to write to him, I had no idea who he was – much less how important his role would be in my story _._ I simply knew what the Red Cross had told me: he was the person who could set my feet on the path I would need to follow.
I understood that Dr Lilienthal would be expecting my letter, so I wrote openly and honestly, explaining that I had always wanted to know where I came from but that I had never known how or where to start.
When I posted the letter I was so excited I wanted to get in the car and drive to Mainz the next day. But something told me that I must wait: whatever information this man had, he would surely need time to pull it together. And so I resolved to be patient and use the time to search through the documents I had found in Gisela's room. I felt tantalisingly close to discovering the story of how I came to be fostered by Gisela and Hermann and it was frustrating to still be in the dark. But I had waited fifty years before embarking on this quest: a few more weeks wouldn't kill me.
I dug out the box of papers. In the years since I had found them I'd never even taken a look at anything other than the diary. Now I began to look closely at the sheaf of fading documents Gisela had kept with it.
The first was a small and slightly dog-eared pink slip. It was a vaccination certificate, dated 19 January 1944 and signed at Kohren-Sahlis, near Leipzig: it showed that Erika Matko, born on 11 November 1941 in a place called St Sauerbrunn, had been inoculated against smallpox.
The date was significant: January 1944 was several months before I had been fostered by Gisela and Hermann. But other than indicating that the signatory was a doctor, nothing else on the form showed where the vaccination had taken place, or at whose request. What organisation had been based at Kohren-Sahlis? And, for that matter, where exactly was St Sauerbrunn? A second certificate recorded further vaccinations. On the reverse side was an official stamp that read: _'Lebensborn Heim Sonnenwiese Kohren-Sahlis'._
_Heim_ meant a children's home: that much I knew from my earliest days, and it certainly fitted with Herman and Gisela having fostered me. But what was Lebensborn? I had never heard the word before.
The next document was even more puzzling. Dated 4 August 1944, it appeared to be a kind of contract-cum-receipt for my foster parents.
The family Hermann von Oelhafen, of Gentz Strasse 5, Munich, has on 3 June 1944, taken into their home the ethnic German girl Erika Matkow [sic], born 11 November 1941. Because she is a child of German stock, on the orders of the Reichsführer she is to be brought up in a German family.
There will be no provision for the maintenance of the child from either side: the child herself has no assets or revenue. The foster parents alone shall be responsible for her support.
The certificate had apparently been issued in Steinhöring. This, I knew, was a small village not far away from Munich, but there was no other information about the organisation that had created it. The only clue was the letterhead at the top of the paper, almost obscured by holepunch holes and the passage of time: _'Der Reichskommissar für die Festigung deutschen Volkstums, Stabshauptamt L'._ I had no idea what this could be: a little research revealed it to be the office of the Reichs Commissioner for Strengthening German Nationhood, a Nazi organisation. What exactly the office did was not immediately clear.
At the bottom of the document was the signature of a Doctor Tesch, who described himself as a Sturmbannführer. Anyone who had grown up in Germany after the war knew that word: it was a paramilitary rank in the Third Reich, equivalent to a major in the regular army but almost exclusively reserved for use by members of the SS. Why would an officer in Heinrich Himmler's reviled Death's Head* organisation have had anything to do with my foster care? I looked again at the certificate: it said that I had been handed over to a German family 'on the orders of the Reichsführer'. That was Himmler again. Bafflingly, it looked as though Hitler's second-in-command, the most feared man in Nazi Germany, had played some kind of role in my childhood.
I was desperate to ask Gisela what all this meant – and, indeed, why she had kept these documents from me for so many years. But Gisela was in Gran Canaria and, by this stage, in the last throes of her dementia. I knew I would get no help from her.
A week had now passed since I sent Georg Lilienthal my letter: I wondered if he was away from his office or whether he was for some reason unwilling to share with me what the Red Cross said he knew – or at least suspected – about my history. In the interim, I decided to begin my own investigations. I wrote to the German state archives (the Bundesarchiv) to ask if they held any documents bearing my name or that of Erika Matko.
I assumed, naively, that the Bundesarchiv would reply quickly: how difficult could it be, in this age of computerised databases, to run a simple check on my names? I was about to discover one of the paradoxes of the new Germany: while the new state was committed to uncovering the terrible sins committed by the rulers of the old East German state, and zealous in rooting out of public life those who had been involved with its secret police, the Stasi _,_ it was much less willing to face up to the crimes committed by Hitler's Thousand Year Reich.
In part this was a legacy of the early post-war years. Konrad Adenauer, West Germany's first chancellor, had vehemently opposed much of the Allied powers' work on denazification, and had pushed for the release of those convicted of war crimes at the Nuremberg trials. He had even appointed, as his right-hand man in government, Hans Globke: a politician who had drafted anti-Semitic laws for Hitler in 1938.
From the outset, no one wanted to look too closely at the past, and many years later at the end of the twentieth century, despite its proud position as the driving force of the European Union, Germany still had skeletons in its historical cupboard – skeletons it was neither ready nor willing to rattle.
The Berlin Wall had not been the only barrier separating Germany from itself. If the nation was now reunited, our collective memory was still decidedly patchy. Over the coming months I would discover that anything relating to the mysterious Lebensborn programme seemed to spark repeated bouts of amnesia. There had been very little published about it, and what information was available suggested a story of national shame and a legacy still shrouded in secrecy.
As I waited for Georg Lilienthal and the Bundesarchiv to respond to me, I thought back to the telephone call from the Red Cross. The woman had seemed reluctant to give me any information: had she been trying to warn me about the problems I would face when she asked whether I really wanted to investigate my past? Perhaps, but however difficult the task might be, I was determined to try. I did not realise then, as I tentatively began my personal quest, that I would also be embarking on a painful journey into Germany's troubled history, as well as that of a country it had once invaded and plundered.
_______
* The SS cap badge was a Death's Head (skull and crossbones): because of this and their role in administering the death camps, they were known as SS-Totenkopfverbände – literally, Death's Head units or regiments.
## SEVEN | SOURCE OF LIFE
_'The eternal law of nature to keep the race pure is the legacy that the National Socialist movement has bestowed upon the German people for all time.'_
NAZI PROPAGANDA FILM, 1935
There was no such place as St Sauerbrunn.
With little else to go on, I returned to the very first record of my existence: the little pink slip of paper showing that I had been vaccinated against scarlet fever and diphtheria. As it documented my birthplace as St Sauerbrunn, that seemed the most logical place to start. But although I searched through atlases and historic maps of Germany and all the countries Hitler's armies had invaded, there was no town or village with that name.
The closest match was the Austrian spa town of Bad Sauerbrunn, close to the border with Hungary. At the start of 2000, I found the address of the Austrian Ministry of Foreign Affairs and wrote a lengthy letter asking if they could help me locate any record of a family called Matko anywhere in the vicinity of Bad Sauerbrunn.
I was now becoming a little impatient. I had not received a reply – much less any information – from the Bundesarchiv, and I was still waiting for Georg Lilienthal to deliver the information he had supposedly found about me during his research into Lebensborn.
Frustrated, I began searching for information about this mysterious-sounding organisation. What struck me immediately was how little seemed to have been published. More than fifty years after the end of the war, the terrible history of the Third Reich and its crimes had been analysed and picked over in meticulous detail, and yet a Google search for Lebensborn produced only a few, largely repetitive results.
Ostensibly, the Lebensborn Society (literally translated, _Lebensborn_ means Fount, or Source, of Life) was founded in 1935 as some sort of welfare organisation, funded by the Nazi Party, to run maternity homes across Germany; it was set up in response to what was rapidly becoming a demographic crisis for the new Reich. When Hitler came to power in the 1930s, the country's population had been falling for decades. In 1900, the statistics showed an average rate of births per thousand of 35.8; by 1932 that had dropped to 14.7. From the outset, the Nazi regime set out to stop and then reverse the trend. They began with slogans – 'Restoring the family to its rightful place' was typical – and then introduced financial incentives such as marriage loans, child subsidies and family allowances to encourage large families. A cult of motherhood was also formally established: every year on the birthday of Hitler's own mother, fertile women were awarded the Honour Cross of the German Mother. Those who produced more than four children were given a bronze medal; more than six earned silver; and gold was awarded to those with more than eight.
When this didn't produce results quickly enough, new laws were introduced to ban the advertisement and display of contraceptives and Germany's pioneering birth control clinics were shut down (in the 1920s Germany had been a world leader in developing contraceptive devices such as the IUD). Abortions were criminalised as 'acts of sabotage against Germany's racial future'.
That phrase, 'racial future', was my first clue to the reality hiding behind the seemingly innocuous Lebensborn Society. Although the ostensible aim of the homes was to allow women who might otherwise abort their pregnancy to give birth in safety and in secret – thus helping to boost Germany's population – they weren't open to everyone.
I was, of course, aware of the Nazis' obsession with race: it was the altar on which Hitler and his regime had sacrificed more than six million Jews. What I hadn't encountered was the extraordinary and convoluted web of organisations that had been established to safeguard the 'purity' of the German race. As I continued my research, I could feel myself being pulled down the rabbit hole of National Socialist madness. At its heart was the sinister figure of Heinrich Himmler.
Himmler had joined the Nazi Party in August 1923, three years after its birth. He was not one of its early fanatics – his membership number was 14,303 – but within six years he had taken charge of its most powerful paramilitary organisation, the Schutzstaffel, more infamously known by its initials: SS.
As Reichsführer-SS, Himmler began creating a parallel, and ultimately much more powerful, organisation to control and monitor the Nazi Party. He had long been interested in the then-fashionable quasi-science of eugenics and became obsessed with the idea of a mystical past in which a Nordic race of pure-blooded warriors had conquered much of Europe. He began reorganising the SS to be the vanguard of a reborn race of Aryan 'supermen'. Under his direction applicants were vetted for their racial 'qualities': he described the process as being 'like a nursery gardener trying to reproduce a good old strain which has been adulterated and debased; we started from the principles of plant selection and then proceeded quite unashamedly to weed out the men whom we did not think we could use for the build up of the SS'.
In 1931, he created a separate department within the SS to ensure his 'plant selection' ran smoothly: Das Rasse-und-Siedlungshauptamt-SS _,_ or RuSHA. A literal translation would be 'SS Race and Settlement Main Office'; what it meant in practice was an organisation dedicated to safeguarding the 'racial purity' of the Schutzstaffel. One of its duties was to oversee the marriages of SS personnel: on Himmler's personal orders, RuSHA only issued a permit to marry after detailed background investigations had proved that both partners had an uninterrupted racial pedigree showing them to have come from pure Aryan blood-stock as far back as 1800.
As I read further, I discovered that the Lebensborn Society had been formed and fostered under RuSHA's banner. In a circular issued on 13 December 1936, Himmler had set out both the lineage and the aims of his new organisation:
The Lebensborn Society is under the direct personal control of the Reichsführer-SS. It is an integral part of the Race and Settlement Head Office and its objects are:
1. To support racially and genetically valuable large families.
2. To accommodate and look after racially and genetically valuable expectant mothers who, after careful investigation of their families and those of the fathers of their children by RuSHA, can be expected to give birth to equally valuable children.
3. To look after those children.
4. To look after the mothers of those children.
Even to me – a German woman, born during the war, who had lived her whole life in a country trying to come to terms with the legacy of Hitler's twisted vision – this sounded the stuff of madness. In German we have a very expressive word for this sort of fantastic lunacy: _unglaublich_ , meaning unbelievable. How could anyone 'prove' their racial or genetic value – and what in any event did such a bizarre concept mean in practice?
I was not, it emerged, alone in being confused. I found a succession of references to rumours which had grown up about Lebensborn facilities. Some of these dated back to the war years and suggested that ordinary Germans had become worried by stories that these ostensible maternity homes were in fact SS stud farms: places where the cream of Himmler's brigades were introduced to suitable Aryan women in order to breed racially valuable babies for the Reich. The gossip was false but the secrecy that surrounded Lebensborn ensured that the rumours had persisted over the years. There was even an entire genre of Nazi exploitation films and books dedicated to mythologising the programme: one typical example, a movie made in 1961 by a German director and widely available on the Internet, had the English title 'Ordered to Love' and the screaming subtitle 'Frauleins Forced into Nazi Breeding!'
I felt ashamed and horrified. I understood that the SS stud farm stories were no more than absurd fantasies (and as often as not, cynical attempts to sell tawdry films and novels), but if this was what the world knew – or thought it knew – about Lebensborn _,_ was it any surprise that modern Germany was unwilling to talk openly about it? Perhaps this explained why my requests for information from the Bundesarchiv were still unanswered, two months on. Alone, I had little chance of being able to investigate Lebensborn, much less to uncover its role in my origins. I needed help, but there seemed to be a wall of silence surrounding anything to do with this corner of Nazi history.
In February 2000, my hopes were dashed further. The Austrian government finally replied to my letter about Matko family records in Bad Sauerbrunn: there was no such record and never had been. My journey seemed to be over almost before it had started. If I hadn't come from Austria, where had I been born?
And then, a few days later, a letter arrived from Georg Lilienthal in Mainz. For the first time it contained clues – solid, historical information – about Lebensborn and how it fitted into my own story. He wrote cautiously, hinting once again at painful secrets lying in wait for me.
Dear Frau von Oelhafen,
I would first like to thank you for the trust you have placed in me with your letter because it's all about the question of your identity. Therefore, I am also glad that Ms Fischer of the German Red Cross in her conversation with you was very careful... I have to apologise to you. My response to your letter took a long time. And while you were waiting for a sign from me, you might have come to doubt whether your request was the right thing to do. I can reassure you.
My long silence was partly due to external reasons (too little time to find the documents together and write): but on the other hand I was also aware that the answer would not be easy because I know what it could mean to you. That's why I have been writing my letter on and off since early January. That is what has led me to outline your presumptive fate so soberly and in a seemingly emotionless way. I did not want to influence your feelings with my feelings.
Now for your request. As you write, the fact that you have two names (Erika Matko and Ingrid von Oelhafen) has long been known. I assume that you have therefore always wondered what it's all about. Apparently your foster parents were not completely open with the little they knew about you.
After reading the letter, I could not have told you exactly what my feelings were: there was anxiety and apprehension, but these were also shot through with excitement.
I knew, of course, that neither Hermann nor Gisela had ever told me the truth about my origins. I had, to some extent, convinced myself that if this hadn't been the result of the tensions of post-war life, it must have been because they didn't really know my history. Lilienthal's letter was the first time I had to confront the possibility that my foster parents might have deliberately withheld information.
And then came the revelation I had been waiting for and half-expecting. Lilienthal's research had found the name Erika Matko in some long-forgotten records of the Lebensborn programme. She had been raised in one of its children's homes: a place called Sonnenweise (literally: 'Sunny Meadow') at Kohren-Sahlis. Since I was – or once had been – Erika Matko, that meant I was a Lebensborn baby. What's more, his investigations had convinced him that Hermann and Gisela had deliberately concealed this information from me.
There was now solid documentary evidence linking me to this bizarre and evidently still-shameful Nazi organisation: one which had been under the direct control of the SS. And yet my overall response was exhilaration rather than shock. It seemed incredible, but it also offered the chance finally to find out more about who I was and where I had come from. And in a way, the revelation also brought me a little peace. Although I did not yet understand the true nature of Lebensborn, I could now let go of one of the worries which I had lived with ever since I discovered that I had been fostered.
If, as my initial research indicated, Lebensborn was a political programme in which the demands of the Nazi regime overrode the feelings of those it ruled, perhaps the reason my real parents had given me up was also political, not (as I had feared) the much more upsetting idea that they simply hadn't wanted to keep me. That realisation brought me some comfort, but also a hint of fear. In some way I had been involved with an organisation which, nearly sixty years later, still evoked fear and loathing. I mentally added the SS to the growing list of Nazi groups I would need to investigate.
Lilienthal's letter contained more surprises. Realising that I knew little about how Lebensborn had operated, he explained how children had arrived at Sonnenweise _._ Some had been born in Lebensborn maternity homes, then brought to Kohren-Sahlis as part of Himmler's programme to increase the population of the Reich. Others, though, had apparently been kidnapped.
Children born in this maternity home were German children born illegitimately in the Lebensborn programme for foster care or for adoption. But there were also children at Kohren-Sahlis who were trafficked from the occupied countries of Germany and who were designated for Germanisation.
I had never heard of 'Germanisation'. Why would the Nazis traffic children from the countries they had invaded? I had always been taught that Hitler and his henchmen viewed the people of many of these conquered states quite literally as 'sub-human'. And how did this fit with my background?
Lebensborn worked with German foster families, with the intention of later adoption after the victorious end of the war. The fall of the Third Reich prevented these plans from being realised. Most of these foreign children returned to their home countries. However, some remained in Germany with their foster families.
There were various different reasons for this. Some of the foster parents cared for their foster families. Some of the foster parents concealed the foreign origin, even from the children themselves, for fear that the children might be removed again or that they would have a yearning to return home. Ultimately, they were afraid to lose the love and affection of their foster children. Also, they wanted to protect the children from hostility and integration difficulties.
These were often the reasons that the children were not adopted after the war, as well as the fact that they often lacked the necessary papers.
Some of the Allies did not want to send the children back to their home countries against their will, and they remained in the German family with the authorities of their home countries in agreement because they had no biological family left.
And then Lilienthal dropped his biggest bombshell.
Frau von Oelhafen, is it your belief that you might not be a child of German parents? I have known your name 'Erika Matko', and the name of your foster parents 'von Oelhafen', for many years from documents in the Bundesarchiv. I have researched Lebensborn for over twenty years and I know many of the fates of Lebensborn children.
Their names are mentioned in lists that were created by Lebensborn for children to be Germanised from Poland, Yugoslavia and Czechoslovakia (in the Lebensborn management they were called only _Ost-Kinder_ ) and in the records and statements of former Lebensborn employees.
Although I can present to you no papers (such as a birth certificate), giving you ultimate peace of mind, I do have documents that seem to show that you probably originated in Yugoslavia.
After you have read my letter, you may ask: what to do now? I cannot give you the answer to this. But if you want to continue to search for your identity, I will be happy to assist you. You can always contact me.
Kidnapping, Germanisation, _Ost-Kinder_ : these words and notions were so alien to me, so far away from the assumptions I had made when I began my investigation, that I didn't know what to make of them. Although the Austrian authorities had been unable to find any trace of a Matko family in or near Bad Sauerbrunn, I had still believed that the search for my past would somehow take me to Austria. It had been a comforting thought in a way: the time I had spent at Innsbruck meant the country felt familiar. Nor was there any language barrier: German was the national language. Now it seemed I had to start again from square one – and in a language I had never even heard spoken. Even worse, Yugoslavia itself had ceased to exist: the last of the former Iron Curtain countries had disintegrated in a bloody civil war before splintering into a series of smaller new states. Where – how – would I begin?
I decided to take Georg Lilienthal at his word. I wrote to him, asking for guidance. Throughout my journey into the past I have been very fortunate to find people who were willing to give their time and share their expertise to assist me from one stumbling step to the next. Dr Lilienthal was the first, and probably the most important of my guides. He told me I needed to write to two German ministries in Berlin – Foreign Affairs and Internal Affairs.
He helped me compose the letters, each of which explained my situation and set out my belief that I had been brought into the Lebensborn programme from the former Yugoslavia: I requested assistance in making contact with their counterparts in Eastern Europe.
My requests fell on deaf ears. Both ministries sent abrupt and unhelpful replies, saying that they could not do anything for me: the only thing they could suggest was that I write to the government of Slovenia – the new nation that had emerged in the central part of Yugoslavia, once controlled by Hitler's Reich.
Around the same time, I received a reply to my original enquiry at the Bundesarchiv. This too was unhelpful: the state archives insisted that they held nothing relevant to my past. A pattern was developing: no government institution seemed interested in helping me investigate my past. Since I knew that Georg Lilienthal had already found documents relating to Erika Matko and the Sonnenweise home in those very same archives, it was clear that German officials were reluctant to talk about Lebensborn _._ Over the next few months it was a reluctance I would encounter over and over again.
Georg Lilienthal pointed me toward two other, lesser-known collections of documents where, he said, I might find information about Lebensborn _._ And he agreed to use his own contacts to find out who I should write to in Slovenia.
Looking back, I realise that this was the pivotal point in my investigation: from here on there would be no turning back. Once I began digging into boxes of dusty papers, stored in archives across modern Germany, there was no way of knowing what skeletons I might disturb, what secrets I might unearth. Such is the benefit of hindsight. At the time I didn't stop to think about what I was doing: for so long I had avoided thinking about my past, but now I was determined to find out whatever was known about me, and by extension about those who had raised me. If that meant asking questions that made people uncomfortable – well, so be it.
## EIGHT | BAD AROLSEN
_'Adolf Hitler has led the German people to the realisation that the Nordic race is the most creative, valuable race on earth. Therefore, caring for the valuable Nordic blood is their most important task.'_
HEINRICH HIMMLER, _R ACIAL POLITICS_
(1943 SS PUBLICATION)
Bad Arolsen is a small, picture-postcard German town. For more than 250 years it was owned and ruled by the Princes of Waldeck-Pyrmont, then a sovereign principality stretching across the rich agricultural heartlands of Hesse and Lower Saxony. This aristocratic family constructed a large baroque-style stately home and drew up plans to build the town around it in a mathematically perfect grid of streets. But when they ran out of money, the grandiose scheme was only half-completed: to compensate, the undeveloped sections were landscaped with shrubbery.
Die Grosse Allee is the main street, running one perfectly straight mile from east to west and lined with 880 German oak trees in strict military formation. Exactly halfway down is an unprepossessing piece of post-war architecture, set back from the road behind long hedge walls so as to be almost unnoticeable to the casual visitor. It is the archive of the International Tracing Service. Here, spread haphazardly over several floors and spilling out into satellite buildings, more than thirty million individual files record the fate of those who fell victim to the National Socialist criminal enterprise.
It is a cliché of modern history that the Nazis were painstaking record-keepers. But the 26,000 linear metres of original documents and 232,710 metres of microfilm housed at the ITS bear witness to this thoroughness and, according to Georg Lilienthal, somewhere in the vast piles of paperwork there was probably a record of how I came to be part of the Lebensborn programme.
I wrote to the archive in the early spring of 2000, asking for its help in locating any document that would help me investigate my origins. In theory, this should have been a straightforward request: it was, after all, exactly what ITS was established to do. But, as I was finding, theory and practice remained a long way apart – and what often separated them was politics.
In 1943, the Supreme Allied Headquarters in Europe asked the international section of the British Red Cross to set up a registration and tracing service for missing persons. Even by that mid-stage of the war, Washington and London had begun to plan for its aftermath, conscious that by the end there would be a vast population of the displaced or the disappeared. The Central Tracing Bureau was established in February 1944: as the war shifted eastwards into each territory successively liberated from the German armies, it moved from London to Versailles, then on to Frankfurt before finally arriving at Bad Arolsen in 1946. Here its researchers set about creating an archive of Nazi documents.
The records came from every corner of the former Reich. Allied forces had rescued them from concentration and death camps or captured them from Wehrmacht field offices and Nazi central registries. Each individual piece of paper was analysed: from them the CTB was able to begin reconstructing the fate of tens of millions of men, women and children who had been taken for slave labour, imprisoned, or murdered in the Holocaust.
From the outset the Allies had two, sometimes conflicting, aims for this unprecedented exercise. The first was to prepare reliable documentary evidence for use at the Nuremberg War Crimes Tribunal: for the first time in history, the surviving leaders of a country were to be put on public trial for the newly defined offences of crimes against humanity, conspiracy to wage aggressive war and the industrial-scale murders of Jews and Eastern Europeans (among many others).
The second, longer-term ambition was to create a mechanism to enable the survivors of the war – and especially of the Holocaust – to find their families and, if possible, eventually to reunite them. And so the CTB began building from the captured files a central name index of every single person they could determine to have been a victim of the Nazis' reign of terror.
Whether or not the Allies realised the scale of the task when they began it, they were soon overwhelmed by the sheer volume of cases. The central name index alone would come to house the individual fates of fifty million people. Behind each handwritten index card was a mound of paper.
As the years passed, responsibility for managing and funding this Herculean effort was passed from one organisation to another. In July 1947, the newly formed United Nations' International Refugee Organisation took over administration of the bureau, changing its name to the International Tracing Service. Less than four years later, it was handed back to the Allied High Commission for Germany – the body set up by America, Britain and France to run their sectors of the former Reich. When the occupied status of Germany was repealed in 1954, ITS was hived off to the International Committee of the Red Cross, which promptly insisted on appointing its own manager to run all daily operations, who, for good measure, had to be a Swiss citizen. It was a sorry catalogue of financial and administrative buck-passing which ensured that the ITS was destined to become the Cinderella of the vast post-war archives mission.
The situation worsened with the implementation in 1955 of the Bonn Agreement, which formally ratified the new nation of West Germany. One clause in this document prohibited the publication of any data that could harm former victims of the Nazis or their families. However well intentioned, the instruction effectively shut the Bad Arolsen archive off from public scrutiny: historians and journalists were not permitted to examine its contents, and although individual victims of the tyranny were theoretically able to ask for any relevant information, this too became caught up in the _Realpolitik_ of modern Europe.
At the start of 2000 – just as I made my request for help – the German parliament was under pressure to set up a fund to compensate an estimated one million survivors of the Nazis' forced and slave labour programme. These were men and women who had been shipped from Eastern Europe to toil in the factories that kept Hitler's war machine running. Soon the Bundestag passed a law establishing a Remembrance, Responsibility and Future Foundation ( _Stiftung Erinnerung, Verantwortung und Zukunft_ ) that would make payments to those who could prove they had been affected. The evidence they needed was primarily held at the ITS: it was almost instantly flooded with applications, and all other enquiries were either ignored or not properly processed. Among them was my letter: I received a brief, and as it would turn out completely inaccurate, reply to the effect that there was no trace of me in the files.
Seven years would pass before the ITS archives were opened to full public scrutiny: lost time that would have a terrible impact on the search for my biological family. But to explain the origins of Lebensborn, I need to step away from my own chronology to pull back the veil of secrecy which then surrounded Bad Arolsen. Among the millions of documents captured from the Nazi war machine were many of Heinrich Himmler's personal papers. These were sent to the ITS, where separate folders were opened, each covering the myriad organisations the Reichsführer had set up, as well as the bizarre and obsessive belief system that underpinned them _._
The pernicious idea that one race was superior to another by virtue of the purity of its blood had begun in the last decades of the nineteenth century. By the early 1920s an entire 'science' based on this had spread across Europe and the western world. Eugenics held that since certain peoples were of higher quality than others, it was naturally right to improve the overall human genetic strain by promoting higher reproduction among those from the superior race or class and, by extension, reducing reproduction by those less well favoured. At the time such thinking was advocated by prominent English novelists, including H.G. Wells, Marie Stopes (the founder of modern birth control) and two American presidents, Woodrow Wilson and Theodore Roosevelt.
Eugenics societies sprang up, often funded by wealthy American foundations, to promote (in the words of a 1911 Carnegie-supported research paper) 'the Best Practical Means for Cutting off the Defective Germ-Plasm in the Human Population'. Sterilisation and euthanasia were the most popular suggested methods.
It was a belief system and a climate tailor-made for the Nazis. It supported their spurious belief that Germans were the true descendants of a breed of Aryan (sometimes called Nordic) supermen whose destiny was once again to rule the world. In 1925 Hitler had promulgated this concept in his autobiographical Nazi manifesto, _Mein Kampf._
The products of human culture, the achievements in art, science and technology with which we are confronted today are almost exclusively the creative product of the Aryan. That very fact enables us to draw the not unfounded conclusion that he alone was the founder of higher humanity and was thus the very essence of what we mean by the term 'man'.
What we must fight for is to safeguard the existence and reproduction of our race and our people, the sustenance of our children and the purity of our blood...
Four years later, he followed this up in a speech to a party rally.
If Germany were to get a million children a year and remove 700,000 to 800,000 of the weakest people, the final result might be an increase in strength.
It was a refrain taken up by the man who became the Führer's most powerful henchman. When Himmler was appointed head of the SS that same year, he told his senior officers:
Should we succeed in establishing our Nordic race again in and around Germany... and from this seed bed produce a race of 200 million, then the world will belong to us. We are called, therefore, to create a basis on which the next generation can create history.
One of the first pieces of legislation passed by Hitler was the Law for the Prevention of Hereditarily Diseased Offspring. This required doctors to register every case of hereditary illness among their female patients of childbearing age. Failure to comply was punishable by substantial fines. The opening paragraphs of the new law set out both the problem (as the Nazis saw it) and its primary cause.
Since the National Revolution [the quasi-legal putsch by which Hitler gained the power to rule by decree], public opinion has become increasingly preoccupied with questions of demographic policy and the continuing decline in the birth rate.
However, it is not only the decline in population which is a cause for serious concern but equally the increasingly evident genetic composition of our people.
Whereas traditionally healthy families have for the most part adopted a policy of having only one or two children, countless numbers of inferiors and those suffering from hereditary conditions are reproducing without restraint, allowing their sick and disadvantaged offspring to be a burden on the community.
The solution was – to Nazi thinking – obvious: sterilisation. A system of 181 Genetic Health Courts was set up to order the enforced neutering of those deemed substandard. A measure of the programme's immediate effect was the volume and outcome of appeals: in less than a year almost 4,000 people tried to overturn the decisions of the sterilisation authorities. Just 41 were successful. Five years later, by the start of the Second World War, at least 320,000 people had been forcibly sterilised under the legislation.
But if the draconian new law addressed the perceived problem of 'inferiors' polluting or weakening the nation's blood-stock, it did not define just what that blood-stock should be. In September 1935, a leading Nazi doctor called Gerhard Wagner announced in a speech that the government would soon introduce a 'law for the protection of German blood' _._ Within days this was codified into the Nuremberg Laws.
These introduced four official categories of human beings in the National Socialist state. People with four German grandparents were classified as 'German or kindred blood'; those who had one or two Jewish grandparents were deemed to have come from 'mixed blood' and were placed – in order of descending value – into two classes of _Mischling_ ; while anyone descended from three or four Jewish grandparents was irredeemably Jewish.
Only those who were formally registered as being the product of 'German or related blood' were now 'racially acceptable' and granted the status of _Reichsbürger_ (citizens of the Reich). _Mischlings_ were placed in the lesser category of _Staatsangehörige_ (state subjects). Jews were from that point on deprived of all citizenship rights, and marriage between Aryans and non-Aryans was outlawed.
The Nazis proceeded to formalise these race classifications. A new set of official documents, _Der Ariernachweis_ (the Aryan Certificate), were introduced to prove that the holder was a true member of the Aryan Race. Those able to satisfy the requirement that their racial ancestry dating back to 1800 showed that 'none of their paternal nor their maternal ancestors had Jewish or coloured blood' were granted a _Grosser Ariernachweis_. Others who could only produce seven birth or baptism certificates (covering themselves, their parents and grandparents), as well as three marriage certificates from their parents and grandparents, were provided with a 'lesser' document, the _Kleiner Ariernachweis._
Two other pieces of paperwork became vital for life in the Nazi state. An _Ahnenpass_ was a certificate, drawn from church records, which recorded the racial characteristics of a person's ancestors: quite literally, an 'ancestors' passport'. It was often supplemented by an _Ahnentafel_ – a carefully tabulated version of the ancestral family tree.
The Nuremberg Laws and the racial certificates that flowed from them were the foundations of the Nazis' determination to arrive at a 'final solution' for the extermination of the Jewish population. But they were also the key cornerstones of the flip-side of that policy: the programme to create a new Master Race of pure-blooded Aryans who would rule Hitler's Thousand Year Reich. The organisation which was to deliver that outcome was Lebensborn, and its architect was Heinrich Himmler _._
Himmler's papers contained his own explanation for establishing Lebensborn. His motive was, he claimed, benign and caring.
I have created the Lebensborn homes because I believe it is not right that an unfortunate girl who expects a child out of wedlock is kicked around by everybody... by all these paragons of virtue, of male and female gender, who feel entitled to condemn her and to mistreat her. I cannot think it right that she is being punished when the state does not provide the facility for help.
Every woman in these homes is addressed by her Christian name. One is Frau Maria and the other Frau Elisabeth – or whatever her name is. Within the homes nobody asks whether they are married or unmarried: we simply educate them, protect them and look after these mothers.
Even if this were true, I had to remind myself that Lebensborn homes were not open to every woman who found herself unexpectedly pregnant. Jewish women and _Mischlings_ were excluded because they were seen as racially worthless.
As war loomed, the Reichsfüher's papers revealed a change in the purpose of the Lebensborn programme _._ No longer was it driven purely by the desire to increase pure Aryan blood-stock within the German population. By October 1939, Himmler had looked into the near future and seen a major threat to his plans for a future master race.
Every war involves a tremendous loss of the best blood. Many victories won by force of arms have inflicted a shattering defeat for a nation's vitality and blood. But the sadly necessary deaths of the best men – deplorable though they are – is not the worst of this. Far more severe is the absence of the children who were never born to the living during the war, or to the dead after it.
And so he issued a revolutionary order to the men under his command. In a proclamation marked 'secret' and issued to every member of the SS and police, the Reichsführer instructed them to fulfil their sacred duty to the Reich by fathering its next generation, whether or not they were married to the mothers.
Berlin, 28 October 1939
Beyond the limits of bourgeois laws and conventions, which are perhaps necessary in other circumstances, it can be a noble task for German women and girls of good blood to become even outside marriage, not light-heartedly but out of a deep moral seriousness, mothers of the children of soldiers going to war of whom fate alone knows whether they will return or die for Germany.
During the last war, many a soldier decided from a sense of responsibility to have no more children during the war so that his wife would not be left in need and distress after his death. You SS men need not have these anxieties; they are removed by the following regulations:
1. Special delegates, chosen by me personally, will take over in the name of the Reichsführer-SS, the guardianship of all legitimate and illegitimate children of good blood whose fathers were killed in the war.
We will support these mothers and take over the education and material care of these children until they come of age, so that no mother and widow need suffer want.
2. During the war, the SS will take care of all legitimate and illegitimate children born during the war and of expectant mothers in cases of need. After the war, when the fathers return, the SS will in addition grant generous material help to well-founded applications by individuals.
SS-Men and you mothers of these children which Germany has hoped for, show that you are ready, through your faith in the Führer and for the sake of the life of our blood and people, to regenerate life for Germany just as bravely as you know how to fight and die for Germany.
The order did not merely authorise free sex; it demanded it. Racially pure men and women were ordered to procreate, whether or not they were married, so that the nation's stock of 'good blood' could be safeguarded. There was to be neither financial penalty for producing illegitimate children nor social stigma.
It is hard to overstate the radical nature of Himmler's decree. Although the Nazis had been in power for six years and had done much to undermine the country's family-based traditional foundations, Germany was still a religiously conservative society. Sex outside marriage was taboo and neither the public nor the churches appeared ready to abandon their social mores _._
Even representatives of the Nazi Party and the Wehrmacht reacted badly to the Reichsführer's new population policy. Yet Himmler stood firm. Three months after his 'procreation order', he issued an unyielding and unrepentant statement to his forces.
Office of the Reichsführer-SS and Chief of the German Police
Berlin, 30 January 1940
**_SS Order for the whole of the SS and Police_**
You are aware of my order of 28 October 1939, in which I reminded you of your duty if possible to become fathers of children during the war.
This publication, which was conceived with a sense of decency and was received in the same sense, states and openly discusses actual problems. It has led to misconceptions and misunderstandings on the part of some people. I therefore consider it necessary for every one of you to know what doubts and misunderstandings have arisen and what there is to say about them.
Objection has been taken to the clear statement that illegitimate children exist, and that some unmarried and single women and girls have always become mothers of such children outside marriage and always will.
There is no point in discussing this; the best reply is the letter from the Führer's Deputy to an unmarried mother, which I enclose together with my order of 28 October 1939.
The Deputy Führer was Rudolf Hess. On Christmas Day 1939, the Nazi Party's daily paper, _Völkischer Beobachter,_ had published an open letter to a notional unmarried mother in which Hess set out the new morality.
The National Socialist philosophy of life has given the family the role in the State to which it is entitled. However, in times of special national emergency special measures can be instituted, which are different from our basic principles. In wartime, which involves the death of many of our best men, every new life is of special importance to the nation. Hence, if racially unobjectionable young men going on to active service leave behind children who pass on their blood to future generations through a girl of the right age and similar healthy heredity... steps will be taken to preserve this valuable national wealth.
By calling on Hess, then more established in the hierarchy of Hitler's regime, Himmler was doubtless giving himself some political cover. But his own command of the SS was absolute and his faith in its fundamental importance to the next generation unshakeable.
The worst misunderstanding [of my original order] concerns the paragraph which reads: 'Beyond the limits of bourgeois laws and conventions ...' According to this, as some people misunderstand it, SS men are encouraged to approach the wives of serving soldiers. However incomprehensible to us such an idea may be, we must discuss it.
What do those who spread or repeat such opinions think of German women? Even if, in a nation of 82 million people, some man should approach a married woman from dishonourable motives or human weakness, two parties are needed for seduction: the one who wants to seduce and the one who consents to being seduced.
Quite apart from our own principle that one does not approach the wife of a comrade, we think that German women are probably the best guardians of their honour. Any other opinion should be unanimously rejected by all men as an insult to German women.
For all the Reichsführer's protestations of outrage, this was far from an outright denial of the charge that he was promoting sex outside marriage. Fears about lax morals in Nazi organisations had been growing for several years: in the summer of 1937, several thousand copies of a privately printed open letter to Goebbels, the Party's propaganda minister, had circulated throughout the country. Signed with the pseudonym 'Michael Germanicus', it pointedly referred to promiscuity throughout the National Socialist movement:
... the sexual excesses in country homes and Hitler Jugend [Hitler Youth] camps; the bad camp morals and the Bund Deutscher Mädel [League of German Maidens] girls made 'young mothers'...
Neither the imprisonment of those caught in possession of the open letter nor Himmler's defence of his 'procreation order' ever completely suppressed this deep-seated public anxiety that Lebensborn homes were being used for sexual liaisons between SS officers and suitable Aryan mates. Himmler's own statements sometimes added fuel to this unfounded rumour, notably his description of the role of the SS in the process: 'We only recommended genuinely valuable, racially pure men as Zeugungshelfer [procreation helpers]'. It was easy, with hindsight, to see where the 'SS stud-farm' myth of Lebensborn began.
But rumours aside, the central role of the SS in the project was beyond doubt. Himmler's statement of January 1940 set out its parental role in the Lebensborn programme.
The question has been raised as to why the wives of the SS and police are looked after in a special way and not treated the same as all the others. The answer is very simple: because the SS through their willingness to make sacrifices and through comradeship have raised the necessary funds, through voluntary contributions from leaders and men, which have been paid for years to the Lebensborn organisation.
Following this statement all misunderstandings should have been cleared up. But it is up to you SS men, as at all times when ideological views have to be put across, to win the understanding of German men and women for this sacred issue so vital to our people and which is beyond the reach of all cheap jokes and mockery.
To properly understand Lebensborn, I needed to delve into the history and nature of an organisation which, more than fifty years after the end of the war, retained the power to instil fear and loathing. I had to immerse myself in the Schutzstaffel.
## NINE | THE ORDER
_'One basic principle must be the absolute rule for the SS. We must be honest, decent, loyal and comradely to members of our own blood and nobody else.'_
HEINRICH HIMMLER, SPEECH TO SS OFFICERS,
6 OCTOBER 1943
Wewelsburg Castle sits on a steep bluff above the rolling hills and dense forests of North Rhine-Westphalia. It was originally built for the medieval prince-bishops who ruled the _Landkreis_ of Paderborn; it was they who created its unique triangular layout of three round towers connected by massive stone walls.
In November 1933, Heinrich Himmler was touring the region. Since taking charge of the SS, he had been searching for a suitable location both to house an ideological training school and to become its spiritual headquarters. When he saw Wewelsburg, he immediately decided to commandeer it.
The Reichsführer had grandiose plans for his acquisition. His obsession with Germany's past had convinced him that Westphalia was the heartland of the (entirely fictional) tradition of Aryan supermen. When he formally took control of the castle in September 1934, the _Völkischer Beobachter_ informed its readers that a lavish ceremony had been held to mark the opening of an SS school dedicated to researching early Germanic history and mythology as the basis for 'ideological and political training'.
The Nazi daily paper did not report Himmler's real motivation: to create a stronghold that would, in his own words, be 'the centre of the world after the Final Victory'. Since the organisation responsible for delivering this triumph was the Schutzstaffel, Wewelsburg must be transformed into a fortress that served and glorified its mystical bonds of brotherhood.
The SS began life as a small, rag-tag paramilitary force, set up to guard Hitler in the roughhouse era of the 1920s when armed Nazis fought street battles with their political opponents in the streets of southern Germany. When Himmler was promoted to its head in 1926, he was determined to transform the organisation. New and deliberately sinister-looking black uniforms replaced the previously favoured provincial _Stiefelhosen_ ; new rules were imposed, banning smoking and instituting military drill sessions.
By the time he ascended to the formal rank of Reichsführer-SS three years later, membership had risen from a few hundred to five thousand. Himmler drew up new criteria for recruitment. All applicants had to be at least 1.7 metres (5 feet 6 inches) tall and those who wished to enter its basic ranks had to sign on for four years, rising to twelve for NCOs and twenty-five for would-be officers. Despite these strict demands, tens of thousands of men applied.
But the height requirement and years of promised service were only the beginning. From the moment he assumed control of the SS, Himmler was determined to form its ranks exclusively with those of sound racial pedigree. He drew up a grading system by which specially appointed 'race experts' would assess the _Erscheinungsbild_ (physical appearance) of each applicant, before sorting them into one of five categories: 'Pure Nordic' was at the top, followed, in descending order of worth, by 'Predominantly Nordic', 'Light Alpine with Dinaric, or Mediterranean Additions', 'Predominantly Eastern' and 'Mongrels of Non-European Origin'. Only those placed in the top three groups were considered for membership of the new SS: the remainder were rejected out of hand.
This was just the first hurdle. Those who passed the racial test were then subjected to a rigorous assessment of their other physical attributes. On a scale of one to nine, those who were assessed in the four top categories were deemed automatically acceptable; those whose fitness or physique condemned them to rungs seven or worse were shown the door; and the middle-ranking levels of five and six were graciously allowed to become SS men if their zeal for the Nazi cause outweighed their physical inadequacy.
There was, however, one criterion that was rigidly enforced. Every potential member, regardless of rank, had to be able to provide documentary proof of his racial ancestry. For enlisted men this pedigree had to stretch back to 1800; officers were required to provide evidence of their heritage from the mid-1700s. Just as ordinary citizens of the National Socialist state would soon be issued with certificates proclaiming them to be greater or lesser Aryans, every member of the SS carried a _Sippenbuch_ – genealogical documentation attesting to their historic racial 'health'.
In writing this I have struggled to express the horrific nature of the Nazi philosophy. It is easy to reach for words like 'obscene' or 'grotesque', but how does one get beyond clichés to truly convey the horror of such ideas? As a German woman raised in Hitler's Reich, I have always been acutely aware of where this obsession with race ended: in global war and devastation, of course, but also in the extermination camps of Auschwitz, Treblinka and Bergen-Belsen. I am conscious that mere words cannot do it justice, but also that I cannot shy away from this history to which my own past is inextricably bound.
The Reichsführer was not simply intent on creating a force of racially pure men: in his mind, the SS was to be the foundation of a new generation, the begetters of a Master Race. This plan had first been articulated by the man Himmler appointed to head up his 'Race and Settlement' organisation, RuSHA. Walther Darré, a former chicken-breeder who had returned to Germany from Argentina, wrote a manifesto entitled 'Blood and Soil'. In 1929, it was printed by the Nazi Party's own publishing house.
From the human reservoir of the SS we shall breed a new nobility. We shall do it in a planned fashion and according to biological laws – as the noble-blooded of earlier times did it instinctively.
Himmler, who had also once been a chicken farmer, wholeheartedly endorsed this agricultural analogy, noting that by adopting Darré's principles it would be possible to 'attain the kind of success in the human sphere that one has had in the realm of animals and livestock'.
But the Reichsführer was also clear that this attempt to create an entire new generation must be kept within the SS as an organisation: he had no intention of allowing members of his precious brotherhood to mate outside its strictly guarded bonds. To ensure this, in 1932 he issued a ten-point Engagement and Marriage Decree to every member of the Schutzstaffel.
1. The SS is an association of German men of Nordic determination selected on special criteria.
2. In accordance with National Socialist ideology and in the realisation that the future of our Volk [people] rests upon the preservation of the race through selection and the healthy inheritance of good blood, I hereby institute the 'Marriage Certificate' for all unmarried members of the SS, effective January 1, 1932.
3. The desired aim is to create a hereditarily healthy clan of a strictly Nordic German sort.
4. The marriage certificate will be awarded or denied solely on the basis of racial health and heredity.
5. Every SS man who intends to get married must procure for this purpose the marriage certificate of the Reichsführer-SS.
6. SS members who marry despite having been denied marriage certificates will be stricken from the SS; they will be given the choice of withdrawing.
7. Working out the details of marriage petitions is the task of the 'Race Office' of the SS.
8. The Race Office of the SS is in charge of the 'Clan Book of the SS', in which the families of SS members will be entered after being awarded the marriage certificate or after acquiescing to the petition to enter into marriage.
9. The Reichsführer-SS, the leader of the Race Office, and the specialists of this office are duty bound to secrecy on their word of honour.
10. The SS believes that, with this command, it has taken a step of great significance. Derision, scorn, and incomprehension do not move us; the future belongs to us!
To assure this future, prospective couples seeking Himmler's blessing had to complete an exhaustive questionnaire, detailing the colour of their hair, eyes, skin and physical attributes, and to which, even more bizarrely, they had to affix photographs of themselves in bathing costumes.
In case the purpose behind the process was unclear, the Reichsführer explicitly set out his reasoning in a directive to the SS about its responsibility to raise the new generation.
A marriage with few children is little more than an affair. I hope members of the SS, and especially its leaders, will set a good example. Four children is the minimum necessary for a good and healthy marriage.
Himmler also had a plan for those SS men who did not – or could not – produce offspring: the Lebensborn programme. In 1936, just nine months after establishing the secretive society, he placed it under the direct control of the SS. And he made clear that childless officers would be expected to help Lebensborn place at least some of the babies born in its homes.
In the event of childlessness it is the duty of every SS leader to adopt racially valuable children, free of hereditary illnesses, and inculcate them in the spirit of our philosophy.
That sentence sent a chill through my bones.
The relationship between Lebensborn and the SS was not simply a matter of bureaucratic control. In the first Lebensborn prospectus, Himmler explicitly described the deep intertwining of his ostensibly benevolent society with the black-uniformed Schutzstaffel _._
The expenses in carrying out [Lebensborn's] tasks will be met in the first instance by member's subscriptions. Every SS leader attached to head office is honour-bound to become a member. Subscriptions are graded according to the SS leader's age, income and number of children...
If he is still childless at twenty-eight, a higher subscription will be due. At the age of thirty-eight his second child should have arrived: if not, his subscription will again be increased.
If at the appropriate ages further children have failed to appear, corresponding increases in the subscription will again become payable. Those who believe they can escape their obligations to the nation and the race by remaining single will pay subscriptions at a higher level that will cause them to prefer marriage to bachelorhood.
I found it hard to reconcile the compassionate image of maternity homes with the evil reputation of the SS. How could anyone – even someone as blinkered and racially-obsessed as Himmler – not have understood that the sinister Death's Head regiments would engender fear and suspicion, not warmth and confidence? The answer, as it turned out, was that he didn't care. At the same time as he handed control of Lebensborn to the Order, he wrote:
I know there are people in Germany who feel ill when they see our black tunic. We understand this: we do not expect to be liked by too many people.
The more I read, the more I understood that the SS was ultimately a clan, insular and secretive, moulded to fit Himmler's belief in a modern order of Teutonic knights, forever striving for the Holy Grail of racial purity. Under the Reichsführer's direction, Wewelsburg Castle was reconstructed to reflect his obsession. Rooms were named after characters from mythology – one was called 'Grail', another 'King Arthur', and in the crypt two special chambers were created. _Der OberGruppenführersaal_ , the lower of the pair, was the location for the mystical rituals Himmler drew up for the twelve most senior SS leaders. Around a central eternal flame, beneath a swastika carved into the arched roof, the Reichsführer planned to hold ceremonies to celebrate death. The worship of death was central to the SS ideology, hence the Death's Head badge on their caps. It stemmed from Himmler's belief in a mystical _Götterdämmerung_ – an apocalyptic vision of the destruction of the world in fire and water before its rebirth in a new and purified form.
The contrast was absurd. This was the organisation that was supposed to be in charge of nurturing and safeguarding new life in the Lebensborn homes. The Nazi regime had full confidence in its 'inevitable' success, however. Two years after the start of the war Hitler had publicly declared:
I do not doubt for a moment that within one hundred years or so from now all the German elite will be a product of the SS, for only the SS practices racial selection.
Elsewhere, Dr Gregor Ebner, a family doctor-turned-SS-officer and the man appointed by Himmler as Chief Medical Officer for the Lebensborn programme, estimated that: 'thanks to the Lebensborns, in thirty years' time we shall have 600 extra regiments'. I looked again at his prediction. A quick calculation revealed that a regiment was normally somewhere between 500 and 700 men. Six hundred new regiments composed entirely of children born in Lebensborn homes? Even at the lowest estimate that would mean 300,000 babies.
Could there really be hundreds of thousands of people like me – children of the Lebensborn programme living throughout Germany?
## TEN | HOPE
_'Beware of what you wish for in youth, because you will get it in middle life.'_
JOHANN WOLFGANG VON GOETHE
From the moment the German Red Cross asked if I was interested in finding out about my family, I had thought of little else. But as the months crawled past with no response to the letters I had written, I came to accept that I had been searching for much longer. Looking back, my whole life seemed to have been overshadowed by secrecy. No matter how hard I worked, no matter how much I gave of myself to the poor, damaged children who came to my practice, nothing could free me from the unhappiness of not knowing who I was. And so I had longed and hoped and dreamed.
The phone call from the Red Cross had broken the spell. No longer was I half-asleep, seeing snatches of my past only in dreams: the promise of solid, reliable information had woken me. And I yearned all the harder for it.
'Be careful what you wish for', warned Germany's most famous poet, writer and statesman. Perhaps I should have listened to Goethe.
The letter arrived in October 2000. It was from a Jože Goličnik, the director of an archive in Maribor, Slovenia's second city and the capital of the Lower Styria region. I had heard that an old repository of parish documents was held there and, having heard nothing back from the government of Slovenia, had decided to try my luck with the church. I'd written purely on chance, with little hope that it would yield anything useful. But Mr Goličnik said he had found a record of my family.
The father of Erika Matko is Johann Matko from Zagorje ob Savi. Her mother came from Croatia. Mr Johann Matko lived in Sauerbrunn and was a glass-maker.
Sauerbrunn. It existed. Not in Austria but in Slovenia – or, more accurately, in the old Yugoslavia. I was so happy that, quite spontaneously (and most unlike me), I literally burst into song, full of relief and excitement. Of course I still had to locate Sauerbrunn, which was not likely to be called that now. The fall of communism had been slower in Yugoslavia than elsewhere across the eastern bloc, but when it happened it brought civil war in its wake. As the smoke cleared from the bloody years in which Serbs fought Croats, Bosnians, Montenegrins and all the other nationalities Tito had welded into a unified republic in the 1940s, new nations rose from the ashes: many changed the names of their towns and cities.
Mr Goličnik's letter had contained a clue, though. Johann Matko had been a glass-maker. If I could find a region that had contained major glass factories, I stood a chance of tracking down Sauerbrunn and finding its new name. Better still, attached to the letter was an actual copy of the parish records, including the Matkos' birth dates. Johann had been born on 12 December 1904; his wife – Helena Haloschan – was eleven years younger, born in St Peter, Croatia, on 8 August 1915.
I wrote again to the government of Slovenia, updating my original request with the information from Maribor. But I decided my best bet was to contact the German Red Cross. They had effectively instigated my investigation and I felt hopeful that if anyone could help me track down the Matkos and Sauerbrunn, it would be their staff. I sent off the latest in my growing volume of letters. And then I began to research glass-making in the former Yugoslavia.
It didn't take me long to discover that glass had been a regional specialty of Lower Styria for more than three centuries. From the 1700s onwards, factories had sprung up producing highly prized and beautifully crafted lead crystal. The centre of this tradition was the town of Rogaška Slatina. And its previous name? Sauerbrunn. I had tracked down the birthplace of Erika Matko; I had found my home.
I cannot describe the elation I felt at that moment. At long last it seemed as though I could almost reach out and touch my biological parents – surely soon I would be able to do so in real life.
Be careful, said Goethe.
My optimism lasted only a few weeks before reality intruded. The Red Cross replied to say they had no information about anyone called Matko from Yugoslavia in its records of those whom the Nazis had captured or killed. The letter advised that the organisation was unable to carry out any research in the archives of former Yugoslavian countries and warned that if I chose to do my own research, there was a very high probability I would discover that Erika Matko's parents were dead, and that they had not died of natural causes. The message was clear: whatever records of their existence might once have existed, my parents were likely to have been killed by the Nazis after Hitler's armies invaded Yugoslavia and, most likely, any trace of them would have disappeared at that point.
The note from the Red Cross was not the worst of it. In February 2001 I received a letter that dashed all my hopes of ever finding my family. The Slovenian government had been neither quick nor helpful in the months since I had first written asking for information. Now, when it did finally send a substantive response, the words hit me like a blow to the stomach.
We wish to inform you that, according to the local administration of Rogaška Slatina, they have discovered [records of] an Erika Matko, born on November 11, 1941. But this woman is still living inside Slovenia: therefore the assumption that Ingrid von Oelhafen was born as Erika Matko is wrong.
The generation from which my parents (whoever they were) came had known more physical suffering than I would ever experience, but still it seemed to me that the cruellest pain of all was that of being offered hope, tentatively reaching out to grasp it – and then seeing it snatched away.
I sat at the table in my flat, the letter in my hand, as my dreams dissolved in front of me. Please believe me: this was not merely self-pity. I had known for decades that I was not really Ingrid von Oelhafen. I had salved that wound with the knowledge, derived from the few scraps of paper that had travelled with me through the years, that I was once Erika Matko. Now I was neither Ingrid nor Erika. I was, truly, no one.
When the shock wore off I took time to reflect on the toll this quest was taking. I forced myself to look at how the ups and downs of my investigation were affecting me and realised that I had spent a whole year riding an emotional roller coaster, soaring high one minute, plunging down the next. Was it really worth the pain? I had made a life – a successful and generally happy life – as Ingrid von Oelhafen, regardless of my true origins, and I had official German papers which said that I was Ingrid. Really, what did it ultimately matter that I might – or might not – have once been called Erika Matko? Would it make me happier to continue pursuing the mystery of the Matko family and the country where this Erika had started life?
I decided that the answer was no. I bundled my letters and research notes into a file and put it away in a drawer. I resolved to forget about them, at least for the time being. Even when the archivists at Bad Arolsen later wrote to me with the news that they had, after all, found some documents relating to Erika Matko and Lebensborn, I simply filed the letter away with the other documents.
Months flew by, then a full year. I buried myself in work and studying music. I had been learning to play the flute; now I practised harder, immersing myself in the notes and melodies of the classical composers.
By the time another envelope dropped onto my doormat, a year and a half had passed since I had put away the folder marked 'Erika Matko'. Perhaps if the letter had been from anyone else it might have joined the others. But this note was from Georg Lilienthal and it contained an invitation. For the first time ever, a group of Lebensborn children was to meet: would I like to come?
Would I? Honestly, I was not sure. My journey so far had been full of dead ends, false trails and seemingly insurmountable obstructions. Did I really want to risk opening up old wounds all over again? And even if the answer was yes, what did I actually have to contribute? I took out my abandoned file of paper, with its obscure Nazi documents and contradictory modern correspondence: what could I really tell anyone? I agonised about it, turning the whole business over and over in my mind.
In the end, I realised that I had no choice. The questions about where and who I came from had been part of my life ever since the day Frau Harte had told me I was not Hermann and Gisela's real daughter. They had been at the back of my mind, pressing down on my emotions – and perhaps shaping my actions – for more than fifty years. Hiding from them simply didn't work: I would have to go to the meeting.
In October 2002, I packed up my car and began the long drive south: it was 260 kilometres from my home in Osnabrück to the town where the meeting was being held. I was sixty-one years old, and it was time to learn about my childhood.
## ELEVEN | TRACES
_'We aren't perfect. We've got all the same illnesses and disabilities as other people.'_
RUTHILD GORGASS, LEBENSBORN CHILD
Between Cologne and Frankfurt, the small town of Hadamar sits on the southern edge of the Westerwald, the long, low mountain range running down the eastern bank of the river Rhine.
It is known today for its highly regarded institutes devoted to forensic and social psychiatry, and for a stark obelisk commemorating the victims of the Nazis' Aktion T-4 euthanasia programme, which had been based in the town. Through their research, Georg Lilienthal and other historians had revealed that between 1941 and 1945, thousands of disabled or otherwise 'undesirable' men, women and children were brought to Hadamar to be sterilised or put to death.
Although Aktion T-4 officially ended in 1941, the programme had, in fact, continued until the Nazis' surrender in 1945. In total, nearly 15,000 German citizens were sent to Hadamar's hospital: most were subsequently murdered in a gas chamber. This was the town where I was to meet the other Lebensborn children.
They were not, of course, children any longer. Like me, the twenty men and women sitting around the room that morning in October were in their sixties and close to retirement. As I took my place, I was very nervous. One by one, we introduced ourselves: when it came to my turn I made myself speak the single sentence I had rehearsed. 'My name is Ingrid von Oelhafen. I don't know anything.' And then I burst into tears.
My new companions were kind and caring. Each was much further into their personal investigations than I, and because they had been through the same emotions they understood my anxiety. As they told their stories, the callous brutality of the Lebensborn programme became clearer to me; and though each new revelation was shocking, learning the truth also somehow put me at ease.
Ruthild Gorgass had been one of the first Lebensborn children to look for others who had been born or brought up in the programme. She was around my age; tall with blue eyes and a brush of short blond hair. She was a physiotherapist too and, like me, she had inherited a diary kept by her mother, which had helped her understand the story of her birth. I liked her immediately and felt comforted by her presence.
Her story was also a good introduction to Lebensborn. Ruthild's father was forty-nine when she was born. He had been a lieutenant in the German army during the First World War. In 1916 he was badly injured at the battle of Verdun, his back and chest a mass of shrapnel splinters.
In the 1930s he had become a committed Nazi and by the start of the Second World War he was a big shot in the chemical industry. He was also married with a teenage son. Despite this, at some point he met and began an affair with Ruthild's mother, who was a clerk in the Leipzig Chamber of Commerce, eighteen years younger than him. Just before Christmas 1941, she found that she was pregnant. Her position fitted precisely Himmler's original aim for Lebensborn: both her parents were dead, she was unmarried and carrying an illegitimate child and therefore at risk of the opprobrium of her family and prejudice from her community. Above all, her child's father was a card-carrying Nazi, and both he and Ruthild's mother were able to demonstrate their genealogical racial purity. In the summer of 1942, the two of them made the 170-kilometre journey from Leipzig to Wernigerode, a small town deep in the spectacular Harz Mountains of Saxony. There, in the heartland of old Germany, Himmler had established a Lebensborn maternity unit. In August 1942, Ruthild was born.
Heim Harz, I learned, was one of twenty-five Lebensborn homes established across Germany and in the countries its armies overran. There were nine homes in Germany itself, two in Austria, eleven in Norway, and one each in Belgium, Luxembourg and France. Often they occupied buildings taken from Hitler's political enemies or wealthy Jewish families: the organisation's central headquarters in Munich had belonged to the writer and exiled anti-Nazi activist, Thomas Mann. Some of the premises were furnished with property confiscated from people who had been sent to the death camps, and each was equipped with state-of-the-art medical equipment to ensure that Himmler's precious pure-blood babies were delivered safely into the world.
And they were. In 1939, Dr Gregor Ebner, Lebensborn's chief medical officer, sent a report to Himmler detailing the success of the programme. More than 1,300 pregnant women had applied to give birth in the homes. Racial and hereditary health examinations had reduced this number by half, so that a total of 653 mothers-to-be were admitted. The neo-natal mortality rate for Germany as a whole at the time was 6 per cent: in the Lebensborn homes this figure was cut in half.
The births are very easy, without many complications. This is attributable to the racial selection and the quality of the women we get.
Their success came at a price, however. Ebner reported that the cost per mother was a substantial 400 Reichsmarks _._ But, he noted, 'that isn't much of a sacrifice if you can save a thousand children of good blood'.
Blood was all-important. Lebensborn was charged with ensuring a racially selected future master race to rule over the global empire of Hitler's Thousand Year Reich. There was even a slogan that encapsulated the duty of the women who gave birth in the homes: ' _Schenkt dem Führer Ein Kind_ ' (give a child to the Führer) _._
The physical health of the Lebensborn mothers may have been uppermost in Himmler's mind, but he was also determined to monitor and guide their political wellbeing. To ensure that they left the homes even more zealous than when they arrived, women were required to attend three sessions of ideological 'education' every week of their stay. During these classes they watched propaganda films, read chapters from _Mein Kampf_ , listened to radio lectures and took part in communal singing of party anthems.
Staff members were themselves carefully monitored and instructed to complete detailed questionnaires about each of the mothers under their care. These RF-Fragebogen (the initials stood for Reichsführer) recorded every aspect of the women's personalities, from their behaviour in the home to their bravery (or otherwise) during birth and their commitment to the National Socialist cause: each document was sent to Berlin, marked for the personal attention of the Reichsführer-SS.
This was not just a bureaucratic nicety. Even in the midst of the war – at a time when he was overseeing wholesale murder in the death camps and the entire apparatus of the Nazi terror throughout Europe – Himmler applied himself devotedly to these questionnaires, deciding, on a case-by-case basis, whether a woman would be allowed to give birth to a second child in Lebensborn at any point in the future. In fact, he supervised every aspect of life in the homes, from the trivial to the absurd. On one occasion he instructed his personal aide, SS-Stamdartenführer Rudolph Brandt, to write to the head of Lebensborn demanding that a record be kept of nose shapes.
The Reichsführer-SS wants a special card index to be kept of all mothers and parents having a Greek nose or the rudiments of one. As an example of the type required, you should refer to the mother in Questionnaire L6008, Frau I.A.
Himmler's hands-on control extended to diet. He issued a stream of memos, instructing cooks on the correct way to steam vegetables and demanding that the homes' supervisors make the women eat porridge – apparently because he had identified this as a vital factor in forming the racially admirable characteristics of the English aristocracy. For good measure he insisted on the application of regular doses of cod liver oil, much to the evident disgust of the recipients. He regularly visited the homes, checking upon the progress of the women and their children. So complete was his involvement that babies born on Himmler's birthday were formally registered as his godchildren and received a special memento – a silver cup, engraved with his name as well as that of the baby.
I found these bizarre details of life in Lebensborn homes bewildering. How did the second most powerful man in the Reich find the time to control day-to-day life in twenty-five maternity homes?
But beyond the oddities, the stories told by the Lebensborn children that day in Hadamar revealed the programme's darker elements. Ruthild told me that she and other children underwent a quasi-religious naming ceremony in which they were dedicated to Hitler and the brotherhood of the SS. This _Namensgebung_ ritual was a distorted version of the traditional Christian baptism, with an altar draped in a swastika flag and a bust or photo of the Führer in pride of place. In front of a congregation made up of Lebensborn staff and black-uniformed SS officers, mothers like Ruthild's promised that their children would be raised as good National Socialists: they then handed over their babies to an SS man who intoned a 'blessing'. There appeared to be different versions of this liturgy in different homes, but the essence of each was the same.
We believe in the God of all things
And in the mission of our German blood
Which grows ever young from German soil.
We believe in the race, carrier of the blood,
And in the Führer, chosen for us by God.
An SS dagger was held over the baby and the senior officer read out a formal welcome to the brotherhood of the SS.
We take you into our community as a limb of our body. You shall grow up in our protection and bring honour to your name, pride to your brotherhood and inextinguishable glory to your race.
How could a mother hand over her precious baby to the care – if that's what it was – of an organisation like the SS? What parent could do something so horrific? As I had told the gathering right at the start, the only thing I knew about my origins was that I had been a baby in the Lebensborn home at Kohren-Sahlis: had I too been dedicated to the service of the Nazis?
There were more terrible revelations to come. I already knew what Himmler was planning with the Lebensborn programme. But I had not realised just how far his organisation would go to ensure that the new _Herrenrasse_ – this master race – was free of any physical defect.
They called them _Kinderfachabteilung_. Literally translated, this means 'children's ward'. It sounds such an innocent phrase, but it wasn't. Under the Aktion T-4 euthanasia programme, babies born in the Lebensborn homes with developmental delay, disease or mental disabilities were killed.
Jürgen Weise was born in the Lebensborn home at Bad Polzin on 5 June 1941. The head of Lebensborn – a Nazi named Max Sollman – ordered that Jürgen be taken to a _Kinderfachabteilung_ in Brandenburg, near Berlin. There he was given tranquillisers and deliberately left untended and unfed. On 6 February 1942, the little boy died; he was eight months old.
Jürgen Weise was not the only disabled baby to be murdered in the name of racial purity and strength. In 2002, when we met, research into this was at an early stage, hampered by the reluctance of staff at official archives to allow access to Nazi-era documents.
But the Brandenburg _Kinderfachabteilung_ had been exposed several years before, and there was convincing evidence that 147 babies were murdered there – including an unknown number from the Lebensborn homes _._
I struggled to take all this in. I had dedicated my life to disabled children. I had seen the joy that my efforts brought to them and to their parents. I had felt the love that comes from helping children like Jürgen. What sort of heartless bureaucrat could so easily extinguish such precious life?
Perhaps my reaction sounds naive. History has told us that the Nazis ruthlessly and quite openly murdered millions of Jews in Auschwitz and Bergen-Belsen and the other camps: why would the deaths of a few babies, born in secret and hidden from public view, matter to men like Himmler and Hitler? But according to their own twisted ideology these children were special: they were in the Lebensborn homes because their parents had been examined and ultimately proven to be suitable blood-stock for the master race.
I looked around at the men and women who had also been part of the Lebensborn programme. I assessed each body, searched each face for evidence that these survivors of Himmler's experiment were truly super-human beings. Were they taller, stronger, healthier than anyone else?
Ruthild answered my unspoken question. She took off her glasses, rubbed her eyes and said: 'We aren't perfect. We've got all the same illnesses and disabilities as other people.'
What, then, was it all for? Himmler's great dream was a generation of super-Aryans who would be so strong, so flawless that they would grow up to become the natural aristocracy of the National Socialist state and the lesser nations it ruled. Yet his scheme seemed to have produced nothing more notable than the group of perfectly ordinary men and women seated around me.
There were, though, two striking characteristics shared by most of these Lebensborn children: deep emotional hurt and a palpable sense of shame. The former stemmed from a problem I was very familiar with. Each of us who began life in that clandestine programme had grown up with the pain of not knowing the truth about our birth.
This secrecy was both deliberate and carefully managed from the outset. Doctors and staff in the Lebensborn homes were required to swear an oath of silence which committed them to respecting 'the honour of pregnant women, whether they conceived before or after marriage', and in June 1939, Himmler had issued an order to protect the identity of illegitimate children born in the programme.
Following an agreement between the genealogical office Reich Minister of the Interior and the L organisation, it is possible to maintain secrecy about the origin of illegitimate children born in Lebensborn homes for an unlimited period. The Reich office will provide a certificate confirming the child's Aryan descent. This certificate can be produced by a child born in a Lebensborn home when they start school, for the Hitler Youth and for institutions of higher education, without the slightest difficulty arising.
This determination to throw a cloak of confidentiality over all aspects of the Lebensborn children extended to the records kept of their delivery. Himmler's organisation set up a special registry office for recording the births: this was kept separate from any other office of the Reich and operated in total secrecy. Mothers' names might be shown in the files, but the identity of the father was generally deliberately omitted.
And many of these deliberately redacted records had themselves disappeared: in the final days of the war, with Allied forces closing in on the homes, Lebensborn staff destroyed much of the organisation's paperwork. As a result, most of the children born within Lebensborn grew up not knowing who their fathers were – and, unless their mothers broke the bonds of secrecy, completely unable to find out.
In particular, this affected the children who had been handed over to foster parents. But even those who, like Ruthild, were kept by their biological mothers, often found it impossible to prise out the information. Many mothers were very vague about their time in Lebensborn homes; others refused point-blank to discuss it.
I knew how that felt. Although I didn't yet fully understand how I fitted into the story of Lebensborn, I was familiar with parental walls of secrecy: as Georg Lilienthal had warned me, Gisela undoubtedly withheld much of what she knew throughout my life.
Why did other mothers do this too? The reason was the second characteristic evident in many of the Lebensborn children sitting beside me in Hadamar. Shame is a powerful emotion, and the political climate in post-war Germany was hardly conducive to honesty about involvement with an organisation as reviled and feared as the SS.
One of the men in our group talked openly about the guilt and the shame that had blighted his life. His story opened my eyes to another aspect of the Lebensborn programme. Hannes Dollinger had grown up in Bavaria, where the couple he thought of as his parents owned an inn. But after he started school, he heard rumours that he was a foundling. He asked his parents whether this was true, but they refused to answer and when he persevered they punished him and forbade him from ever raising the subject again.
It was not until he was fifty that he learned the truth. Just as Frau Harte had once broken the news to me that Hermann and Gisela were not my real parents, a former employee of Hannes' family told him on her deathbed that he had been adopted. That alone was a shock, but the story of how he came to Bavaria was worse.
Norway was the northernmost country occupied by Hitler's army. The Wehrmacht invaded in April 1940 and from then until the end of the war, Norway was run by a collaborationist government which enthusiastically did the Nazis' bidding.
Himmler had for many years viewed the largely blond and blue-eyed local population as de facto Aryans. He and his officials actively encouraged liaisons between SS or Wehrmacht troops and Norwegian women, establishing a network of Norwegian Lebensborn homes in which the resulting babies were born, then shipped back to the Reich and handed over to suitable couples either for adoption or fostering.
The legacy of this collaboration was long and bitter. Unlike the desperate bonfires built by Lebensborn staff across Germany, in Norway the SS never managed to destroy its files. As a result, after the war thousands of Lebensborn babies and their mothers were identified and faced the fury of their countrymen. Women and their children were harassed by their neighbours or schoolmates. Police arrested between 3,000 and 5,000 women who had slept with German soldiers and marched them off to internment camps. The head of Norway's largest mental hospital publicly stated that women who had mated with Germans were 'mental defectives' and declared that 80 per cent of their children were retarded.
Hannes discovered that he was one of these children. He began researching his origins and found that his real name was Otto Ackermann and that he was born in September 1942 in a Lebensborn home near Oslo. From there he had been sent like a parcel across Germany, first to a Lebensborn home in Klosterheide, near Berlin, then on to Kohren-Sahlis, the home in which I had been raised.
Eventually, after being transported to the Lebensborn home in what was now Poland, he was handed over to his adoptive parents in Bavaria. It took him many years to retrace this long and complicated route. Finally he managed to obtain the name of his biological mother, but by the time he discovered this, she had died. His father, a Wehrmacht soldier, had been killed in the last months of the war and his adoptive parents had also passed away.
In many ways, Hannes was typical of our generation of Germans – ironically, since by birth he wasn't actually German. He was then a local government official and a stickler for doing things by the book. He informed the federal government of his real name and that he was originally Norwegian, and asked to change his identity documents to make them accurate. For his trouble, the government declared him stateless – and by law stateless people were forbidden from employment in any public office. It took two long and difficult years before he was offered German citizenship. Even then, to get it he had to give up his original real name.
It was heartbreaking to listen to Hannes' story. So much of it mirrored my own life – the home at Kohren-Sahlis, the problem of being declared stateless – but his experiences seemed to have been much worse. I began to feel almost lucky, and perhaps grateful, that I knew so little about where I had come from.
But at the same time the question was still hanging over me. I had learned a great deal about the Lebensborn programme and about life in its homes, but I did not know how I fitted in to this history. The documents I had found in Gisela's room showed that I had been fostered as part of something called 'Germanisation'. Neither Ruthild's story nor Hannes' contained anything to shed light on this mysterious word.
And then another member of the group stood up to speak, and I began to see the worst horror of Himmler's terrible experiment – and how I had come to be a part of it.
Folker Heinecke was six months older than me. He was a big, well-dressed man who had made a small his fortune as a shipping broker in Hamburg and London. Though he was financially well off, for much of his life he had been deeply troubled by the knowledge that he had been raised in a Lebensborn home, before being put up for adoption by the home in 1943 before the age of three.
My first memory is of being in a room with thirty other children. I remember these people coming in, while we were lined up like pet dogs to be chosen for a new home. The people were to be my parents. They went away and came back a day later. My 'mother' apparently wanted a girl, but my 'father' wanted a boy who would be able to take on his family business in the future. I laid my head on his knee and that did it for him – I was to be their son.
Folker's new family was financially secure and well connected. Adalbert and Minna Heinecke were fanatical Nazis and owned a successful Hamburg shipping company. Adalbert was also deaf and, under Lebensborn's strict rules, should not have been allowed to foster – let alone adopt – one of its precious children.
But Adalbert was also a personal friend of Heinrich Himmler: both the Reichsführer and Martin Bormann (Hitler's personal secretary and one of the most powerful men in the Nazi regime) visited the family's home.
Like many other Germans, the Heineckes kept a small flock of hens. As Himmler had once been a chicken farmer and was a firm believer in applying the principles of poultry breeding to the human species, it was only natural that he and Adalbert talked as they studied the family's birds. When they had finished, Himmler agreed to rubber-stamp Folker's adoption.
Folker remembered a happy childhood, insulated from hunger or want due to his family's wealth. Even at the height of the Allied assault on Germany, when he watched RAF bombers weave through the flak and searchlights to launch raids into enemy territory, his chief memory was of finding the war exciting. It was not until after the war that he discovered he had been adopted.
One of the local kids I was playing with said: 'You know you're a bastard, don't you, they're not your real mum and dad.' But back then I didn't really know what that meant.
His parents never talked to Folker about where he had come from or how he came to be adopted. When his father retired, he took over the family shipping business and enjoyed a successful career.
In 1975, after his parents died, he found among his father's papers a series of official documents he had never seen before. These recorded that he had been born at Oderberg in Upper Silesia: this area had been annexed into Hitler's Reich, but after the war it had been transferred back into the territories of the new republic of Poland. The papers also indicated (falsely, as it turned out) that both of his biological parents had died – hence the need for his adoption.
The discovery prompted Folker to investigate his origins. He approached the German Red Cross, the British Army of Occupation, the American authorities and more than thirty other agencies and church offices: slowly he began piecing together the confusing jigsaw of his past. But Poland was then still locked away behind the Iron Curtain, and it was difficult even for a man of his wealth to gain access to its archives. It wasn't until the fall of communism and the restructuring of Eastern Europe after 1989 that he finally unearthed the truth.
By 1941, Himmler's great hope that the Lebensborn programme would produce tens of thousands of racially pure babies was fading fast. In part this was due to the rigorous selection criteria, which led more than half the pregnant women who applied to give birth in the network of homes to be rejected.
Nor had the SS lived up to its leader's expectations: far from meeting the minimum requirement of fathering at least four children, the birth rate stayed stubbornly around an average of 1.5 per man. The 600 new battalions of Lebensborn babies predicted by its chief medical officer Gregor Ebner were a long way off – if they were achievable at all.
The Thousand Year Reich needed its future warriors to survive. Hitler's vision had always been for a total and global war, followed by permanent occupation of conquered lands. But by 1941 the war was claiming thousands of German lives a week. The Lebensborn homes could not hope to fill the gap. And so Himmler decided on a new strategy: he issued secret instructions to his troops and officials to kidnap 'racially valuable' children from the countries they overran.
The wholesale stealing of children – could it really be true? Shockingly, it was: there was even a recording of Himmler giving a private speech to SS officers in which he justified the policy.
What good blood there is of our kind in these peoples, we will take in; we will steal the children if necessary and bring them up here with us.
The name given to this plan was Germanisation. It was the word on my documents, which I had never understood: now I began to learn what it meant in practice.
Folker's tragedy was that at the age of two he looked German: he had blond hair, blue eyes and looked for all the world like a pure-blooded Aryan boy. Because of this he was snatched from his parents by SS officers and taken to a medical institute for full racial assessment.
I was measured everywhere – head size, body size – and doctors checked to be sure that I had no 'Jewish Aspects'. When I passed those tests the Nazis declared I was capable of being Germanised and shipped me off to a Lebensborn home in the Fatherland.
After a brief stay in Bad Polzin, he was then sent on to Kohren-Sahlis. There were no completely reliable dates for his arrival there, but from what he had discovered it sounded as though we must have been in the home at the same time. The possibility excited me and I tried hard to unearth any memories from the back of my mind, but I could recall nothing more about the place. It was so frustrating to meet someone with whom I might have shared my earliest years and yet to be unable to dredge up any recollection of the place or people.
There were other similarities in our stories. Folker's documents indicated that he had been picked up from Kohren-Sahlis by the Heinecke family. That was where Hermann and Gisela had come to collect me. Had I been lined up for inspection, one of the 'pet dogs', as Folker put it, to be examined by my prospective foster parents?
Folker's investigations had also revealed that Lebensborn frequently gave new identities to the kidnapped foreign babies shipped to its homes and issued false documents proclaiming them to be either German orphans or ethnically Aryan children of the German diaspora: _Volksdeutsche_. Once again I recognised the word Folker used: it was on the papers I found when clearing out Gisela's room.
The clues were beginning to add up. According to my documents I had been born Erika Matko, a _Volksdeutsches Mädchen._ I had been brought from Sauerbrunn to the Lebensborn home at Kohren-Sahlis for Germanisation _,_ before being handed to the von Oelhafens to be raised as a 'real' German girl. I was part of the scheme to ' _Schenkt dem Führer ein Kind_ ' _._ I was one of Hitler's children.
It was horrific – and yet for the first time in years I felt I was finally getting close to solving the mystery of where I came from.
The biggest question I had been struggling with was whether I really was (or had once been) Erika Matko. The Slovenian government's response to my letter had seemed to prove that I could not be, and yet the few original documents I had been able to find all showed the opposite.
Folker Heinecke's story suggested an answer to the puzzle. His investigations had led him to believe that the name recorded on his Lebensborn papers might not be genuine and that the birthplace listed might also be false. The Lebensborn head office evidently went to great lengths to erase the original identity of those children stolen from the Reich's occupied territories. Folker had discovered documents in the archives of the Nuremberg War Crimes Tribunal that told the story of a baby named Aleksander Litau, stolen from parents living in Alnova on the Crimean Peninsula. There were strong indications that Folker might have been this child: the dates matched, as did the Lebensborn homes the little boy had been shipped off to. Could I have suffered a similar fate?
Folker had ultimately run into the same bureaucratic stone wall that had previously defeated me: the documents he needed to confirm or disprove this theory were almost certainly somewhere in the ITS files in Bad Arolsen, but the archive was still not yet fully open. It was, he told us, painful to be so close and yet still so far away from the truth.
All I really want is to find the grave of my biological mother and father. I don't want to end up driven bitter and mad by wondering what might have happened to them. I just want to know who I was and what I might have been if things hadn't turned out the way they did.
I have to keep searching to find something that might lead me to who my parents really were and where they are buried. Then I will have done my duty as a son. I will have honoured my real parents.
I was determined that I too would keep searching and that one day I would track down my true family. Meeting the other Lebensborn children – fellow survivors of such a terrible experiment – gave me renewed strength to restart my own investigations. Now I knew how and where to begin: Nuremberg.
There was one final conversation I needed to have before making the long drive home to Osnabrück. One of the few non-Lebensborn people in the room that day was a man called Josef Focks. I had not heard of him before but he had established a reputation for tracking down documents and information about families who had become separated during or after the war. For his efforts the press had nicknamed him 'the Father Finder'.
I spoke with him briefly and explained my situation. I described the difficulty I had faced in getting information from official archives and told him about the documents I possessed showing me to have once been Erika Matko from St Sauerbrunn, and how this had been explicitly contradicted first by the Austrian authorities and then by the government of Slovenia.
Herr Focks listened and took notes. When I finished, he agreed to help me.
I was grateful, of course, but to be completely honest I was thinking more about the enquiries I would make in Nuremberg than what the Father Finder might unearth.
## TWELVE | NUREMBERG
_'Lebensborn was responsible, amongst other things, for the kidnapping of foreign children for the purpose of Germanisation... numerous Czech, Polish, Yugoslav and Norwegian children were taken from their parents.'_
INDICTMENT: THE NUREMBERG MILITARY TRIBUNALS, CASE 8
It was spring 2003 by the time I set off for Nuremberg, 500 kilometres south.
Nuremberg was the dark heart of National Socialism. Between 1927 and 1938 it was the city where Hitler held spectacular torchlit rallies – serried ranks of tens of thousands of supporters screaming 'Sieg Heil' beneath an ocean of swastika banners, all captured in melodramatic propaganda films – and where the 1935 Race Laws that signalled the start of the Holocaust were promulgated.
For the Nazi Party, Nuremberg's position at the centre of the country symbolised, in some mystical way, the connection between the Third Reich and the supposed Aryan supermen of Himmler's imagination. It was also heavily fortified, which made it one of the last cities to fall to the Allied forces in the final weeks of the war. Despite systematic bombing that destroyed 90 per cent of the medieval centre, the city was only captured after four days of fierce house-to-house fighting.
The three main Allied Powers, the Soviet Union, the United Kingdom and the United States, had long planned to mount public trials of the Nazi leaders, even before the war ended. On 1 November 1943 they published a joint 'Declaration on German Atrocities in Occupied Europe', issuing 'full warning' that as and when the Nazis were defeated, the Allies would 'pursue them to the uttermost ends of the earth... in order that justice may be done'.
For the next eighteen months, as their armies slowly inched to victory, lawyers and politicians from all three countries hammered out a set of innovative legal principles under which leading Nazis could be prosecuted for war crimes and crimes against humanity. When the war ended, the only remaining issue was where to hold the hearings.
Leipzig and Luxembourg were briefly considered and rejected. The Soviet Union favoured Berlin – the 'capital of the fascist conspirators' – as a suitably symbolic location, but the overwhelming destruction suffered by the city made it impractical. The decision to choose Nuremberg was based on two key factors. Its role in the Nazi propaganda machine made it an appropriate site to dispense exemplary justice but, more importantly, its spacious Palace of Justice had survived the war largely intact – and its buildings included a large prison facility.
The surviving leaders of the Third Reich were brought to the cells beneath the courtroom in November 1945. Hitler himself had cheated justice, committing suicide in the Führerbunker amid the flames and ruins of Berlin. Himmler, too, had taken the coward's way out, swallowing a cyanide capsule while in captivity. But twenty-two others, including Reichsmarschall Hermann Göring and Deputy Führer Rudolf Hess, were brought before the International Military Tribunal and arraigned for the crimes of the Nazi regime. Eleven months later the judges, one each from France, Britain, America and the Soviet Union, pronounced the verdicts. Twelve of the accused were sentenced to death; seven received prison sentences ranging from ten years to life in prison; and three were acquitted (two trials did not proceed, one defendant having killed himself and another declared unfit for trial). On 16 October 1946, the executions were carried out in a gymnasium attached to the court building.
It was to this forbidding complex that I was heading on a spring morning in 2003. The reason for my journey was not the famous trial itself, but a much less celebrated set of proceedings held in the same building.
Although the Allies had initially planned to hold a lengthy series of joint-power tribunals, the looming Cold War and the freeze in relations between East and West made this impossible. While the main trial was still in progress, the United States took a unilateral decision to mount subsequent hearings for so-called 'second-tier Nazis' on its own. The result was a series of twelve separate prosecutions, between 1946 and 1949, in which a total of 183 defendants were indicted. Among them were the leaders of Lebensborn _._
Within days of returning home from Hadamar, I wrote to the office that kept records of the Nuremberg tribunals. Given my previous experience of German official archives, I was not expecting much in the way of a response, and so I was pleasantly surprised to receive a letter from the archivists which said that there was an entire box full of relevant documents and asked whether I would like to examine them.
The papers turned out to be from Case 8 of the subsequent prosecutions. This was formally titled _The United States of America vs. Ulrich Greifelt, et al_ , but was more usually referred to as the RuSHA trial, since all fourteen defendants had held senior positions in the Rasse-und-Siedlungshauptamt [RuSHA]: the Race and Settlement Main Office Himmler had established to safeguard the 'racial purity' of the SS and which had then gone on to administer Lebensborn.
I began by examining the official indictment, served on 10 March 1948. It was long and detailed – fourteen closely typed foolscap pages setting out charges under three separate headings: crimes against humanity, war crimes and membership of the SS, which had been declared a criminal organisation. It started with a bleak, uncompromising assertion:
Between September 1939 and April 1945, all the defendants herein committed Crimes Against Humanity... The object of this programme was to strengthen the German nation and the so-called 'Aryan' race...
The SS Main Race and Settlement Office (RuSHA) was responsible, amongst other things, for racial examinations. These racial examinations were carried out by RuS leaders... or their staff members called racial examiners (Eignungsprüfer) in connection with... [the] kidnapping of children eligible for Germanisation... Lebensborn was responsible, among other things, for the kidnapping of foreign children for the purpose of Germanisation.
It was chilling to read these words. Here, in the cold legal language of a trial, was the essence of the organisation that had cared for me – if that was the right word – in my earliest years. The indictment went on to set out both the motivation for the programme, and the countries in which it had operated.
An extensive plan of kidnapping 'racially valuable' alien children was instituted. This plan had the two-fold purpose of weakening enemy nations and increasing the population of Germany. It was also used as a method of retaliation and intimidation in occupied countries. During the war years numerous Czech, Polish, Yugoslavian and Norwegian children were taken from their parents or guardians and classified according to their 'racial value'.
I made a note of the countries identified in the charges: since the documents I had found in Gisela's papers indicated that I had been brought to the Reich for Germanisation, it seemed likely that I had originally come from Czechoslovakia, Poland, Yugoslavia or even Norway – though I somehow doubted that last was a real possibility. Perhaps there would be more solid clues in the rest of the papers in the Case 8 trial folder.
Before I could get to them, the list of defendants' names caught my eye. Four senior officials of Lebensborn _–_ three men and one woman _–_ had been in the dock at Nuremberg, their roles and ranks spelled out clearly.
MAX SOLLMAN – Standartenführer (colonel) in the SS; Chief of Lebensborn.
GREGOR EBNER – Oberführer (senior colonel) in the SS. Chief of the main health department of Lebensborn.
INGE VIERMETZ – Deputy chief of main department A of Lebensborn.
It was the fourth name that stopped me in my tracks.
GUNTHER TESCH – Sturmbannführer (major) in the SS; Chief of the main Legal Department of Lebensborn.
I knew that name. Sturmbannführer Tesch had signed the document recording the contract under which I had been handed over to Hermann and Gisela. The man who had arranged my fostering had been indicted at Nuremberg for crimes against humanity.
I turned back to the disconcertingly large box of papers in front of me. The trial had lasted fifty-seven days, examined almost 2,000 exhibits and heard evidence from 116 witnesses for either prosecution or defence: the printed record ran to a staggering 4,780 pages. I began to wonder whether the three days I had planned to spend in Nuremberg would be enough.
The chief prosecutor, an American military lawyer called Telford Taylor, began by laying out the background to the charges. Just as I had learned in Hadamar, from the start Lebensborn's kidnapping and Germanisation programme had been an integral part of the Nazis' plan to eradicate those they regarded as 'lesser races'.
With the launching of the wars of aggression by the Third Reich, it became possible to put these noxious principles into practice. By the middle of 1940, a very definite plan was being effectuated. This is shown by the top-secret document which Himmler wrote, entitled 'Reflections on The Treatment of Peoples of Alien Races in the East'...
I had never heard of this document – but then, as Brigadier General Taylor went on to tell the court, it was so secret that Himmler gave instructions that it was not to be copied or shown to anyone beyond a small inner circle of Hitler and the most senior party leaders. After quoting a section in which the Reichsführer pronounced his hope that 'the concept of Jews will be completely extinguished', Taylor read into the trial record a section of Himmler's blueprint for stealing those deemed pure Aryan children.
The parents of such children of good blood will be given the choice to give away their child – they will then probably produce no more children so that the danger of this sub-human people of the East obtaining a class of leaders, which, since it would be equal to us, would also be dangerous for us, will disappear...
If we acknowledge such a child as of our blood, the parents will be notified that the child will be sent to a school in Germany and that it will remain permanently in Germany.
Poland was the first country overrun by the Nazis. It became the testing ground for Lebensborn kidnappings. The court was shown a letter, dated 18 June 1941, in which Himmler spelled out his instructions.
I would consider it right if small children of Polish families who show especially good racial characteristics were apprehended and educated by us in special children's institutions and children's homes, which must not be too large.
After half a year the genealogical tree and documents of descent of those children who prove to be acceptable should be procured. After altogether one year it should be considered to give such children as foster children to childless families of good race.
Six months later he issued a new decree detailing how kidnappings were to work in practice, and how the wheat of prospective 'racially valuable' children was to be separated from the chaff of their parents – many of whom, he recognised, would be active opponents of the Nazi occupation.
Politically heavily incriminated persons will not be included in the resettlement action [the Reichsführer's euphemism for wholesale stealing of children]. Their names are also to be submitted by the Higher SS and police leaders to the competent State Police Main Office for the purpose of transfer to a concentration camp... In such cases the children are to be separated from their parents...
The Higher SS and police leaders are to pay particular attention that the Germanisation of the children does not suffer as the result of detrimental influence by the parents.
Should such detrimental influence be determined to exist, and should it be impossible to eliminate them through coercive measures by the State Police, accommodations are to be found for the children with families who are politically and ideologically above reproach and ready to take in the children as wards, without reservation and out of love for the good blood present in the children and to treat them as their own children.
Also included in the trial record were Himmler's directives on the fate of individual Polish families who did not meet the Nazi's standards for racial 'value':
Brunhilde Muszynski is to be taken into protective custody. Her two children, aged four and seven years, are to be sterilised and lodged somewhere with foster parents.
Ingeborg von Avenarius is also to be taken into protective custody. Her children too are to be lodged somewhere with foster parents, after sterilisation.
The court was presented with the transcript of a speech Himmler gave in October 1943, in which he justified these horrific orders:
I consider that in dealing with members of a foreign country, especially some Slav nationality, we must not start from German points of view and we must not endow these people with decent German thoughts and logical conclusions of which they are not capable, but we must take them as they really are.
Obviously in such a mixture of peoples there will always be some racially good types. Therefore I think that it is our duty to take their children with us, to remove them from their environment, if necessary by robbing or stealing them. Either we win over any good blood that we can use for ourselves, and give it a place in our people, or we destroy this blood.
Thus were the children of Poland – at least the blond, blue-eyed ones who conformed to the Nazis' belief in 'good Aryan characteristics' – snatched from their families and transported to holding camps. Here trained 'racial examiners' set up offices to measure, prod and evaluate thousands of youngsters brought to them by the SS. Their assessment was final and unchallengeable. A ruling issued by RuSHA explicitly stated:
The racial sentence once passed... by an expert may not be altered by any office. The judgment of an expert is an expert diagnosis just like that of a physician.
The chosen children were then handed over to Lebensborn _._ In the box of files there were records: closely typed sheaves of paper recording the names of those taken from their families and shipped to the network of Lebensborn homes across Germany. I closed my eyes and imagined the scene: railway stations crowded with thousands of unaccompanied children, stuffed like cattle into trucks and carriages, without anyone to care for them on their journey.
At that point, strangely, a fragment of memory came to me: one I had never remembered before and yet which somehow I felt to be true. I was very little; I was on a train, sitting on the floor with another small child. We were sharing a blanket, each of us trying to pull it from the other. I lost the battle and, as the train ploughed on through long, dark tunnels, I felt terribly cold. Was it real? Had reading the accounts of what had happened in Poland awakened a recollection that had lain dormant in my subconscious for sixty years?
The more I read in the Nuremberg files, the more uneasy and unsettled I felt. In my mind's eye I saw the trial unfolding, its focus shifting from Poland to a little village in what was then Czechoslovakia.
Marie Doležalová was fifteen when she stood on the witness stand in 1947 and gave evidence about what had happened to her five years earlier. On the morning of 9 June 1942, ten trucks filled with SS and Gestapo troops rolled into Lidice, a farming hamlet near Prague.
Two weeks earlier, Czech partisans had assassinated SS- Obergruppenführer Reinhard Heydrich, Himmler's protégé and the man charged with ruling this corner of the Reich's conquered lands. Hitler demanded mass reprisals: the raid on Lidice was specifically ordered because the village was suspected of having links to the men who killed Heydrich.
Armed soldiers jumped out of their vehicles and rounded up Lidice's entire population. Every adult man – 173 of them, including Marie's father – was lined up and shot against the wall of a barn. Their bodies were laid out in seventeen rows in an orchard before the village was burned to the ground.
The women of Lidice, almost 200 in number, some heavily pregnant, were transported to Ravensbrück concentration camp. Their children were snatched from them: 184 youngsters were pushed into buses and transported to a former textile factory in Łodz. On Himmler's staff orders they were not fed, and were forced to sleep on cold dirt floors without blankets.
Once RuSHA's 'race examiners' arrived in Łodz, they assessed each child for signs of Aryan qualities. They 'failed' 103 children: of them, seventy-four were immediately handed over to the Gestapo for onward transportation to the extermination camp at Chelmo, seventy kilometres away. Here they were gassed to death in specially adapted killing trucks. Just seven children were selected as suitable candidates for Germanisation. Marie Doležalová was one of them.
When she arrived at the children's home, she found herself among many other children from different countries. She was forced to learn German and was punished if she was caught speaking Czech. Lebensborn eventually handed her over to an approved German family. Her foster parents were kind, giving her two new dresses to mark her arrival with them, but she was encouraged to forget where she had come from.
After the war ended, the handful of Lidice women who had survived the massacre and the concentration camp began searching for their missing children. A year later – just before she gave evidence at Nuremberg – Marie was reunited with her mother, who was by then dying. As she stood at her mother's bedside, she realised that she couldn't remember a word of her native language.
All of this Marie Doležalová told the judges at Nuremberg. As I read her testimony, I put myself in her place. Perhaps I too had come from a village burned to the ground by Himmler's troops, one of the so-called 'racially valuable' lucky ones saved from the death camps by the promise of blond hair or blue eyes. But if so, where exactly had I been stolen from? And was there any hope that, like Marie, I might one day meet my real mother before she died?
And then I found the lists. They were tattered grey sheets of foolscap, prepared by Lebensborn staff in 1944; almost sixty years later the type had faded, making them only just legible. Each was divided into four columns. The first column was an alphabetical register of names and, since the adjacent column showed birth dates from the early 1940s, it was clear that this was some kind of register of children. The third column was headed 'transferred to', and in the final line there was a date against each entry.
There were 473 children identified in these documents. Halfway down one I read the following:
Matko, Erika.
[Born on] 11.11.41.
[Transferred to] Oberst Hermann von Oelhafen, Munich, Gentzstrasse 5.
[On] 3.6.44.
I had found my original name.
It had to be genuine: not only were these official court records but the address shown for Hermann – and the date on which I was handed over to him – was correct.
I sat back, the list in front of me. I was surprised to find that I was not emotional: ever since I had received the letter from the Slovenian government telling me that I could not be Erika Matko, because that person was still living in the Rogaška Slatina area, I had felt lost. Now as I looked at this fading Lebensborn list, I felt my purpose and true identity coming back to me.
Accompanying the lists were two sworn statements by former Lebensborn staff who had been interrogated by investigators for the Nuremberg prosecutors. The first was a woman called Maria-Martha Heinze-Wissede, who had worked in the Lebensborn head office _._ On 9 August 1948, she had been shown the documents and identified the origin of some of the children. Erika Matko was one of them.
From the lists before me, I recognised the following names of Yugoslavian children...
... Erika MATKO
These children, I know only their files a little, since they had already been transferred to ... German families by Lebensborn.
As was clear from the documents, they were called 'bandit children', and Lebensborn took them over from Volksdeutsche Mittelstelle [VoMi] ... As far as I remember, the Lebensborn took these children from a Volksdeutsche Mittelstelle camp in the district of Bayreuth.
My heart raced. There it was in black and white: I had been brought from Yugoslavia and passed to Lebensborn by this VoMi organisation.
A quick search revealed VoMi to have been another of the confusing and overlapping bodies answering to Himmler: it was set up before the war, ostensibly to manage the interests of the _Volksdeutsche_ – ethnic Germans who lived outside the borders of Nazi Germany. But once Hitler's armies overran Poland, Czechoslovakia and Yugoslavia, VoMi settled half a million German volunteers in the conquered territories, simultaneously shipping out or imprisoning the rightful occupants. It was essentially a precursor to what we now call ethnic cleansing, and the organisation's involvement in my origins did not bode well for the fate of my biological family.
Who were they? There was a clue, a thrilling hint, in Maria-Martha Heinze-Wissede's affidavit. The other Yugoslavian youngsters and I were listed as 'bandit children'. In Nazi terminology this meant partisan fighters. I felt a surge of pride: our fathers were rebels, they had opposed the Nazi occupiers. How brave they must have been. I doubted that in their position I would have found the courage to fight back against Hitler's armies.
The second witness statement was from a former Lebensborn clerk called Emilie Edelmann. She had joined the organisation in 1939 and worked within it until the very end, rising to a position that gave her responsibility for the care of children being readied for placement with foster families. On 3 April 1948, she too told her American interrogators that children had been snatched from Yugoslavia, and filled in a few of the missing details of my journey to a Lebensborn home. She described these kidnapped children as _Südost-kinder_ – VoMi-speak for those taken from the south-eastern territories conquered by Germany.
I re-read everything obsessively, desperate to be certain. It was unequivocal: here in the Nuremberg files was definitive proof that I had been one of at least twenty-five infants who in 1942 and 1943 were kidnapped and transported from Yugoslavia to the Fatherland. I had been taken to a VoMi holding camp at Werdenfels in southern Germany before being shipped on to the Sonnenwiese home at Kohren-Sahlis and then eventually given to the Oelhafen family in Munich.
I had only one question left to answer before I left Nuremberg: how did the trial end – what punishment was meted out to the senior Lebensborn officials in the dock?
Astonishingly, although most of the top RuSHA officials had been convicted and sentenced to lengthy terms in prison, the four Lebensborn defendants had been acquitted of crimes against humanity and war crimes. The three men had been found guilty of membership of the SS – as a woman, Inge Viermetz was excluded from its ranks – but none spent a single additional day behind bars. Despite all the evidence presented to them, the judges had reached the incredible conclusion that Lebensborn had been no more than 'a welfare organisation'.
I was furious. I had read the evidence and I had listened to the harrowing accounts from my fellow survivors at the meeting in Hadamar. I knew the truth now, and it made me more determined than ever to discover more about how I had originally fallen into Lebensborn's clutches.
Whatever the government of Slovenia thought, I had definitely come from there. I just had to prove it.
## THIRTEEN | ROGAŠKA SLATINA
_'I would be very, very grateful if you could answer some questions about your childhood... I am not asking out of mere curiosity...'_
LETTER TO ERIKA MATKO, JUNE 2003
I found a large number of people throughout Germany with the surname Matko. I wrote to each address, asking if they knew anything about my background or that of Hermann and Gisela von Oelhafen. It was a succession of shots in the dark but, to my surprise, letters began trickling back: each thanked me for contacting them and wished me well but none of them were able to help with my investigation.
In the meantime, Josef Focks was busy. The 'Father Finder' was not deterred by the results of my letters to the German Matkos; instead he expanded his geographical search.
Josef Focks is one of the people without whom I would never have found the truth about my past. He was a former army officer who had been seconded to NATO in the 1980s. During a posting to Norway he first encountered the stories of children fathered by German troops and the plight of those who had been born in the Lebensborn programme. Moved by their pain and the sense of shame that had blighted their lives, he offered to help them track down their families.
From the outset, he ran into the problem that the Lebensborn officials had deliberately concealed many of the fathers' names. Ironically it was the logistical difficulty of genealogical searching in the days before the Internet and online records that led him to find innovative solutions. He made use of local contacts (he found that taxi drivers were good sources of information), dug into obscure archives and the libraries of old newspapers, and even visited cemeteries to examine the names carved into headstones. Gradually he developed a way to unlock the puzzle.
By the time I met him, he had taken on more than a thousand cases, not all of them Lebensborn children, successfully tracing family members in most of them. His investigations had led him across Germany and as far afield as America and Australia, and his office in Bonn was stuffed with innumerable files, each one bulging with paper. All of this he did without charging a penny for his time: he had long since retired from the army and lived on his state pension, helping people like me for nothing more than the reward of easing our pain. I cannot thank him enough.
It was Josef who found the most promising Matkos. He had tapped up one of his contacts, a woman whose mother had been taken from Yugoslavia by the Nazis as slave labour: with her help, he discovered contact details for several Matkos still living in or near Rogaška Slatina. They seemed to be an extended family: some were my age or slightly older, others clearly a generation younger. Most promising of all, one of them was named Erika.
Josef unearthed an address for her and also the telephone number for a Maria Matko, who, he thought, might be a relation. We agreed that he would phone Maria and that I would write to Erika.
I sat down at the computer and thought about what to say. It was not an easy letter to compose: I knew nothing about this woman nor the country in which she lived. In the end, I decided to speak openly about my need to discover the truth about my past. I told myself that since many of the unrelated Matkos in Germany who I had contacted out of the blue had taken the trouble to reply even though they could not help me, this person bearing my name and who lived in the place I knew I came from might also be moved by my plea for help.
Osnabrück, Germany
16 February, 2003
Dear Mrs Matko,
I am writing to you today about a very personal matter and hope that you can help me. The problem, of course, is that I do not speak Slovenian and I cannot hope that you speak German. But I hope that there is somebody who can help you to translate my letter.
For some years I have been researching my biological parents and during this research I found out very strange things, which have made me very anxious and disturbed, but I know that I must keep going on.
My foster parents picked me up from the Lebensborn home 'Sonnenwiese'. There I was given two vaccination documents in which my name is shown as Erika Matko, born in St Sauerbrunn. I don't have any other documentation about my early life. I don't know the circumstances which brought me to the home. My foster parents didn't give me any information.
Ten years ago I didn't even know that I was a Lebensborn child. The Red Cross couldn't find out any information about my identity. I asked Dr Georg Lilienthal, who is a researcher about Lebensborn and has published a book about the subject.
He gave me the idea that maybe I'm a member of the group of kidnapped children, and my origins lie in Yugoslavia.
Now in the course of my research I have discovered you. I don't ask from curiosity but I only want to know how this double identity has occurred. Did you ever live in a Lebensborn home or were you lucky enough to spend your whole life in Rogaška Slatina?
I would be very, very grateful if you could answer my questions about your childhood.
Best wishes, Ingrid von Oelhafen
There was nothing more I could say or do. I posted the letter, hoping that something in it would strike a chord with this other Erika Matko.
In the meantime, Josef had made progress. He got in touch with Maria Matko and they had a good conversation via phone with the help of a translator, as he didn't speak Slovenian and she could not understand German. She was apparently my age and had spent her whole life in Rogaška Slatina.
From what Maria told Josef, she was the matriarch of the extended Matko family, which had once been involved with the anti-Nazi partisan movement. She was slightly hazy on the details, remembering only that one member of the family had been executed by the Nazis and that she had heard a story, long ago, that three children might have been kidnapped in the early 1940s. It sounded close to the likely Matko family history that I was looking for, and even more promising was the news that she knew the mysterious Erika very well.
But it was the final piece of information that threw me. Herr Focks had persuaded Maria that she should meet me – and that she should bring Erika with her. I was instantly nervous: I wanted desperately to go, but the prospect terrified me. What if these were the 'wrong' Matkos, and the trip turned out to be a wild goose chase? I would, I knew, be devastated. And even if these people were my relatives, that didn't mean a meeting would go well; perhaps they would be hostile or somehow resent me, which would be even worse.
The Father Finder was having none of it. He pushed and pushed until I agreed to his plan. This involved flying first to Munich, then on to Ljubljana, the Slovenian national capital. From there I would find a taxi to take me the eighty kilometres to Celje, the main town in the region.
There was an additional reason for going: every autumn a handful of survivors of the Nazis' kidnapping and deportation programme met in Celje. Josef had arranged for me to join them before heading on the next day to Rogaška Slatina, where I was to meet Maria in a cafe. Nor was I to go alone: he had asked a friend of his who spoke Slovenian to accompany me as my translator.
I knew precious little about Slovenia or its history at that point. I didn't even know how the country had come into being after the breakup of Yugoslavia. As the date for my departure approached, I started reading up, hoping to gain some insight into what life might have held for me had I not been stolen for the Lebensborn programme _._
Yugoslavia had been one of the first countries to overthrow its German conquerors. Under the leadership of Josip Tito, the partisans were the most effective anti-Nazi resistance force in occupied Europe; by the middle of 1943 their activity had grown from running sporadic guerrilla raids to causing major military defeats and inflicting heavy casualties that Hitler's army could ill afford. By the start of 1944 they had managed to push the Wehrmacht out of the Serbian regions; a year later all German troops were expelled.
They achieved this with only limited support from the Soviet Union and although Tito's post-war regime was unashamedly communist – a single-party state with little tolerance of dissent or democracy – for the next twenty-five years the country was the most independent of all the Soviet Union's satellites behind the Iron Curtain. It began to distance itself from Moscow in 1948, determined instead to forge its own brand of socialism. It felt free to criticise the Kremlin and the West in equal measure and was one of the founders of the Non-Aligned Movement – the group of states which defiantly refused to ally themselves with either side in the Cold War.
But there were always tensions beneath the surface. The new nation was welded together from six separate and frequently hostile republics: Serbia, Croatia, Bosnia-Herzegovina, Macedonia, Montenegro and Slovenia. Each of these had very different ethnic, religious and political histories. What held them together was the inspirational figure of Josip Tito. His death in 1980 precipitated an unravelling of the whole country.
Serbs had always been the largest ethnic group in Yugoslavia and prior to the Second World War had been the most dominant force in the Kingdom of Yugoslavia. With Tito gone, Serbian communist leader Slobodan Milošević sought to restore this historic supremacy. The other republics, especially Slovenia and Croatia, denounced this power grab but were unable to stop it.
Industrial action by ethnic Albanian miners in Kosovo in 1989 was the spark that ignited the simmering tension. Slovenia and Croatia supported the Albanian miners and the strikes turned into widespread demonstrations demanding a Kosovan republic. This angered Serbia's leadership, which proceeded to use police force against the miners before sending in the Federal Army to restore order.
In January 1990, an extraordinary Congress of the League of Communists of Yugoslavia was convened. Since the country was a one-party state, this was effectively the ruling body for all of the Federal Republic. The meeting degenerated into an argument between Slovenia and Serbia about the future of the nation: in the end the League dissolved itself. The writing was on the wall for the future of Yugoslavia.
The immediate outcome was a constitutional crisis. Fuelled by a toxic rise in ethnic-based nationalism and inspired by the fall of communism across the rest of Eastern Europe, five of the republics demanded independence and an end to Serbian dominance. The stage was set for war.
What followed was Europe's worst conflict since the Second World War, and one that once again raised the spectre of crimes against humanity. Over the next decade, at least 140,000 people died in the fighting. Hundreds of thousands more – possibly millions – endured ethnic cleansing, rape as a weapon of war, concentration camps and mass bombing.
The first of these dirty wars broke out in Slovenia. In December 1990, 88 per cent of the population voted for full independence from the disintegrating federal republic, knowing that to do so would inevitably lead to an attempted invasion by the Serb-dominated Yugoslav People's Army. The fledgling Slovenian government secretly reorganised its antiquated territorial defence force into a well-trained and equipped guerrilla army and the partisan resistance which had once driven Hitler's troops from the country was effectively reborn.
The Slovenes knew that they stood no chance in a conventional battle: the YPA was simply too big and too powerful. So the country prepared for a campaign of guerilla warfare – a return to the resistance tactics of blowing up bridges and small close-quarter attacks in the towns and villages of their nascent nation.
At the same time, Slovenia sought help from the European Community and the United States. Neither was prepared to recognise the country's independence since they found it more convenient to deal with a single federation rather than a series of small states. The rebuff emboldened the Serbs and made a full-blown civil war inevitable.
The first shot was fired by the YPA on 27 June 1990 in the small village of Divača, just seventy-five kilometres from Ljubljana. That same afternoon, Slovenian soldiers shot down two Yugoslavian army helicopters.
Over the next ten days, the fighting moved westward towards Ljubljana, then on past the capital and into the eastern heartland around Celje and Rogaška Slatina. A ceasefire was announced on 6 July: Slovenia won its independence, though at the cost of at least sixty-two deaths and almost 330 wounded. By contrast with the ensuing conflicts in Croatia, Bosnia and Kosovo, this was a small war, but it was the first time since the Nazis had been expelled that Slovenians had their freedom. In some rather inexplicable way, I felt proud.
At the end of September 2003, I flew to Munich. Josef Focks had arranged for me to meet his translator at the airport so we could fly on to Ljubljana together. But by the time our flight was called he had not arrived and I boarded the plane alone. I was already nervous about who or what I would find in Slovenia and, since I spoke only German, I felt vulnerable and exposed. Fortunately the translator managed to get a message to the plane and asked a stewardess to tell me that he had been held up in traffic and would catch a later flight and meet me at Ljubljana.
I waited all day in the airport. I had no signal on my mobile phone, no one seemed to speak German and I could not work out how to use the local payphones. All I could do was sit and hope that my contact would turn up.
By the time he finally arrived it was mid-evening and I was in something of a state. But I had no time to dwell on my feelings: there was to be a meeting of the stolen Slovenian children that evening in the primary school at Celje. This town, I gathered, had been called Cilli during the German occupation, and had been both a centre for partisan resistance and the site of Nazi reprisals.
As we drove through the countryside, I looked out of the window, trying to take in the landscape, the land of my birth. I had wondered beforehand if seeing it for the first time in almost seventy years would prompt some memories: it felt disappointing to find that it did not.
I knew very little about the people I was meeting that evening and was surprised to discover that the event in Celje was a very different gathering of stolen children from my first encounter. They had started searching for one another as far back as 1962, determined to tell their stories to the (then) Yugoslavian public.
The men and women I met that evening were all in their eighties – between ten and fifteen years older than me. They were the leaders of what had become an officially supported group of survivors, and their accounts filled in some of the gaps in my knowledge.
Throughout 1942 a total of 654 children, from babies to eighteen-year-olds, were snatched from their families by the Nazis and shipped off to a succession of camps across the Reich. Most of the older ones – at least those who survived the rigours of slave labour or attempted Germanisation – had been brought back home at the end of the war. By the time I arrived in Celje, only around 200 were still alive.
Despite their age, their memories were strong and they were determined that the world should not forget what had been done to them. Two speakers stood up in the primary school to give testimony. I sat silently: even if I had felt able to trust the translator to speak for me, I knew too little to make any worthwhile contribution. But my presence had been noted. After the meeting was over, three people came over to speak to me. Each had been stolen from Celje in August 1942 and, to my complete astonishment, each one said they recognised me.
The first woman was seventeen when she was caught in a round-up of children from the area and held for two days in the primary school by SS troops. The children ranged from small infants to eighteen-year-old teenagers: because they were separated from their mothers, the older ones were ordered to look after the babies. This warm and emotional elderly woman told me that the babies were constantly crying. Her task had been to clean the smallest ones, and she specifically remembered washing me.
I couldn't believe what I was hearing. I would have been less than a year old when the round-up happened; how on earth could someone recognise me more than sixty years later? But somehow this woman believed that she did. It was astonishing. I had come to Slovenia hoping only to find some evidence that tied me to the country. Instead I had come face to face with someone who claimed that she could place me in Celje on the day that I was stolen – and who said she had actually held and looked after me when I was a baby.
On reflection it seemed more probable that the woman had been told I was coming and this had prompted her to remember me. But either way it was a connection.
The next person I spoke to was a man around the same age. He had been fourteen on the day of the kidnapping, and he was able to tell me a bit more about what had happened to the stolen children on the day after the round-up. He told me that we were transported 150 kilometres north to the holding camp at Frohnleiten in Austria. He was adamant that he had seen me there, and that the name he knew me by was Erika Matko.
Another elderly lady chimed in then, confirming what the man said. She had been kidnapped from Celje and shipped to Frohnleiten: she too remembered me there and that my name was Erika Matko.
I suddenly felt intensely happy. After so long, after so many disappointments, I had first-hand evidence of who I had once been and where I had come from. It was an extraordinary sensation.
I did not have time that night to press my new contacts for further details of the kidnapping. I would have to wait to find out more about the events of August 1942 and how I had been caught up in them.
The next day, we moved on to the Maribor region. Ahead of my meeting with Maria, Josef Focks had arranged for me to meet two other people whose surname was Matko. He had also equipped me with sterile test tubes and cotton buds so that if these Matkos were willing, I could take samples of their saliva, which we would have analysed to see whether we shared any genetic similarities. In this way I could discover whether we were related.
Our first stop was a village near Rogaška Slatina. As we drove closer, I looked out of the window at the green and densely wooded landscape, hoping to see something familiar, but I didn't recognise anything. The village was clearly very poor: the woman I was due to meet was eighty and lived with her forty-year-old son. Both seemed puzzled by my visit but the old lady agreed to give me a saliva sample. Her son was more hostile and refused. Neither of them were able to tell me much about their family background and I left thinking that if we were related it was probably only distantly.
The next Matko was a thirty-year-old hairdresser. She willingly gave me a sample for DNA analysis but, again, had little information to help me on my journey.
Finally it was time to meet Maria Matko. Herr Focks had arranged our rendezvous in a little cafe in Rogaška Slatina, and she had promised to bring the mysterious other Erika with her. But the moment I walked in, I could see that Maria was alone. I felt a sharp pang of disappointment.
Maria herself turned out to be warm and helpful, and a vital link in unlocking the chains of my past. She was seventy-three years old and not a Matko by birth; she had married into the family. Her husband was called Ludvig and he had two sisters: Tanja, who was older than him, and Erika, who was younger. Both Ludvig and Tanja were dead, but Erika was still alive, if rather infirm. Maria said that in the end she had been unwilling to meet me.
It was plain from our conversation that none of the extended Matko family thought I was related to them. Maria was the most open-minded, but even she was sceptical. At that point I was beginning to share their doubts.
But then Maria gave me the details that once again raised my hopes. The parents of Ludvig, Tanja and Erika had been called Johann and Helena – the very names I had been given three years before by the archivist in Maribor. What's more, Johann had been imprisoned by the Nazis for resistance activity: that seemed to fit what Georg Lilienthal had told me about my background.
Over the preceding years there had been a pattern to all my attempts to discover the truth: one piece of new information would emerge to lift my spirits and make me believe that I could find the answers I sought. But it would always be followed by a letter or conversation that dashed those hopes and sent me back into despair. And so it proved that day in Rogaška Slatina: no sooner had my expectations been raised than they were lowered once again. Maria produced two photographs. One was of Helena, taken in 1964: she was looking directly at the camera, a solid-looking woman with dark hair and a strong but kindly face. The second picture was of Erika, and my immediate reaction was that she looked just like Helena. That suggested she was Helena's daughter – and if so, then surely it meant that I could not have been.
When we parted, I felt a little depressed. The only thing buoying my spirits was that Maria agreed to meet me again, and promised to talk to the other members of the Matko family to see if they would cooperate with me.
The next day I went to her apartment. She was babysitting her granddaughter, and talked openly about the family history. She told me that Ludvig, Tanja and Erika had lived with their parents until Johann was arrested by the Nazis and imprisoned in a concentration camp. In the early summer of 1942 he had been released and returned to the family home, after which point her knowledge ended. Johann's brother, Ignaz, had not been so lucky: he too was a partisan and arrested by the Germans, but he was shot by a firing squad.
During our conversation, Maria's son Rafael joined us. Rafael was in his forties, solidly built and balding. Although he was polite, he clearly did not believe I was related to them. It was obvious that the family was close-knit and protective of one another.
The one exception was, apparently, Erika. Maria told me that her sister-in-law had never married, though she did have a son, and had been so ill throughout her life that she had never worked. It was plain that for all the family bonds – Erika was usually invited to Sunday lunches – the two women weren't particularly close.
By the time I took my leave, I had convinced myself that although this family seemed to share some remarkable similarities with the Matkos I was searching for, they were probably not my blood family. Nonetheless, before I set off I explained about my DNA testing plans. Rafael kindly agreed to give me a saliva sample and, after some hesitation, so did his cousin Marko.
My final appointment was in Maribor. I went to the museum, which maintained a special section dedicated to the remembrance of what the Nazis had done in Slovenia. In addition to rounding up and executing 'bandits' – partisan fighters like Ignaz and Johann Matko – all Slovenian books were burned, the language was banned and anyone caught speaking it was severely punished. Himmler's plan to subjugate the local population was put into effect brutally and efficiently.
Perhaps once I would have been shocked by what I saw in the museum, but everything I had learned over the previous three years about Lebensborn and its operations had inured me to the routine cruelty of the Nazi occupation. Instead I was preoccupied by trying to process what I had discovered during my four days in Slovenia. I had arrived uncertain about how I fitted into the country's history in general, and into the story of the Matko family in particular. With each contradictory new piece of information I had, by turns, been convinced that I was Erika Matko from Rogaška Slatina and then certain that I could not be her. By the time I got back on the plane in Ljubljana, I was completely confused.
I had been away from my physiotherapy practice in Osnabrück for too long. My patients needed me, and I needed them. It was time to go home: to be Ingrid von Oelhafen once again.
## FOURTEEN | BLOOD
_'Scientific analysis is 93.3 per cent certain...'_
DNA TEST RESULTS, OCTOBER 2003
How do we put together the jigsaw puzzle of our lives? It is not easy, even when none of the pieces are missing and we have the picture on the box to compare it with. How much harder when we lack the solid, clear-cut corners to build out from.
This was how the story of my childhood looked to me when I arrived home from Slovenia. I had dozens of individual pieces but they were oddly shaped, sometimes overlapping, sometimes contradictory. There seemed to be no way to fit them together to reveal the full picture.
I couldn't decide whether or not to believe the survivors of the Celje kidnappings, so convinced that I was the Erika Matko they remembered. Surely it wasn't possible that someone would recognise in the face of a seventy-two-year-old woman the features of a nine-month-old baby they had known only in the most traumatic of circumstances? Plus their accounts were at odds with the reaction of Maria Matko and her son: they plainly did not believe I was part of their family. And always in the background was the shadowy figure of the other Erika, who had still not replied to my letter and who had deliberately avoided meeting me in Rogaška Slatina.
The answer, I hoped, lay in the swabs and test tubes I had brought home with me. Saliva analysis would provide genetic fingerprints, which are as reliable as a blood test in determining family relationships. The irony of this was not lost on me: the Lebensborn experiment had been based on the Nazis' belief in blood as the determining factor of human worth. Himmler's obsession with blood and bloodlines was the reason I had been plucked from my family – whoever they were – in Yugoslavia and reborn as a German child. It had shaped the course of my life from that day onward. Now I was trying to use it to unravel the tangled web Lebensborn had spun.
The saliva swabs had not been given without a great deal of anxiety. There had been a dispute within the Matko family about me taking them: some of the younger members of the family had been adamantly opposed to the idea of scientific tests to establish whether I was one of them. They worried, I think, primarily about the stress this could cause to Erika, who was not in good health. Others, though, believed that it was important to find out one way or the other. After much discussion, I had been given the family's blessing to have their samples analysed. I set about finding a laboratory to test them for me.
Only at this point did I discover that it was not going to be easy – or cheap. The science of DNA testing really began in 1985 and was then both rudimentary and the exclusive domain of law enforcement authorities. Although it had since been refined and made more commercially available, it was still expensive.
All of the cotton buds I had collected in Slovenia had to be processed so that the individual DNA of each person could be isolated; I also gave a sample for comparison. Although 99.9 per cent of human DNA sequences are the same for everyone, there are enough differences to make it possible to distinguish one individual from another. What the scientists would look for were places in these sequences called 'loci'. Where two people are related by blood, these loci are very similar to each other; in samples from biologically unrelated donors, they look completely different.
I used my savings to pay for the tests. It meant tightening my belt and not taking holidays for the foreseeable future, but there was simply no alternative. I carefully parcelled up the swabs and sent them off to the laboratory in Munich. It took several months for the results to come back. What they revealed was both the answer to the big puzzle and a new mystery.
I looked first at the analysis of the samples provided by the hairdresser and the old lady. As I had suspected, neither had any genetic relationship to me. I allowed myself a guilty feeling of relief: it was clear that the old lady lived in some poverty, which was sad to see. All along I had hoped that my biological family had not suffered; it was painful to imagine that my real mother might have had such a hard life.
The next set of results were for Rafael Matko, the son of Ludvig and Maria. When I read them, my guilty feelings were transformed into pure happiness.
Scientific analysis shows that Ingrid von Oelhafen and Rafael Matko are relatives in the second degree... it is 93.3 per cent certain that Ingrid von Oelhafen is the aunt of Rafael Matko.
There it was – the evidence I had been seeking for so long. If I was Rafael's aunt, that meant I was Ludvig's sister, and therefore the daughter of Johann and Helena Matko. I had found the vital piece of the jigsaw puzzle: I was unquestionably Erika Matko from St Sauerbrunn/Rogaška Slatina.
It is hard to convey what this news meant to me. Unless you have lived your life as I had, haunted by never knowing who I was and where I came from, I don't think it is possible to fully understand the overwhelming elation: it was like being liberated. I felt as though the weight of sixty years had been lifted from me.
Then, as always seemed to happen, the other test results dragged me back into uncertainty. Analysis of the saliva sample given by Marko Matko – Rafael's cousin – had yielded what appeared to be a completely contradictory outcome: it showed that to a 98.8 per cent degree of certainty I was not related to Marko.
This simply didn't make sense. I looked again at the Matko family tree to remind myself of what I already knew: Johann and Helena had three children, Tanja, Ludvig and Erika. Ludvig's son was Rafael and the tests proved that I was his aunt: therefore I must be Erika. But the same set of results showed that Tanja's son, Marko, was not my biological relative. No matter how I arranged these jigsaw pieces, I could not get them to fit. If I was Ludvig and Tanja's sister, why wasn't I Marko's aunt? The Matko family seemed to be surrounded by secrets
The person most likely to have the answers was the mysterious other Erika. She was the last living first-degree blood relative – at least in theory. She had been raised by Helena and Johann and had grown up with Tanja and Ludvig. But she was still ignoring me and I had to assume by this point that she was not prepared to help me. It was immensely frustrating; I could not understand why she was being so obstructive.
I tried to focus on the positives. I knew for certain that I was – or had once been – Erika Matko, daughter of Johann and Helena, and that Ludvig at least was my brother. Quite how Tanja and the other Erika fitted into the picture was still a mystery, but at least I could be certain who my biological parents were. That was a very real comfort. But I was still no closer to solving the puzzle of how I had been taken from them. It would be another four years before those pieces fell into place.
## FIFTEEN | PURE
_'To be rooted is perhaps the most important and least recognised need of the human soul.'_
SIMONE WEIL, _T HE NEED FOR ROOTS_, 1949
Wernigerode lies on the eastern edge of the Harz mountains in the German heartland of Saxony. It is a quiet, picturesque town, in which half-timbered houses flank the Holtemme river and horse-drawn carts still rumble through cobbled streets. It looks and feels like the setting for a fairy tale: the sort of place where the Brothers Grimm might have based their stories.
But Wernigerode has another, altogether less cosy history. On the top of a steep hill just outside the town lie the ruins of Heim Harz, one of the network of Lebensborn homes.
In late summer 2005, I drove to Wernigerode to take part in the creation of a new organisation. Lebensspuren ('Traces of Life') was the first formal attempt by those of us who had been born or reared under Himmler's Master Race programme to band together: our aim was both to provide much-needed support and to begin the process of bringing Lebensborn into the public gaze, free from the prejudice and shame that prevented others from understanding what had been done to us in its homes.
It was a long journey. The road stretched more than 260 kilometres through the woods and fields of central Germany: as I drove I had time to reflect on how I had got here. It was more than five years since I had begun the search for my roots. I had learned so much during that time, and yet I still knew relatively little.
In the ten months since I had received the scientific tests that proved who I was, I had made no real progress in discovering how I had been brought into the Lebensborn programme, nor did I fully understand the extent of the experiment itself. In this I was far from alone. The meeting in Hadamar had been a first step for the handful of Lebensborn children to come together and share our stories. Each of us had a little part of the overall jigsaw, but even together we could not complete the full picture.
The title of our new organisation was a deliberate twist on Lebensborn: where that had been, in Himmler's vision and language, the Fount of Life, our association was to be the way for its survivors to explain it. But I was also aware of a possible play on words: that middle syllable, 'pur', was an acknowledgement of the Nazis' obsession with racial purity which lay at the root of all our problems.
We chose a particular quotation to head our articles of association:
Uprootedness is by far the most dangerous disease to which human society is exposed. Whoever is uprooted, uproots others. Who is rooted himself, doesn't uproot others. To be rooted is perhaps the most important and least recognised need of the human soul.
It came from the French philosopher and activist, Simone Weil. She had fought fascism in Germany in the early 1930s and later as a volunteer on the Republican side during the Spanish Civil War. In 1943 she wrote a book called _The Need for Roots_ , examining the social, cultural and spiritual malaise undermining western society; the quotation we selected from it perfectly encapsulated the story of our lives.
I liked Guntram Weber the moment I met him at the Lebensspuren meeting. We were staying in the same guesthouse and we shared a common interest in working with young people: Guntram was a creative writing teacher specialising in helping disadvantaged youngsters. He was two years younger than me, but his face bore witness to the pain he had experienced throughout his life. When he stood up to tell his story, his eyes filled with tears as he described his struggle to find the truth about his origins as well as the overwhelming desire to run away from it.
He had grown up in an outwardly normal post-war German family, living with his parents and two siblings – an older sister and a younger brother. But behind closed doors it was a different story.
As a child I remember sensing that I wasn't quite normal. Relatives seemed to treat me awkwardly and it gradually became clear that the man I called Father was actually my stepfather. Of course I wanted to find out who my real father was, but the subject was taboo in our house.
Relatives had been well drilled by my mother to hide the truth behind vague statements. 'It was the war,' they would say. 'Things were very confusing. We didn't see much of each other – you will have to ask your mother.'
It wasn't until he was thirteen that Guntram's mother agreed to discuss the issue.
'Well, Guntram,' she said, 'You are old enough to know the truth about your father now.' Then she gave me a name, told me when his birthday was and that she had married him in 1938 on a beautiful sunny day and that they had driven to church in a horse-drawn cart.
During the war he had been a truck driver for the Luftwaffe, far away from the front, who had died driving over a landmine in Yugoslavia. She added that he certainly wasn't involved in killing anyone.
But there were no documents and no photos of this man, and when I pressed her, my mother she said she didn't want to say any more about him because it was too painful.
It was a plausible story. Guntram was a little suspicious, but the climate of German society during the 1950s actively discouraged awkward questions. Many children were told lies about what their parents did in the war and, as I knew from my own experience, it wasn't the 'done thing' to challenge them.
Curiosity and uncertainty gnawed away at Guntram. Sometimes he thought about confronting his mother about his doubts, but he never managed to do so. The lack of any photos or documents led him to question the story of the non-combatant Luftwaffe driver: instead he began worrying that his father had been a Nazi and that this was the reason for his family's secretive behaviour. He began inspecting his facial features in the mirror and poring over history books in the school library, searching for photos of soldiers who could be his father or for women concentration camp guards who looked like his mother. For one terrible period he even convinced himself that Joseph Goebbels, Reich Minister for Propaganda and one of Hitler's most devoted followers, might be his father. A year or so later he made a disturbing discovery.
My mother had a strongbox in the bottom right-hand corner of her wardrobe. One afternoon when she was out, I decided to look in. I had terrible qualms about doing this; I knew I was breaking the trust between us and she was my only security in the world. But I felt I had no choice.
Inside the trunk, Guntram found the first clue to his identity: a small silver cup. It bore a profoundly unsettling inscription.
We were a fairly poor family at the time. Like many others, my mother had lost everything during the war, so to find a silver object in the house was extremely unusual. I picked it up carefully and discovered my Christian name on it; my second name, though, was shown as 'Heinrich'. And then I turned it over and saw the writing on the other side. It read: 'From your godfather, Heinrich Himmler.'
Guntram desperately wanted to ask his mother about the cup. But, like Gisela with me, she was secretive – and he knew how badly she would react to the revelation that he had been rummaging through her things. It remained an unspoken and unsettling mystery.
In 1966 he first heard the word Lebensborn. His older sister needed her birth certificate in order to get married and was surprised to discover that she didn't have one. When she questioned their mother she was obstructive and told her daughter she didn't know where it was.
An enquiry at her place of birth turned up the unexpected news that Guntram's sister was the illegitimate child of an army officer. Her records were still intact and showed that she had been born in a Lebensborn home. That led to the revelation that Guntram had also been a Lebensborn child. Rather than question his mother further though, he chose instead to get away, and moved to the United States. He stayed there for eight years and had a family of his own, putting aside questions about his past.
But when his partner died in a car crash, he returned to Germany with his son. Before long the uncertainty about his roots began nagging at him again and finally, in 1982, he decided to confront his mother during a long car journey where, as he put it, 'she could not escape from me'. He pulled off the road and forced his mother to talk to him.
My mother was angry but she uttered three sentences that I will never forget. First she said: 'I don't want to talk about that.' Then she tried to stop me digging into my past: 'People will throw dirt at you,' she told me. Eventually, when she saw that I would not be put off, she made a promise to write the whole story down for me. I believed her and felt better, trusting that she would give me the truth.
Sadly she never did. For Guntram's mother the truth was simply too difficult to speak about: she died two years later, taking her secrets to the grave and leaving Guntram both frustrated and angry. As she had once told him in an unguarded moment: 'The relationship between mother and child is a power struggle.' Guntram felt he had been powerless in that struggle.
It wasn't until 2001, when he was fifty-eight, that Guntram discovered who his father was – not, as his mother had told him, a young soldier who died honourably, but an SS major-general who oversaw the deaths of tens of thousands of people while stationed in what is now western Poland. He had been convicted of war crimes and sentenced to death by a Polish court in 1949, but had escaped to Argentina, where he died in 1970.
My father was a war criminal. He was a man who allowed himself everything. And the SS enabled him to live that way. I assume my mother fell in love with a powerful military man. He died peacefully and at his funeral his old comrades stood beside his grave and raised their right arms in the Nazi salute. I knew then that a racist is always a racist.
There was a bitter irony to Guntram's description of his life. As a Lebensborn baby, his 'racially pure' genes were supposed to have ensured that he grew up strong and confident – a future leader of the Master Race. Instead he had suffered for more than sixty years from feelings of low self-esteem, loneliness and uncertainty. The only thing that helped, he told us in Wernigerode, was finding other Lebensborn children.
It has been a huge relief for me, although I haven't been able to shake this feeling of inadequacy. Maybe in ten years it will be gone. It's important that other children in Germany and abroad hear about this group because it could help us all.
I agreed wholeheartedly with Guntram: Lebensspuren needed to be publicly visible so that other men and women who had been through the Lebensborn programme could make contact with us and perhaps find some solace. But in 2005 our group was not ready to do that: we met privately – partly due to the sense of shame still attached to our past.
To some extent Helga Kahrau exemplified the dichotomy we all faced, that of needing support and acceptance while simultaneously struggling with the painful reality of her birth. A tall and forceful woman, with her blond hair dyed a striking red, Helga had been born into the heart of the Nazi regime. During the war her mother, Margarete, had been a secretary in the offices of Hitler's top aide, Martin Bormann, and of Joseph Goebbels. She had memories of growing up in privilege and comfort, surrounded by important-looking men in crisp uniforms.
In the decades that followed the end of the Third Reich, Margarete refused to talk to Helga about the war, much less tell her about the father she had never known. It was only after Margarete's death in 1993 that Helga began to examine her family's past. She was horrified by what she discovered.
Margarete was a fervent Nazi who barely knew Helga's father. He was a German army officer and they met in June 1940 at a party celebrating Hitler's conquest of France. They had a one-night stand, which left Margarete pregnant. She was a perfect candidate for Himmler's Master Race programme: politically committed, racially pure and expecting an illegitimate child from an equally Aryan German soldier. Nine months later, Helga was born in the main Lebensborn home at Steinhöring, outside Munich.
When Helga was three months old, Margarete left the home and returned to work in Goebbels' Propaganda Ministry. Helga was handed over to foster parents: her new father was a high-ranking Nazi official in the occupied Polish city of Łodz. Here, Helga believed, he helped oversee the gassing of thousands of Jews at the nearby Chelmno concentration camp.
I spent the first four years of my life raised and tutored by the Nazi elite. I was involved, in a fundamental way, with murderers.
At the end of the war, unlike many Lebensborn children, Helga was sent back to Munich to live with Margarete. Here she encountered the irony of Himmler's obsession with Nordic features. Although the city and its surrounding regions were the birthplace of Nazism, most Bavarians are dark-haired: the very racial characteristics which Lebensborn valued ensured that Helga stuck out.
I was big, blond and Aryan, different from the southern Germans, and everyone asked me where I had come from. I couldn't answer them.
The only document Helga possessed was a cryptic birth certificate from an 'SS Mothers Home'. It showed her mother's name but not that of her father. Nor was Margarete Kahrau willing to help her daughter understand. She deliberately concealed the truth, saying only that Helga's father had been a soldier who died in the war. She was also reluctant to discuss what she had done in the service of Hitler's Reich: like most Germans of her generation, Margarete preferred to forget about the Nazis.
When Margarete died in 1993, Helga began investigating. She discovered Nazi files that provided detailed information about her foster father and the crimes he committed in the service of the Final Solution. But there was nothing in the documents about her biological father. Then, in 1994, she received a phone call. The man told her that he had been a Wehrmacht officer in Paris, that he had met Margarete for a single night of passion. He was now suffering from terminal cancer and wanted to reach out to his daughter. It was a bittersweet moment for Helga: she had found her real father at last, but only as he was dying. She decided to make the most of the time they had left and devoted herself to nursing him round the clock.
Her father had enjoyed a very successful post-war career in property, which had made him a multi-millionaire. As his eldest child, Helga might have expected to inherit at least some of his estate. But when he died she encountered another legacy of her birth. Her father left no will: shortly after his funeral Helga received a letter from his lawyers stating that because she was illegitimate, she could legally inherit nothing.
Since then, Helga had found some solace in visiting her birthplace, the Lebensborn home at Steinhöring. But she never came to terms with her identity and worried constantly that people would assume that she, like her mother and stepfather, was a Nazi.
I grew up on the side of the murderers. Being a Lebensborn child is still a source of shame.
Shame – the word that has blighted the lives of so many of those who had been part of Himmler's plan to create a new Master Race. The more I heard from those who had been born into Lebensborn, rather than kidnapped to strengthen it, the more I felt lucky to have been one of the _Banditenkinder_ , the child of courageous partisans who fought against Nazi rule.
Gisela Heidenreich, a tall, strikingly Aryan-looking woman from Bavaria, was four years old when she first encountered shame: she overheard her uncle describe her as 'an SS bastard'. She was a family therapist, a fact I noted with interest. There seemed to be a common trait in Lebensborn children: by accident or design many of us had chosen careers in which we helped others overcome problems, while struggling with our own.
Gisela described vividly the confusion she had suffered throughout her life and the web of lies that blighted her childhood. Her mother was Emilie Edelmann, the Lebensborn secretary who had given evidence at Nuremberg and who had been responsible for finding foster parents to take children stolen from the occupied countries.
Like so many of those who had been part of Lebensborn, Emilie was secretive and deceitful. At first she had led Gisela to believe that she was not her real mother but her aunt. Later she admitted that this was not true and told her daughter that she fell pregnant following an affair with a married man. The SS packed her off to occupied Norway, to give birth in the Lebensborn home near Oslo. Several months later, Emilie brought Gisela back to Germany.
Throughout her childhood, Gisela's mother refused to answer any questions about the war. Only after Emilie's death did Gisela discover the depths of her mother's involvement with the Nazis. She found a bundle of love letters written to Emilie by Horst Wagner, the director of 'Jewish Affairs' in the Reich's Foreign Office and its link-man with the SS. In this role, he helped carry out the round-ups, deportation and extermination of both German and foreign Jews. Emilie's love affair with Wagner grew to the point where the couple considered formally making him Gisela's stepfather. Their relationship continued when the Reich fell and he was arrested by the Americans and held for trial at Nuremberg. But before he could be brought to justice, he fled down one of the infamous 'Ratlines' – clandestine escape routes for Nazi war criminals – to South America.
Shocking though the relationship was, it did not bring Gisela any closer to discovering who her real father was. She continued to search for him and, many years later, she eventually tracked him down. He was the head of the SS officer school at Bad Toelz in Bavaria. Her own reaction to their reunion both surprised her and helped her understand how so many Germans were able to live with knowledge of the crimes committed by the Nazis:
When I first met him it was on a station platform. I ran into his arms and all I thought was 'I've got a father'. In that instant I sanitised the person I knew my father was. And I never asked him what he did. My own reaction – that of an educated adult with knowledge of the Lebensborn programme – has helped me to understand how people in those days just put the blinders on and ignored the terrible things that were happening.
Gisela brought to our group a determination to rehabilitate the image of Lebensborn children. In part this was due to her experience of the postwar treatment of Norwegian babies born in the programme's homes. As I had heard at the first meeting in Hadamar, Norwegian hatred of the occupying German armies led to discrimination against the 8,000 children like Gisela born in its Lebensborn homes. At first the post-war government in Oslo tried to have all the children shipped to Germany. When that plan failed, many of them were locked away in mental institutions or children's homes.
Gisela believed that this hatred and persecution was driven by Norway's national guilt at being occupied, the shame of its leaders having collaborated with the Nazis and, above all, the wildly inaccurate rumours about Lebensborn homes being 'SS stud farms'. Three years earlier, the Norwegian government had quietly paid on average €24,000 compensation to each of the children it had victimised. Now, Gisela argued, the time had come to end the lies and the discrimination.
It's high time to tell the truth. There's been too much talk about Nazi babies, women being kept as SS whores and tall, blond people being bred. The Holocaust was about extinguishing so-called lesser races. Lebensborn was the reverse side of this coin: the idea was to further the Aryan race by whatever means were available.
What I have learned is that I, and every other Lebensborn child, have a feeling of deep uncertainty about our identity. This has to stop.
The stories of these 'pure' Aryan children were harrowing. But theirs was only half the picture: there were others, like me, at the inaugural Lebensspuren meeting who had been forced, not born, into Himmler's programme. Their accounts helped me understand how the process had worked.
What happened to Barbara Paciorkiewicz was typical. She was born in 1938 in Gdynia, near Gdansk in Poland. Her family name was Gajzler but, because her mother had died and her father had disappeared, she and her sister were separated, each sent to live with one of their grandparents.
Gdansk was in the part of Poland under German occupation and the Nazis had renamed it Danzig. In 1942, when Barbara was four, the Youth Welfare Office issued instructions for all children to be brought to the regional youth welfare office in Łodz. Her grandmother took Barbara and was forced to leave her there.
There were a lot of children in the centre. Each was measured – heads, chests and hips – and weighed on scales. Their faces were photographed from three angles. The people making these measurements were some of Himmler's race examiners: they were looking for racially suitable children to Germanise. Barbara had blond hair and looked Nordic. She was shipped off to a succession of different homes in Łodz.
Here I was subjected to more tests – always there were more tests. We were forbidden upon pain of punishment to speak Polish. Every one of us was crying.
Barbara's account of her kidnapping was similar to others I had heard. But her recollections of life in the Lebensborn home at Bad Polzin gave me new information about what my own experience might have been like.
This is where my memories truly begin. I can remember exactly where we were kept, the conditions there, and the treatment we received. There was a separation between us stolen children and those who had been born in the home.
The stolen children were kept on the ground floor of the building. The babies born in the home were kept on the floor above and we were never allowed to mix with them; nor were the staff who looked after these babies allowed to mix with the staff in charge of us. It was as if there was a hierarchy of our value: apparently the Nazis viewed the babies as more important than us children brought in for Germanisation.
In the home we were constantly given medical tests – I think it must have been every day. We were in a big room on the ground floor which had a large semi-circular wall of windows. I have been there since and this room still exists: it looks almost the same. There was a very sinister atmosphere in the room and we were individually taken into a side room and given injections by a doctor. I fear now that these were to tranquillise us: I cannot see any other reason for it. We were terrified of these injections. All the children were crying in that room: no one ever laughed.
Even for the precious Aryan babies like Guntram Weber, the regime in Lebensborn homes was cold to the point of severity. They were separated from their mothers immediately after birth, and kept apart for the next twenty-four hours. Thereafter they were allowed just twenty minutes together every four hours: even during that brief period of contact the SS staff strongly discouraged mothers from caressing or talking to their children.
The older children were monitored constantly, and reports made about their behaviour. Uncleanliness, bedwetting, farting, nail-biting and masturbation (which older boys were told on arrival was forbidden) were enough to ensure expulsion: these rejects were shipped off to forced education camps where they were brutalised or used as slave labour.
This Spartan regimen was intended to produce strong and ruthless future leaders for the Master Race. But children need love, not unyielding discipline: Barbara Paciorkiewicz remembered clearly how the rules frequently produced the opposite effect to Himmler's objective.
The children often reacted to them by wetting their beds. In the mornings when this was discovered, the children were beaten for this: even if only one had wet the bed, all of us were punished.
Himmler's overall plan was the same for both types of Lebensborn children: wherever possible, for the duration of the war, they were to be handed over to carefully vetted foster parents who would raise them as model Aryans. After Germany's eventual victory, the boys were to be sent to elite schools – the network of SS-run _Nationalpolitische Erziehungsanstalten_ – where they would receive a strong physical and political education. The girls were to be sent to schools run by the Bund Deutscher Mädel (the female equivalent of the Hitler Youth) and trained to become housewives and mothers.
Barbara had also uncovered the way in which Lebensborn deliberately obscured the true identities of children stolen from the occupied territories. First it cut an essential link to their home country by forbidding them to speak their own language; then it told would-be foster parents that the children were the orphans of fallen German soldiers. The men who ran the programme – those who had been indicted and then acquitted at the Nuremberg trial – knew this to be a lie but Himmler's orders made it clear that for youngsters like Barbara and me, every single trace of our previous life in Poland or Yugoslavia was to be erased.
Barbara's foster parents were from Lemgo in North Rhine-Westphalia. The Rossmanns were in their fifties and had two grown-up sons who had been drafted into the Wehrmacht: they had also had a daughter who had died of scarlet fever when she was nine. Neither was, as far as Barbara had been able to discover, a member of the Nazi Party. Herr Rossman was the director of a school and his wife was a housewife.
Although they were good and kind people who longed for a child to replace the daughter they had lost, even as a young child Barbara felt out of place in her new family. She suffered from bad nightmares in which unknown men came in through an open window to steal her.
There was always an air of uneasiness at home. When I entered a room, everyone stopped talking. And I always asked myself: 'What is it about me that makes this happen?'
Back in Poland, Barbara's grandmother had never given up hope of finding her. At the end of the war she contacted the Red Cross: it located documents showing where Barbara was living. Not long after, she was taken from her foster family and placed in a temporary children's home run by the British Army. Six months later, she was put on a train to Poland. She was eleven years old and had never been told about her biological parents, nor that she was anything other than a normal German child.
I was very frightened and confused: I still thought the Rossmanns were my real parents and I didn't know anything about Poland. It didn't mean anything to me. I didn't speak Polish and I didn't even know I had a grandmother. It all seemed like a terrible journey into the unknown.
I realised that I had never given any thought to what had happened to other stolen children after the end of the war. It had never occurred to me that some children might have been traced and sent back to a country they could not remember. Barbara's story made me wonder whether, had I been given the choice, I would have wanted to be returned to Yugoslavia.
The journey to Poland took a terribly long time. The train sometimes stood for days in a siding. And then, suddenly, I heard people around me shout out 'Poland, Poland' and everyone was happy. I felt no joy when I arrived.
We were taken to a Red Cross camp at Katowice. It was chaos with people running around shouting out names. My grandmother had sent my uncle and he was shouting, but I had no idea who he was or what he was saying because he was speaking Polish. It was only when he shouted 'Gajzler/Rossmann' that I realised. And I also realised at that moment that I had completely lost my identity.
In the ruins of post-war Poland, a country with every reason to hate Germany, Barbara felt confused and isolated. Her uncle and his wife had been in a Nazi forced labour camp together with older children stolen from Gdynia but who were not deemed racially valuable.
Barbara was taken to visit the former Stutthof concentration camp near Gdansk. Her uncle forced her to look at the piles of children's shoes and the gas chambers: 85,000 people had died here and he wanted her to see and understand what the Germans had done. But she still thought of herself as German. She found it impossible to believe what her uncle said.
I thought German people were all good: that was how I had been brought up. It was even worse in school: the children played games in the playground in which Hitler was the bad guy. Because I was German, I was always picked as Hitler: but I didn't mind – in fact, in my ignorance, I shouted out with pride that Hitler was my uncle.
I always thought that a terrible mistake had been made: that I had been mistaken for someone else, and this was why a good German girl like me was in this strange place.
Barbara Paciorkiewicz spoke quietly and with dignity. But in her story I recognised the truth of our Lebensspuren motto. As Simone Weil had realised, all of us were still desperate to find and connect with our roots. Not being able to – because Lebensborn had destroyed our original identities and because our foster families often erected a wall of secrecy – had eaten away at the fabric of our lives for decades. Barbara spoke for all of us when she said how it had affected her, and how important our new organisation could be.
All my life I never felt good enough – nor did I know really who I was or where I was truly from. It hurts very much inside. I've always wanted to ask questions but until recently there was no one to ask. Now I want to talk about this, even though it hurts, so the world does not forget the terrible idea of stealing children and racial tests. It must never happen again.
The question was how to achieve this. There were only a few dozen of us at our first meeting – a fraction of the number of children born or kidnapped into Lebensborn – and already it was clear that there were differences between us. Some people felt we should step out of the shadows and hold a press conference; others wanted to concentrate on building a memorial in Wernigerode, on the site of the old Lebensborn home. I sensed then that there would be difficulties in the years ahead.
## SIXTEEN | TAKEN
_'At 6.30 a.m. about 430 children aged between one and eighteen years were brought by cars to the railway track. The children had only such hand luggage as they could carry themselves. For breakfast they got black coffee and a small piece of bread.'_
GERMAN RED CROSS MEMO, AUGUST 1942
In October 2007, the final pieces of the jigsaw fell into place. The previous two years had been busy. I was still working in my physiotherapy practice (though beginning to think about retiring) and Lebensspuren occupied much of my spare time.
We went public in 2006: around forty Lebensborn children attended our second meeting in Wernigerode, along with journalists from the German and international press. Articles began appearing in quality newspapers and the BBC broadcast a story around the world about what they called _Hitler's Children_. Guntram Weber, Gisela Heidenreich and I answered a never-ending stream of questions, convinced that openness was vital to educate the public about the truth of Himmler's Master Race experiment.
The publicity seemed to work. Gradually it became possible to discuss Lebensborn in public and the more it was talked about, the more enquiries Lebensspuren received from people who suspected they might have been part of the programme. The following year, more than sixty people came to our annual meeting in Wernigerode.
It would be pleasant, though probably naive, to believe it was our frankness that had helped open the archives, which had previously proved so resistant to giving out information. But for whatever reason, formerly unhelpful organisations finally began to open their files. The most important was the International Tracing Service in Bad Arolsen.
The ITS had long been criticised for refusing fully to open up its millions of files. It had claimed, with the backing of the German government, that federal law required one hundred years to pass from the creation of records to their public release. It was an odd argument given that, due to its multinational funding and oversight arrangements, ITS was not technically subject to German law. Its critics alleged that a desire to repress information about the Holocaust in Germany was the real reason for its secrecy.
The fact that in January 2000 all eleven governments sitting on the International Commission of the ITS had endorsed a call for the opening of Nazi archives worldwide seemed to support this view. In March 2006, the US Holocaust Memorial Museum publicly accused the International Tracing Service and the International Committee of the Red Cross of obstruction: 'The ITS and the ICRC have consistently refused to cooperate... and have kept the archive closed.'
Two months later, the ITS announced that it would finally open its archives in the autumn of 2007.
For seven years I had waited to see the documents it held about me and my family. Two requests had produced only an acknowledgement that there were relevant files but that it needed time 'to evaluate' them, followed by a letter in 2003 asking me for any information I had discovered myself.
When I finally saw them, the documents told me a great deal about both my biological family and Gisela von Oelhafen. The first was a set of papers detailing what had happened to Johann Matko: they showed the dates he had been arrested for partisan activity and held in Mauthausen concentration camp. His name appeared on several lists of political prisoners made by the Nazis: I could not see any reason why the ITS had withheld these from me for so long.
The second tranche of documents came from Lebensborn files: these showed that Ingrid von Oelhafen was once known as Erika Matko from St Sauerbrunn in Yugoslavia. They even included a copy of an insurance policy that Lebensborn had taken out in my name. I had wasted valuable time trying to locate St Sauerbrunn in Austria, while all along the ITS had withheld clear evidence showing I came from Yugoslavia.
The third bundle of records I saw was a series of letters from various branches of the Red Cross dating back to the immediate post-war years. These showed that in 1949 two separate agencies had been searching for me, with a view to returning me to my real family. The first was Caritas, the international Catholic relief organisation, which was responsible for overseeing the work of all welfare organisations accredited to the United Nations. Caritas officials had been tracking children stolen from Nazi-occupied countries – and in particular from Yugoslavia – and must have found my name on the list of transports. They had written to the Yugoslavian Red Cross asking for help in locating me.
At the same time the International branch of the Red Cross, which then had overall responsibility for tracking down and repatriating displaced persons, was also searching for me. It, too, had written to the Yugoslavian Red Cross in an attempt to confirm my true identity.
Surprisingly, given the difficulties caused by the Cold War and their focus on creating a new, unified country, Yugoslavian officials responded. They did not have much in the way of documents themselves – most Nazi files had been destroyed in the fighting or in desperate last-minute bonfires – but they were able to issue an urgent request to the International Tracing Service office in Hamburg. It asked for social workers from the German branch of the Red Cross to visit the von Oelhafen family in Hamburg to ascertain whether I was living with them and whether I was Erika Matko.
I thought back; in 1949 I was not in Hamburg but in the children's home at Langeoog to which Gisela had sent me the day after we escaped to the west. Unless my foster parents chose to help them, the officials would have had no chance of finding and interviewing me. The final document showed that Gisela, at least, was not terribly forthcoming.
GERMAN RED CROSS,
Central Zone, Hamburg
25 October, 1950
To: The International Tracing Service, Child Search Branch, Arolsen
Subject: MATKO, Erika, born on 11/11/1941 in St Sauerbrunn
The wife of von Oelhafen was visited by us. We could not get information from the husband because he does not currently live in Hamburg.
Frau von Oelhafen has been away on a trip for some time. She picked up in person the child identified above from the children's home Kohren-Sahlis at Leipzig and therefore should be able to provide the best information.
However, the only thing she has on paper is a vaccination certificate from Kohren-Sahlis, of which we enclose a copy. Frau von Oelhafen drove to Kohren-Sahlis at the instigation of The Lebensborn Society in Munich. There she was told that Erika Matko was an ethnic German child.
At that point the child Dietmar Holzapfel had already been living with the von Oelhafens for half a year: he had been picked up from the Municipal baby nursing home at Munich. The father of this child is missing at Stalingrad.
Erika MATKO will continue to remain as a foster child with Frau von Oelhafen. An adoption of the child is not intended. Frau von Oelhafen is always happy to make additional information available. She also has a self-interest in learning about the origin of the child, so that she can answer the child's questions. However, she cannot contribute to any further details, having no documents about the child.
The von Oelhafens also fled from the Russians, and in doing so lost their belongings. Frau von Oelhafen remembers that she probably received a transfer note [for Erika Matko] from the children's home at Kohren-Sahlis. This was lost while she was on the run.
We regret that we can currently give you no further information. We would be very grateful if you would inform us of the outcome of your negotiations in St Sauerbrunn.
Without Gisela's cooperation – and because Lebensborn had destroyed all records of my true identity – there was no hope of finding my biological parents, much less returning me to them. All the organisations involved abandoned their efforts at this point.
I was astonished. Gisela had never once told me that the Red Cross had contacted her – not even during the years when I was struggling to obtain official documents from the German government. Nor had she ever even hinted that she knew my original identity. Worst of all, she had misled the social workers: she had told them that the 'transfer note', given to her and Hermann when I was handed over to their care by Lebensborn, had been lost when we escaped from the Russian sector in 1947. Yet I had found that same document when I cleared out her room in the 1990s.
I find it hard, even now, to speak badly of Gisela. Deep down I still want to be loved by her, even though she is long gone. But I have tried to put aside these emotions, to recognise this for what it was: a betrayal. She knowingly hid the truth from me. I was glad – happy, even – to know that someone had been looking for me all those years ago, but I was very hurt to learn how that search had been obstructed by the woman I had once called my mother.
That same month I made my second visit to Slovenia. I organised my trip to coincide with the annual meeting of stolen children in Celje, and I arranged to travel on from there to Rogaška Slatina, where I was to meet up again with Maria Matko and her family. I was also being filmed for a German television programme: the reporter had arranged for me to spend a day with local historians in Maribor.
I don't know whether it was the presence of the cameras or simply that the Slovenian authorities had managed to unearth more evidence, but my welcome there was warmer than it had been two years earlier, and the meeting much more productive. At last I learned exactly what had happened in Celje – and the truth about Erika Matko.
The occupation of Yugoslavia in the spring of 1941 began well for Germany. Its troops advanced swiftly through the country and in just ten days the Yugoslav High Command capitulated. On 16 April, the day of surrender, the Gestapo moved into Celje and began to arrest local anti-Nazi partisans. Three days later, Himmler himself arrived to inspect the town's ancient prison, Stari Pisker. In the coming months it would be used for the torture and execution of hundreds of resistance fighters.
But unlike other countries overrun by the German Blitzkrieg, Yugoslavia was never fully conquered. Resistance was organised by the charismatic Josip Tito. On 4 July 1941, he issued a secretly printed call to rise up against the Nazi occupiers.
Peoples of Yugoslavia: Serbs, Croats, Slovenes, Montenegrins, Macedonians and others! Now is the time, the hour has struck to rise like one man, in the battle against the invaders and hirelings, killers of our peoples. Do not falter in the face of any enemy terror. Answer terror with savage blows at the most vital points of the Fascist occupation bandits. Destroy everything – everything that is of use to the Fascist invaders. Do not let our railways carry equipment and other things that serve the Fascist hordes... Workers, Peasants, Citizens, and Youth of Yugoslavia... battle against the Fascist occupation hordes who are striving to dominate the whole world.
This led to an intensive guerrilla campaign against the Germans. By September 1941, there were at least 70,000 resistance fighters active in Yugoslavia. Tito's partisans engaged in classic hit-and-run tactics and when the Germans launched a major counter-offensive against the rebels, they simply retreated into the mountains. Hitler's response to this (and partisan activity across all the occupied countries) was the Nacht und Nebel decree. The phrase meant 'night and fog': in practice, it was an order to murder anyone who dared oppose Nazi rule. On 7 December, the instructions were sent out to commanders in the field.
Within the occupied territories, communistic elements and other circles hostile to Germany have increased their efforts against the German State and the occupying powers...
The amount and the danger of these machinations oblige us to take severe measures as a deterrent. First of all the following directives are to be applied:
1. Within the occupied territories, the adequate punishment for offences committed against the German State or the occupying power which endanger their security or a state of readiness is, on principle, the death penalty.
2. The offences listed in paragraph I as a rule are to be dealt with in the occupied countries only if it is probable that sentence of death will be passed upon the offender, or at least the principal offender, and if the trial and the execution can be completed in a very short time. Otherwise the offenders, at least the principal offenders, are to be taken to Germany.
3. Prisoners taken to Germany are subject to military procedure only if particular military interests require this. Should German or foreign authorities enquire about such prisoners, they are to be told that they have been arrested but that the proceedings do not allow any further information.
4. The commanders in the occupied territories and the court authorities within the framework of their jurisdiction are personally responsible for the observance of this decree.
The same day, Himmler issued instructions to his Gestapo and SS forces.
After lengthy consideration, it is the will of the Führer that the measures taken against those who are guilty of offences against the Reich or against the occupation forces in occupied areas should be altered. The Führer is of the opinion that in such cases penal servitude or even a hard labour sentence for life will be regarded as a sign of weakness. An effective and lasting deterrent can be achieved only by the death penalty or by taking measures that will leave the family and the population uncertain as to the fate of the offender. Deportation to Germany serves this purpose.
It was a deliberate rejection of the laws of war. No longer would the Geneva Convention or any other regulations protect the civilian population of occupied territories. On 12 December, Feldmarschall Wilhelm Keitel, Chief of the German Supreme High Command of German armed forces, issued his own re-enforcement of Hitler's decree to Wehrmacht troops throughout the expanded German Reich.
Efficient and enduring intimidation can only be achieved either by capital punishment or by measures by which the relatives of the criminals do not know the fate of the criminal.
In order to nip disorders in the bud the sternest measures must be applied at the first sign of insurrection. It should also be taken into consideration that in the countries in question a human life is often valueless. In a reprisal for the life of a German soldier, the general rule should be capital punishment for 50–100 communists. The manner of execution must have a frightening effect.
But the campaign failed to terrorise the population into submission. Tito's Partisan Army grew in numbers and effectiveness. By the middle of 1942, Himmler issued orders for an even greater crackdown on resistance. On 25 June he put Obergruppenführer Erwin Rösener, Head of SS forces in the region, in charge of anti-resistance operations and ordered him to murder or imprison all families suspected of involvement with the rebels.
Rösener planned six separate _Aktionen_ , or campaigns, in the Lower Styria area, centred on Maribor. In the first of these, on 22 July 1942, 1,000 people were arrested and brought to Celje. The men were separated from their families and one hundred of them were lined up against the walls of Stari Pisker, then summarily executed by firing squad. The killings were photographed by Nazi cameramen as a warning against partisan activity. The images were processed at a local photographer's studio: he made extra copies in secret and hid them until the end of the war. It was sobering to hold these pictures in my hands. Among the murdered men that day was Ignaz Matko. Was he one of the bodies lying against the courtyard wall or being lifted on to a stretcher by those who would be shot next?
Some adults were held as hostages, to ensure compliance from the surrounding towns and villages with future _Aktionen_. The remainder were shipped to concentration camps like Auschwitz, where they were murdered or worked and starved to death. Their children were transported to Frohnleiten, in Austria, for racial examinations. Those deemed suitably Aryan – there were not many – were taken away for Germanisation, the rest were sent to 'education camps' where they were treated brutally and suffered from hunger and disease.
The second _Aktion_ was scheduled for the beginning of the following month. All families in the nearby villages were ordered to report to the school at Celje on 3 August. Johann and Helena Matko were among the hundreds of families who arrived at the schoolyard that morning with their three children, Tanja, Ludvig and Erika. Heavily armed soldiers quickly separated them into three groups: men, women and children, who were pulled from their parents and taken inside.
Once again, a photographer recorded the event. One picture showed families lined up against the outside wall; another captured the moment when the troops separated them. In this frame a woman in a headscarf was being restrained by a Wehrmacht officer while a second soldier, his rifle slung from his shoulder, stood in front of a mother carrying a baby: she seemed to be pleading with him. A third photograph was shot inside the school: in a crude wooden structure, lined with straw, children and babies were being undressed by unidentified helpers. One little boy was struggling; the faces of others, though, were blank.
Looking at these pictures I felt two quite separate emotions. Somewhere in the crowded yard or among the babies in the schoolroom was Erika Matko: these grainy black and white photographs recorded the day I was stolen from my family. My first reaction was fear. I do not wish to sound melodramatic but I felt a shiver run through my body, a sensation of loneliness and vulnerability. But there was anger, too, at the people who had treated children like this. These soldiers had snatched babies from their mothers and imprisoned them in what amounted to a cattle stall.
Throughout my life I have tried to hide my emotions, to bury them so deeply that the feelings of abandonment and powerlessness could not rise up to engulf me. Those photographs stripped away my defences. I was Erika once again.
The other children and I were kept in the schoolroom for two days. We were crudely assessed for racial value: blond hair, light-coloured skin and blue eyes were taken as a sign of Aryan blood, while brown hair and dark skin or eyes were the marks of worthless Slavonic ancestry. I was designated valuable while Tanja and Ludvig were rejected. They were sent outside and handed back to our parents.
In total, 430 children were judged to be Aryan. We were held in the schoolroom and then taken to the train station. The older children carried babies like me in baskets. The German Red Cross (DRK) was on hand to supervise this children's transport. The DRK's leadership were committed Nazis – even to the point of wearing uniforms and ceremonial daggers – but below them was a civilian army of volunteers. I saw a report by one of these workers, a woman named Anna Rath.
At 6.30 a.m. about 430 children aged between one and eighteen years were brought by cars to the railway track. The children had only such hand luggage as they could carry themselves. For breakfast they got black coffee and a small piece of bread.
The embarkation went smoothly. At 10.30 a.m., a delay of one hour was announced. The train started but only travelled until 2.45 p.m. The children then had to endure a four-hour wait: during this time neither the Red Cross nor officials from the National Socialist Peoples' Welfare Organisation provided any food for them. Fresh water was only brought by a DRK helper who accompanied the train.
Once they arrived in Frohnleiten, the DRK helpers and the young children (two to five years of age) had to walk to the resettlement camp, carrying suitcases and bundles. The children were half-naked and hungry, some with very dirty nappies: this was because there were no clothes to change them. They were screaming and crying.
When we arrived [at the camp], there was another delay for the children because no food was ready for them. They had to stay outside in the courtyard and in a meadow. Finally, at 5 p.m. the children were allowed to go into the eating room. The sixteen DRK accompanying assistants, although tired and weary themselves (they had been on duty since 4.30 a.m.), had to look after the children because the all the camp staff, save for four people, were on vacation.
This was how the Nazis treated the children they stole. This was how my life in their hands began. No wonder, then, that I have struggled all my life with a longing to be loved.
In Frohnleiten there were more tests. Himmler's race examiners prodded and poked us, measuring and recording our every characteristic before assigning us to one of four categories. The top two ensured a place in a Lebensborn home; the others guaranteed a ticket to a re-education camp. Among those assessing the kidnapped Yugoslavian children was Inge Viermetz, the female official who had been tried and acquitted at Nuremberg: she was operating on written instructions to 'take only young children who have not yet reached school age'. But it appeared Lebensborn was not the only organisation that wanted us. Officials from VoMI, the agency Himmler had set up to 'protect' ethnic Germans living outside the old Reich, wanted their share of the children who were to become the future leaders of the Master Race. According to testimony given at Nuremberg: 'a real competition for these children arose between VoMI and Lebensborn, and it was thanks to Frau Viermetz's efforts that Lebensborn won in the end'.
I looked at all this evidence and marvelled again at the decision of the judges at Nuremberg to acquit Viermetz and the other senior Lebensborn officials. She had quite literally handled stolen goods: she had decided the fates of hundreds of children, from all over the occupied countries – sending some into a programme that erased their identities and the rest to camps where many died. How could this woman have been declared innocent?
I was moved from Frohnleiten to another camp, at Werdenfels, near Regensburg in Bavaria. At the end of 1942, Lebensborn sent Emilie Edelmann – Gisela Heidenreich's mother – to supervise further racial selection tests. She must have decided that I was good enough: the forms were signed and I was shipped off to Kohren-Sahlis.
Ours was not the last transport of Yugoslavian children. The documents I saw in Maribor revealed that there had been four further _Aktionens_ in which hundreds more families were summoned to Celje for racial examination, sifting and separation. They too were divided up between Lebensborn homes and re-education camps.
The officials who showed me these papers had one more revelation for me. Johann and Helena had arrived at the schoolyard with three children. When they were permitted to leave, the records showed that they went home with three children – Tanja and Ludvig, and a baby girl called Erika. I knew that my sister and brother had been handed back to my parents, but who was this other child? Somehow Erika Matko had been simultaneously on the train to Frohnleiten and on the journey back to Rogaška Slatina with Johann and Helena. It made no sense.
It was Maria Matko who led me towards the answer. We met in her house the day after my visit to Maribor. After the initial shock of the DNA results she had accepted that I was her sister-in-law. Now, with her help, I was finally able to piece together what had happened to me.
On the day that the children were handed back to Johann and Helena, the Nazis executed several suspected partisans in Celje prison. Witnesses recalled that the families waiting outside the schoolyard heard the volleys of shots. The children of those murdered men and women were being held with me inside the schoolroom: some were now orphans.
When Tanja and Ludvig were returned to her, Helena must have complained that her third child was missing. Perhaps to appease her, or because they did not know what to do with it, the Germans gave her an orphaned baby. This was the girl who grew up as Erika Matko.
I was torn between pain, anger and bewilderment. My mother must have known that the baby thrust into her arms was not her own. It wouldn't have looked the same nor would it have smelled right, in that indefinable way that mothers know the smell of their own baby. So how could she have accepted this cynical substitution?
The only explanation I could imagine was that she did not dare to argue; that the sound of the firing squads made her fearful for her own life and those of her husband and children. But rational understanding was one thing; emotion quite another. For almost sixty years I had struggled with not knowing who I really was; for the past seven years I had been on a long and harrowing journey to unravel the mystery of my past and discover how I had been transformed from Erika Matko to Ingrid von Oelhafen.
Now I knew. And it didn't help at all.
## SEVENTEEN | SEARCHING
_'What are we doing? I asked myself. What in God's name are we doing?'_
GITTA SERENY: FORMER UNRRA
CHILD WELFARE OFFICER
I was angry at everyone. Angry with Hitler and Himmler for the orders to kidnap me in the first place; angry with Inge Viermetz and the Lebensborn officials for concealing my true identity and reinventing me as a German child; angry with the soldiers who had given my parents another child in my place. I hated what the Nazis had done to me and to all the other victims of their obsession with pure blood and the Aryan master race. All the resentment and hurt I had suppressed in years gone past was rising to the surface.
My rage was also focused on those much closer to home. Gisela and Hermann von Oelhafen had been willing accomplices in this wretched scheme. They surely should have realised that Lebensborn was not to be trusted: even in wartime Germany there had been enough information – and indeed rumour – about the organisation for them to have had doubts about the provenance of a baby it was offering for fostering.
Then there was Gisela's strange maternal ambivalence: sending me away to foster homes hardly suggested someone who was committed to bringing up a child in a warm and loving environment. Above all, her evasiveness and deceptions about my origins had hampered my search for the truth: how much easier my life would have been if she had only told me where I came from. I knew from my friends in Lebensspuren that some women who fostered Lebensborn children had been open and honest, and that this had eased some of their anxieties. Why did Gisela choose not to talk to me?
But it was the actions of my biological family that hurt the most. I could just about understand why Helena and Johann Matko had accepted the baby handed to them by the Nazis. A family of known partisans could not – especially on a day when their compatriots were being executed – have done anything else. I put myself in their place and tried to imagine the fear of a knock on the door and the discovery of a Gestapo or SS officer standing outside. Although I was haunted by the thought that this other Erika grew up to live what should have been my life, safe in my mother's love, I could not condemn them for it. What I could not accept was that Helena had carried on with the lie long after the end of the war. Barbara Paciorkiewicz's story had shown me that some families of kidnapped children had been determined to bring home their stolen youngsters, yet Helena had lived with the knowledge that her real baby was somewhere in Germany. How could she have carried on without once trying to find me?
I would have loved to ask her myself. But Helena died in 1994: at that time I had not even found the documents Gisela had hidden from me, much less discovered that my roots lay in Slovenia. The actions of my real and my foster mother had conspired to rob me of the chance to seek answers.
My emotions needed a lightning rod: someone living who I could blame for my position. Erika Matko – the other Erika – became the focus of all my anger and pain. Her refusal to meet me, even to answer my letter, enraged me: it seemed so callous. The feeling that she had stolen my life gnawed away at me. Maria had told me that she had been ill much of her life and as a result had never worked. I thought about how hard I had worked to build my physiotherapy business and my struggles with German bureaucracy, and compared this to the way Erika had apparently been supported by her government. And my anger grew.
My friends tried to reason with me. It was not, they rightly pointed out, Erika's fault that she had been given my identity. As a child she could not have known how our lives were swapped and nor could she have done anything about it. And after the war, in Tito's Yugoslavia, surely it would not have been safe for Helena and Johann to reveal any brush with the German occupiers: the communists were not always careful about the innocence or otherwise of those tainted by any form of involvement with the Nazis. Most likely she never knew the truth about her origins; perhaps no one except my parents knew.
Other people urged me to imagine what it must have been like for Erika when I first contacted her. She was then over sixty and suffering from a severe heart condition: it must have been an enormous shock for a complete stranger suddenly to appear and challenge everything she knew about her family. Could I not sympathise?
I could not. I was too consumed by the injustice of everything that had happened to allow myself to feel sorry for a woman whose life I had surely turned upside down.
It took a long while for the anger to dissipate. As months, and then years, passed, I slowly gained enough distance to analyse the situation more clearly. I began to consider what an alternate history would have looked like for me. I thought again of Barbara's story of being taken away from her German foster family and the only home she could really remember; I made myself imagine myself on the train with her to Poland, and feel her bewilderment.
I knew from meeting the group of stolen children in Celje that some of those kidnapped from Yugoslavia had been returned to their families. There had even been a court case to set a precedent for these repatriations. Ivan Petrochik had been snatched by an SS detachment in 1943 when he was less than two years old. His father had been shot by the Gestapo and his mother sent to a concentration camp: he was given the label _Banditenkind_ and handed by Lebensborn to a German family.
His mother survived the war and searched for seven years to trace him. In 1952, a court ordered that Ivan be returned to Yugoslavia: he was eleven and had been raised for most of that time as a German child.
Ivan and Barbara's stories made me wonder how the process of repatriation worked and what effect it had on those involved. In 2014, I found some of the answers.
Gitta Sereny was a highly respected journalist and author. She had been born in Vienna in 1921, the daughter of an Austrian aristocrat and a former actress from Hamburg. When she was thirteen, her parents sent her away to boarding school in England, but her train was delayed in Nuremberg and she witnessed one of the Nazis' mass rallies. It left an indelible mark on her and when she finished school she moved to France to help orphans suffering under the German occupation. She also worked with the French Resistance.
When the war ended, she joined the United Nations Relief and Rehabilitation Administration (UNRRA), helping to repatriate the millions of displaced people scattered across the former Reich. She was later assigned to the Child Tracing Department: fifty-three years later she published an account of her experiences in a now-defunct magazine.* In it she described the repatriation of a young boy and girl from Germany to Poland, and for the first time I truly understood what being returned to Yugoslavia would have meant for me.
The process began with Sereny visiting the foster home. It was a traditional Bavarian single-storey farmhouse, its windows uncurtained and with only two dim lights showing the way to the front door.
Sereny had prepared for the visit by examining the region's population records in the local mayor's office. Six people were registered as living at the farm: the farmer and his wife, both in their mid forties, and his elderly parents. There were also two young children – a boy and a girl.
She was keenly aware of the potential distress her visit, and the uncomfortable questions she needed to ask, would cause. It was vital to see the children in the family surroundings, but she hoped that before the interview went too far the youngsters would be sent to bed.
Her reception was distinctly chilly. The family was sat around the kitchen table and deliberately declined to stand up when Sereny stepped inside. Although the farmer, his wife and the two children shook her proffered hand – the boy uneasily, the girl enthusiastically – the grandfather refused to do so, hiding his hand behind his back and gruffly demanding what this intruder wanted.
The children were called Johann and Marie. Both were officially six years old and both had blue eyes and blond hair – the boy's cut short and crudely, the girl's longer and neatly braided.
Sereny explained that she only wanted to talk with the family for a short time. To ease the frosty atmosphere she gave each of the children a chocolate bar – a precious gift in the austerity of post-war Germany. It provoked, though, a mixed reaction.
It was when the little girl, beaming, said, 'Danke', and I stroked her face, that the farmer's wife said sharply, 'Geht zu Bett' [Go to bed], and the two children shot up to obey.
The little girl hugged her mother and reached out for her father's hand. The little boy politely, but formally, bade his parents goodnight then gave Sereny a suspicious look before kissing his grandfather. Then the farmer took the two children away and put them to bed, holding them tightly as he did so.
In 1945 there were 8,500 'unaccompanied children of United Nations and assimilated nationality' registered in the tracing services' files. Within months, tens of thousands of new names were added, sometimes accompanied by snapshot photographs or physical descriptions, all of them kidnapped from the east for Himmler's Germanisation programme. Marie and Johann were among them.
Gitta expressed her disbelief at this situation:
Who would have taken babies or toddlers away from mothers?... How could anyone, even bigots gone mad, believe they could discern 'racial values' in young, undeveloped children? Above all how, in practice, could there now be large numbers of foreign children – at least some of whom would have to be old enough to have memories – living, basically in hiding, within the German community?
The farmer was hostile when Sereny began asking questions. He said that their son had been killed by the Red Army during the siege of Stalingrad; his sister had died four years earlier in a road accident. They had fostered Johann and Marie to replace their lost children.
It was plain that this family loved the children and Sereny tried to reassure the farmers that she understood this, whilst simultaneously insisting that they must disclose everything they knew about the youngsters' origins. When she asked about their biological parents, the farmer's wife said that they had died, but was very vague about who had given her this information. Sereny pushed harder, explaining to the family that many Eastern European parents were searching for children who had been stolen from them.
'East?' said the grandfather and, repeating it, virtually spat out the hated word: 'East? Our children have nothing to do with "east". They are German, German orphans. You need only look at them.' And there it was: 'You need only look at them.'
Somebody had indeed once looked at them. Just as had happened in Celje, villagers around the city of Łodz had been instructed to bring their children to the Youth Welfare Office where the race examiners had done their work and shipped the chosen children off to Lebensborn. Johann and Marie's parents had been searching for them ever since and had photographs to support their claim. UNRRA decided the children were to be returned to them.
Gitta Sereny was deployed away from the area shortly afterwards. Then, in the summer of 1946, she was sent to work in a Children's Centre in Bavaria. To her surprise – and dismay – she found Johann and Marie were being held there. They were plainly struggling to cope with their removal from the farmer and his wife: both had deep shadows under their eyes and their skin was unhealthily pale. Sereny was shocked by their condition.
Marie was scrunched up in a chair, her eyes closed, the lids transparent, her thumb in her mouth, but Johann raced up as soon as he saw me, and shouting hoarsely, 'Du! Du! Du!' ['You! You! You!'], hit out at me with feet and fists...
The staff at the centre had seen all of this before: the children's pitiful state was, they told Sereny, all too typical of other youngsters who had been taken away from their German foster families prior to being sent back to their countries of birth. Many, including Johann and Marie, had to be kept in the unit after their official repatriation date; it seemed the only way to ease the pain of what was, after all, the second separation in their young lives and to prepare them for the overpowering expectations of their biological parents. Experience showed that these reunions placed a terrible mental strain on already traumatised children.
This was a caring and thoughtful approach – but for Johann and Marie it had clearly failed. The young boy was already showing signs of aggression, whilst his sister had effectively reverted to babyhood: she wet the bed frequently and would only eat when fed from a bottle.
Later that night, the resident psychiatrist suggested that Sereny try feeding Marie with the bottle.
She lay there, her eyes shut, the only movement in her lips, which sucked, and in her small throat, which swallowed. I held her until she was asleep. It helped me but, I fear, not her.
What are we doing? I asked myself. What in God's name are we doing?
Now I understood. This would have been my fate, had I been sent back to Rogaška Slatina. I cannot believe that I would have understood what was happening any more than Johann and Marie grasped why they were taken from the only family they could remember. Now, at last, I wasn't angry any more.
_______
* _Talk_ Magazine. Reproduced in the Jewish Virtual Library, 2009. The magazine ceased publication several years ago and Gitta Sereny died in 2012.
## EIGHTEEN | PEACE
_'My identity might begin with the fact of my race but it didn't, couldn't, end there.'_
BARACK OBAMA _, D REAMS FROM MY FATHER:
A STORY OF RACE AND INHERITANCE_, 1995
What is identity and how is it formed? Does identity shape the person – or is it the other way round?
This is not, as it might seem, merely an exercise in abstract philosophy. As my journey ended, it was the question I had to face up to. I knew now who I was – or had once been; I was less sure about what this meant.
Identity is much more than merely the answer to the question 'who am I?'. It is also about personality. I was struggling to understand how I had become the person I was today. Was I simply the product of the first years of my life as a Lebensborn child? Was my past to blame for my shyness, my lack of confidence and my desire to put the needs of others – of children especially – above my own? In other words, was the course of my life set in stone by Himmler? That, after all, was what he intended: we Lebensborn babies were supposed to fulfil his vision of a new and uniform generation of the German Master Race.
Surely I was just as much the product of my own choices. Genetics may dictate hair and skin colour, but identity must involve an element of free will. I had chosen to devote my life to working with disabled children; chosen not to get married and have a family of my own. These were my decisions – not the ineluctable result of the Lebensborn programme.
Perhaps those who have never endured the uncertainty of not knowing who they really are, are rarely troubled by these existential questions. And yet which of us hasn't, in our darker moments, returned to a particular moment of our lives and wondered what would have happened had events played out differently?
In Shakespeare's _Hamlet_ , Ophelia says, 'We know what we are, but not what we may be.' I could not help but dwell on what might have been. What if I had failed the racial examination that day in Celje? What would my life have been, growing up as Erika Matko? Would I have had the opportunity of a rewarding career, or would my horizons have been limited – as seemed to have been the case with the other Erika – by my environment? If Gisela had been more honest and if the Cold War had not intervened, I would have been reunited with my biological parents: what would that have meant for the course of my life? I asked myself whether I would have been better off had the Nazis left me with my family or whether, in some twisted irony, they did me an eventual favour.
The annual meetings of Lebensborn children exacerbated this uncertainty. The tensions I had detected at our first gathering had grown steadily over the years until Lebensspuren was riven by arguments. All of us had been damaged by our involvement in the Master Race experiment; all of us struggled to come to terms with our personal histories. By 2014, many of those who had joined together to create a supportive environment had walked out or drifted away to form new and smaller groups: I was one of them.
But that year I made two trips which helped me to find some peace. The first was a visit to a former Lebensborn employee, Anneliese Beck. Now ninety-two and almost blind, when I arrived at her home near Frankfurt she greeted me with tea and stollen.
Frau Beck had worked at the Sonnenwiese home in Kohren-Sahlis at the time when I was held within it. She did not remember me: there were 150 youngsters living there, and I had not been in the group for which she was responsible. But she was able to tell me a great deal about the daily routine at Sonnenwiese and to help me understand what my life would have been like. She showed me a photograph of her with some of the children. I was pleased to see that they were nicely dressed and clearly well fed. And she was adamant that despite the circumstances and the presence of the SS, for the most part our time in Kohren-Sahlis was happy and comfortable.
Sitting with Frau Beck helped me fill one of the last remaining gaps in my knowledge. I had no memories of Sonnenwiese _,_ and though I often tried, I was unable to visualise the years I spent there: no matter how much I forced myself to think about it, all I could see was a dark hole. Now that hole was filled and the walls that protected me from my memories were beginning to crumble. I sensed that the final stage would be to travel to Kohren-Sahlis, to walk inside the buildings there: that would, I felt sure, unlock my mind. I was not yet strong enough to go, but I knew that in the coming years I would make that journey.
In October I returned to Slovenia. I went first to Rogaška Slatina where, in a pretty public park, I paid my respects at the memorial to the men and women who were shot between 1941 and 1945. More than one hundred names were carved into the stone: I looked for my Uncle Ignaz and when I found him I traced the letters with my fingertips.
Later, Maria showed me the house where I was born, before taking me to the cemetery, sitting on the top of a hill, where my parents, my grandmother, my brother and my sister are all buried. I laid flowers on the graves and lit candles for my sister and brother, while Maria and her niece cleaned the paths around the stones. I had expected to be overwhelmed by a sense of loss and so I was surprised to discover that, aside from the normal sadness of visiting a graveyard, I felt very little.
It was a similar story later in the afternoon when Maria invited me back to her apartment for Slovenian coffee and homemade blueberry liqueur with other members of the family. The atmosphere was warm, and everyone was hospitable and open. The Matkos had come to accept me as one of their own, and they gave me photographs of my parents, siblings, nephews and nieces. But although I was grateful for the love and generosity of the family I had longed for, I felt like a child among them. My overriding emotion was anxiety, the sort I had always endured when faced with an exam.
The following day I went to the civil registry office to look for records relating to my parents' marriage. The official there dug out a large book in which every local birth was documented. Together we found the page that recorded my arrival; it also revealed that Johann and Helena were married in 1938, several years after my sister Tanja and brother Ludvig were born. It was a clue to the puzzle of the DNA tests: these had shown that while Ludvig's son, Raphael, was definitely my nephew, I was equally certainly not a blood relative of Tanja's son, Marko. Given that both Tanja and Ludvig had been born before my parents' wedding, the most likely explanation was that Tanja had a different father. The Matko family seemed to have more than its fair share of secrets.
The last remaining mystery was the other Erika. She had still not responded to my letters nor, according to Maria, was she willing to speak to me in person.
I thought long and hard about what I should do. In the end I decided that I would go to her apartment and confront her. I had her address, which turned out to be on the fourth floor of a run-down block in a poor area of Rogaška Slatina. I knew Erika would be at home: the Matkos had told me she was too ill to walk down the stairs and so stayed inside every day. I found her letterbox and the bell with her name on it. I wanted to press it, to be invited upstairs and to see this enigmatic woman with my own eyes. I wanted to hug her, to speak with her and to demand answers. I wanted, above all, to find the peace of mind that would come with confronting my other self.
But I did none of these things. As I stood on the street outside the apartment block, I realised that my emotions were not merely unproductive but corrosive. I had no right to force my own needs on a sick and vulnerable woman who – just as much as me – had been a victim of the Nazis and of Lebensborn. I knew I had to learn not just to understand but to forgive. I walked quietly away.
Two days later, after a last meeting with my Slovenian family, I went home to Osnabrück. As I settled back into my life there, I reflected on what the previous fifteen years had taught me. I seemed to have travelled a great distance, yet my journey had really been a giant circle and I had arrived back at the very place where I began.
It had not been easy or painless, but I was glad – I _am_ glad – to know the truth about Lebensborn and how I came to be caught up in it. I draw comfort from the extended 'family' of those who were born or kidnapped into Himmler's experiment; in the years since that first meeting in Hadamar, hundreds of us have discovered what we were searching for.
I am certain that I was once a Yugoslavian child called Erika Matko. I am certain that I was stolen from my family and I am grateful to have had the chance to be reunited with them. I wish, of course, that I could have met my biological mother; I wish I could remember her love for me, and I will always regret not having had the chance to ask her about her life or about why she didn't search for me after the war.
I do not feel close to the Matkos in the way that families should feel close. Too much has happened; too great a separation of time and of place. There is a gulf between us that is more than simply language. I realise that I can no more understand what it means to have grown up in Yugoslavia than I can understand Slovenian. In fact I feel much greater kinship with my late step-brother, Hubertus. We were not blood relations but in our relationship lies the ultimate defeat for the Nazis' ideology: blood is not all-important.
I can smile at that now. How did it take me so long to see something so obvious? I have spent a lifetime working with children burdened with physical or mental disabilities; I have seen how love and patience can overcome these challenges. Nurture can always find a way to beat nature; the hammer does not necessarily shape the hand.
For years I had allowed my life to be overshadowed by the search for something that could not be found. There is for all of us, I believe, a gap between what we want and what we can have, and regret flourishes in that space. I spent too long trapped in a disappointing No Man's Land between dreams and reality. I lost sight of the fundamental truth that we are not defined by the facts of our birth but rather by the choices we make throughout our life.
Mahatma Gandhi once said: _'_ The best way to find yourself is to lose yourself in the service of others.' It has taken me all my life to understand this. Yet although I am back exactly where I started, if I had not embarked on my journey I don't think I would ever have worked this out. I now know who I was and who I am. Erika Matko was a Lebensborn baby, stolen from Yugoslavia, who disappeared in the madness of the Lebensborn programme. Ingrid von Oelhafen is a German woman, a physical therapist who has brought comfort to generations of children.
My name was Ingrid von Oelhafen. It was also Erika Matko. Ingrid is German; Erika is from Yugoslavia. Both of them were me. But now? Now I am Ingrid Matko-von-Oelhafen. As I always have been.
## AFTERWORD
_'That men do not learn very much from the lessons of history is the most important of all the lessons of history.'_
ALDOUS HUXLEY
This is a story of what happened more than seventy years ago. It would be easy to think of it only as history: easy, but wrong.
Since 1945, the world has not known a global conflict. Nor has there been a criminal enterprise on the scale of the Third Reich, nor an ideology that so openly worships the mystic importance of pure blood. But the key words are 'global' and 'openly'. The twisted creed that one person is inherently superior to another by virtue of his race has not disappeared. Nor have the wars fought because of it.
From Southeast Asia to the Middle East, from Africa to the Balkans, there have been those convinced that neighbouring peoples, races or nations are inherently inferior, that these post-Nazi Untermenschen are 'other' and therefore less deserving of respect, food, land or life. In the two generations since the Lebensborn experiment died in the rubble of a devastated Europe, the world has known a succession of smaller, more localised conflicts. Most have had, at their root, a version of Himmler's belief in greater and lesser races.
This book is a personal memoir as well as an examination of history. It has been written at a time when the world is fracturing into ever greater hostility between nations, regions or religions. Some of this hostility blossoms into nasty little wars: one ethnic group hacking another to pieces, one branch of a belief system blowing up those it deems to be unworthy in the eyes of its God.
In Europe particularly, and at its borders with countries that were once behind the Iron Curtain, politicians toy with nationalism, stoking the fires of hatred based on racial or historical inferiority. Not since 1945 has the continent – and beyond it, the world – been so dangerously divided.
The lesson of history is that no one learns the lesson of history. It is time we began.
Ingrid aged nearly three with Dietmar, the boy she believed to be her brother.
Hermann and Gisela von Oelhafen with Ingrid and Dietmar, Bandekow, summer 1944.
Ingrid aged eleven at Bad Salzuflen.
The Lebens-born home, Sonnenwiese, in 1942, from a wartime postcard.
Ingrid aged twenty-one with Hubertus, her foster-mother's son.
The Lebensborn logo, from the Lebensborn Society brochure, circa 1939.
The Nuremberg Laws' Racial Identification chart, 1935, showing the racial classifications between Germans, ' _Mischelinge_ ' ('crossbreeds') and Jews. Only those with four German grandparents were considered to be of German blood.
The four most senior Lebensborn officials, photographed before their trial at Nuremburg in 1947.
Heinrich Himmler inspects SS troops.
A Lebensborn baby is handed over to the care of the SS during the _Namensgebung_ 'baptism' ritual inside a Lebensborn home.
An SS officer intones the _Namensgebung_ liturgy, with the baby lying in front of an altar dedicated to Adolf Hitler.
A boy's head is measured using calipers during a racial examination, 1937.
The SS flag flies outside a Lebensborn home.
The schoolyard, Celje, former Yugoslavia, August 1942. Mothers plead in vain with German soldiers as their children are taken away for racial examination. Ingrid and her parents were among them;
Stari Pisker prison, Celje, former Yugolsavia, summer 1942. Partisans are lined up against a wall prior to being shot;
Stari Pisker prison, Celje, former Yugolsavia, summer 1942. The partisans' bodies lie where they fell.
Inside the school, Celje, former Yugoslavia, August 1942. The children are held in makeshift enclosures, packed with straw, as they wait for their racial examinations.
Ingrid Matko-von-Oelhafen today, with the first official document of her existence, a vaccination certificate issued by Lebensborn.
## ACKNOWLEDGEMENTS
Finding my roots has been a long and rocky road, but I have met many wonderful people who accompanied me along the way.
I particularly want to say thank you to my best and most long-standing friend, Dorothee Schlüter. She has been with me from the first timid steps I took in the search for my origins. She has supported me mentally and emotionally and has been deeply involved as I progressed. Thanks are also due to Jutta Schröder, who has always been there with help and care.
I must express my gratitude to Dr Georg Lilienthal for guiding me and to Josef Focks, who pushed me over and over again (as I hesitated) until at last I agreed to go to Slovenia.
To my friends in Lebensspuren, where I first met other children of the Lebensborn programme: you, above all, know how important you have been.
For their company and support on my trips to Rogaška Slatina, I thank my friends Ute Grünwald, Ingrid Rätzmann and Helga Lucas. And I am grateful to my Slovenian family for being so friendly and open.
I owe a special debt to Dr Dorothee Schmitz-Köster. From the moment I met her, she has been a great help. She not only encouraged me to believe that my story could be written but also contributed her extensive knowledge of Lebensborn. She was very good and sensible company on my last trip to Slovenia.
When Tim suggested this book, I made myself examine all the stages of my life. So much had been unknown and troublesome, but as we worked together I found the darkness which had enveloped me gradually disappearing. I also discovered that in the process of writing I was able to 'talk' to Helena, Johann and even Erika Matko. I was able, on the page, if not in reality, to ask 'why'. I did not necessarily find all the answers. But these conversations (some of them heated!) helped me to forgive and to love life as it is.
Ingrid Matko-von-Oelhafen
Osnabrück, April 2015
This book grew out of a film I made in 2013. I had heard of Lebensborn several years earlier and had tried, unsuccessfully, to persuade various television networks to commission a documentary about it. Finally, Channel 5 agreed to fund a sixty-minute film: I am indebted to its commissioning editor, Simon Raikes, for seeing the importance of the story and backing it.
I met Ingrid while researching the programme: she agreed to be filmed and was immensely generous when I was unable, for reasons of space, to include her story within the documentary. She was also kind enough to listen when I subsequently suggested that we should write a book about her extraordinary and brave journey to discover the truth about Lebensborn and her past.
Neither that film nor this book could have emerged without the efforts and encouragement of Dr Dorothee Schmitz-Köster. The Lebensborn children have no greater champion than Dorothee, and her commitment to telling their stories in her own books (sadly published only in Germany) has been crucial in bringing Himmler's shadowy organisation into the light.
Many sources were helpful in cross-checking and verifying the Lebensborn survivors' stories, including: 'Nazi "Selective Breeding"', _The Times_ (14 December 1943); 'Hitler's Children', Joshua Hammer, _Newsweek International_ (19 March 2000); 'Nazi Past Haunts "Aryan" Children', Kate Bissell, BBC News website (13 June 2005); 'Sixty of Hitler's children meet', Associated Press (5 November 2006); 'Eight people, products of the Lebensborn programme to propagate Ayran traits, met to exchange their stories', Mark Landler, _New York Times_ (7 November 2006); 'Nazi program to breed master Race', David Crossland, _Der Spiegel_ (7 November 2006); 'Documents detail Nazis' drive for racial purity', Melissa Eddy, Associated Press (6 April 2007); 'Man kidnapped by SS discovers true identity', _Daily Telegraph_ (6 January 2009); 'Stolen by the Nazis: The tragic tale of 12,000 blue-eyed blond children taken by the SS to create an Aryan super-race', _Daily Mail_ (9 January 2009); 'Stolen Children', Gitta Sereny, _Talk_ Magazine (2009); 'Third Reich poster child', Titus Chalk, _ExBerliner_ (22 November 2010); Tone Ferenc: Documents relating to the Nazi occupation of Yugolsavia at http://karawankengrenze.at/ferenc/index.php?r=documentshow&id=249
Our thanks are also due to our British publishers, Elliott & Thompson, for so enthusiastically supporting this book, and to our editor there, Olivia Bays: her cool-headed advice significantly improved our manuscript.
Similarly, Andrew Lownie is the very model of a perfect literary agent. His initial guidance, and thereafter his relationship with publishers across the world, has ensured that this story will be read in countries as far apart as Finland, Italy and the United States.
Finally, I could not write without the love and support of my partner, Mia Pennal. After a lifetime of searching, I was lucky enough to be found. _Cursum Perficio_ : my journey ends here.
Tim Tate
Wiltshire, April 2015
## SELECT BIBLIOGRAPHY
Bessel, Richard, _Germany 1945: From War to Peace_ (London: Simon & Schuster, 2009)
Clay, Catrine and Michael Leapman, _Master Race: The Lebensborn Experiment in Nazi Germany_ (London: Coronet, 1995)
Hillel, Marc and Clarissa Henry, _Of Pure Blood_ (New York: McGraw-Hill, 1976)
Owen, James, _Nuremberg: Evil On Trial_ (London: Headline, 2006)
Schmitz-Köster, Dorothee, _Lebenslang Lebensborn_ (Munich: Piper Verlag, 2012)
Taylor, Frederick, _Exorcising Hitler: The Occupation and Denazification of Germany_ (New York: Bloomsbury, 2011)
Weale, Adrian, _The SS: A New History_ (London: Little, Brown, 2010)
## INDEX
_Aktionen_ 179–180,
Aktion T4 ,
Aller, river 23–24
Allies 5–8, 10–13, , , , , , , , ,
Supreme Allied Headquarters in Europe
Alnova
America _see_ United States
Andersen, Ingrid ('Eka') , , , , , , , 48–50,
Andersen family , ,
Ansbach
Argentina ,
Aryans 65–66, , 82–83, , , , , , , , , , , , , , , , 180–181,
Auschwitz ,
Australia
Austria , , , , , , , , , ,
Avenarius, Ingeborg von
Bad Arolsen 75–88, , , ,
Bad Polzin , ,
Bad Salzuflen , 32–34, ,
Bad Sauerbrunn , ,
Bad Toelz
Bahrdorf
Bandekow 8–11, , ,
_Banditenkinder_ ,
Bavaria , 111–112, , ,
Bayreuth
Beck, Anneliese
Belgium ,
Bergen-Belsen ,
Beria, Lavrenti
Berlin , , , , , , , , ,
Wall 51–53,
Black Forest 45–46
Blitzkrieg ,
bombing , , , , ,
carpet
RAF
Bonn
Agreement
Bormann, Martin ,
Bosnia-Herzegovina ,
Brandenburg 108–109
Brandt, Rudolph
Britain , , ,
British Army , ,
Budapest
Bund Deutscher Mädel ,
Bundesarchiv , , , ,
Bundestag
camps vii, , , , , , , , , , , , , 182–183
concentration , , 13–14, , , , , , , ,
_see also Schweigelager_ and named camps
Caritas
Celje (Cilli) , , , 141–144, , , , , , , , ,
Central Tracing Bureau _see_ International Tracing Service
Chelmo ,
Churchill, Winston
Cold War vii, , , , ,
Cologne ,
communism , , , , ,
Congress of the League of Communists of Yugoslavia
Constance, Lake
Crimea
Croatia , ,
Czechoslovakia , , ,
Danzig 164–165
Darré, Walther
Death's Head regiments , 95–96
denazification , ,
Deutsche Frauenschaft ,
Dietmar _see_ Holzapfel, Dietmar Divača
Doležalová, Marie 130–131
Dollinger, Hannes 111–113
Dömitz
Ebner, Gregor , 105–106, ,
Edelmann, Emilie , ,
Elbe, river ,
Emi, Sister
Engagement and Marriage Decree 92–93
England , ,
ethnic cleansing , ,
eugenics
European Union
fascism , ,
Final Solution , ,
First World War , ,
Focks, Josef , 135–136, 138–139, , 144–145
_Fragebogen_ ,
France , , , , , , , ,
Frankfurt , ,
Frings, Josef
Frohnleiten , , , ,
Führerbunker ,
Gdansk _see_ Danzig
Gdynia 164–165
Gehrendorf
Geneva Convention
German Democratic Republic , ,
Germanisation 70–72, , 116–117, , 124–128, , ,
Germany viii, 4–8, 10–14, , , , , , , , , 50–53, 59–61, 63–67, , , , , , , 88–90, , , , , , , , , , , 135–137, , , , , , , , , , , , ,
re-unification
Supreme Command
Gestapo , , , , , ,
Globke, Hans
Goebbels, Joseph , , , , ,
Goethe, Johann Wolfgang von , ,
Goličnik, Jože
Gordonstoun
Gorgass, Ruthild 103–105, , , ,
Göring, Hermann ,
Gran Canaria 54–55,
Graz
Greifelt, Ulrich
Hadamar , , , , , , , ,
Haloschan, Helena
Hamburg , 28–29, , , , , 44–46, 48–50, , , ,
Hanover ,
Harte, Emmi , 35–37, ,
Harte, Karl , ,
Harz Mountains
Heidenreich, Gisela 162–164, ,
Heiligenholz
Heim Harz
Heinecke, Folker 113–118
Heinze-Wissede, Maria-Martha 132–133
Hertfordshire
Hess, Rudolf , ,
Hesse
Heydrich, Reinhard
Himmler, Heinrich , , , , , , 65–66, , , 78–80, 82–90, 92–96, 105–107, 109–110, 112–16, 121–23, 126–128, 130–133, , , , , , , , 164–166, , , , , , , ,
Hitler, Adolf vii, viii, , , , , 10–12, , , , , , 63–65, , , , , , , , , , , , , , , 115–117, , , , , , , , , , , 177–179,
_Mein Kampf_ ,
Hitler Youth , , ,
Holland
Holocaust vii, , , ,
Holzapfel, Dietmar , 20–22, , , 27–29, 31–33, , ,
Honecker, Erich
Hungary
Huxley, Aldous
Innsbruck ,
International Refugee Organisation (IRO)
International Tracing Service (ITS) 76–78, , ,
Iron Cross
Iron Curtain , , , , , ,
Italy
Jews , , , , 81–83, , , , , ,
Jodl, Alfred ,
Kahrau, Helga 159–161
Kahrau, Margarete 159–161
Katowice
Keitel, Wilhelm
_Kinderfachabteilung_ 108–109
Klosterheide
Kohren-Sahlis , , , 112–113, , , , 174–175, ,
Kosovo ,
Langeoog 28–31, ,
Lauenburg
Lebensspuren , , , , , ,
Leipzig , , ,
Chamber of Commerce
Lemgo
Lidice 130–131
Lilienthal, Georg , , , , , 69–73, , , , ,
Litau, Aleksander
Ljubljana , ,
Lobetal
Loccum
Łodz , , 164–165,
London , ,
Lower Styria , ,
Lübtheen
Ludwigslust
Luftwaffe
Lurker, Otto ,
Luxembourg ,
Maas, river
Macedonia
Magdeburg 22–23
Mainz ,
Mann, Thomas
Maribor , , , , , ,
Marshall Plan
Master Race , , , , , 159–161, , , , ,
Matko family , , , , , 135–36, , 144–50, ,
Matko, Erika , , , 136–138, 145–147, 149–150, , , , , 186–187, 196–197
Matko, Helena , , , , , 183–185, ,
Matko, Ignaz , , , ,
Matko, Johann , , 145–147, , , , , 183–185, ,
Matko, Ludvig , 145–146, , , 180–181, 183–184
Matko, Maria , , , 144–147, , , , , 195–196
Matko, Marko ,
Matko, Pieter
Matko, Rafael , ,
Matko, Tanja , 145–146, , 180–181, 183–184
Mauthausen ,
Mecklenburg ,
Memel, river
Milošević, Slobodan
_Mischlings_ 81–83
Montenegro
Moscow , , , , , ,
Munich , , 58–59, , , , , , , , 174–175
Muszynski, Brunhilde
Nacht und Nebel decree
_Namensgebung_ 107–108
NATO
NKVD ,
Non-Aligned Movement
North Rhine-Westphalia ,
North Sea
Norway , , , , ,
Nothelfer 28–29
Nuremberg , 121–134,
bombing of
Laws 81–82
trials , , , , 121–123, , , , , 131–134, , , , 182–183
Obama, Barack
Occupation Zones , , ,
American , , , ,
British , , , ,
Soviet , 13–15, 22–25,
Oder, river
Oderberg
Oelhafen, Gisela von , , 14–17, 19–22, , , 29–31, , , , , 44–45, , , , , , , , , , , , , , 111–113, , , , , , , , , 185–186,
accident 45–48
and dementia ,
as physiotherapist , 48–49,
death
journal 19–20, 23–25, , 55–56,
moves to Gran Canaria
starts training
Oelhafen, Hermann von 9–10, 12–14, , , , 31–37, , , , , , , , , , 111–112, , , , , ,
Oelhafen, Hubertus von , , , 46–48, ,
Oelhafen, von, family , , , , ,
Oslo , ,
Osnabrück , , , , ,
_Ossis_
_Ost-Kinder_ 71–72
Paciorkiewicz, Barbara 164–169, 186–188
Paderborn
Paris
partisans , , , , , ,
Partisan Army
_see also_ resistance
_Persilschein_
Petrochik, Ivan
pogroms vii
Poland , , , , , , , , , , , , , ,
Potsdam
Agreement
Race Laws (1935)
rape 14–16, ,
Rath, Anna
'Ratlines'
ration cards 15–16,
Ravensbrück
Red Army 14–16, ,
Red Cross , , , , , , , 173–175
British
German (DRK) , 56–57, , , , , , 181–182
International Committee of the ,
Regensburg
Reims
Remembrance, Responsibility and Future Foundation
resistance , , , , ,
_see also_ partisans
Rhine, river
Ribbentrop, Joachim von
Rogaška Slatina , , , 98–100, , , , 135–48, , , , , , , , ,
Roosevelt, Franklin Delano ,
Roosevelt, Theodore
Rösener, Erwin
Rossmanns 167–68
RuSHA (Rasse-und-Siedlungshauptamt) , 92–93, 123–124, , ,
Russia
_see also_ Soviet Union
Saxony , , ,
Schiller, Friedrich
_Schweigelager_ 13–14,
Serbia
Sereny, Gitta , ,
Silesia
Slovenia 72–73, , , , , 139–142, , , , , , ,
Socialist Unity Party
Sollman, Max ,
Sonnenweise 69–70, , ,
South America
Soviet Military Administration (SMAD) 13–15,
Soviet Union , , , ,
Soviet army ,
_see also_ Russia
Spanish Civil War
SS , , , , 65–67, , , 83–96, 106–108, , , , , 123–124, , , , , , , , , , , , , , , ,
St Peter
St Sauerbrunn _see_ Rogaška Slatina
Stalin, Joseph ,
Stalingrad
Stari Pisker ,
Stasi
Steinhöring , ,
Stopes, Marie
Stutthof
Switzerland
Taylor, Telford
Tesch, Gunther , ,
Tito, Josip , 139–140, 176–177, ,
Treblinka
Truman, Harry ,
Tyrol, Austrian
United Kingdom ,
United Nations , , ,
_see also_ UNRRA
United States , , , , , , , , , , ,
UNRRA , ,
_Untermenschen_ ,
Verdun, battle of
Versailles
Vienna
Congress of
Viermetz, Inge , , 182–183,
_Völkischer Beobachter_ ,
_Volksdeutsche_ ,
VoMI (Volksdeutsche Mittelstelle) 132–134, 182–183
Wagner, Gerhard
Wagner, Horst
Waldeck-Pyrmont, Princes of
Washington , , ,
Weber, Guntram 155–159, ,
Wehrmacht , , , , , , , , , , , ,
Weil, Simone , ,
Weise, Jürgen 108–109
Wells, H.G.
Werdenfels ,
Wernigerode , , , 170–172
_Wessis_
Westerwald
Westphalia
Wewelsburg 89–90,
Wilson, Woodrow
Wunstorf
Yugoslavia , , , , , , , , , , , , , , , , , , , , , , ,
Congress of the League of Communists
Yugoslav High Command
Yugoslav People's Army
Zagorje ob Savi
First published 2015 by
Elliott and Thompson Limited
27 John Street
London WC1N 2BX
www.eandtbooks.com
epub: 978-1-78396-121-4
MOBI: 978-1-78396-122-1
Copyright © Ingrid von Oelhafen and Tim Tate 2015
All rights reserved. No part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, photocopying, recording or otherwise) without the prior written permission of the publisher. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages.
_Plate section picture credits:_
Page 1: © Ingrid Matko-von-Oelhafen; Page 2 (bottom): © Ingrid Matko-von-Oelhafen; Page 3 (bottom): Markus Moors, Wewlesburg Castle Museum; Page 4: © US National Archives; Page 5: © Bundesarchiv, Germany; Page 6 (top and bottom, right): © Bundesarchiv, Germany; Page 6 (bottom, left): © bpk/Fritz Carl; Page 7 (top): © Josip Pelikan/Museum of Recent History, Celje; (middle and bottom): © Museum of Recent History, Celje; Page 8 (top): © Josip Pelikan/Museum of Recent History, Celje; (bottom): © Tim Tate
Every possible effort has been made to locate and credit copyright holders of material reproduced in this book. The authors and publisher apologise for any omissions or errors, which can be corrected in future editions.
A catalogue record for this book is available from the British Library.
Jacket design: Jem Butcher Design
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{"url":"https:\/\/openplantpathology.github.io\/hagis\/news\/index.html","text":"### Minor changes\n\n\u2022 Sample names in calculate_diversities() are not required to be numeric values. Previously, this column was converted to numeric so if character values were present, these values became NA. This allows for greater flexibility when analysing the data as sample names are often more descriptive than just a numerical value.\n\n\u2022 Spelling corrections in code comments\n\n\u2022 Clean up CITATION file\n\n\u2022 README is now more complete with information and links to the MPMI paper\n\n### Minor changes\n\n\u2022 Improved documentation formatting\n\n\u2022 Update roxygen details\n\n\u2022 Fixes incomplete end of line in test-create_binary_matrix.R\n\n\u2022 Add wordlist of allowed words for spellchecking\n\n\u2022 More consistent code styling in vignettes\n\n\u2022 Prefer \u201c_\u201d to \u201c*\u201d for italics\n\n\u2022 More verbose handling of importing data.table as a whole package using \u201cR\/utils-data.table.R\u201d in place of \u201cR\/zzz.R\u201d\n\n### Minor Changes\n\n\u2022 Use ape, vegan, dplyr and vidiffr packages conditionally\n\n\u2022 Remove covr from Suggests\n\n\u2022 Better documentation formatting\n\n### Major Changes\n\n\u2022 Add new function, create_binary_matrix() to format data for exporting beta diversity matrices representing the pathotype of each isolate. Users can export a binary pathotype data matrix which could then be used to visualize beta-diversity of pathotypes using vegan or ape in R\n\n\u2022 Add new vignette, \u201cBeta-diversity Analyses\u201d, to illustrate the use of the new functionality\n\n### Minor Changes\n\n\u2022 Use ROxygen 7.1.1\n\n\u2022 Spell check and correct spelling errors\n\n### Minor Changes\n\n\u2022 Update citation with full MPMI citation\n\n\u2022 Fix issue in CITATION file where nasapower was referred to in text\n\n\u2022 Use ROxygen 7.0.0\n\n\u2022 Remove an extra \u201c\/\u201d in the CITATION\u2019s DOI\n\n### Defunct functions\n\n\u2022 plot() is now defunct. Use autoplot() to plot hagis objects in place of plot(). This is to avoid the side-effect of generating and displaying a plot every time plot() is called, which can be troublesome when using ggplot2 themes since it created two plots, the original with the base theme and the new themed plot\n\n### Minor Changes\n\n\u2022 Rename output column N_susc to N_virulent_isolates\n\n\u2022 Don\u2019t round results from summarize_gene() or calculate_complexities() before returning values to user\n\n\u2022 Implement fix suggested by @zkamvar to ensure that the user-input data is not changed from a data.frame or tibble object to a data.table object in the R session\n\n\u2022 Add ability to sort graph x-axis in ascending or descending order based on the y-axis values rather than only by gene or complexity.\n\n\u2022 Move example data set into internal data and provide documentation for them\n\n\u2022 Provide documentation for how diversity indices are calculated along with mathematical notation where possible to display\n\n\u2022 Calculate Shannon and Simpson indices internally rather than rely on vegan to reduce number of Dependencies\n\n\u2022 Replace the term field with column in documentation\n\n\u2022 Test coverage now 100 %\n\n\u2022 Add funding agencies to DESCRIPTION field\n\n\u2022 Initial CRAN release\n\n\u2022 Completely new R-package format rather than just Rmd and script files\n\n\u2022 Initial release of Rmd and script files by A. McCoy and Z. Noel","date":"2022-10-07 22:36:03","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\": 0.1712261140346527, \"perplexity\": 12625.831512327395}, \"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-2022-40\/segments\/1664030338280.51\/warc\/CC-MAIN-20221007210452-20221008000452-00142.warc.gz\"}"}
| null | null |
Q: Execute multiple semi-colon separated query using mysql Prepared Statement I am trying to create a stored procedure in mysql which creates a new table on every request copies the content from another table and extracts the required data and finally drops the table. The stored procedure is quite large so I cant have EXECUTE after every query and thus I am trying to execute the query all together in a semicolon separated format. But on final execution I get Error Code: 1064.
Is the approach I am trying possible, or is there a better approach.
SET tableName = (SELECT CONCAT("table",(UNIX_TIMESTAMP(NOW()))));
SET @tquery =CONCAT('CREATE TABLE `',tableName,'` (select pt.* from post_table pt join on user u on pt.user_id=u.id where pt.client="client",pt.group="group");');
SET @tquery = CONCAT(@tquery,' SELECT * FROM ',tableName,';');
SET @tquery = CONCAT(@tquery,' DROP TABLE ',tableName,';');
PREPARE stmt FROM @tquery;
EXECUTE stmt;
DEALLOCATE PREPARE stmt;
A: No, it is not possible. PREPARE / EXECUTE stmt can execute only one query at a time, many statements cannot be combined. See documentation: http://dev.mysql.com/doc/refman/5.0/en/prepare.html
... a user variable that contains the text of the SQL statement. The text must represent a single statement, not multiple statements.
Anyway, to simplify your code I would create a simple procedure:
CREATE PROCEDURE exec_qry( p_sql varchar(100))
BEGIN
SET @tquery = p_sql;
PREPARE stmt FROM @tquery;
EXECUTE stmt;
DEALLOCATE PREPARE stmt;
END
/
and I would call this procedure in the main procedure, in this way:
CALL exec_qry( 'CREATE TABLE t2 AS SELECT * FROM test');
CALL exec_qry( 'SELECT * FROM t2');
CALL exec_qry( 'SELECT count(*) FROM t2');
CALL exec_qry( 'SELECT avg(x) FROM t2');
CALL exec_qry( 'DROP TABLE t2');
Take a look at a demo: http://www.sqlfiddle.com/#!2/6649a/6
|
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{"url":"https:\/\/www.physicsforums.com\/threads\/a-little-fun.575242\/","text":"A little fun\n\n1. Feb 7, 2012\n\nsquareroot\n\nHey , can you give me any idea of a cool physics avatar for yahooM?I tried the one with Schrodinger's cat is DEAD\/ALIVE , but you can't understand the writing.I need something physics-related and funny, something to observe even in a small picture.\n\nTY guys!\n\n2. Feb 8, 2012\n\nMansonn\n\nthat's really good joke....funny....","date":"2018-11-13 01:07:16","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\": 0.8298406004905701, \"perplexity\": 4549.157400957476}, \"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-2018-47\/segments\/1542039741176.4\/warc\/CC-MAIN-20181113000225-20181113022225-00552.warc.gz\"}"}
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\section{Introduction}
Quantum metrics on C*-algebras, formally provided by generalized Lipschitz seminorms called \emph{Lip-norms} \cite{Rieffel98a,Rieffel99}, are the seeds for a new analytic framework which brings techniques from metric geometry in C*-algebra theory and provides a new tool set for problems from mathematical physics, such as finite dimensional approximations of quantum space-time \cite{Rieffel00,Rieffel01,Latremoliere05,Latremoliere13c,Rieffel10c,Rieffel15,Latremoliere16}, or perturbations of quantum metrics \cite{Latremoliere15b,Latremoliere15c}. Quantum compact metric spaces form a natural category \cite{Latremoliere15b}, whose morphisms are Lipschitz in an appropriate sense. In this paper, we prove that any *-morphism between two quantum compact metric spaces is actually Lipschitz if and only if it is compatible with the domains of the Lip-norms. In particular, *-automorphisms which preserve the domain of a Lip-norm must be bi-Lipschitz, and thus all Lip-norms with a common domain are actually equivalent. We then explore three related problems: we show that the topology of pointwise convergence on the group of Lipschitz automorphisms of a quantum compact metric space may be metrized using Lip-norms, and that our previous work on quantum perturbations naturally provide new examples of compact classes of quantum compact metric spaces for the quantum propinquity. We also construct the noncommutative generalization of the Lipschitz distance.
A compact quantum metric space is a generalization of a Lipschitz algebra, inspired by the work of Connes \cite{Connes89, Connes} and formalized by Rieffel \cite{Rieffel98a, Rieffel99}:
\begin{notation}
The space of self-adjoint elements in a C*-algebra ${\mathfrak{A}}$ is denoted by $\sa{{\mathfrak{A}}}$, while the state space of ${\mathfrak{A}}$ is denoted by ${\mathscr{S}}({\mathfrak{A}})$. The unit of a unital C*-algebra ${\mathfrak{A}}$ is denoted by $1_{\mathfrak{A}}$. Last, we denote the norm on a normed vector space $E$ by $\|\cdot\|_E$ by default.
Last, the diameter of a metric space $(E,\mathrm{d})$ is denoted by $\diam{E}{\mathrm{d}}$.
\end{notation}
\begin{definition}[\cite{Rieffel98a, Rieffel99}]\label{qcms-def}
A pair $({\mathfrak{A}},{\mathsf{L}})$ is a \emph{quantum compact metric space} when ${\mathfrak{A}}$ is a unital C*-algebra and ${\mathsf{L}}$ is a seminorm defined on a dense subspace $\dom{{\mathsf{L}}}$ of $\sa{{\mathfrak{A}}}$, such that:
\begin{enumerate}
\item $\{a\in\dom{{\mathsf{L}}} : {\mathsf{L}}(a) = 0\} = {\mathds{R}}1_{\mathfrak{A}}$,
\item the \emph{{Mon\-ge-Kan\-to\-ro\-vich metric}} $\Kantorovich{{\mathsf{L}}}$ defined for any two $\varphi,\psi \in {\mathscr{S}}({\mathfrak{A}})$ by:
\begin{equation*}
\Kantorovich{{\mathsf{L}}}(\varphi,\psi) = \sup\left\{ |\varphi(a) - \psi(a)| : a\in\dom{{\mathsf{L}}}, {\mathsf{L}}(a)\leq 1 \right\}\text{,}
\end{equation*}
metrizes the weak* topology on ${\mathscr{S}}({\mathfrak{A}})$.
\end{enumerate}
If $({\mathfrak{A}},{\mathsf{L}})$ is a quantum compact metric space, then ${\mathsf{L}}$ is called a \emph{Lip-norm} on ${\mathfrak{A}}$.
\end{definition}
The classical picture behind Definition (\ref{qcms-def}) is provided by a pair $(C(X),\mathsf{Lip})$ of the C*-algebra of a compact metric space $(X,\mathsf{d})$ and the Lipschitz seminorm $\mathsf{Lip}$ associated to the distance function $\mathsf{d}$. A generalization of quantum compact metric spaces to the quantum locally compact setting was proposed in \cite{Latremoliere05b,Latremoliere12b}.
We may define a category whose objects are quantum compact metric spaces, and whose morphisms are a special type of *-morphisms between the underlying C*-algebras. There are at least two natural ideas. We may require that a Lipschitz morphism be a *-morphism which is also continuous with respect to the Lip-norms. Formally, if $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ are two quantum compact metric spaces, and $\varphi : {\mathfrak{A}}\rightarrow{\mathfrak{B}}$ is a *-morphism, this first approach to Lipschitz morphism consists in requiring that there exists $C > 0$ such that ${\mathsf{L}}_{\mathfrak{B}}\circ\varphi \leq C{\mathsf{L}}_{\mathfrak{A}}$. This relation imposes that $\varphi$ must be unital or null. Indeed, ${\mathsf{L}}_{\mathfrak{B}}\circ\varphi(1_{\mathfrak{A}}) \leq C {\mathsf{L}}_{\mathfrak{A}}(1_{\mathfrak{A}}) = 0$ so $\varphi(1_{\mathfrak{A}})\in{\mathds{R}}1_{\mathfrak{B}}$; since $\varphi$ is a *-morphism, this leaves us with $\varphi(1_{\mathfrak{A}}) \in \{ 0, 1_{\mathfrak{B}} \}$. In this paper, we will work with unital *-morphisms.
Alternatively, we may require that the dual map associated to a unital *-morphism be a Lipschitz map between the state spaces equipped with their respective {{Mon\-ge-Kan\-to\-ro\-vich metric}}. Continuing with our notations, we would thus ask that there exists $C > 0$ such that for all $\mu, \nu \in {\mathscr{S}}({\mathfrak{B}})$, we have:
\begin{equation*}
\Kantorovich{{\mathsf{L}}_{\mathfrak{A}}}(\mu\circ\varphi, \nu\circ\varphi) \leq C \Kantorovich{{\mathsf{L}}_{\mathfrak{B}}}(\mu, \nu)\text{.}
\end{equation*}
In general, these two notions of a Lipschitz morphisms are not equivalent, owing to the fact that the {{Mon\-ge-Kan\-to\-ro\-vich metric}} does not allow the recovery of the Lip-norm from which it was defined. After all, many Lip-norms may give the same {{Mon\-ge-Kan\-to\-ro\-vich metric}}.
However, among all Lip-norms which provide a given {{Mon\-ge-Kan\-to\-ro\-vich metric}}, there is a particular one: the largest among all of them, which is characterized as being lower semi-continuous with respect to the norm of the underlying C*-algebra. In \cite{Rieffel99}, the study of this problem led to the notion of a closed Lip-norm, though the context there was more general (the underlying space was not a C*-algebra but a more general object called an order-unit space, which may not be complete, leading to some important subtleties).
For our purpose, it is thus natural to work with lower semi-continuous Lip-norms. In this context, our two notions of Lipschitz morphisms coincide. So we summarize our notion by:
\begin{definition}[{\cite{Latremoliere15b}}]
Let $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ be two quantum compact metric spaces, with ${\mathsf{L}}_{\mathfrak{A}}$ and ${\mathsf{L}}_{\mathfrak{B}}$ lower semi-continuous with respect to the norms of, respectively, ${\mathfrak{A}}$ and ${\mathfrak{B}}$. A unital *-morphism $\varphi : {\mathfrak{A}} \rightarrow {\mathfrak{B}}$ is \emph{$k$-Lipschitz} for some $k \geq 0$ when ${\mathsf{L}}_{\mathfrak{B}}\circ\varphi \leq k {\mathsf{L}}_{\mathfrak{A}}$, or equivalently:
\begin{equation*}
\mu \in {\mathscr{S}}({\mathfrak{B}}) \longmapsto \mu\circ\varphi \in {\mathscr{S}}({\mathfrak{A}})
\end{equation*}
is a $k$-Lipschitz map from $({\mathscr{S}}({\mathfrak{B}}),\Kantorovich{{\mathsf{L}}_{\mathfrak{B}}})$ to $({\mathscr{S}}({\mathfrak{A}}),\Kantorovich{{\mathsf{L}}_{\mathfrak{A}}})$.
\end{definition}
It is easy to check that indeed, composition of Lipschitz morphisms is again Lipschitz, and the identity morphism is $1$-Lipschitz, so we have indeed defined a category. It is also easy to check that a $k$-Lipschitz morphism between two classical compact metric spaces is indeed of the form $f \in C(Y)\mapsto f\circ \theta$ with $\theta : X\rightarrow Y$ is $k$-Lipschitz.
In this paper, we investigate a third approach of Lipschitz morphisms. If $\dom{{\mathsf{L}}_{\mathfrak{A}}}$ and $\dom{{\mathsf{L}}_{\mathfrak{B}}}$ are the domains of ${\mathsf{L}}_{\mathfrak{A}}$ and ${\mathsf{L}}_{\mathfrak{B}}$, then a *-morphism may satisfy $\varphi(\dom{{\mathsf{L}}_{\mathfrak{A}}}) \subseteq \dom{{\mathsf{L}}_{\mathfrak{B}}}$. This appear to be a weaker notion, but as we shall see, it is again equivalent to the notion of a Lipschitz morphism. This reinforces that our notion of a category of quantum compact metric spaces is indeed appropriate.
Our paper then continues with an observation regarding sets of uniformly equivalent Lip-norms. We note that such sets are naturally compact for the quantum propinquity. The quantum Gromov-Hausdorff propinquity \cite{Latremoliere13} is a special member of the family of Gromov-Hausdorff propinquities \cite{Latremoliere13b,Latremoliere14}, which are all noncommutative analogues of the Gromov-Hausdorff distance \cite{Gromov, Gromov81} extending the topology of the latter to quantum compact metric spaces. The Gromov-Hausdorff propinquity provides a framework for the geometric study of classes of quantum compact metric spaces. We have established, for instance, the continuity of various natural families of C*-algebras such as quantum tori \cite{Latremoliere13c} or certain AF-algebras \cite{Latremoliere15d}. We constructed finite dimensional approximations for these spaces as well, answering some informal statements in mathematical physics. Another recent advance was the generalization of Gromov's theorem to our propinquity \cite{Latremoliere15}, providing us with an insight into the topological properties of various sets of compact quantum metric spaces.
In general, proving that a class of quantum compact metric spaces is indeed totally bounded for the propinquity may be subtle. For instance, \cite{Latremoliere15} relies on finite dimensional approximations, which are themselves a challenge. Nonetheless, as seen in \cite[Theorem 6.3]{Latremoliere15d}, our generalized Gromov's theorem can be put to use. Another approach taken in \cite{Latremoliere15d} was the construction of continuous maps from some compact spaces onto classes of quantum compact metric spaces; this method also applies to quantum tori and their finite dimensional approximations \cite{Latremoliere05,Latremoliere13c} and we shall see that it applies to conformal deformations as well in this paper. In this paper, we take yet another road to establish the compactness of some interesting classes of quantum compact metric spaces, obtained as perturbations of given quantum metrics.
Prior to the introduction of noncommutative Gromov-Hausdorff distances \cite{Rieffel00,kerr02,Latremoliere13,Latremoliere13b,Latremoliere14}, the idea of perturbations for a quantum metrics seemed to largely rely on the informal idea that certain algebraic expressions are qualitatively close to some original metric. We recently formalized this idea by actually establishing bounds on how far, in the sense of the propinquity, certain particular perturbations actually are. Examples of such perturbations include conformal deformations \cite{Ponge14} of quantum metrics arising from certain spectral triples \cite{Latremoliere15b}, leading to twisted spectral triples introduced by Connes and Moscovici \cite{Connes08} . We also brought curved quantum tori of Sitarz and D\k{a}browski \cite{Sitarz13,Sitarz15} into our program \cite{Latremoliere15c}. We shall see that a core ingredient of the constructions of these perturbations provide a uniform equivalence between Lip-norms, which in turn gives a compactness result.
We note that, besides the Gromov-Hausdorff distance, a standard extended metric between compact metric spaces is the Lipschitz distance. We provide in this paper a construction for the noncommutative version of the Lipschitz distance, which fits very well with the picture of Lipschitz morphisms presented in this paper.
The last section of this paper presents a natural metric on Lipschitz morphisms, built from quantum metrics.
\section{Equivalence of Lip-norms and Lipschitz morphisms}
Quantum metrics defined on the same domain are, in fact, equivalent, under a natural technical condition, which may always be assumed to no cost to the underlying metric structure. This observation is the subject of the following theorem, and constitutes our main result.
\begin{theorem}\label{equiv-thm}
Let $({\mathfrak{A}},{\mathsf{L}})$ be a quantum compact metric space, with ${\mathsf{L}}$ lower semi-continuous with domain $\dom{{\mathsf{L}}}$. Let $\mathsf{S}$ be a seminorm on $\dom{{\mathsf{L}}}$ such that:
\begin{enumerate}
\item $\mathsf{S}$ is lower semi-continuous with respect to $\|\cdot\|_{\mathfrak{A}}$,
\item $\mathsf{S}(1_{\mathfrak{A}}) = 0$.
\end{enumerate}
Then there exists $C > 0$ such that for all $a\in\dom{{\mathsf{L}}}$:
\begin{equation*}
\mathsf{S}(a) \leq C {\mathsf{L}}(a)\text{.}
\end{equation*}
\end{theorem}
\begin{proof}
Our proof proceeds in three steps. First, we prove that the domain of a lower semi-continuous seminorm can be made naturally into a Banach space. Then, we use the open mapping theorem to show that different lower semi-continuous seminorms defined on the same domain give rise to equivalent Banach norms with the construction in our first step. Last, we conclude our theorem.
\begin{step}
Let $\mathsf{S}$ be a lower semi-continuous seminorm defined on some dense subspace $\dom{\mathsf{S}}$ of $\sa{{\mathfrak{A}}}$. Let $\|\cdot\|_{\mathsf{S}} = \|\cdot\|_{\mathfrak{A}} + {\mathsf{S}}$. We first check that $\dom{{\mathsf{S}}}$ is a Banach space for the norm $\|\cdot\|_{\mathsf{S}}$.
\end{step}
It is straightforward that $\|\cdot\|_{\mathsf{S}}$ is a norm on $\dom{{\mathsf{S}}}$.
Let $(a_n)_{n\in{\mathds{N}}}$ be a Cauchy sequence in $\dom{{\mathsf{S}}}$ for $\|\cdot\|_{\mathsf{S}}$. Thus $(a_n)_{n\in{\mathds{N}}}$ is a Cauchy sequence for $\|\cdot\|_{\mathfrak{A}}$, which is complete, so $(a_n)_{n\in{\mathds{N}}}$ converges to some $a \in \sa{{\mathfrak{A}}}$ for $\|\cdot\|_{\mathfrak{A}}$.
We also observe that $(a_n)_{n\in{\mathds{N}}}$ is Cauchy, hence bounded, for ${\mathsf{S}}$; thus there exists $M > 0$ such that ${\mathsf{S}}(a_n)\leq M$ for all $n\in{\mathds{N}}$. As ${\mathsf{S}}$ is lower semi-continuous with respect to $\|\cdot\|_{\mathfrak{A}}$, we obtain that ${\mathsf{S}}(a) \leq \liminf_{n\rightarrow\infty} {\mathsf{S}}(a_n) \leq M$. Thus $a\in \dom{{\mathsf{S}}}$.
Let $\varepsilon > 0$. Since $(a_n)_{n\in{\mathds{N}}}$ is Cauchy for ${\mathsf{S}}$, there exists $N\in{\mathds{N}}$ such that for all $p,q\geq N$ we have ${\mathsf{S}}(a_p-a_q) \leq \varepsilon$. Since ${\mathsf{S}}$ is lower semi-continuous with respect to $\|\cdot\|_{\mathfrak{A}}$, we thus have, for all $p\geq N$:
\begin{equation*}
{\mathsf{S}}(a-a_p)\leq \liminf_{q\rightarrow\infty} {\mathsf{S}}(a_p-a_q) \leq \varepsilon\text{.}
\end{equation*}
Thus $\lim_{p\rightarrow\infty} {\mathsf{S}}(a-a_p) = 0$. This proves that $\lim_{n\rightarrow\infty} \|a-a_n\|_{\mathsf{S}} = 0$ as desired.
\begin{step}
If ${\mathsf{L}}$ and $\mathrm{S}$ be two lower semi-continuous seminorms on some common dense subspace $\dom{{\mathsf{L}}}$ of $\sa{{\mathfrak{A}}}$, then the norms:
\begin{equation*}
\|\cdot\|_{{\mathsf{L}}} = \|\cdot\|_{\mathfrak{A}} + {\mathsf{L}} \text{ and }\|\cdot\|_{\mathsf{S}} = \|\cdot\|_{\mathfrak{A}} + \mathsf{S}
\end{equation*}
are equivalent.
\end{step}
The norms $\|\cdot\|_{{\mathsf{L}}}$ and $\|\cdot\|_{\mathsf{S}}$ both make $\dom{{\mathsf{L}}}$ into a Banach space, by step 1.
We begin with a simple observation. Let $(x_n)_{n\in{\mathds{N}}}$ be a sequence in $\dom{{\mathsf{L}}}$ which converges for both $\|\cdot\|_{{\mathsf{L}}}$ and $\|\cdot\|_{\mathsf{S}}$. Let $x \in \dom{{\mathsf{L}}}$ be the limit of $(x_n)_{n\in{\mathds{N}}}$ for $\|\cdot\|_{{\mathsf{L}}}$ and $y \in \dom{{\mathsf{L}}}$ be the limit of $(x_n)_{n\in{\mathds{N}}}$ for $\|\cdot\|_{\mathsf{S}}$. We note that in particular, $(x_n)_{n\in{\mathds{N}}}$ converges to both $x$ and $y$ for $\|\cdot\|_{\mathfrak{A}}$. Thus $x=y$.
Let now $\|\cdot\|_\ast = \|\cdot\|_{{\mathsf{L}}} + \|\cdot\|_{\mathsf{S}}$. If $(x_n)_{n\in{\mathds{N}}}$ is a Cauchy sequence for $\|\cdot\|_\ast$, then the sequence $(x_n)_{n\in{\mathds{N}}}$ is Cauchy for both $\|\cdot\|_{{\mathsf{L}}}$ and $\|\cdot\|_{\mathsf{S}}$ and thus converges for both these norms, since they are complete; by our previous observation, $(x_n)_{n\in{\mathds{N}}}$ has the same limit $x\in\dom{{\mathsf{L}}}$ for both these norms. Hence, $(x_n)_{n\in{\mathds{N}}}$ converges to $x$ for $\|\cdot\|_\ast$, i.e. $\left(\dom{{\mathsf{L}}},\|\cdot\|_\ast\right)$ is a Banach space.
Now, since $\|\cdot\|_{{\mathsf{L}}}\leq \|\cdot\|_\ast$, the open mapping theorem \cite{Brezis83} implies that there exists $k > 0$ such that $\|\cdot\|_\ast\leq k\|\cdot\|_{{\mathsf{L}}}$. We then conclude easily that $\|\cdot\|_{\mathsf{S}}\leq k \|\cdot\|_{{\mathsf{L}}}$. Similarly, for some $k' > 0$, we have $\|\cdot\|_{{\mathsf{L}}}\leq k' \|\cdot\|_{\mathsf{S}}$.
\begin{step}
We now conclude our theorem.
\end{step}
We thus are given a lower semi-continuous Lip-norm ${\mathsf{L}}$ on ${\mathfrak{A}}$, and some seminorm $\mathrm{S}$ on the domain of ${\mathsf{L}}$, with $\mathrm{S}(1_{\mathfrak{A}}) = 0$ and $\mathrm{S}$ lower semi-continuous with respect to $\|\cdot\|_{\mathfrak{A}}$.
Using our previous step, there exists $k > 0$ such that for all $a\in\dom{{\mathsf{L}}}$, we have:
\begin{equation*}
\begin{split}
\mathsf{S}(a) &= \|a\|_{\mathsf{S}} - \|a\|_{\mathfrak{A}} \\
&\leq k\|a\|_{{\mathsf{L}}} - \|a\|_{\mathfrak{A}} \\
&\leq k\left(\|a\|_{\mathfrak{A}} + {\mathsf{L}}(a)\right) - \|a\|_{\mathfrak{A}} \\
&= (k-1)\|a\|_{\mathfrak{A}} + k{\mathsf{L}}(a)\text{.}
\end{split}
\end{equation*}
Let $D = \diam{{\mathscr{S}}({\mathfrak{A}})}{\Kantorovich{{\mathsf{L}}}}$ (which is finite since ${\mathsf{L}}$ is a Lip-norm). Let $a\in\dom{{\mathsf{L}}_{\mathfrak{A}}}$. As ${\mathsf{L}}$ is a Lip-norm, there exists $t\in{\mathds{R}}$ such that:
\begin{equation}\label{equiv-thm-exp1}
\|a+t1_{\mathfrak{A}}\|_{\mathfrak{A}} \leq D {\mathsf{L}}(a)\text{.}
\end{equation}
Indeed, let $\mu\in{\mathscr{S}}({\mathfrak{A}})$. For all $\nu\in{\mathscr{S}}({\mathfrak{A}})$ then:
\begin{equation*}
|\mu(a)-\nu(a)|\leq {\mathsf{L}}(a)\Kantorovich{{\mathsf{L}}}(\mu,\nu) \leq {\mathsf{L}}(a)D\text{.}
\end{equation*}
Of course, ${\mathsf{L}}(a+t1_{\mathfrak{A}}) = {\mathsf{L}}(a)$ and $\mathsf{S}(a+t1_{\mathfrak{A}}) = \mathsf{S}(a)$ since $\mathsf{S}(1_{\mathfrak{A}}) = 0$. Thus:
\begin{equation*}
\begin{split}
\mathsf{S}(a) &= \mathsf{S}(a + t1_{\mathfrak{A}}) \\
&\leq (k-1)\|a + t1_{\mathfrak{A}}\|_{\mathfrak{A}} + k {\mathsf{L}}(a + t1_{\mathfrak{A}}) \\
&\leq (k-1) D {\mathsf{L}}(a) + k{\mathsf{L}}(a) = ((k-1)D + k) {\mathsf{L}}(a)\text{.}
\end{split}
\end{equation*}
This concludes our proof, with $C = ((k-1)D+k)$.
\end{proof}
Theorem (\ref{equiv-thm}) has the following consequences:
\begin{corollary}\label{Lipschitz-cor}
Let $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ be two compact quantum metric spaces whose Lip-norms are lower semi-continuous. Let $\varphi : {\mathfrak{A}}\rightarrow{\mathfrak{B}}$ be some unital *-morphism such that $\varphi(\dom{{\mathsf{L}}_{\mathfrak{A}}}) \subseteq \dom{{\mathsf{L}}_{\mathfrak{B}}}$. Then there exists $C > 0$ such that:
\begin{equation*}
{\mathsf{L}}_{\mathfrak{B}}\circ\varphi \leq C {\mathsf{L}}_{\mathfrak{A}}
\end{equation*}
i.e. $\varphi$ is a $C$-Lipschitz morphism.
\end{corollary}
\begin{proof}
Let $\mathsf{S} = {\mathsf{L}}_{\mathfrak{B}}\circ\varphi$. We note that $\mathsf{S}$ is a lower semi-continuous seminorm which takes finite values on $\dom{{\mathsf{L}}_{\mathfrak{A}}}$ by assumption. Moreover $\mathsf{S}(1_{\mathfrak{A}}) = 0$. Thus our corollary follows from Theorem (\ref{equiv-thm}).
\end{proof}
\begin{corollary}\label{bi-Lipschitz-cor}
Let $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ be two quantum compact metric spaces whose Lip-norms are lower semi-continuous. If $\varphi : {\mathfrak{A}}\rightarrow{\mathfrak{B}}$ is a *-isomorphism such that $\varphi(\dom{{\mathsf{L}}_{\mathfrak{A}}}) = \dom{{\mathsf{L}}_{\mathfrak{B}}}$, then there exists $C > 0$ such that:
\begin{equation*}
C^{-1} {\mathsf{L}}_{\mathfrak{A}} \leq {\mathsf{L}}_{\mathfrak{B}}\circ\varphi \leq C {\mathsf{L}}_{\mathfrak{A}}\text{.}
\end{equation*}
\end{corollary}
\begin{proof}
This follows from Corollary (\ref{Lipschitz-cor}).
\end{proof}
\begin{corollary}\label{auto-Lipschitz-cor}
Let $({\mathfrak{A}},{\mathsf{L}})$ be a quantum compact metric space where ${\mathsf{L}}$ is lower semi-continuous. Let $\alpha$ be a *-automorphism of ${\mathfrak{A}}$. The following two assertions are equivalent:
\begin{enumerate}
\item $\alpha(\dom{{\mathsf{L}}}) = \dom{{\mathsf{L}}}$,
\item there exists $C > 0$ such that $C^{-1} {\mathsf{L}}\circ\alpha \leq {\mathsf{L}} \leq C {\mathsf{L}}\circ\alpha$.
\end{enumerate}
\end{corollary}
\begin{proof}
Assume (2) first. Let $a\in\dom{{\mathsf{L}}}$. Then:
\begin{equation*}
{\mathsf{L}}(\alpha^{-1}(a)) \leq C {\mathsf{L}}(\alpha(\alpha^{-1}(a))) = {\mathsf{L}}(a) < \infty\text{,}
\end{equation*}
hence $\alpha^{-1}(a) \in \dom{{\mathsf{L}}}$. Thus $a = \alpha(\alpha^{-1}(a)) \in \alpha(\dom{{\mathsf{L}}})$, i.e. $\dom{{\mathsf{L}}} \subseteq \alpha(\dom{{\mathsf{L}}})$. The converse inclusion is proven similarly.
Assume (1). Then (2) follows from Corollary (\ref{bi-Lipschitz-cor}).
\end{proof}
\begin{corollary}\label{equiv-cor}
Let ${\mathfrak{A}}$ be a unital C*-algebra and ${\mathsf{L}}_1$, ${\mathsf{L}}_2$ be two lower semi-continuous Lip-norms on ${\mathfrak{A}}$. The following assertions are equivalent:
\begin{enumerate}
\item ${\mathsf{L}}_1$ and ${\mathsf{L}}_2$ are equivalent,
\item $\dom{{\mathsf{L}}_1} = \dom{{\mathsf{L}}_2}$.
\end{enumerate}
\end{corollary}
\begin{proof}
Apply Corollary (\ref{auto-Lipschitz-cor}) to the identity automorphism of ${\mathfrak{A}}$.
\end{proof}
\section{Compactness of classes of perturbations of {quasi-Leibniz quantum compact metric space s}}
To present the result in this section, we introduce a simple metric on Lip-norms over a fixed C*-algebra.
\begin{definition}\label{HausLip-def}
For any two lower semi-continuous Lip-norms ${\mathsf{L}}$, ${\mathsf{L}}'$ on a unital C*-algebra ${\mathfrak{A}}$, we define $\HausLip{{\mathfrak{A}}}({\mathsf{L}},{\mathsf{L}}')$ by:
\begin{equation*}
\HausLip{{\mathfrak{A}}}({\mathsf{L}},{\mathsf{L}}') = \Haus{\|\cdot\|_{\mathfrak{A}}}\left(\left\{a\in \sa{{\mathfrak{A}}} : {\mathsf{L}}(a)\leq 1\right\}, \{a\in \sa{{\mathfrak{A}}}: {\mathsf{L}}'(a)\leq 1\} \right)\text{,}
\end{equation*}
where $\Haus{\|\cdot\|_{\mathfrak{A}}}$ is the Hausdorff distance induced on closed subsets of ${\mathfrak{A}}$ by the norm $\|\cdot\|_{\mathfrak{A}}$.
\end{definition}
We begin by observing that our distance $\HausLip{{\mathfrak{A}}}$ is indeed finite. Let us start by recalling the following fundamental characterization of quantum compact metric spaces proved by Rieffel in \cite{Rieffel98a}, and akin to a noncommutative Arz{\'e}la-Ascoli theorem:
\begin{theorem}[{\cite{Rieffel98a, Rieffel99}}]\label{Rieffel-az-thm}
Let ${\mathfrak{A}}$ be a unital C*-algebra and ${\mathsf{L}}$ a seminorm defined on a dense subspace $\dom{{\mathsf{L}}}$ of $\sa{{\mathfrak{A}}}$ and such that $\{a\in\dom{{\mathsf{L}}}:{\mathsf{L}}(a) = 0\} = {\mathds{R}}1_{\mathfrak{A}}$. The following assertions are equivalent:
\begin{enumerate}
\item $({\mathfrak{A}},{\mathsf{L}})$ is a quantum compact metric space,
\item $\{a\in\dom{{\mathsf{L}}} : {\mathsf{L}}(a) \leq 1, \varphi(a) = 0\}$ is totally bounded in norm for some state $\varphi\in{\mathscr{S}}({\mathfrak{A}})$,
\item $\{a\in\dom{{\mathsf{L}}} : {\mathsf{L}}(a) \leq 1, \varphi(a) = 0\}$ is totally bounded in norm for all states $\varphi\in{\mathscr{S}}({\mathfrak{A}})$,
\item $\{a\in\dom{{\mathsf{L}}} : {\mathsf{L}}(a)\leq 1, \|a\|_{\mathfrak{A}}\leq 1\}$ is totally bounded in norm.
\end{enumerate}
In particular, if ${\mathsf{L}}$ is lower semi-continuous, then ${\mathsf{L}}$ is a Lip-norm if, and only if any of the sets above are compact.
\end{theorem}
\begin{lemma}\label{finite-HausLip-lemma}
Let ${\mathfrak{A}}$ be a unital C*-algebra. For any two lower semi-continuous Lip-norms ${\mathsf{L}}_1$ and ${\mathsf{L}}_2$ on ${\mathfrak{A}}$, and for any state $\varphi \in {\mathscr{S}}({\mathfrak{A}})$, we have:
\begin{multline*}
\HausLip{{\mathfrak{A}}}({\mathsf{L}}_1,{\mathsf{L}}_2) \leq \Haus{\|\cdot\|_{\mathfrak{A}}}(\{a\in\sa{{\mathfrak{A}}}:\varphi(a) = 0, {\mathsf{L}}_1(a) \leq 1\}, \\ \{a\in\sa{{\mathfrak{A}}}:\varphi(a)=0, {\mathsf{L}}_2(a)\leq 1\}) \text{.}
\end{multline*}
In particular, $\HausLip{{\mathfrak{A}}}({\mathsf{L}}_1,{\mathsf{L}}_2)$ is finite.
\end{lemma}
\begin{proof}
Let $\varphi \in {\mathscr{S}}({\mathfrak{A}})$. By Theorem (\ref{Rieffel-az-thm}), $\{a\in\sa{{\mathfrak{A}}} : \varphi(a) = 0, {\mathsf{L}}_j(a)\leq 1\}$ are compact for $j=1,2$. Thus the Hausdorff distance (for the norm of ${\mathfrak{A}}$) between these two sets is finite; let us denote it by $d$.
Let $a\in\sa{{\mathfrak{A}}}$ with ${\mathsf{L}}_1(a)\leq 1$. There exists $b \in \sa{{\mathfrak{A}}}$ with ${\mathsf{L}}_2(b) \leq 1$ and $\varphi(b) = 0$ such that $\|a-\varphi(a)1_{\mathfrak{A}} - b\|_{\mathfrak{A}} \leq d$. Thus $\|a - (b + \varphi(a)1_{\mathfrak{A}})\|_{\mathfrak{A}} \leq d$, and we note that ${\mathsf{L}}_2(b + \varphi(a)1_{\mathfrak{A}}) = {\mathsf{L}}_2(b) \leq 1$. As the argument is symmetric is ${\mathsf{L}}_1$ and ${\mathsf{L}}_2$, we have shown our lemma.
\end{proof}
Our goal is to establish a new sufficient condition for certain classes of quantum compact metric spaces to be totally bounded for the \emph{quantum propinquity}. We refer to \cite{Latremoliere13,Latremoliere13b,Latremoliere14,Latremoliere15,Latremoliere15b} for the definition of the quantum propinquity and some of its properties. We briefly recall from \cite{Latremoliere13} the notion of a bridge, as it will be used in our next proof, and provide a characterization of the quantum propinquity.
Let ${\mathfrak{A}}$ and ${\mathfrak{B}}$ be two unital C*-algebras. A \emph{bridge} $({\mathfrak{D}},\omega,\pi_{\mathfrak{A}},\pi_{\mathfrak{B}})$ from ${\mathfrak{A}}$ to ${\mathfrak{B}}$ is a unital C*-algebra ${\mathfrak{D}}$ and two unital *-monomorphisms $\pi_{\mathfrak{A}} : {\mathfrak{A}} \hookrightarrow{\mathfrak{D}}$ and $\pi_{\mathfrak{B}}:{\mathfrak{B}} \hookrightarrow{\mathfrak{D}}$, as well as an element $\omega\in{\mathfrak{D}}$ such that for at least one state $\varphi$ of ${\mathfrak{D}}$, we have $\varphi(\omega d) = \varphi(d \omega) = \varphi(d)$ for all $d\in{\mathfrak{D}}$. The set of all such states of ${\mathfrak{D}}$, denoted by ${\mathscr{S}}_1({\mathfrak{D}}|\omega)$, is the $1$-level set of ${\mathfrak{D}}$.
Now if $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ are two quantum compact metric spaces, then we can associate a number, called the length, to a bridge $\gamma = ({\mathfrak{D}},\omega,\pi_{\mathfrak{A}},\pi_{\mathfrak{B}})$ from ${\mathfrak{A}}$ to ${\mathfrak{B}}$. We first define the \emph{reach} of $\gamma$ as the Hausdorff distance between $\{ \pi_{\mathfrak{A}}(a)\omega : a\in\sa{{\mathfrak{A}}}, {\mathsf{L}}_{\mathfrak{A}}(a)\leq 1\}$ and $\{ \omega\pi_{\mathfrak{A}}(b) : b\in\sa{{\mathfrak{B}}}, {\mathsf{L}}_{\mathfrak{B}}(b)\leq 1\}$ for $\Haus{\|\cdot\|_{\mathfrak{D}}}$. We then define the \emph{height} of $\gamma$ as the maximum of the Hausdorff distance, for $\Haus{\Kantorovich{{\mathsf{L}}_{\mathfrak{A}}}}$, between ${\mathscr{S}}({\mathfrak{A}})$ and $\{\varphi\circ\pi_{\mathfrak{A}} : \varphi\in{\mathscr{S}}_1({\mathfrak{D}}|\omega)\}$, and the Hausdorff distance for $\Haus{\Kantorovich{{\mathsf{L}}_{\mathfrak{B}}}}$, between ${\mathscr{S}}({\mathfrak{B}})$ and $\{\varphi\circ\pi_{\mathfrak{B}} : \varphi\in{\mathscr{S}}_1({\mathfrak{D}}|\omega)\}$.
The length $\bridgelength{\gamma}{{\mathsf{L}}_{\mathfrak{A}},{\mathsf{L}}_{\mathfrak{B}}}$ of the bridge $\gamma$ is the maximum of its reach and its height. The quantum propinquity is constructed from bridges, although it requires a few technical steps. In particular, the quantum propinquity is defined on classes of {\Qqcms{F}} for an \emph{admissible function $F$}, i.e. a function $F : [0,\infty)^4\rightarrow[0,\infty)$ which is increasing for the product order on $[0,\infty)^4$ and such that $F(x,y,l_x,l_y) \geq xl_y + y l_x$ for all $x,y,l_x,l_y \geq 0$. Given such a function, a {\Qqcms{F}} $({\mathfrak{A}},{\mathsf{L}})$ is a quantum compact metric space such that for all $a,b \in \sa{{\mathfrak{A}}}$ we have:
\begin{equation*}
\max\left\{ {\mathsf{L}}\left(\frac{a b + b a}{2}\right), {\mathsf{L}}\left(\frac{a b - b a}{2i} \right) \right\} \leq F(\|a\|_{\mathfrak{A}},\|b\|_{\mathfrak{A}},{\mathsf{L}}(a),{\mathsf{L}}(b)) \text{.}
\end{equation*}
The following result characterizes the quantum propinquity.
\begin{theorem-definition}[\cite{Latremoliere13}]\label{def-thm}
Let $\mathcal{L}$ be the class of all {\Qqcms{F}s} for some admissible function $F$. There exists a class function $\qpropinquity{F}$ from $\mathcal{L}\times\mathcal{L}$ to $[0,\infty) \subseteq {\mathds{R}}$ such that:
\begin{enumerate}
\item for any $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}), ({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}) \in \mathcal{L}$ we have:
\begin{equation*}
0\leq \qpropinquity{F}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) \leq \max\left\{\diam{{\mathscr{S}}({\mathfrak{A}})}{\Kantorovich{{\mathsf{L}}_{\mathfrak{A}}}}, \diam{{\mathscr{S}}({\mathfrak{B}})}{\Kantorovich{{\mathsf{L}}_{\mathfrak{B}}}}\right\}\text{,}
\end{equation*}
\item for any $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}), ({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}) \in \mathcal{L}$ we have:
\begin{equation*}
\qpropinquity{F}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) = \qpropinquity{F}(({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}),({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}))\text{,}
\end{equation*}
\item for any $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}), ({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}), (\alg{C},{\mathsf{L}}_{\alg{C}}) \in \mathcal{L}$ we have:
\begin{equation*}
\qpropinquity{F}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),(\alg{C},{\mathsf{L}}_{\alg{C}})) \leq \qpropinquity{F}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) + \qpropinquity{F}(({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}),(\alg{C},{\mathsf{L}}_{\alg{C}}))\text{,}
\end{equation*}
\item for all $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}), ({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}) \in \mathcal{L}$ and for any bridge $\gamma$ from ${\mathfrak{A}}$ to ${\mathfrak{B}}$, we have:
\begin{equation*}
\qpropinquity{F}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}), ({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) \leq \bridgelength{\gamma}{{\mathsf{L}}_{\mathfrak{A}},{\mathsf{L}}_{\mathfrak{B}}}\text{,}
\end{equation*}
\item for any $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}), ({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}) \in \mathcal{L}$, we have $\qpropinquity{F}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) = 0$ if and only if $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ are isometrically isomorphic, i.e. if and only if there exists a *-isomorphism $\pi : {\mathfrak{A}} \rightarrow{\mathfrak{B}}$ with ${\mathsf{L}}_{\mathfrak{B}}\circ\pi = {\mathsf{L}}_{\mathfrak{A}}$, or equivalently there exists a *-isomorphism $\pi : {\mathfrak{A}} \rightarrow{\mathfrak{B}}$ whose dual map $\pi^\ast$ is an isometry from $({\mathscr{S}}({\mathfrak{B}}),\Kantorovich{{\mathsf{L}}_{\mathfrak{B}}})$ into $({\mathscr{S}}({\mathfrak{A}}),\Kantorovich{{\mathsf{L}}_{\mathfrak{A}}})$,
\item if $\Xi$ is a class function from $\mathcal{L}\times \mathcal{L}$ to $[0,\infty)$ which satisfies Properties (2), (3) and (4) above, then $\Xi(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}), ({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) \leq \qpropinquity{F}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}))$ for all $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ in $\mathcal{L}$,
\item the topology induced by $\qpropinquity{F}$ on the class of classical metric spaces agrees with the topology induced by the Gromov-Hausdorff distance.
\end{enumerate}
\end{theorem-definition}
We connect our new distance between Lip-norms on a fixed C*-algebra and the propinquity easily.
\begin{proposition}\label{HausLip-qprop-prop}
Let ${\mathfrak{A}}$ be a unital C*-algebra. If ${\mathsf{L}}_1$ and ${\mathsf{L}}_2$ are two $F$-quasi-Leibniz lower semi-continuous Lip-norms on ${\mathfrak{A}}$ for some admissible function $F$, then:
\begin{equation*}
\qpropinquity{F}(({\mathfrak{A}},{\mathsf{L}}_1),({\mathfrak{A}},{\mathsf{L}}_2)) \leq \HausLip{{\mathfrak{A}}}({\mathsf{L}}_1,{\mathsf{L}}_2)\text{.}
\end{equation*}
\end{proposition}
\begin{proof}
We simply use the bridge $({\mathfrak{A}},1_{\mathfrak{A}},\mathrm{id},\mathrm{id})$ where $\mathrm{id}$ is the identity on ${\mathfrak{A}}$.
\end{proof}
The purpose of this section is to establish the fact that uniformly equivalent Lip-norms, as defined in the hypothesis of the next proposition, provide totally bounded classes of quantum compact metric spaces for the metric $\HausLip{}$ and thus for the quantum propinquity, whenever applicable.
\begin{proposition}\label{totally-bounded-prop}
Let $({\mathfrak{A}},{\mathsf{L}})$ be a quantum compact metric space where ${\mathsf{L}}$ is lower semi-continuous. If $\Xi$ is a set of lower semi-continuous Lip-norms on ${\mathfrak{A}}$ for which there exists $C > 0$ such that, for all $\mathsf{Lip} \in \Xi$, we have ${\mathsf{L}} \leq C \mathsf{Lip}$, then $\Xi$ is totally bounded for $\HausLip{{\mathfrak{A}}}$ (and therefore, when applicable, for the quantum propinquity as well).
\end{proposition}
\begin{proof}
We fix $\mu \in {\mathscr{S}}({\mathfrak{A}})$. By assumption, for all $\mathsf{Lip} \in \Xi$, we have:
\begin{equation*}
\left\{ a\in\sa{{\mathfrak{A}}} : \mathsf{Lip}(a) \leq 1, \mu(a) = 0 \right\} \subseteq \left\{ a\in\sa{{\mathfrak{A}}} : {\mathsf{L}}(a)\leq C, \mu(a) = 0 \right\}\text{.}
\end{equation*}
Now, since ${\mathsf{L}}$ is a lower semi-continuous Lip-norm, the set:
\begin{equation*}
\alg{L} = \left\{ a\in\sa{{\mathfrak{A}}} : {\mathsf{L}}(a)\leq C, \mu(a) = 0 \right\}
\end{equation*}
is compact for $\|\cdot\|_{\mathfrak{A}}$ by Theorem (\ref{Rieffel-az-thm}). Thus by Blaschke's Theorem, the hyperspace of the closed subsets of $\alg{L}$ is compact for the Hausdorff distance $\Haus{\|\cdot\|_{\mathfrak{A}}}$. We conclude our proof using Lemma (\ref{finite-HausLip-lemma}) (and Proposition (\ref{HausLip-qprop-prop}) for the quantum propinquity conclusion).
\end{proof}
The main application of Proposition (\ref{totally-bounded-prop}) in this paper concerns certain perturbations we have established in \cite{Latremoliere15,Latremoliere15b}. These perturbations were constructed using \cite[Lemma 3.79]{Latremoliere15b}, which encapsulates a recurrent computation when estimating the quantum propinquity between two {quasi-Leibniz quantum compact metric space s}. When using this lemma, one will typically obtain uniformly equivalent families of Lip-norms, thus the following examples will be typical.
\begin{notation}
If ${\mathscr{H}}$ is some Hilbert space and $T$ is a bounded linear operator on ${\mathscr{H}}$, then we denote the norm of $T$ by $\opnorm{T}{{\mathscr{H}}}$.
\end{notation}
\begin{example}[Bounded pertubations]
\begin{proposition}[{\cite[Proposition 3.82]{Latremoliere15b}}]
Let ${\mathfrak{A}}$ be a unital C*-algebra, $\pi$ a unital faithful *-representation of ${\mathfrak{A}}$ on some Hilbert space ${\mathscr{H}}$, and $D$ a self-adjoint, possibly unbounded operator on ${\mathscr{H}}$ such that setting:
\begin{equation*}
{\mathsf{L}} : a\in{\mathfrak{A}} \longmapsto \opnorm{[D,\pi(a)]}{{\mathscr{H}}}
\end{equation*}
and allowing for ${\mathsf{L}}$ to assume the value $\infty$, the pair $({\mathfrak{A}},{\mathsf{L}})$ is a {{Lei\-bniz} quantum compact metric space}.
Let ${\mathfrak{B}}$ be the C*-algebra of all bounded linear operators on ${\mathscr{H}}$.
For any $\omega \in \sa{{\mathfrak{B}}}$, we define:
\begin{equation*}
D_\omega = D+\omega\text{ and }{\mathsf{L}}_\omega : a\in{\mathfrak{A}} \mapsto \opnorm{[D_\omega, \pi(a)]}{{\mathscr{H}}}\text{.}
\end{equation*}
The pair $({\mathfrak{A}},{\mathsf{L}}_\omega)$ is a {{Lei\-bniz} quantum compact metric space} for all bounded self-adjoint $\omega$ on ${\mathscr{H}}$ and, moreover:
\begin{equation*}
\omega \in \sa{{\mathfrak{B}}} \longmapsto ({\mathfrak{A}},{\mathsf{L}}_\omega)
\end{equation*}
is continuous for the quantum Gromov-Hausdorff propinquity $\qpropinquity{}$.
\end{proposition}
We shall henceforth refer to the proof of \cite[Proposition 3.82]{Latremoliere15b}. We observe that key to that argument is the observation that for all $a\in\sa{{\mathfrak{A}}}$ we have:
\begin{equation*}
{\mathsf{L}}(a) \leq (1 + r\|\omega\|_{\mathfrak{B}})\|\omega - 1_{\mathfrak{B}}\|_{\mathfrak{B}} {\mathsf{L}}_\omega(a)
\end{equation*}
and thus we see that if $\Omega = \{ \omega \in \sa{{\mathfrak{B}}} : \|\omega\|_{\mathfrak{B}} \leq C\}$ for some $C > 0$, then the set of quantum metrics:
\begin{equation*}
\{{\mathsf{L}}_\omega : \omega \in \Omega \}
\end{equation*}
is totally bounded by Proposition (\ref{totally-bounded-prop}) for both $\HausLip{{\mathfrak{A}}}$ and $\qpropinquity{}$. Yet $\Omega$ is not totally bounded unless ${\mathfrak{B}}$ is finite dimensional.
\end{example}
A deeper example of a non trivial class of totally bounded quantum metrics obtained from perturbations and \cite[Lemma 3.79]{Latremoliere15} is given by curved quantum tori, where, once more, the space of parameters is not totally bounded. These examples come from mathematical physics and were treated in \cite{Latremoliere15c} from the metric perspective.
\begin{example}[Curved quantum tori]
\begin{theorem}[{\cite[Theorem III.1]{Latremoliere15c}}]\label{ctori-thm}
Let ${\mathfrak{A}}$ be a unital C*-algebra, and let $\alpha$ be a strongly continuous ergodic action of a compact Lie group $G$ on ${\mathfrak{A}}$. Let $n$ be the dimension of $G$. We endow the dual $\alg{g}'$ of the Lie algebra $\alg{g}$ of $G$ with an inner product $\inner{\cdot}{\cdot}$, and we denote by $C$ the Clifford algebra of $(\alg{g}',\inner{\cdot}{\cdot})$. Let $c$ be a faithful nondegenerate representation of $C$ on some Hilbert space ${\mathscr{H}}_C$.
We fix some orthonormal basis $\{e_1,\ldots,e_n\}$ of $\alg{g'}$, and we let $X_1,\ldots,X_n \in \alg{g}$ be the dual basis. For each $j\in \{1,\ldots,n\}$, we define the derivation $\delta_j$ of ${\mathfrak{A}}$ via $\alpha$ and $X_j$, by:
\begin{equation*}
\partial_j : a \longmapsto \lim_{h\rightarrow 0} \frac{\alpha_{\exp(h X_j)}(a) - a }{h}
\end{equation*}
wherever defined. Let ${\mathfrak{A}}^1$ be the common domain of $\partial_1,\ldots,\partial_n$, which is a dense *-subalgebra in ${\mathfrak{A}}$.
Let $\tau$ be the unique $\alpha$-invariant tracial state of ${\mathfrak{A}}$. Let $\rho$ be the representation of ${\mathfrak{A}}$ obtained from the Gel'fand-Naimark-Segal construction applied to $\tau$ and let $L^2({\mathfrak{A}},\tau)$ be the corresponding Hilbert space. As ${\mathfrak{A}}^1$ is dense in $L^2({\mathfrak{A}},\tau)$, the operator $\partial_j$ defines an unbounded densely defined operator on $L^2({\mathfrak{A}},\tau)$ for all $j \in \{1,\ldots,n\}$.
Let ${\mathscr{H}} = L^2({\mathfrak{A}},\tau) \overline{\otimes} {\mathscr{H}}_C$ where $\overline{\otimes}$ is the standard tensor product for Hilbert spaces. We define the following representation of ${\mathfrak{A}}$ on ${\mathscr{H}}$:
\begin{equation*}
\pi(a) : b \otimes f \longmapsto \rho(a)b \otimes f\text{.}
\end{equation*}
Let:
\begin{equation*}
H = \begin{pmatrix}
h_{11} & \cdots & h_{1n}\\
\vdots & & \vdots \\
h_{n1} & \cdots & h_{nn}
\end{pmatrix}
\end{equation*}
where for all $j, k \in \{1,\ldots,n\}$, the coefficients $h_{jk}$ are elements in the commutant of ${\mathfrak{A}}$ in $L^2({\mathfrak{A}},\tau)$, and where $H$ is invertible as an operator on ${\mathscr{H}}' = \oplus_{j=1}^n L^2({\mathfrak{A}},\tau)$. We denote the identity over ${\mathscr{H}}'$ by $1_{{\mathscr{H}}'}$.
We define:
\begin{equation*}
D_H = \sum_{j=1}^n \sum_{k=1}^n h_{kj} \partial_k \otimes c(e_j)
\end{equation*}
so that for all $a\in{\mathfrak{A}}$:
\begin{equation*}
[D_H,\pi(a)] = \sum_{j=1}^n \sum_{k=1}^n h_{kj} \rho(\partial_k(a)) \otimes c(e_j)\text{.}
\end{equation*}
We define:
\begin{equation*}
{\mathsf{L}}_H(a) = \opnorm{[D_H,\pi(a)]}{{\mathscr{H}}}\text{.}
\end{equation*}
Then:
\begin{enumerate}
\item $({\mathfrak{A}},{\mathsf{L}}_H)$ is a {{Lei\-bniz} quantum compact metric space},
\item if we set:
\begin{equation*}
H' = \begin{pmatrix}
h_{11}' & \cdots & h_{1n}'\\
\vdots & & \vdots \\
h_{n1}' & \cdots & h_{nn}'
\end{pmatrix}
\end{equation*}
where $h_{jk}'$ lies in the commutant of $\rho({\mathfrak{A}})$ for all $j,k \in \{1,\ldots,n\}$ and where $H'$ is invertible as an operator on ${\mathscr{H}}'$, then:
\begin{multline*}
\qpropinquity{}(({\mathfrak{A}},{\mathsf{L}}_H),({\mathfrak{A}},{\mathsf{L}}_{H'})) \leq n \max\left\{ \opnorm{1_{{\mathscr{H}}'} - H' H^{-1}}{{\mathscr{H}}'}, \opnorm{1_{{\mathscr{H}}'}-H H'^{-1}}{{\mathscr{H}}'} \right\}\\ \times \left[1 +
\frac{1}{2}\max\left\{\left( 1 + n\opnorm{1-H^{-1}}{{\mathscr{H}}'}\right)^{-1},\right.\right. \\ \left.\left. \left( 1 + n\opnorm{1-H'^{-1}}{{\mathscr{H}}'}\right)^{-1}\right\}\diam{{\mathscr{S}}({\mathfrak{A}})}{\Kantorovich{{\mathsf{L}}}} \right]\text{.}
\end{multline*}
\end{enumerate}
\end{theorem}
We thus may apply Proposition (\ref{totally-bounded-prop}) when we restrict the parameter space $\mathcal{H}_C = \{H:\opnorm{H}{{\mathfrak{B}}},\opnorm{H^{-1}}{{\mathfrak{B}}}\leq C\}$ for any $C>0$. Of course this set itself is not totally bounded, yet the space $\{{\mathsf{L}}_H : H \in \mathcal{H}_C \}$ is totally bounded for both $\HausLip{{\mathfrak{A}}}$ and $\qpropinquity{}$.
\end{example}
We conclude this section with another example of a compact class of quantum compact metric spaces obtained from perturbations. We include this example as another compact class of quantum metric spaces, derived from \cite[Lemma 3.79]{Latremoliere15}, though in this case it does not require Proposition (\ref{totally-bounded-prop}).
\begin{example}[Conformal Perturbations]\label{conformal-ex}
We proved in \cite{Latremoliere15b} that small conformal perturbations of quantum metrics are indeed close for the quantum propinquity. We recall the result here to fix our notations.
\begin{theorem}[{\cite[Theorem 3.81]{Latremoliere15b}}]\label{conformal-thm}
Let ${\mathfrak{A}}$ be a unital C*-algebra, $\pi$ a faithful unital *-representation of ${\mathfrak{A}}$ on some Hilbert space ${\mathscr{H}}$ and $D$ be a not necessarily bounded self-adjoint operator on ${\mathscr{H}}$ such that if ${\mathsf{L}} : a\in\sa{{\mathfrak{A}}} \longmapsto \opnorm{[D,\pi(a)]}{{\mathfrak{B}}}$ then $({\mathfrak{A}},{\mathsf{L}})$ is a {{Lei\-bniz} quantum compact metric space}.
Let $\mathrm{GLip}({\mathfrak{A}})$ be the set of all invertible elements $h$ in $\sa{{\mathfrak{A}}}$ with ${\mathsf{L}}(h) < \infty$. For any $h\in\mathrm{GLip}({\mathfrak{A}})$, we define $D_h = \pi(h)D\pi(h)$, $\sigma_h : a\in{\mathfrak{A}}\mapsto h^2 a h^{-2}$ and:
\begin{equation*}
{\mathsf{L}}_h : a\in \sa{{\mathfrak{A}}} \longmapsto \opnorm{D_h\pi(a) - \pi(\sigma_h(a))D_h}{{\mathfrak{B}}} \text{.}
\end{equation*}
Then $({\mathfrak{A}},{\mathsf{L}}_h)$ is a {\Qqcms{\left(\|h^2\|_{\mathfrak{A}}\|h^{-2}\|_{\mathfrak{A}}, 0 \right)}} and moreover, if $(h_n)_{n\in{\mathds{N}}}$ is a sequence in $\mathrm{GLip}({\mathfrak{A}})$ which converges to $h \in \mathrm{GLip}$ and such that:
\begin{equation*}
\lim_{n\rightarrow\infty} {\mathsf{L}}(h_n^{-1}h) = \lim_{n\rightarrow\infty} {\mathsf{L}}(h_n h^{-1}) = 0\text{,}
\end{equation*}
then:
\begin{equation*}
\lim_{n\rightarrow\infty} \qpropinquity{F_M}\left(({\mathfrak{A}},{\mathsf{L}}_{h_n}),({\mathfrak{A}},{\mathsf{L}}_h)\right) = 0 \text{,}
\end{equation*}
where $M\geq \sup_{n\in{\mathds{N}}} \|h_n^2\|_{\mathfrak{A}}\|h_n^{-2}\|_{\mathfrak{A}}$ and $F_M : x,y,u,v \in [0,\infty) \mapsto M(x v + y u)$.
\end{theorem}
Let $K_1, K_2, K_3 > 0$ and define:
\begin{equation*}
\Omega_{K_1,K_2,K_3} = \left\{ \omega \in \sa{{\mathfrak{A}}} : {\mathsf{L}}(\omega)\leq K_1, \|\omega\|_{{\mathfrak{A}}} \leq K_2, \|\omega^{-1}\|_{\mathfrak{A}}\leq K_3 \right\}\text{.}
\end{equation*}
Then $\Omega_{K_1,K_2,K_3}$ is compact for $\|\cdot\|_{\mathfrak{A}}$ since ${\mathsf{L}}$ is a lower semi-continuous Lip-norm. Theorem (\ref{conformal-thm}) shows that conformal perturbations are continuous for the quantum propinquity, and thus, using our notations, $\{{\mathsf{L}}_\omega : \omega \in \Omega_{K_1, K_2, K_3}\}$ is compact.
\end{example}
\section{Lipschitz distance between compact quantum metric spaces}
The Lipschitz distance between compact metric spaces \cite{Gromov} provides a distance between homeomorphic compact metric spaces based upon bi-Lipschitz isomorphisms, and thus it is natural to define it in this paper in light of our study of Lipschitz morphisms.
This section provides the noncommutative generalization of the Lipschitz metric, which in essence is a metric on Lip-norms with common domains. The quantum Lipschitz distance is complete and dominates the quantum propinquity when working on appropriate classes of {quasi-Leibniz quantum compact metric space s}. The Lipschitz distance also provides natural examples of totally bounded classes for the quantum propinquity, and thus compact classes for the dual propinquity \cite{Latremoliere13b}.
\begin{notation}
Let $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ be two quantum compact metric spaces and let $\varphi : {\mathfrak{A}}\rightarrow{\mathfrak{B}}$ a unital *-morphism. We denote by $\mathsf{dil}(\varphi)$ the Lipschitz seminorm of the dual map $\varphi : \mu \in ({\mathscr{S}}({\mathfrak{B}}), \Kantorovich{{\mathsf{L}}_{\mathfrak{B}}}) \mapsto \mu\circ\varphi \in ({\mathscr{S}}({\mathfrak{A}}),\Kantorovich{{\mathsf{L}}_{\mathfrak{A}}})$, i.e.:
\begin{equation*}
\mathsf{dil}(\varphi) = \sup\left\{ \frac{\Kantorovich{{\mathsf{L}}_{\mathfrak{B}}}(\mu\circ\varphi,\nu\circ\varphi)}{\Kantorovich{{\mathsf{L}}_{\mathfrak{A}}}(\mu,\nu)} : \mu,\nu \in {\mathscr{S}}({\mathfrak{B}}),\mu\not=\nu \right\}\text{,}
\end{equation*}
with the understanding that this quantity may be infinite. We refer to this quantity as the \emph{dilation factor}, or just \emph{dilation} of the given Lipschitz morphism.
\end{notation}
\begin{remark}
If $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ are two quantum compact metric spaces with lower semicontinuous Lip-norms, and if $\varphi : {\mathfrak{A}}\rightarrow{\mathfrak{B}}$ a unital *-morphism, then $\mathsf{dil}(\varphi) = \inf\left\{ C > 0 : {\mathsf{L}}_{\mathfrak{B}}\circ\varphi \leq C {\mathsf{L}}_{\mathfrak{A}} \right\}$ with the usual convention that $\inf\emptyset = \infty$.
\end{remark}
\begin{definition}
The \emph{Lipschitz distance} between two quantum compact metric spaces $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ is:
\begin{multline*}
{\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) = \\
\inf\left\{ \max\left\{ \left|\ln(\mathsf{dil}(\varphi))\right|, \left| \ln(\mathsf{dil}(\varphi^{-1}))\right| \right\} : \varphi : {\mathfrak{A}}\rightarrow{\mathfrak{B}} \text{ is a *-isomorphism} \right\}\text{,}
\end{multline*}
with the conventions that $\inf\emptyset = \infty$ and $\ln(\infty) = \infty$.
\end{definition}
\begin{proposition}
If $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ are two quantum compact metric spaces with lower semicontinuous Lip-norms. Then:
\begin{multline*}
{\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) = \\
\inf\left\{\max\left\{ \left|\ln(\mathsf{dil}(\varphi))\right|, \left|\ln(\mathsf{dil}(\varphi^{-1}))\right|\right\} \middle\vert \begin{array}{l}
\varphi : {\mathfrak{A}} \rightarrow{\mathfrak{B}} \text{ is a *-isomorphism} \\
\varphi(\dom{{\mathsf{L}}_{\mathfrak{A}}}) = \dom{{\mathsf{L}}_{\mathfrak{B}}}
\end{array}\right\} \text{,}
\end{multline*}
with the convention that $\inf\emptyset = \infty$.
\end{proposition}
\begin{proof}
This follows from Corollary (\ref{bi-Lipschitz-cor}).
\end{proof}
A natural class for the study of the Lipschitz distance is given by the following definitions, which includes all {quasi-Leibniz quantum compact metric space s}: we simply require lower semi-continuity of the quantum metrics, as it fits the general framework of this paper, and we require that the domain of the Lip-norm is a Jordan-Lie algebra, to retain a minimum amount of information on the multiplicative structure of the C*-algebra.
\begin{definition}
A compact quantum metric space $({\mathfrak{A}},{\mathsf{L}})$ is \emph{Jordan-Lie} when ${\mathsf{L}}$ is lower semi-continuous, and its domain $\dom{{\mathsf{L}}}$ is a Jordan-Lie subalgebra of $\sa{{\mathfrak{A}}}$.
\end{definition}
The Lipschitz distance between Jordan-Lie quantum compact metric spaces is actually achieved, as established in the following lemma. This observation will prove useful in establishing that the Lipschitz distance in indeed a distance up to quantum isometry.
\begin{lemma}\label{LipschitzD-reached-lemma}
If $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ are two Jordan-Lie compact quantum metric spaces such that ${\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) < \infty$ then there exists a *-isomorphism $\varphi : {\mathfrak{A}}\rightarrow{\mathfrak{B}}$ such that:
\begin{equation*}
\max\left\{ |\ln(\mathsf{dil}(\varphi))|, |\ln(\mathsf{dil}(\varphi^{-1}))| \right\} = {\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) \text{.}
\end{equation*}
\end{lemma}
\begin{proof}
Suppose that ${\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) = C$ for some $C \geq 0$. There exists a sequence of *-isomorphism $(\varphi_n)_{n\in{\mathds{N}}}$ such that for all $n\in{\mathds{N}}$ we have:
\begin{equation*}
C^{-1} \exp\left(-\frac{1}{n+1}\right){\mathsf{L}}_{\mathfrak{B}}\circ\varphi_n \leq {\mathsf{L}}_{\mathfrak{A}} \leq C \exp\left(\frac{1}{n+1}\right){\mathsf{L}}_{\mathfrak{B}}\circ\varphi_n \text{.}
\end{equation*}
Let $a \in \sa{{\mathfrak{A}}}$ with ${\mathsf{L}}_{\mathfrak{A}}(a) < \infty$. Since $\|\varphi_n(a)\|_{\mathfrak{B}} = \|a\|_{\mathfrak{A}}$ and ${\mathsf{L}}_{\mathfrak{B}}\circ\varphi_n(a) \leq 2C {\mathsf{L}}_{\mathfrak{A}}(a)$ for all $n\in{\mathds{N}}$, we conclude that $(\varphi_n(a))_{n\in{\mathds{N}}}$ admits a convergent subsequence since ${\mathsf{L}}_{\mathfrak{B}}$ is a lower semicontinuous Lip-norm. Let $\varphi_\infty(a)$ be its limit.
Since $\{ a \in \sa{{\mathfrak{A}}} : {\mathsf{L}}_{\mathfrak{A}}(a)\leq n, \|a\|\leq n \}$ is compact for the norm $\|\cdot\|_{\mathfrak{A}}$, hence separable for all $n\in{\mathds{N}}$, so is:
\begin{equation*}
\dom{{\mathsf{L}}_{\mathfrak{A}}} = \left\{a\in\sa{{\mathfrak{A}}} : {\mathsf{L}}_{\mathfrak{A}}(a) < \infty \right\} = \bigcup_{n\in{\mathds{N}}} \left\{a\in\sa{{\mathfrak{A}}} : {\mathsf{L}}_{\mathfrak{A}}(a)\leq n, \|a\|_{\mathfrak{A}}\leq n \right\}\text{.}
\end{equation*}
Let $\alg{F}$ be a countable dense subset of $\{a\in\sa{{\mathfrak{A}}}:{\mathsf{L}}_{\mathfrak{A}}(a) < \infty \}$. A diagonal argument proves that there exists a subsequence $(\varphi_{f(n)})_{n\in{\mathds{N}}}$ such that for all $a\in\alg{F}$ we have $(\varphi_{f(n)}(a))_{n\in{\mathds{N}}}$ converges uniformly to $\varphi_\infty(a)$ (see \cite[Theorem 5.13]{Latremoliere13}).
Moreover, if $a\in\sa{{\mathfrak{A}}}$ with ${\mathsf{L}}_{\mathfrak{A}}(a)<\infty$, then for all $\varepsilon>0$, there exists $a_\varepsilon \in \alg{F}$ with $\|a-a_\varepsilon\|<\frac{\varepsilon}{3}$. Let $N\in{\mathds{N}}$ be such that for all $p,q \geq N$, we have $\|\varphi_{f(p)}(a_\varepsilon) - \varphi_{f(q)}(a_\varepsilon)\|_{\mathfrak{B}} \leq \frac{\varepsilon}{3}$. Thus for all $p,q\geq N$, we have:
\begin{equation*}
\begin{split}
|\varphi_{f(p)}(a) - \varphi_{f(q)}(a)| &\leq |\varphi_{f(p)}(a-a_\varepsilon)| + |\varphi_{f(p)}(a_\varepsilon) - \varphi_{f(q)}(a_\varepsilon)| + |\varphi_{f(q)}(a-a_\varepsilon)| \\
&\leq \frac{\varepsilon}{3} + \frac{\varepsilon}{3} + \frac{\varepsilon}{3} = \varepsilon\text{.}
\end{split}
\end{equation*}
Thus $(\varphi_{f(n)}(a))_{n\in{\mathds{N}}}$ converges as well, since it is a Cauchy sequence in ${\mathfrak{A}}$ which is complete. Its limit is denoted once more by $\varphi_\infty(a)$.
Note that since for all $n\in{\mathds{N}}$ and for all $a\in\dom{{\mathsf{L}}_{\mathfrak{A}}}$, we have $\|\varphi_n(a)\|_{\mathfrak{B}} = \|a\|_{\mathfrak{A}}$, we also have $\|\varphi_\infty(a)\|_{\mathfrak{B}} = \|a\|_{\mathfrak{A}}$. We thus have defined an isometric map $\varphi_\infty : \dom{{\mathsf{L}}_{\mathfrak{A}}} \rightarrow\sa{{\mathfrak{B}}}$. Moreover, as a pointwise limit of Jordan-Lie morphisms, $\varphi_\infty$ is also a Jordan-Lie morphisms on $\dom{{\mathsf{L}}}$.
Now ${\mathsf{L}}_{\mathfrak{B}}$ is lower semi-continuous and, for all $n\in{\mathds{N}}$ we have ${\mathsf{L}}_{\mathfrak{B}}\circ\varphi_n(a) \leq C \exp(\frac{1}{n+1}) {\mathsf{L}}_{\mathfrak{A}}(a)$. Thus ${\mathsf{L}}_{\mathfrak{B}}\circ\varphi_\infty(a) \leq C {\mathsf{L}}_{\mathfrak{A}}(a)$. Thus $\mathsf{dil}(\varphi_\infty) \leq C$.
Thus $\varphi_\infty$ extends by continuity to a Jordan-Lie morphism from $\sa{{\mathfrak{A}}}$ to $\sa{{\mathfrak{B}}}$. Our argument is now concluded in the same manner as \cite[Claim 5.18, Theorem 5.13]{Latremoliere13} and proves that $\varphi_\infty$ extends to a unital *-morphism from ${\mathfrak{A}}$ to ${\mathfrak{B}}$ with $\mathrm{dil}(\varphi_\infty) \leq C$.
The same method may be applied to construct some subsequence of $\left(\varphi_n^{-1}\right)_{n\in{\mathds{N}}}$ converging pointwise on $\dom{{\mathsf{L}}_{\mathfrak{B}}}$ to some *-morphism $\psi_\infty$ on ${\mathfrak{B}}$ with ${\mathsf{L}}_{\mathfrak{A}}\circ\psi_\infty \leq C {\mathsf{L}}_{\mathfrak{A}}$. Up to extracting further subsequences, we shall henceforth assume that both $(\varphi_{f(n)})_{n\in{\mathds{N}}}$ and $(\varphi_{f(n)}^{-1})_{n\in{\mathds{N}}}$ converge pointwise to, respectively $\varphi_\infty$ on $\dom{{\mathsf{L}}_{\mathfrak{A}}}$ and $\psi_\infty$ on $\dom{{\mathsf{L}}_{\mathfrak{B}}}$. It is then immediate to check that $\varphi_\infty\circ\psi_\infty$ is the identity of $\dom{{\mathsf{L}}_{\mathfrak{B}}}$ and $\psi_\infty\circ\varphi_\infty$ is the identity on $\dom{{\mathsf{L}}_{\mathfrak{A}}}$. Then by construction, $\psi_\infty\circ\varphi_\infty$ is the identity on ${\mathfrak{A}}$ and $\varphi_\infty\circ\psi_\infty$ is the identity on ${\mathfrak{B}}$. Thus $\varphi_\infty$ is a *-isomorphism from ${\mathfrak{A}}$ to ${\mathfrak{B}}$.
In particular, we also obtain that ${\mathsf{L}}_{\mathfrak{A}}\circ \varphi_\infty^{-1} \leq C {\mathsf{L}}_{\mathfrak{B}}$ and thus $\mathsf{dil}(\varphi^{-1})\leq C$. As we may not have both $\mathsf{dil}(\varphi) < C$ and $\mathsf{dil}(\varphi^{-1}) < C$, since $C$ is the infimum of the dilations of such *-isomorphisms, the lemma is proven.
\end{proof}
We now establish that the Lipschitz distance is indeed, a distance up to quantum isometry, and that it dominates the quantum propinquity.
\begin{theorem}\label{LipschitzD-distance-thm}
The Lipschitz distance is an extended metric up to quantum isometry on the class of Jordan-Lie quantum compact metric spaces. Explicitly, for all $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$, $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ and $({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}})$ Jordan-Lie compact quantum metric spaces, we have:
\begin{enumerate}
\item ${\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) \in [0,\infty]$, and is finite if and only if there exists a *-isomorphism $\varphi : {\mathfrak{A}} \rightarrow {\mathfrak{B}}$ such that $\varphi(\dom{{\mathsf{L}}_{\mathfrak{A}}}) = \dom{{\mathsf{L}}_{\mathfrak{B}}}$,
\item ${\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{D}})) \leq {\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) + {\mathsf{LipD}}(({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}),({\mathfrak{D}},{\mathsf{L}}_{\mathfrak{B}}))$,
\item ${\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) = {\mathsf{LipD}}(({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}),({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}))$,
\item ${\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) = 0$ if and only if $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ are isometrically isomorphic, i.e. there exists a *-isomorphisms $\varphi : {\mathfrak{A}}\rightarrow{\mathfrak{B}}$ such that ${\mathsf{L}}_{\mathfrak{B}}\circ\varphi = {\mathsf{L}}_{\mathfrak{A}}$ on $\sa{{\mathfrak{A}}}$. Such a map $\varphi$ is called a \emph{quantum isometry} or an \emph{isometric isomorphism}.
\item if $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and $({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})$ are two {\Qqcms{F}s} then:
\begin{multline*}
\qpropinquity{F}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}})) \leq \\
\left|1-\exp\left({\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}}),({\mathfrak{B}},{\mathsf{L}}_{\mathfrak{B}}))\right)\right| \\
\left(\frac{1}{2} + \max\left\{\diam{{\mathscr{S}}({\mathfrak{A}})}{\Kantorovich{{\mathsf{L}}_{\mathfrak{A}}}}, \diam{{\mathscr{S}}({\mathfrak{B}})}{\Kantorovich{{\mathsf{L}}_{\mathfrak{B}}}}\right\}\right) \text{.}
\end{multline*}
\end{enumerate}
\end{theorem}
\begin{proof}
The function ${\mathsf{LipD}}$ is valued in $[0,\infty]$ by definition, and finite if and only if there exists a bi-Lipschitz isomorphism between its two arguments, by Corollary (\ref{bi-Lipschitz-cor}). It is symmetric in its two arguments by construction. The triangle inequality follows from simple computations as well.
The last assertion of our proposition follows immediately from \cite[Proposition 3.80]{Latremoliere15b}.
If there exists an isometric isometry between two compact quantum metric spaces, then their Lipschitz distance is null. Only the converse of this observation requires our assumption that the domain of Lip-norms be Jordan-Lie algebras. We simply apply Lemma (\ref{LipschitzD-reached-lemma}).
\end{proof}
We now prove that closed balls for the Lipschitz distance are compact classes for the dual propinquity.
\begin{theorem}\label{LipschitzD-compact-thm}
Let $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ be a {\Qqcms{F}} for some admissible function $F$. If $R \geq 0$ then the class $\mathcal{B}$ of {\Qqcms{F}s} in the closed ball of center $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and radius $R$ for the Lipschitz distance ${\mathsf{LipD}}$ is totally bounded for the quantum Gromov-Hausdorff propinquity.
Therefore, the closure of $\mathcal{B}$ for the dual propinquity is compact.
\end{theorem}
\begin{proof}
For all $({\mathfrak{B}},{\mathsf{L}})$ within Lipschitz distance $R$ of $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$, there exists by definition a *-isomorphism $\varphi_{\mathfrak{B}} : {\mathfrak{A}} \rightarrow {\mathfrak{B}}$ which maps the domain of ${\mathsf{L}}_{\mathfrak{A}}$ into the domain of ${\mathsf{L}}$. We set ${\mathsf{L}}' = {\mathsf{L}}\circ\varphi_{\mathfrak{B}}$ and note that ${\mathsf{L}}'$ is a lower semicontinuous Lip-norm with the same domain as ${\mathsf{L}}_{\mathfrak{A}}$. Moreover, by definition of the Lipschitz distance, we have ${\mathsf{L}}_{\mathfrak{A}} \leq \exp(R) {\mathsf{L}}'$. Last, $({\mathfrak{A}},{\mathsf{L}}')$ and $({\mathfrak{B}},{\mathsf{L}})$ are isometrically isomorphic, thus their propinquity is zero, and so is their Lipschitz distance.
Thus by Proposition (\ref{totally-bounded-prop}), the closed ball $\mathcal{B}$ of center $({\mathfrak{A}},{\mathsf{L}}_{\mathfrak{A}})$ and radius $R$ is totally bounded for $\HausLip{{\mathfrak{A}}}$. By Proposition (\ref{HausLip-qprop-prop}), the subclass of {\Qqcms{F}s} in the closed ball $\mathcal{B}$ is totally bounded for the quantum propinquity. The rest of the theorem follows from the completeness of the dual propinquity \cite{Latremoliere13b} and the dominance of the quantum propinquity over the dual propinquity.
\end{proof}
We conclude this section by proving that the Lipschitz distance is indeed complete.
\begin{theorem}
The distance ${\mathsf{LipD}}$ is complete on the class of Jordan-Lie quantum compact metric spaces.
\end{theorem}
\begin{proof}
As in the proof of Theorem (\ref{LipschitzD-compact-thm}), we can assume that we are given a sequence $({\mathsf{L}}_n)_{n\in{\mathds{N}}}$ of lower semi-continuous Lip-norms on some unital C*-algebra, such that $({\mathfrak{A}},{\mathsf{L}}_n)_{n\in{\mathds{N}}}$ is a Cauchy sequence for ${\mathsf{LipD}}$.
By Theorem (\ref{LipschitzD-distance-thm}), the sequence $\alg{L}_n = \{ a \in \sa{{\mathfrak{A}}} : {\mathsf{L}}_n(a) \leq 1 \}$ is Cauchy for $\HausLip{\|\cdot\|_{\mathfrak{A}}}$. As the latter metric is complete, $(\alg{L}_n)_{n\in{\mathds{N}}}$ converges to some $\alg{L}$.
Let $\varepsilon \in (0,1)$. There exists $N\in{\mathds{N}}$ such that for all $n\geq N$ we have:
\begin{equation*}
(1-\varepsilon){\mathsf{L}}_q \leq {\mathsf{L}}_p \leq (1+\varepsilon) {\mathsf{L}}_q
\end{equation*}
for all $p,q \geq N$. In other words, $\alg{L}_q \subseteq (1+\varepsilon) \alg{L}_p$ and $\alg{L}_p \subseteq \frac{1}{1-\varepsilon}\alg{L}_q$ for all $q, p \geq N$. Now $(1+\varepsilon)\alg{L}_n$ is closed for all $n\in{\mathds{N}}$, thus the hyperspace of its closed subsets is complete and thus closed for the Hausdorff distance $\Haus{\|\cdot\|_{\mathfrak{A}}}$. Consequently, $\alg{L} \subseteq (1+\varepsilon) \alg{L}_n$ for all $n\geq N$. Thus ${\mathsf{L}}_n \leq (1+\varepsilon){\mathsf{L}}$.
On the other hand, let $a\in\alg{L}_n$ for $n\geq N$. For $p\geq N$ we have $a\in\frac{1}{1-\varepsilon}\alg{L}_p$ so $a \in \frac{1}{1-\varepsilon}\alg{L}$ since $ \frac{1}{1-\varepsilon}\alg{L}$ is the Hausdorff limit of $s = \left( \frac{1}{1-\varepsilon}\alg{L}\right)_{p \geq n}$ and thus contains all the limits of convergent sequences obtained by picking one element in each set of $s$.
We conclude ${\mathsf{LipD}}(({\mathfrak{A}},{\mathsf{L}}),({\mathfrak{A}},{\mathsf{L}}_n)) \leq \ln(1+\varepsilon)$ for $n\geq N$. Our theorem is now proven.
\end{proof}
\section{A metric for pointwise convergence on the automorphism group of a quantum compact metric space}
We now introduce a new metric on the automorphism group of a quantum compact metric space. Our motivation is given by Theorem (\ref{equiv-thm}), and in particular Corollary (\ref{auto-Lipschitz-cor}), as well as our new understanding of compactness for Lip-norms with a shared domain.
\newcommand{\KantorovichLength}[1]{{\mathrm{mk}\ell_{#1}}}
\begin{definition}
Let $({\mathfrak{A}},{\mathsf{L}})$ be a quantum compact metric space. For any *-automorphism of ${\mathfrak{A}}$, we define:
\begin{equation*}
\KantorovichLength{{\mathsf{L}}}(\alpha) = \sup\left\{ \|\alpha(a) - a\|_{\mathfrak{A}} : a\in\dom{{\mathsf{L}}}, {\mathsf{L}}(a)\leq 1\right\}\text{.}
\end{equation*}
\end{definition}
\begin{proposition}
Let $({\mathfrak{A}},{\mathsf{L}})$ be a quantum compact metric space.
\begin{enumerate}
\item $\KantorovichLength{{\mathsf{L}}}$ is a length function on $\mathrm{Aut}({\mathfrak{A}})$ which metrizes the topology of pointwise convergence,
\item $\diam{\mathrm{Aut}({\mathfrak{A}})}{\KantorovichLength{{\mathsf{L}}}} \leq \diam{{\mathscr{S}}({\mathfrak{A}})}{\Kantorovich{{\mathsf{L}}}}$
\item If ${\mathsf{L}}$ is closed, $\Xi$ is a subset of $\mathrm{Aut}({\mathfrak{A}})$ and there exists $C > 0$ such that:
\begin{equation*}
\forall \alpha \in \Xi \quad {\mathsf{L}}\leq C {\mathsf{L}}\circ\alpha
\end{equation*}
then $\Xi$ is totally bounded for $\KantorovichLength{{\mathsf{L}}}$.
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{claim}
$\KantorovichLength{{\mathsf{L}}}$ is a length function on the group $\mathrm{Aut}({\mathfrak{A}})$.
\end{claim}
if $\mathrm{id}_{\mathfrak{A}}$ is the identity of ${\mathfrak{A}}$ then $\KantorovichLength{{\mathsf{L}}}(\mathrm{id}_{\mathfrak{A}}) = 0$. Now let $\alpha\in\mathrm{Aut}({\mathfrak{A}})$.
If $\KantorovichLength{{\mathsf{L}}}(\alpha) = 0$ then $\|a-\alpha(a)\|_{\mathfrak{A}} = 0$ for all $a\in\dom{{\mathsf{L}}}$ with ${\mathsf{L}}(a)\leq 1$. Thus $\|a-\alpha(a)\|_{\mathfrak{A}} = 0$ for all $a\in\dom{{\mathsf{L}}}$, and by continuity of $\alpha$ and density of $\dom{{\mathsf{L}}}$, we conclude that $\|a-\alpha(a)\|_{\mathfrak{A}} = 0$ for all $a\in\sa{{\mathfrak{A}}}$. Thus by linearity, $\alpha(a) = a$ for all $a\in{\mathfrak{A}}$.
Moreover, since $\alpha$ is an isometry of $({\mathfrak{A}},\|\cdot\|_{\mathfrak{A}})$, we have for all $a\in{\mathfrak{A}}$:
\begin{equation*}
\|a-\alpha(a)\|_{\mathfrak{A}} = \|\alpha(\alpha^{-1}(a) - a)\|_{\mathfrak{A}} = \|\alpha^{-1}(a)-a\|_{\mathfrak{A}}
\end{equation*}
from which it follows that $\KantorovichLength{{\mathsf{L}}}(\alpha) = \Kantorovich{{\mathsf{L}}}\left(\alpha^{-1}\right)$.
Let now $\beta\in\mathrm{Aut}({\mathfrak{A}})$. For all $a\in{\mathfrak{A}}$:
\begin{equation*}
\begin{split}
\|\alpha\circ\beta(a) - a\|_{\mathfrak{A}} &\leq \|\alpha\circ\beta(a) - \alpha(a)\|_{\mathfrak{A}} + \|\alpha(a) - a\|_{\mathfrak{A}}\\
&\leq \|\alpha(\beta(a) - a)\|_{\mathfrak{A}} + \|\alpha(a) - a\|_{\mathfrak{A}}\\
&\leq \|\beta(a) - a\|_{\mathfrak{A}} + \|\alpha(a) - a\|_{\mathfrak{A}}\text{,}
\end{split}
\end{equation*}
from which we conclude:
\begin{equation*}
\KantorovichLength{{\mathsf{L}}}(\alpha\circ\beta) \leq \KantorovichLength{{\mathsf{L}}}(\beta) + \KantorovichLength{{\mathsf{L}}}(\alpha)\text{.}
\end{equation*}
\begin{claim}
The topology induced by $\KantorovichLength{{\mathsf{L}}}$ is the topology of pointwise convergence.
\end{claim}
Let $(\alpha_j)_{j\in J}$ be a net in $\mathrm{Aut}({\mathfrak{A}})$ which converges pointwise to some $\alpha \in \mathrm{Aut}({\mathfrak{A}})$ over ${\mathfrak{A}}$, where $(J,\succ)$ is a directed set. Let $\varepsilon > 0$.
Let $\varphi \in {\mathscr{S}}({\mathfrak{A}})$. Since $\alg{B} = \{a\in\dom{{\mathsf{L}}} : {\mathsf{L}}(a)\leq 1,\varphi(a) = 0\}$ is totally bounded, there exists $\alg{F}\subseteq\alg{B}$ with $\alg{F}$ finite and such that for all $a\in\alg{B}$ there exists $f(a) \in \alg{F}$ with $\|a-f(a)\|_{\mathfrak{A}} \leq \frac{\varepsilon}{3}$.
For each $a\in\alg{F}$, there exists $j_a \in J$ such that if $j\succ j_a$, we have $\|\alpha_j(a)-\alpha(a)\|_{\mathfrak{A}}\leq \frac{\varepsilon}{3}$. As $J$ is directed and $\alg{F}$ is finite, there exists $j'\in J$ with $j' \succ j_a$ for all $a\in\alg{F}$. Thus, if $a\in\alg{B}$ and $j\succ j'$, then:
\begin{multline*}
\|\alpha_j(a)-\alpha(a)\|_{\mathfrak{A}} \leq \|\alpha_j(a) -\alpha_j(f(a))\|_{\mathfrak{A}} + \|\alpha_j(f(a)) - \alpha(f(a))\|_{\mathfrak{A}} \\ + \|\alpha(f(a)) - \alpha(a)\|_{\mathfrak{A}}
\leq \varepsilon\text{,}
\end{multline*}
where we used that all automorphisms are isometries. Thus, for all $j\succ j'$ we have:
\begin{equation*}
\KantorovichLength{{\mathsf{L}}}(\alpha_j^{-1}\circ\alpha) \leq\varepsilon\text{.}
\end{equation*}
Conversely, let $(\alpha_j)_{j\in J}$ be a net in $\mathrm{Aut}({\mathfrak{A}})$ converging for $\KantorovichLength{{\mathsf{L}}}$ to $\alpha\in\mathrm{Aut}({\mathfrak{A}})$. Let $a\in\sa{{\mathfrak{A}}}$ and $\varepsilon > 0$.
Since $\dom{{\mathsf{L}}}$ is dense, there exists $b\in\dom{{\mathsf{L}}}$ such that $\|a-b\|_{\mathfrak{A}} \leq \frac{\varepsilon}{3}$. Now, by assumption, there exists $j_0 \in J$ such that for all $j\succ j_0$, we have $\KantorovichLength{{\mathsf{L}}}(\alpha_j^{-1}\circ\alpha)\leq\frac{\varepsilon}{3({\mathsf{L}}(b)+1)}$. Thus $\|\alpha_j(b)-\alpha(b)\|_{\mathfrak{A}} \leq \frac{\varepsilon}{3}$ and thus for all $j\succ j_0$:
\begin{equation*}
\|\alpha_j(a) - \alpha(a)\|_{\mathfrak{A}} \leq \|\alpha_j(a)-\alpha_j(b)\|_{\mathfrak{A}} + \|\alpha_j(b)-\alpha(a)\|_{\mathfrak{A}} + \|\alpha(a) - \alpha_j(a)\|_{\mathfrak{A}} \leq \varepsilon\text{,}
\end{equation*}
as desired.
\begin{claim}
Assume ${\mathsf{L}}$ is closed. The set $\{\alpha\in\mathrm{Aut}({\mathfrak{A}}) : {\mathsf{L}}\circ\alpha \leq C{\mathsf{L}} \}$, for some fixed $C > 0$, is totally bounded and closed for $\KantorovichLength{{\mathsf{L}}}$.
\end{claim}
Fix $\varphi \in {\mathscr{S}}({\mathfrak{A}})$.
All automorphisms of ${\mathfrak{A}}$ are isometries and thus $\mathrm{Aut}({\mathfrak{A}})$ is equicontinuous over the compact set $\alg{X} = \{a\in\dom{{\mathsf{L}}}:{\mathsf{L}}(a)\leq 1, \varphi(a) = 0 \}$. Moreover, for all $a\in\dom{{\mathsf{L}}}$ with ${\mathsf{L}}(a)\leq 1$, then ${\mathsf{L}}\circ\alpha(a)\leq C$. Thus, $\alpha(a) \in \{a\in\dom{{\mathsf{L}}}:{\mathsf{L}}(a)\leq C, \varphi(a)=0\}$ for all $a\in\dom{{\mathsf{L}}}$ with ${\mathsf{L}}(a)\leq 1$ with $\varphi\circ\alpha^{-1}(a) = 0$.
Thus, by Arz{\'e}la-Ascoli, the set $\{\alpha\in\mathrm{Aut}({\mathfrak{A}}):{\mathsf{L}}\circ\alpha\leq C{\mathsf{L}}\}$ is totally bounded for the supremum norm over $\alg{X}$.
We note that as automorphisms are unital, the supremum over $\alg{X}$ of the difference of two automorphisms of ${\mathfrak{A}}$ equals to the supremum over $\{a\in\sa{{\mathfrak{A}}} : {\mathsf{L}}(a)\leq 1\}$. Therefore, $\{\alpha\in\mathrm{Aut}({\mathfrak{A}}):{\mathsf{L}}\circ\alpha\leq C{\mathsf{L}}\}$ is totally bounded for $\KantorovichLength{{\mathsf{L}}}$.
Moreover, if ${\mathsf{L}}\circ\alpha_n\leq C{\mathsf{L}}$ for all $n\in{\mathds{N}}$, by lower semicontinuity, ${\mathsf{L}}\circ\alpha\leq C{\mathsf{L}}$ as well. Thus our set is closed. As $\KantorovichLength{{\mathsf{L}}}$ is complete, our proof is complete.
\end{proof}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,945
|
import test_patma
class TestSyntaxErrors(test_patma.TestSyntaxErrors):
def assert_syntax_error(self, code):
try:
exec(test_patma.inspect.cleandoc(code))
print('--- syntax error not raised')
except SyntaxError as exc:
print('syntax error raised', exc)
pass
test = test_patma.TestPatma()
for method in dir(test):
if method.startswith('test_'):
print(method)
m = getattr(test, method)
try:
m()
except Exception as exc:
print(exc)
test = TestSyntaxErrors()
for method in dir(test):
if method.startswith('test_'):
print(method)
m = getattr(test, method)
m()
test = test_patma.TestTypeErrors()
for method in dir(test):
if method.startswith('test_'):
print(method)
m = getattr(test, method)
try:
m()
except Exception as exc:
print(exc)
test = test_patma.TestInheritance()
for method in dir(test):
if method.startswith('test_'):
print(method)
m = getattr(test, method)
try:
m()
except Exception as exc:
print(exc)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,518
|
{"url":"https:\/\/www.overleaf.com\/help\/86-how-do-i-remove-the-frog-picture-from-the-default-paper-template","text":"## How do I remove the frog picture from the default paper template?\n\nYou can delete the frog.jpg image file using the project menu in the editor, and if you then remove the lines:\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{frog.jpg}\n\\caption{\\label{fig:frog}This is a figure caption.}\n\\end{figure}\n\n\nfrom the code, that will remove it from the document.","date":"2017-11-23 09:17:48","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\": 0.46956783533096313, \"perplexity\": 1503.0121083478214}, \"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-2017-47\/segments\/1510934806768.19\/warc\/CC-MAIN-20171123085338-20171123105338-00046.warc.gz\"}"}
| null | null |
'use strict';
const assert = require('assert');
const logger = require('./logger');
const ISO_8601_REGEX = /^\d{4}-\d\d-\d\dT\d\d:\d\d:\d\d(\.\d+)?(([+-]\d\d:\d\d)|Z)?$/i;
describe('logger', () => {
let output;
logger.setDelay(null); // Disable workaround for missing stdout lines in katana-1.0.0
// Test setup
const log = console.log; // backup native call
beforeEach(() => {
output = null; // stdout will be stored here
console.log = (...args) => { // overwrite native call
output = args.join(' ').trim(); // console.log is variadic, this should be too
};
});
afterEach(() => {
output = null; // cleanup
console.log = log; // Restore log here to allow mocha logging test info
});
describe('log()', () => {
// https://github.com/kusanagi/katana-sdk-spec#53-logging
it('should log a message formatted according to the spec', () => {
logger.log(logger.levels.INFO, 'test');
const [timestamp, level, tag, message, requestId] = output.split(' ');
assert.equal(true, ISO_8601_REGEX.test(timestamp));
assert.equal(level, `[${logger.levels.INFO}]`);
assert.equal(tag, '[SDK]');
assert.equal(message, 'test');
assert.equal(requestId, undefined);
});
it('should not output the requestId if not set', () => {
logger.setRequestId(null);
logger.log(logger.levels.INFO, 'test');
const [, , , , requestId] = output.split(' ');
assert.equal(requestId, undefined);
});
it('should log the formatted Request Id if set', () => {
const uuid = '55f33a09-3d00-40ac-8418-4c88c37679cb';
logger.setRequestId(uuid);
logger.log(logger.levels.INFO, 'test');
const [, , , , requestId] = output.split(' ');
assert.equal(requestId, `|${uuid}|`);
});
it('should set level as INFO when the level setting is invalid/unknown', () => {
logger.setLevel('UNKNOWN');
assert.equal(logger.getLevel(), logger.levels.INFO);
});
it('should use INFO level when calling log() with an unknown level', () => {
logger.log('UNKOWN', 'test');
const [, level] = output.split(' ');
assert.equal(level, `[${logger.levels.INFO}]`);
});
});
describe('debug()', () => {
it('should log a DEBUG message when logger level setting is lower or equal DEBUG', () => {
const message = 'test';
logger.setLevel(logger.levels.DEBUG);
logger.debug(message);
const [, level, , loggedMessage] = output.split(' ');
assert.equal(level, `[${logger.levels.DEBUG}]`);
assert.equal(message, loggedMessage);
});
it('should not log a DEBUG message when logger level is higher than DEBUG', () => {
logger.setLevel(logger.setLevel(logger.levels.INFO));
logger.debug('test');
assert.equal(null, output);
});
});
describe('info()', () => {
it('should log a INFO message when logger level setting is lower or equal INFO', () => {
const message = 'test';
logger.setLevel(logger.levels.INFO);
logger.info(message);
const [, level, , loggedMessage] = output.split(' ');
assert.equal(level, `[${logger.levels.INFO}]`);
assert.equal(message, loggedMessage);
});
it('should not log a INFO message when logger level is higher than INFO', () => {
logger.setLevel(logger.setLevel(logger.levels.WARNING));
logger.info('test');
assert.equal(null, output);
});
});
describe('warning()', () => {
it('should log a WARNING message when logger level setting is lower or equal WARNING', () => {
const message = 'test';
logger.setLevel(logger.levels.WARNING);
logger.warning(message);
const [, level, , loggedMessage] = output.split(' ');
assert.equal(level, `[${logger.levels.WARNING}]`);
assert.equal(message, loggedMessage);
});
it('should not log a WARNING message when logger level is higher than WARNING', () => {
logger.setLevel(logger.setLevel(logger.levels.ERROR));
logger.warning('test');
assert.equal(null, output);
});
});
describe('error()', () => {
it('should always log ERROR messages', () => {
const message = 'test';
logger.error(message);
const [, level, , loggedMessage] = output.split(' ');
assert.equal(level, `[${logger.levels.ERROR}]`);
assert.equal(message, loggedMessage);
});
});
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,649
|
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