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A skin patch that improves men's performance, a vibrator and a robot that simulates human mouths at CES
Sex tech returned for another year at the Consumer Electronics Show (CES) with several companies that showcased a range of erotic devices that improve the performance and experience of both men and women.
At a technology conference in Las Vegas, Satisfyer brought a series of vibrators, all of which pair with a smartphone and some use the power of the air to stimulate the body.
Health technology pioneer Lora DiCarlo is back at her third CES since the lifting of the sexual technology ban in 2020, and this year her company offered a microbot that simulates the human mouth and provides gentle massage and pressure to correct sexual dysfunction in women.
Technologies to increase men's sexual performance are also at the conference, with one company introducing a smart patch that prolongs sexual intercourse.
Scroll down for videos
Satisfyer brought a number of vibrators to this year's Consumer Electronics Show and joined several other companies in the field of sexual technology.
Sex technology was banned at CES after the 2019 event, but the organizers came under fire because of the move and returned it to the test in 2020.
Erotic devices seem to remain here.
Satisfyer brings a number of vibrators to CES that pair with a smartphone and allow users to customize their erotic experience
Satisfyer, a repeat CES participant, is back with a huge collection of new vibrators of all different sizes, colors and purposes.
Satisfyer, a repeat CES participant, is back with a huge collection of new vibrators of all different sizes, colors and purposes. The corporate Hot Lover vibrator (pictured) was at the conference, which is a warm-up rabbit vibrator for $ 59.95
The picture shows Pro 2, which uses air to stimulate parts of the body
At the company's booth was the $ 39.95 Signet Ring for a man who also paired with a smartphone.
Each of them connects to the company's companion app, providing users with remote control on their smartphone and allowing them to customize their feelings.
The company exhibited its Hot Lover, a $ 59.95 warm-up rabbit vibrator at the conference.
The $ 39.95 Men's Signet Ring, which also pairs with a smartphone, was on the company's booth along with the Pro 2 vibrator, which sells for $ 59.95.
Lora DiCarlo is awarded for a microrobot that simulates the human mouth
Lora DiCarlo is a well-known name at CES not only for the company's erotic facilities, but also for the incident that occurred in 2020.
Lora DiCarlo demonstrated her Baci-baci, a bio-inspired microrobot that simulates the human mouth and was developed in collaboration with researchers from the Oregon State University College of Engineering.
One of his robotic masseurs won the award that year, but was quickly withdrawn due to his erotic nature.
However, this did not deter Lora DiCarlo from participating in the third year in a row, and this year the company won an award again for its Baci-baci – but this time he was not disqualified.
Baci-baci is a bio-inspired microrobot that simulates the human mouth and was developed in collaboration with researchers from Oregon State University's College of Engineering.
According to the American Sexual Health Association, the largest U.S. study of female sexual dysfunction found that 21 percent of women live with female orgasmic disorder (FOD), an inability to reach orgasm despite arousal.
And the company's founder, Lora DiCarlo, hopes the mini robot can change that.
"Baci-baci (in Italian it means a kiss) represents a whole new category of consumer technology for women's health: an intelligent robot that replicates the feeling and movement of human lips and tongue," the company said.
"Baci-baci (which means" kiss "in Italian) represents a whole new category of consumer technology for women's health: An intelligent robot that replicates the feeling and movement of human lips and tongue," the company said.
"Small but powerful, it provides direct, diverse stimulation of the clitoris to end the frustration of a failed orgasm."
The sexual health of men is represented at the conference this year, which has been lacking so far.
vPatch aims to help men last longer during intercourse by allowing users to control how much the muscles and nerves between the genitals and the anus are stimulated.
Virility Medical, a consumer health care company specializing in sexual well-being, is the first and only FDA-approved wearable skin patch in the world at CES, scientifically designed to delay ejaculation.
Virility Medical, a consumer health care company specializing in sexual well-being, is the first and only FDA-approved wearable skin patch scientifically designed at CES to delay ejaculation.
Because stimulation inhibits rhythmic muscle contractions, ejaculation is delayed. And the patch can be applied hours before contact
Tal Gollan, founder of Virility Medical, said in a statement: "The Virility Patch patch creates a paradigm shift in sexual well-being, prolongs ejaculation time and fulfills the desire to last longer during intercourse.
"The patch improves men's climax time and creates a better outcome for couples."
A patch called vPatch is placed between the genitals and the anus and users turn it on by pressing a button on the device.
The wearable device stimulates the muscles and nerves in this area, which is controlled by an accompanying application.
Because stimulation inhibits rhythmic muscle contractions, ejaculation is delayed.
And the patch can be applied hours before contact.
The Virility Patch will be available for purchase in low and high intensity variants from autumn 2022.
Experts say that super-centuries could live to be 180 years old by the end of the century
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Celtic's last transfer: Matt O'Riley could sign on January 5 | News from the Transfer Center
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Explore, Dream, Discover
365 Blogs Worth Following
Explanation Page
Tag Archives: Park
Wapato Lake's North Loop and Old Entrance
While walking our new pup around Wapato Lake in an effort to socialize him, I found one of the Park's original entrances off of Wapato Lake Drive. Pup and I had taken the long way around the lake, crossing a narrow bridge and passing the largest field of cat tails that I've ever seen. We came out of the minor upper lake path to the site of two entry markers that said "Built by WPA". The history of Wapato Lake is very interesting and can be found here.
Field of cattails
Lake Wapato
This use to be an entrance to Wapato Lake Park
The new dog, Rhody, a Norfolk Terrier
The narrow bridge on the north loop of Wapato Lake, Tacoma
This entry was posted in Historic Sites, Parks, Tacoma, WA State - Pierce County and tagged Bridge, cat tails, cattails, dog, history, norfolk terrier, Park, path, Rhody, Tacoma, trail, Wapato, Wapato Lake, WPA on July 7, 2017 by admin.
The Wreck of the Peter Iredale, Fort Stevens State Park
On October 25, 1906 the four-masted steel sailing vessel, the Peter Iredale, sank near the mouth of the Columbia River. No lives were lost. Dear husband and I had seen the ship before when we stayed at Fort Stevens State Park near Hammond, Oregon about 25 years ago. At that time we had our red chow, Yum, with us and we enjoyed the adventure of camping. And here I was years later with Lilly, our cairn terrier. I had dropped dear husband off at Astoria so that he could peruse the antique stores. There was less of the ship now, but still in all I was impressed that so much of it remained.
Graveyards of the Pacific lists some of the other ships that have gone down in this dangerous part of the Pacific.
While at Fort Stevens, Lilly and I also enjoyed two wildlife view areas, one with a boardwalk/concrete viewing area that overlooked an estuary and another with a viewing platform that overlooked the crashing waves of the Pacific. There is also the fort part of Fort Stevens State Park, which operated from the time of the Civil War through World War II. All of the photos are mine with the exception of the historical view of the ship.
This entry was posted in Historic Sites, Other States - OR, Parks and tagged camping, Fort Stevens, Lilly, Oregon., Pacific Ocean, Park, shipwreck, State Park, viewing platform on April 6, 2017 by admin.
A Barn at Fort Steilacoom Park
Steilacoom Barn
Western State Hospital for the Insane, later shortened to simply Western State, opened in 1871 and soon after included a farm with animals. The farm included several barns and this one, near the entrance to the dog park in what is now Fort Steilacoom Park in Lakewood, is one of them. The farm workers were patients of the hospital and the farm supplied much of the food for the hospital. The farm closed in 1959.
I went in the evening and there was a lovely sunset and a flock of geese.
This entry was posted in Farms and Gardens, Lakewood, Parks, WA State - Pierce County and tagged barns, Fort Steilacoom, historic, Insane, Lakewood, old, Park, twilight, Western State Hospital on September 11, 2016 by admin.
Marlake at the West Hylebos Wetlands Park
Dear husband and I wanted to stretch our legs today so we went to the West Hylebos Wetlands Park in Federal Way. Our intent was to walk along the boardwalk path, but we turned right instead of left and ended up at Marlake (that's what Google Maps says it is called). It is really a beautiful spot and some of the surrounding area must have been an orchard at one point because we found plum, pear and apple trees, as well as grapes and blackberry bushes. The blackberries and plums were delicious! The lake itself has a dock with a bench on it to contemplate life. Many of the trees leading up to the lake were actually labeled and my favorite was a ginkgo tree. Such lovely leaves. The park really is a perfect place to take a walk.
On an amusing note, the handwritten sign that greats visitors asks us to protect the wetlands and no dogs (I get it), bikes (still get it) or Pokemon (what?!). I'm not sure what damage the not really there Pokemon could do. Perhaps they meant no Pokemon players. But there were a bunch of players and they were all respectful and having a good time with their families. Perhaps the highlight for me was that I won my first gym (it's a game thing) and let out a yell of victory. An older teen smiled at me and we talk about the game for a while. He even set it up for me so that I could really win the gym since I obviously don't know what I'm doing. I love that the game gave two very different people a chance to visit and work together. What fun. 🙂
This entry was posted in Federal Way, Parks, WA State - King County and tagged apple, blackberries, blackberry, ducks, Federal Way, fruit, Ginkgo, grapes, Marlake, Park, pear, plums, pokemon, sign, trees, West Hylebos Wetlands Park, wetlands on September 5, 2016 by admin.
There was a famous aquarium at Salter's Point
Dear daughter, her dear friend and I were looking for relieve from the heat on June 5 and we decided on a beach. Dear daughter suggested that beach where one goes on a footbridge over the railroad station. So after my online community and I figured out where that was, Salter's Point at 91 Champion St. in Steilacoom, off we went. The footbridge is now a steep metal affair with gates on both sides of the stairs. It was constructed in 2014 after the prior wooden bridge was damaged and closed. About 60 trains a day pass along the tracks.
I was expecting to find some natural beauty, relief from the heat and perhaps a couple of small crabs. And indeed I found all that, but I also found an interesting history.
The covered picnic area built in 1939 as part of the Federal Works Progress Administration (WPA). Per Wikipedia the WPA "was the largest and most ambitious American New Deal agency, employing millions of unemployed people (mostly unskilled men) to carry out public works projects, including the construction of public buildings and roads. In a much smaller but more famous project, Federal Project Number One, the WPA employed musicians, artists, writers, actors and directors in large arts, drama, media, and literacy projects."
The remains of a marina and store is located at the southerly end of the park. The property was destroyed by fire in 2009. The fire commanded the attention of three fireboats and more than 50 firefighters from eight fire departments. In 1987 there was a homicide at the store (Wang's Maritime Marina); two teens killed the store's owner. In close proximity to the marina ruins was the Soundview Inn and Boathouse, which was a boardinghouse with family style meals. They also had boat rentals.
The site of the Deep Sea Aquarium is also located at Salter's Point. It was constructed by Ed Bair, brother of Bair (Bair Drugstore). The aquarium featured an extensive collection of sea life including a seal that lived under the porch where it could swim depending on the tide. The aquarium was promoted all over the western states, but closed in the 1930s.
Here are some interesting articles about the location.
A first person account about the aquarium
"The evolution of Saltar's Point," Steilacoom Historical Museum Quarterly, XIV (Summer, 1985) p. l, 3-6.
City of Steilacoom Park Info
Fire destroys Steilacoom Marina and Store
This entry was posted in Historic Sites, Parks, Steilacoom, WA State - Pierce County and tagged Aquarium, Bair, beach, Federal Works Progress Administration, fire, historic, marina, New Deal, Park, picnic, Salter's Point, Soundview Inn, Steilacoom, store, WPA on June 29, 2016 by admin.
Washtucna, WA
I drove to Pullman today and along the way I stopped in the town of Washtucna in Adams County. The town has a population of about 200 people. I picked Washtucna because I heard it had a good birding park known as Bassett Park after the first mayor. It was some nice little park with a small creek running through it and one of the local residents came and chatted with me for a while. And then as I was leaving the town I found their original sheriff's office/jail which consisted of a very small wooden building with two jail cells and a front area for the sheriff. There was also the original outhouse, a two seater! Altogether it was a pleasant little diversion on the long drive to Pullman and I'm glad I stopped.
This entry was posted in Historic Sites, Parks, WA State - Adams, WA State - Adams Co., Washtucna and tagged Adams County, birding, birds, farms, jail, outhouse, Park, sheriff, Washington State, Washtucna on April 7, 2016 by admin.
The Guardian Stone, Poulsbo
The Guardian Stone is Poulsbo's newest public art installation and it is really lovely. My first thought was "oh, look, a sword in the stone! King Arthur!", but no. It being Poulsbo, the piece is reflective of Norwegian history. The rock, steel and glass sculpture by Lisa Stirrett was installed in February 2016 at the Muriel Iverson Williams Waterfront Park. The nine foot tall piece is a nod to Swords of the Rock in Norway, though that piece is much larger (see a photo of that too). Really, it is captivating. You should go see it!
This entry was posted in Art -- there is art here!, Parks, Poulsbo, WA State - Kitsap County and tagged art, glass, Guardian Stone, Lisa Stirrett, Muriel Iverson Williams, Norway, Norwegian, Park, Poulsbo, scupture, steel, stone, waterfront park on April 5, 2016 by admin.
The Peace Arch connecting the ties between the US and Canada
The Peace Arch spans the United States and Canadian border and commentates the signing of the Treaty of Ghent in 1814. The arch was dedicated in 1921 and was placed on the US National Register of Historic Places in 1996. Both countries flags fly on the monument. The Peace Arch and the associated area are considered to be an international park and one does not need to have a passport to visit it. The Peace Arch border crossing never closes. It was a pretty quick passage for us, only about 45 minutes.
This entry was posted in BC, Blaine, Canada, Historic Sites, National Historic Register, Parks, WA State - Whatcom County, WA State Grant Co and tagged arch, BC, boundary, British Columbia, international, monument, Park, passport, peace, Peace Arch, Surrey, Whatcom on July 28, 2015 by admin.
The dock at Thea's Park
Thea Park hosted dragon boat races today and dear husband and I managed to see the very end of it. Not the racing part, but there was still a group receiving awards and a booth or two. I wandered on to the dock to see the new still dragon boats, their dragon heads being safely stowed.
This entry was posted in Once in a while events, Parks, Sports and tagged boat, Commencement Bay, dragon, dragon boat, Park, Tacoma, Thea Park, Thea's on May 16, 2015 by admin.
Velodrome at Marymoore Park
Washington State's only velodrome (a cycle-racing track, typically with steeply banked curves) is the Velodrome at Marymoore Park in Redmond, WA. Dear husband and I checked it out today. There were only a couple of folks causally biking the loop on this lovely day. Velodrome racing was a popular sport in the U.S from the early 1800s to the 1930s and has been a part of the Olympic Games since 1896. There are about 26 functioning velodromes across the country. Interesting fact, the bicycles that race the track can go speeds of upward of 50 mph and have no brakes.
While at Marymoore, we also checked out the spectacular rock climbing wall and the turn of the century Windmill in the Willowmoor farmstead area.
This entry was posted in Parks, Redmond, Sports, WA State - King County and tagged bicycle, bicycles, bikes, Marymoore, Marymoore Park, Park, racing, Redmond, rock climbing, scale, track, Velodrome, Willowmoor, Windmill on May 9, 2015 by admin.
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Subject: UFO Sighting
Date/time: Saturday,July 10, 2004 - Dusk
Location: Rt. 550 - Leburn,Kentucky (Knott County)
On July 11, 2004 at 5:06 p.m., a KnottCounty, Kentucky resident called the Cincinnati UFO Hotline (513-588-4548)to report a UFO sighting by 3 witnesses from the previous evening, Saturday,July 10th. His name and phone number are on file, deleted from this reportfor privacy purposes. He wanted to report the sighting and had asked hisson go online to lookup a UFO reporting number.
The witness, 57 years old, is now retiredbut formerly worked with the EPA for 25-years. He said that he was standingoutside shortly after dusk and, as darkness set in, stood talking withhis wife and a friend on property along Route 550 near Leburn, Kentucky(Knott County). He said that besides himself, there were two other witnessesto the sighting; his wife (49-years old) who stood to his left, and hisfriend who works for The State of Kentucky as a Mining Inspector (also49-years old). He said that the other two witnesses were impressed by thesighting as well.
"We were just standing outside talking,"he said, "when all of the sudden there was some kind of flash. This thingcaught my eye, and it looked just like a star blew in half."
The witness said that from his vantagepoint on property along Route 550 near Leburn, he was standing next tohis wife and facing in the direction looking to the northeast. His friendhad his back turned but also quickly looked to see the object(s). The witnesssaid that he couldn't determine height or angle with exacting specifics,but thought it was situated in the sky at an estimated 70 degree to thenortheast and the object itself was fairly high, like a star.
"Something caught my eye, like an implosion,"he said, "and I looked up and there was a flash. I would say the last ofthe flash part that caught my attention.
"I happened to look up and saw thisthing that looked like a star that got real bright and spilt in two, onewent one direction and one went the other direction. The one that wentto the left kind of faded away in about 2 or 3 minutes after travelingstraight across. The one that went to the right continued on in a sort-of-southerlydirection, even somewhat southeast, and after a few minutes it suddenlymade a 90-degree turn and came towards us. It started to come my directionwhere I was observing it from.
"It looked like a just like a star,as best as I could describe it, then after a couple of minutes it fadedout, too."
The witness also said the two objects,traveling in opposite directions, looked identical, resembling two starsin a straight line.
"The right-hand object went toward theeast for several minutes and then it appeared as if it changed directionby about a 90-degree turn and it came toward us for 2 or 3-minutes andthen it faded out."
The witness informs that the largestcommunity near his location would be Prestonsberg, Kentucky, about 35-milesaway. Hazard, Kentucky is about 25-miles distant. He was unsure of anyadditional witness, asking if I knew if anything had been reported.
This appears to be an unusual sightingand the witness adds that he is "not a UFO person" and was curious if therewere other reports. He remarked that the objects appeared star-like inappearance but could not account for the apparent flight performance.
I inquired if the witness knew of anymilitary air zones near this location where aerial flare-training couldbe taking place, and he knew of no Military Operation Areas nearby. Hesaid that he was in the military for years and had the ability to recognizeflares, stressing that the right-hand object 'undertook a certain maneuver'during the event. This reported maneuver seems to complicate any payloador satellite re-entry, meteor/debris or flare explanation.
Filed,
Kenny Young
UFO Research
https://kenny.anomalyresponse.com
UFO Hotline: 859-801-6767
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: What is the "Angular root folder" Im trying to follow a tutorial on how to write a simple angular/mongodb CRUD-Application and im stuck at this part:
"Inside the angular root folder, create one folder called api and go inside that folder. Remember, it will be a completely separate project from Angular. So its node_modules are different from an Angular."
The question basically is where i should create the folder in my angular project :/
Heres the link for more context if needed: https://appdividend.com/2018/11/04/angular-7-crud-example-mean-stack-tutorial/
My proposal:
AngularApp
New Folder
e2e
node_modules
(rest of angular project)
Api
A: If they have different node modules, then they should have different package.json files.
Given that this file is at the root of the project, that's not possible.
You should have ONE FOLDER containing TWO FOLDERS :
*
*api
*client
That's the monorepo structure. For more information, you can take a look at Narwhal's team latest, NxDev
EDIT
Your proposal is
AngularApp
Api
Mine is
MyMeanApp
AngularApp
Api
A: The tutorial you are following is making seperate front end and back end folders in one project.
You can go with this however you can add modules in one folder as well.
Yes, you can create folder like you are creating right now.
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"redpajama_set_name": "RedPajamaStackExchange"
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Q: Best way to remove a leading zero after formatting a date in xslt 1.0 I've got a date format template that I'm passing a date value to in the format YYYYMMDD
The template is the following:
<xsl:template name="formatDate">
<xsl:param name="date" />
<xsl:variable name="year" select="substring($date, 1, 4)" />
<xsl:variable name="month" select="substring($date, 5, 2)" />
<xsl:variable name="day" select="substring($date, 7, 2)" />
<xsl:value-of select="concat($month, '/', $day, '/', $year)" />
</xsl:template>
This would return the string 20131004 as 10/04/2013 which is correct.
What I need to do though is if the $month has a leading zero, to remove it. For example, 20130930 would be 09/30/2013 when I would prefer 9/30/2013.
What's the most efficient way to do that? I could do a choose/when before I set the value of the variable but I'm trying to do it in the proper manner with xslt (I'm still trying to get into it, it's coming along).
Thanks
A: You could utilize number() function
<xsl:variable name="month" select="number(substring($date, 5, 2))" />
<xsl:variable name="day" select="number(substring($date, 7, 2))" />
It should remove leading zero.
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"redpajama_set_name": "RedPajamaStackExchange"
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| 9,452
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Q: F# program flow question with Async and Option I'm trying to wrap my head around how to accomplish the following task in F#. The following is a simplified C# pseudocode equivalent I'm looking to replicate.
var x = await GetXAsync();
if (x == null) return "not found";
var y = await GetYAsync(x);
return y;
My initial F# version looks something like:
task {
let! x = GetXAsync()
match x with
| None -> // need to return a hard-coded value here
| Some x` ->
let! y = GetYAsync(x`)
// More code
// return some value based on y here
}
Obviously this is terrible, but I'm unsure of how to proceed. Should I attempt a full ROP style of programming here, or is there something simpler?
A: In your example, you are returning the "not found" string to indicate that something went wrong from a function that otherwise returns strings. I would not do this, because it will be hard to distinguish between the case where everything worked and the case where it did not.
If the fact that GetXAsync returns null is something that indicates a failure, then I'd just use exceptions. F# async has a nice support for propagating those and you can catch them using try .. with. There is nothing wrong with using exceptions in F# to handle exceptional situations!
exception InvalidX of string
let GetXAsync() = async {
// whetever code that calculates 'failed' and 'result' goes here
if failed then raise (InvalidX "not found")
return result }
Then you can just call the functions and exceptions get propagated automatically.
async {
let! x = GetXAsync()
let! y = GetYAsync(x)
return y }
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Conferences \ Antibodies and Antibody Drug Conjugates \ Agenda \ Rathnam Chaguturu
Rathnam Chaguturu's Biography
Rathnam Chaguturu, CEO, Innovation Czar-iDDPartners
Rathnam Chaguturu is the Founder & CEO of Innovation Czar-iDDPartners, a nonprofit think-tank focused on pharmaceutical innovation. He has more than thirty-five years of experience in academia and industry, managing new lead discovery projects and forging collaborative partnerships with academia, disease foundations, nonprofits, and government agencies.. He is the Founding President of the International Chemical Biology Society, a Founding Member of the Society for Biomolecular Sciences, and Editor-in-Chief of the journal, Combinatorial Chemistry and High Throughput Screening. He serves on several editorial- and scientific advisory boards, recipient of several awards, and is a sought-after speaker at major national and international conferences, passionately advocating the virtues of collaborative partnerships in addressing the pharmaceutical innovation crisis. 'Collaborative Innovation in Drug Discovery: Strategies for Public and Private Partnerships,' edited by Rathnam with the Foreword by Ferid Murad, the 1998 Nobel Prize winner in Medicine, has just been published by Wiley.
Thursday, 29 September 2016 at 09:20
Add to Calendar ▼2016-09-29 09:20:002016-09-29 10:20:00Europe/LondonWelcome AddressAntibodies and Antibody Drug Conjugates in BengaluruBengaluruSELECTBIOenquiries@selectbiosciences.com
Add to Calendar ▼2016-09-29 00:00:002016-09-30 00:00:00Europe/LondonAntibodies and Antibody Drug ConjugatesAntibodies and Antibody Drug Conjugates in BengaluruBengaluruSELECTBIOenquiries@selectbiosciences.com
#AADCIn16 Tweets
Extracellular Vesicles 2023: Drug Delivery, Biologics & Therapeutics
Flow Chemistry Asia 2023
Flow Chemistry European Summit 2023
Location: Rotterdam, The Netherlands
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| 7,652
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{"url":"https:\/\/gaupdate.wordpress.com\/2013\/12\/21\/c-castro-cliff-alg-grand-unification-of-gravity-standard-model-review\/","text":"# C. Castro: A Clifford Algebra Based Grand Unification Program of Gravity and the Standard Model : A Review\u00a0Study\n\nCarlos Castro, 70 Pages. Submitted to Int. J. Mod. Phys. A.\n\nAbstract.\nA Clifford $Cl ( 5, C )$ Unified Gauge Field Theory formulation of Conformal Gravity and $U (4 ) \\times U ( 4 ) \\times U(4)$ Yang-Mills in $4D$, is reviewed, along with its implications for the Pati-Salam group $SU (4) \\times SU(2)_L \\times SU(2)_R$, and $Trinification$ GUT models of $3$ fermion generations based on the group $SU (3)_C \\times SU (3)_L \\times SU(3)_R$. We proceed with a brief review of a unification program of $4D$ Gravity and $SU(3) \\times SU (2) \\times U (1)$ Yang-Mills emerging from $8D$ pure Quaternionic Gravity. A realization of $E_8$ in terms of the $Cl(16) = Cl (8) \\otimes Cl(8)$ generators follows, as a preamble to Tony Smith\u2019s $E_8$ and $Cl(16) = Cl(8) \\otimes Cl(8)$ unification model in $8D$. The study of Chiral Fermions and Instanton Backgrounds in ${\\bf CP}^2, {\\bf CP}^3$ related to the problem of obtaining $3$ fermion generations is thoroughly studied. We continue with the evaluation of the coupling constants and particle masses based on the geometry of bounded complex homogeneous domains and geometric probability theory. An analysis of neutrino masses, Cabbibo-Kobayashi-Maskawa quark-mixing matrix parameters and neutrino-mixing matrix parameters follows. We finalize with some concluding remarks about other proposals for the unification of Gravity and the Standard Model, like string, $M, F$ theory and Noncommutative and Nonassociative Geometry.","date":"2017-09-24 22:47:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5529834628105164, \"perplexity\": 667.2092304387036}, \"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-2017-39\/segments\/1505818690228.57\/warc\/CC-MAIN-20170924224054-20170925004054-00093.warc.gz\"}"}
| null | null |
Конгенитални тортиколис или криви врат. Основна карактеристика је нагиб главе према рамену услед скраћења бочног прегибача главе (m. sternocleidomastoideus). Узрок овоме може бити лош положај фетуса у материци последњих недеља трудноће или повреда мишића у вријеме порођаја. Одмах послије рођења уочава се да су врат и глава нагнути бочно, а брада је окренута према супротном рамену. У неким тежим случајевима настаје асиметрија лица. У случају да је мишић повређен током порођаја, тада долази до прожимања мишићних влакана крвљу. Потом она подлежу фиброзним променама и тада се на мишићу могу запазити палпирати мала задебљања, што изискује дуготрајнију терапију у виду корекције деформитета.
Медицина
Дефектологија
|
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"redpajama_set_name": "RedPajamaWikipedia"
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| 8,619
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{
"redpajama_set_name": "RedPajamaC4"
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| 1,453
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\chapter{Exact Cost VCG Algorithm}
\label{apend:exact}
\begin{lstlisting}[language=ml,label=lis:wp-exact,float=htb,caption={Weakest Precondition Implementation for Exact Costs.}]
let rec wpc s phi =
match s with
(...)
| If (b, s1, s2) ->
let wp1, _ = wpc s1 phi verbose in
let wp2, t = wpc s2 phi verbose in
let v_b = annot_of_bexp b in
let tb = time_bexp b in
(AAnd (AImpl (v_b, wp1), AImpl (ANeg v_b, wp2)), Sum (t, tb))
(* TODO: fix max *)
| For (id, i, a, b, s') ->
let x = Lt (Var i, b) in
let inv = get_oracle id in
let _, tc = wpc s' inv verbose in
( inv, Sum (Mul (Sum (Sub (b, Cons a), Cons 2), time_bexp x),
Mul (Sum (Sub (b, Cons a), Cons 1), tc)))
\end{lstlisting}
\begin{lstlisting}[language=ml,label=lis:vc-exact,float=htb,caption={Verification Condition for \textit{for} loop.}]
let rec vc s phi : annot list =
match s with
(...)
| For (id, _, a, b, s') ->
let inv = get_oracle id in
let x = ALt (Cons a, b) in
AImpl (inv, x)
:: AImpl (AAnd (inv, x), wp s' inv)
:: AImpl (AAnd (inv, ANeg x), phi)
:: vc s' inv
\end{lstlisting}
\chapter{Amortized Cost VCG algorithm}
\label{apend:amortized}
\begin{lstlisting}[language=ml,label=lis:wp-amortized,float=htb,caption={Weakest Precondition Implementation with Amortization - while case.}]
let rec wpc s phi verbose =
match s with
(...)
| While (id, b, _) ->
let inv, f, n, a, p = get_oracle id in
let time_body = Mul (n, a) in
( AAnd (AAnd (inv, AGe (f, Cons 0)), AEq (p, Cons 0))
, Sum (Mul (Sum (n, Cons 1), time_bexp b), time_body) )
\end{lstlisting}
\begin{lstlisting}[language=ml,label=lis:vc-amortized,float=htb,caption={VC Implementation with Amortization - while case.}]
let rec vc s phi : annot list =
match s with
(...)
| While (id, b, s') ->
let inv, f, n, a, p = get_oracle id in
let b = annot_of_bexp b in
let wp, t = wpc s' (AAnd(inv, AAnd (AGt (f, Var "k"), AEq (p, Var "pk")))) false in
AImpl (AAnd (inv, b), ALe (f, n))
:: AImpl (inv, AGe(p, Cons 0))
:: AForall ( "k", AImpl ( AAnd (inv, AAnd (b, AEq (f, Var "k"))), wp) )
:: AGe (Sub (Sum(a, p), Var "pk"), t)
:: AImpl (AAnd (inv, ANeg b), phi)
:: vc s' inv
\end{lstlisting}
\section{Historical Background}\label{sec:pre-history}
As computers became more powerful, programs also became longer and more complex. When programs were still relatively small, flowcharts or extensive testing was enough to prove a program's functionality.
But programs quickly started being so complex that these methods became unreliable and more prone to error.
At the beginning of the second half of the 20th-century, experts started to find vulnerabilities in public distributed software.
Since then, the use of computational systems has grown exponentially, and so did the number of vulnerabilities and their impact. A simple error might have drastic consequences, such as a leak of confidential information, the crash of critical systems, and direct loss of assets.
This problem proved to be enough reason to start thinking about a more reliable way to guarantee the properties of a program and develop tools that help verify these properties.
Formal Verification refers to using mathematical principles to prove the correction of a given specification of a program.
It is hard to pinpoint where it all started, but the works of Robert Floyd~\cite{floyd1967} and Tony Hoare~\cite{hoare1969} were undoubtedly pioneers in the field, and their definitions are still the base of verification tools used today.
\section{Semantics}\label{sec:pre-semantics}
When defining programming languages, we want a way to be capable of reasoning about what programs are doing.
The syntax describes the grammatical rules we must follow to write a program in a language. The syntax allows us to distinguish between languages and identify a program's language.
But if we want to understand what that program is doing, we need to look at its semantics. Semantics is a way to make sense of the meaning of a program and understand what it is trying to accomplish.
There are multiple strategies to analyze the meaning of a program. The most popular ones are operational, denotational, and axiomatic semantics.
Operational semantics focus on what steps we take during the program's execution. In denotational semantics, we do not care about the "how" but only about "what" the program is doing. In axiomatic semantics, we are concerned about evaluating the satisfability of assertions on the program and its variables. We will go more in-depth on how operational and axiomatic semantics work.
\subsection{Operational Semantics}\label{subsec:operational}
Operational semantics describes the meaning of a program by specifying the transitions between states of an abstract state machine.
As we mentioned, unlike with denotational semantics, here we are concerned about \emph{how} the machine changes states with the execution of a statement.
There are two styles of operational semantics
\begin{itemize}
\item Small-step or Structural Operational Semantics
\item Big-step or Natural Semantics
\end{itemize}
In Structural Operational Semantics or Small-step Semantics, we are concerned about every individual transition we take throughout the program's execution.
In Natural Semantics or Big-step semantics, we want to understand how we transition from the initial to the final state. We are concerned about a high-level analysis of how the machine state changes and not about each individual step.
\subsection{Axiomatic Semantics (Hoare Logic)}\label{subsec:pre-hoare}
In 1967 Floyd specified a method that would allow proving properties on programs, such as correctness, equivalence, and termination \cite{floyd1967}. They achieved this by representing programs as flowcharts and associating propositions to each connection on the flowchart. The proof is done by induction on the number of steps. If an instruction is reached by a connection whose proposition is true, then we must leave it with a true condition as well.
In 1969 Hoare wrote a paper where they extended Floyd's logic to prove properties on a simple imperative program \cite{hoare1969}. In this paper, they defined what we now call Hoare (or Floyd-Hoare) triples.
\begin{definition}[Hoare Triples]\label{def:hoare-triple}
A Hoare triple is represented as $$\{P\}Q\{R\}$$
and can be interpreted as "if the assertion $P$ is true before we run program $Q$, then assertion $R$ will be true when the program ends".
\end{definition}
Notice this definition does not offer guarantees over termination. Executing program $Q$ from a state validating $P$, does not have to halt. As long as whenever it does the final state validates $R$. We call assertions in the form $\{P\}Q\{R\}$ \emph{partial correctness assertions}.
Using this definition Hoare specifies a proof system with a set of axioms and inference rules, which allow to prove assertions on any program written in this language. A derivation on this proof system is called a \emph{theorem} and is written as $\vdash \{P\}Q\{R\}$.
As an example, let us look at the assignment axiom. Consider an assignment to variable $x$ of an expression $a$ in the form
$$x := a$$
If an assertion P is true after executing the assignment (when variable x takes the value of expression a) then it has to be true before the assignment if we replace any mentions of $x$ in P by $a$. This is usually represented as $P[a/x]$.
Therefore the assignment axiom is written as
$$\{P[a/x]\}\ x := a\ \{ P \}$$
If, in addition to proving a program specification is correct, we also want to prove the program always halts, we are looking for Total Correctness.
\begin{definition}[Total Correctness]
A total correctness assertion is represented as
$$[P]Q[R]$$
where $P$ and $R$ are assertions and $Q$ is a program. If we execute $Q$ from a state that satisfies $P$ program Q will terminate and the final state will satisfy $R$.
\end{definition}
partial correctness + termination = total correctness
Total correctness is harder to prove than partial correctness and not always possible. But it also gives a stronger guarantee about a programs behavior.
We consider two properties on this proof system, soundness an completeness. Soundness ensures our proof systems generates valid partial correctness assertions. Completeness ensures that our system is capable of deriving every valid assertion.
\begin{definition}[Validity]
We say an assertion \{P\}Q\{R\} is valid if it is true for all possible states.
$$\forall \sigma \in \Sigma_\perp.\ \sigma \models \{P\}Q\{R\}$$
Or we can simply represent it as
$$\models \{P\}Q\{R\}$$
\end{definition}
\begin{definition}[Soundness]
Our proof system is sound if every rule preserves validity.
In other words, for any partial correctness assertion $\{P\}Q\{R\}$
$$\mathrm{if}\ \vdash \{P\}Q\{R\}\ \mathrm{then}\ \models \{P\}Q\{R\}$$
\end{definition}
\begin{proof}
Soundness is proved by structural induction on the statement $Q$.
\end{proof}
\begin{definition}[Completeness]
A proof system is complete if every true assertion $\{P\}Q\{R\}$ can be proved by our system.
$$\mathrm{if}\ \models \{P\}Q\{R\}\ \mathrm{then}\ \vdash \{P\}Q\{R\}$$
\end{definition}
Proving completeness is not as trivial and most of the times not possible. The completeness of the proof system presented by Hoare was established by Cook in 1978~\cite{cook1978}. In this paper he presents a proof of \textit{relative completeness}, that is, assuming our assertion language is complete then the logic presented by Hoare is also complete.
\section{Verification Conditions Generator}
We now have the necessary notation to specify the behavior of a program. Hoare's set of axioms and rules allows proving that program's said behavior.
Manually proving these properties is not only long and tedious but also prone to error. We are missing a mechanized solution we could apply to any program to guarantee its validity.
Dijkstra defined the weakest precondition algorithm in is 1976 book "A Discipline of Programming" \cite{dijkstra1976}.
\begin{definition}[Weakest Precondition]
The weakest precondition is the simplest condition, necessary and sufficient to guarantee the post-condition is true when the program terminates. We use the notation $wp(Q,R)$ where Q is a statement and R is a post-condition.
$$\models \{P\}Q\{R\}\ \textit{iff}\ P \rightarrow wp(Q,R)$$
\end{definition}
Let us consider as an example the statement $Q \equiv x := y + 2$ and we want to prove $R \equiv x \ge 0$ is true after executing Q. The weakest precondition $wp(Q,R)$ would say that as long as $y \ge 2$ is true before executing statement Q, then R will be true after execution.
In 1979, JC King presents the first mechanized algorithm to automatically verify the correctness of a program~\cite{king1970}. His work was based on the definitions provided by Floyd~\cite{floyd1967} and Manna~\cite{manna1969}.
In more recent implementations this algorithm is usually based on Hoare's definition of correctness and it is called a VCG. This algorithm makes use of the weakest precondition function. Given a Hoare triple $\{P\}Q\{R\}$, for the program Q to be correct we must guarantee that $P \rightarrow wp(Q,R)$ is true. This is the condition that needs to be proved in order to guarantee correctness. If our language contains loops we need to satisfy extra conditions, including the preservation of the loop invariant.
The \textit{loop invariant} is an assertion that is satisfied before and after every execution of the loop's body.
\begin{definition}[Verification Condition Generator]
A VCG is an algorithm that when applied to a Hoare triple returns a set of VC. The Hoare triple is derivable in our proof system, if and only if all the generated conditions are valid.
$$\models VCG(\{P\}Q\{R\})\ \textit{iff}\ \vdash \{P\}Q\{R\}$$
\end{definition}
\begin{theorem}[Soundness of VCG]\label{theor:soundness-vcg}
$\models VCG(\{P\}Q\{R\})\ \rightarrow\ \vdash \{P\}Q\{R\}$.
\end{theorem}
\begin{proof}
By induction on the structure of Q.
\end{proof}
\begin{theorem}[Completness of VCG]\label{theor:completeness-vcg}
$\vdash \{P\}Q\{R\}\ \rightarrow\ \models VCG(\{P\}Q\{R\})$.
\end{theorem}
\begin{proof}
By induction in the derivation of $\vdash \{P\}Q\{R\}$.
\end{proof}
Returning to our last example where $wp(x:=y+2, x \ge 0) = y \ge -2$ let us consider the partial correctness assertion $\{y = 0\} x := y + 2 \{x \ge 0\}$. Since in this case the program does not contain any loops there is only one verification condition that needs to be satisfied in order to prove correctness: $(y = 0) \rightarrow (y \ge -2)$. Since this VC is valid, we know our program is correct.
\section{Tool Architecture}
\input{sections/chapter 7/7.1-Architecture.tex}
\section{Implementation Details}
\input{sections/chapter 7/7.2-VCG_Implementation.tex}
\section{Examples}
\input{sections/chapter 7/7.3-Examples.tex}
\section{Future Work}\label{sec:trab}
One of the first improvements we are aiming towards is to develop a single system capturing all of the cost logics together, creating a more cohesive, powerful tool.
We primarily focused on theoretical definitions and guaranteeing a sound logic that produced reasonable bounds. Our implementation is a simple prototype that serves as proof of concept of these definitions. Therefore many improvements can be made to our tool concerning efficiency, transforming it from a conceptual tool to a practical one.
We also defined our logic using a simple language, which allowed us to focus on the cost estimation aspect without having to worry so much about language details. In the future, we want to extend our language with more features, such as functions, to improve the expressiveness of programs.
We would also like to expand the application of our logic to more extensive and complex case studies, namely cryptographic implementations.
Our work started as an adaptation of the EasyCrypt cost logic developed in~\cite{barbosa21}. In the future, we would like to propose an extension to the EasyCrypt tool with our logic.
\section{Axiomatic Semantics Systems}
One way of proving bounds on a program
's resources is by using axiomatic semantics. Other works have already implemented systems that use axiomatic semantics for resource analysis, which differ from our work in multiple ways, from the paradigm of language used to the precision of the derived bounds. This section explains some of the most relevant work in the literature that inspired our definitions. We further subdivided this literature into two categories: classical cost analysis and amortized cost analysis.
\subsection{Classical Cost Analysis}
There is already some work using derivations on the logic presented by Hoare~\cite{hoare1969} for resource analysis. Some estimate orders of magnitude, while others use a more detailed annotation to automate the process but lack ways of optimizing the bounds.
One of the first and still one of the most relevant works on this topic is the one presented by Nielsen~\cite{nielson1987,nielson2007}. The author defines an axiomatic semantics for a simple imperative programming language in this work and extends Hoare's logic so that the proof system would be capable of proving the magnitude of worst-case running time and termination. This system is also proven sound. Even though this work operates on a similar imperative language to the one we defined, it lacks the precision our logic provides since it only allows proving order-of-magnitude.
In 2014, Carbonneaux \emph{et al.}~\cite{carbonneaux2014} presented a system that verifies stack-space bounds of compiled machine code (x86 assembly) at the C level. That is, it derives bounds during compilation from C to assembly. They developed a quantitative Hoare logic capable of reasoning about resource consumption. This work is an extension of the CompCert C Compiler~\cite{leroy2009}. Coq was used to implement and verify the compiler. The work by Carbonneaux focuses on the compilation from C to assembly using quantitative logic, which does not serve the same purpose we are trying to achieve. With our work, we can prove tight bounds on imperative programs using an assisted proof system, where the user can help make the bounds as precise as possible. Also, how default constructors' costs are defined makes our work easy to adjust for a system with different resource usage.
In 2018, Kaminski \emph{et al.} defined a conservative extension of Nielsen's logic for deterministic programs~\cite{nielson1987,nielson2007} by developing a Weakest Precondition calculus for expected runtime on probabilistic programs~\cite{kaminski2018}. Again this work is largely automated, which differs from our user-assisted approach. Since it reasons about probabilistic programs, it faces other challenges than the ones we are interested in this work.
In 2021 a paper was released extending the logic of EasyCrypt~\cite{barbosa21}. EasyCrypt is an interactive tool created to prove the security of cryptographic implementations. One of its core concepts is a set of Hoare Logics, which allow proofs on relational procedures and probabilistic implementations. In this paper, the authors propose an extension to the EasyCrypt tool, allowing to prove properties on the cost of a program. To achieve this, they extended the existing logic to include cost rules. They also implemented a way to define the cost of custom operators. Our work operates on a subset of EasyCrypt's language, but we extended the logic to use the potential method of amortized analysis, increasing the accuracy of the generated bounds.
\subsection{Amortized Cost Analysis}
We will now present some of the literature that, in addition to using axiomatic semantics for static cost analysis, also uses amortized analysis to increase the accuracy of the bounds.
Carbonneaux \emph{et al.}~\cite{carbonneaux2015} continued their previous work~\cite{carbonneaux2014} on deriving worst-case resource bounds for C programs, but they now implemented a system that uses amortized analysis and abstract interpretations.
In~\cite{haslbeck2018}, Haslbeck and Nipkow analyze the works of Nielson\cite{nielson2007}, Carbonneaux \emph{et al.}~\cite{carbonneaux2014,carbonneaux2015} and Atkey~\cite{atkey2011} and prove the soundness of their systems. In this paper, they implement Verification Condition Generators based on Nielsen's logic and Carbonneaux's Logic, proving it sound and complete. They compare all three methodologies and explain some of the limitations of these systems.
\section{Type-Based Systems}
While our system uses a Hoare logic to prove upper bounds on program cost, many existing systems are type-based. Usually, these systems use type inference and type size/cost annotation in order to be able to analyze resource usage statically. Even though these works highly differ from ours, we will briefly mention some of the most relevant works in the field.
In \cite{radicek2018}, the authors present a proof system for cost analysis on functional programs using a fine-grained program logic for verifying relational and unary costs of higher-order programs. The paper~\cite{avanzini2017} presents a fully automated methodology for complexity analysis of higher-order functional programs based on a type system for size analysis with a sound type inference procedure. Hoffman and Jost, \cite{hoffmann2006} defined a type system capable of analyzing heap space requirements during compilation time based on amortized analysis. This work was limited to linear bounds in the size of the input, so they later extended it to polynomial resource bounds~\cite{hoffmann2010}.
In 2017, Hoffman \emph{et al.}~\cite{hoffmann2017} developed a resource analysis system capable of proving worst-case resource bounds. This resource is a user-defined input and can be anything from time or memory to energy usage. This work is an extension of their previous work in Automatic Amortized Resource Analysis (AARA), where they used amortized analysis to derive polynomial bounds for the first time. Their proof system is a type system with inductive type, refined from OCaml's type system.
Atkey~\cite{atkey2011} presents a type-based amortized resource analysis system adapted from Hofmann's work to imperative pointer-manipulating languages. They achieve this by implementing a separation logic extension to reason about resource analysis.
Serrano \emph{et al.}~\cite{serrano2014} introduced a general resource analysis for logic programs based on sized types, i.e., types that contain structural information and lower and upper bounds on the size of the terms. They achieved this by using an abstract Interpretation Framework.
In~\cite{simoes2012}, the authors develop a type-based proof system capable of automatically and statically analyzing heap allocations. This work is an extension of Hoffman's work for a lazy setting.
Vasconcelos \emph{et al.}~\cite{vasconcelos2015} defined a type system capable of predicting upper bounds on the cost of memory allocation for co-recursive definitions in a lazy functional language.
\section{Other Proof Systems}
Some other works on static estimation of resource bounds use other methodologies other than axiomatic semantics or type theory.
In a 2009 paper, the tool COSTA is presented~\cite{albert2009}. COSTA is a static analyzer for Java bytecode. It infers cost and termination information. It takes a cost model for a user-defined resource as input and obtains an upper bound on the execution time of this resource.
In~\cite{gulwani2009} they compute symbolic bounds on the number of statements a procedure executes in terms of its input. They use the notion of counter variables and an invariant generation tool to compute linear bounds on these counters. These bounds are then composed to generate a total non-linear bound.
Brockschmidt \emph{et al.}~\cite{brockschmidt2014} uses Polynomial Rank Functions (PRF) to compute time bounds. Then these bounds are used to infer size bounds on program variables. They consider small parts of the program in each step and incrementally improve the bound approximation.
\chapter{Desenho e Desenvolvimento}\label{chap:devel}
O referente ao ``Your work'' do capítulo~\ref{chap:back}.
No contexto deste template, este capítulo serve de exemplos de uso do \LaTeX e
de algumas regras de tipografia. Note-se que as regras sobre a formatação estão
disponíveis
\href{https://sigarra.up.pt/fcup/pt/conteudos_geral.ver?pct_pag_id=1011511&pct_parametros=pv_unidade=97&pct_grupo=33673&pct_grupo=33670&pct_grupo=33675&pct_grupo=33683&pct_grupo=36711#36711}{Estrutura
e Layout de teses}. Pode-se ver exemplos de tese (para a formatação e conteúdo)
no \href{https://repositorio-aberto.up.pt/handle/10216/9535}{repositório aberto
da UP}.
\section{Exemplo de código}
\begin{lstlisting}[numbers=none,language=java,caption={[CommandDaemonCallsItf]
{CommandDaemon} callback interfaces},label=lis:commandDCallsItfs,float=htb]
public interface CallBackCmdMeasurements { // comment
public abstract void newMeasure(MeasurementBasic measure, int reqId);
public abstract void newMeasuresAggSimp(MeasurementBasic[] measuresAggSimp, 'A string');
}
\end{lstlisting}
É possível referir uma lista de código, por exemplo o bloco de código~\ref{lis:commandDCallsItfs}.
\section{Acrónimos}
Deve-se acrescentar os acrónimos no ficheiro \texttt{acros.tex} e ordená-los
alfabeticamente nesse ficheiro.
Vamos usar o acrónimo \ac{TCP} que deve estar expandido, assim como no
capítulo~\ref{chap:tests}. Os acrónimos devem aparecer expandidos em cada
capítulo (o que está já configurado para esta dissertação).
Ao usar $\backslash$\texttt{acp} o \texttt{acronym} tentará colocar o plural
(acrescentando um s). É possível no ficheiro \texttt{acros.tex} colocar qual o
plural pretendido por exemplo \acp{USF}. Deve-se usar
$\backslash$\texttt{newacroplural\{USF\}\{Unidades de Saúde Familiar\}},
definindo o novo plural. Pode-se opcionalmente definir um novo ``plural'' para a
versão do acrónimo com $\backslash$\texttt{newacroplural[USFes]\{USF\}\{Unidades
de Saúde Familiar\}} (acrónimo no plural incorreto; apenas para exemplo).
Podem usar $\backslash$\texttt{acs} para apenas mostrar o acrónimo,
$\backslash$\texttt{acl} para mostrar a expansão. Ver mais na documentação do
pacote \texttt{acronym}.
\section{Figuras}
Podem ver a figura~\ref{fig:logoFCUP} muito bem. Notem que na lista de figuras
não aparece tudo o que está na legenda, mas apenas o que está entre~\texttt{[]}.
\begin{figure}[htb]
\centering
\includegraphics[scale=.4]{pics/fc_logo}
\caption[FCUP logo velho]{O logo da FCUP antigo}\label{fig:logoFCUP}
\end{figure}
As figuras devem usar o posicionamento [htb] para serem preferencialmente colocadas no local onde foram declaradas.
\subsection{Formato}
Devem colocar as figuras sempre que possível em pdf. Usem o programa (ou aplicação web) para exportar para pdf. Desse modo a qualidade das mesmas é sempre maior.
No caso em que o pdf gerado ocupe uma página A4 inteira (por exemplo nas
ferramentas do MS Office) podem usar o
\texttt{\href{https://www.ctan.org/pkg/pdfcrop?lang=en}{pdfcrop}} no pdf
gerado. Este comando está disponível na maior parte das distribuições de
\LaTeX.
\section{Referências a bibliografia}
Algumas referências para se ver a ordenação~\cite{yaacoub2012} (estão ordenadas pelo apelido do autor). Aqui outra ainda~\cite{etsitr102732}.
Temos aqui o Clausen~\cite{Clausen2003}. E para ver várias~\cite{yaacoub2012, etsitr102732, strunk2007elements}.
No ficheiro bib de referências deve-se colocar entre \{\} o que se quiser manter com maiúsculas no título. Exemplo: \texttt{title = \{Addressing the \{WLAN\} Problem\}} resulta em \texttt{Addressing the WLAN problem}, note-se o \textbf{p} de \emph{problem}.
Para saber os campos do bibtex relevantes para cada tipo de entrada pode-se utilizar um editor de bibtex como o \href{http://www.jabref.org}{JabRef}.
Os meses indicados no ficheiro \texttt{bib} poderão aparecer em inglês, caso se pretenda deve-se colocar o texto em português explicitamente no ficheiro \texttt{bib}.
\section{Capa}
Para a inclusão da formatação correta para a capa, pode-se criar um pdf usando o
pptx disponível
(\texttt{\href{https://sigarra.up.pt/fcup/pt/conteudos_service.conteudos_cont?pct_id=162960&pv_cod=19anaMgwFYkm}{CapaMSc\_PowerPoint.pptx}},
necessário fazer login no Sigarra) e depois incluí-lo com o comando seguinte
(comentado no tex original).
\begin{lstlisting}[numbers=none,language=TeX,caption={[Capas tese] Incluir capas oficiais},label=lis:capasTese,float=htb]
\includepdf[pages=1]{FrontPage-MSc.pdf}
\cleardoublepage
\includepdf[pages=2]{FrontPage-MSc.pdf}
\cleardoublepage
\end{lstlisting}
\section{Tabelas}
Exemplo de uma tabela mais complexa na tabela~\ref{tab:bsnvswsn}.
\begin{table}[htbp]
\caption[BSN vs WSN]{BSN versus WSN
(with input from Latré and Guang )}
\label{tab:bsnvswsn}
\centering
{
\footnotesize
\begin{tabularx}{0.98\textwidth}{|>{\columncolor{gray-cell}}c|X|X|}
\hline
\rowcolor{gray-cell}
& \centering \textbf{BSN} & \centering \textbf{WSN} \tabularnewline
\hline
\begin{sideways} \hspace{-11em} \textbf{Distribution} \end{sideways}
&
\begin{asparaenum}[\bfseries i)]
\item Existence of a \ac{BS};
\item \ac{BS} collects, maintains and processes the data;
\item Nodes will do minimal processing, sending all data to the \ac{BS};
\item Centralized system where \ac{BS} controls all nodes;
\item Node replacement is difficult in in-body sensor nodes;
\item Smaller number of nodes;
\item Nodes need to take biocompatibility, wearability into account.
\end{asparaenum}
&
\begin{asparaenum}[\bfseries i)]
\item A \ac{BS} may or not exist or there may be several \acp{BS} (e.g. mobile nodes
collect info, clustering);
\item As in \ac{BSN}, but also on-demand querying;
\item Nodes will do processing, aggregation to alleviate communication or
correlate results;
\item Distributed system, nodes decide cooperatively;
\item Node replacement is difficult due to location, scale, etc.;
\item (usually) Wide areas covered by large number of nodes.
\item Nodes may need to be environment friendly, indiscernible from surroundings.
\end{asparaenum}
\tabularnewline \hline
\textbf{Comm.}
&
\begin{asparaenum}[\bfseries i)]
\item One hop to \ac{BS};
\item Close range but attenuated by body;
\item Data rates heterogeneous.
\end{asparaenum}
&
\begin{asparaenum}[\bfseries i)]
\item Multi hop through network of sensor nodes;
\item Long(er) range;
\item Data rates homogeneous.
\end{asparaenum}
\tabularnewline \hline
\textbf{Data}
&
This is some text on this cell. The multirow package does not know the height of the cell and can not center the cell to the right. This is because of the X from tabularx.
&
\multirow{ 2}{*}{This is on two rows of the table}
\tabularnewline \cline{1-2}\noalign{\vskip.3pt
\textbf{Energy}
&
Some more text just to show something
&
\tabularnewline \hline
\end{tabularx}
}
\end{table}
Uma tabela de lado exemplificada em~\ref{tab:sensorExample}.
\begin{sidewaystable}
\centering
\begin{threeparttable}
\caption{Sensor examples}\label{tab:sensorExample}
{
\footnotesize
\begin{tabular}{|>{\centering}m{9.5em}|>{\centering}m{6.5em}|>{\columncolor{gray-cell}\centering}m{14em}|m{15em}|c|c|>{\centering}m{6.5em}|}
\hline
\rowcolor{gray-cell}
\textbf{Device} & \textbf{Availability}& \textbf{Sensed} & \centering \textbf{Technology} &
\textbf {Frequency} & \textbf{Data Rate} \tnote{$\mathbf{\propto}$} & \textbf{Energy} \tnote{$\mathbf{\Diamond}$}
\tabularnewline \hline
\multirow{6}{*}{\vspace{-3em}BioHarness BT} &
\multirow{6}{*}{\vspace{-3em}commercial}
& \acs{HR} & detection of \emph{QRS} complex in \acs{ECG} &
1~Hz & 8~bps &
\multirow{6}{5.5em}{\centering 21~h transmitting}
\tabularnewline \cline{3-6}
&
& Breathing rate & conductive elastic measurement of thorax excursion & 1~Hz & 7~bps &
\tabularnewline \cline{3-6}
&
& 3D Accelerometer & variability of a weight reference & 50~Hz \tnote{$\Join$}& 500~bps \tnote{$\Join$} &
\tabularnewline \cline{3-6}
&
& \acs{ECG} & potential difference across electrodes in body & 250~Hz &
2500~bps &
\tabularnewline \cline{3-6}
&
& Gyroscope & angular momentum & 1~Hz & 9~bps&
\tabularnewline \cline{3-6}
&
& Skin temperature & thermistor & 1~Hz & 9~bps&
\tabularnewline \hline
Actigraph GT3X+~\cite{actiGSpecs} &
commercial
& 3D Accelerometer & variability of a weight reference & 30-100~Hz \tnote{$\Join$} &
360-1200~bps \tnote{$\Join$} &
31 days
\tabularnewline \hline
Shimmer Research GSR Sensor &
commercial
& Galvanic skin response & measure skin conductivity & up to 15.9~Hz &
191~bps &
60 $\mu$A
\tabularnewline \hline
\multirow{2}{*}{Nonin Onyx II~\cite{noninOnyxSpecs}} &
\multirow{2}{*}{commercial}
& $\mathrm{O_2}$ saturation (full waveform) & \multirow{2}{15em}{measure light absorption by blood haemoglobin} &
75~Hz & 1200~bps &
\multirow{2}{6.5em}{\centering 2$\times$1.5~V AAA (600 tests)}
\tabularnewline \cline{3-3} \cline{5-6}
&
& $\mathrm{O_2}$ saturation (display format) & &
1~Hz & 8~bps &
\tabularnewline \hline
Medtronic iPro CGM & commercial
& continuous glucose meter & electrochemical detection of glucose through its reaction with glucose oxidase
& 0.1~Hz \tnote{$\lhd$} & 1~bps \tnote{$\rhd$}&
up to 72~h
\tabularnewline \hline
Brain sensor by Nurmikko et al. & in research \tnote{$\ddag$} & brain activity & microelectrode arrays detect neuron firing & \multicolumn{2}{c|}{40~k samples/sec$\times$16 channels \tnote{$\aleph$}} &
12~mW
\tabularnewline \hline
Molecular biomarkers by Ling et al. & research prototype \tnote{$\star$} & serum cardiac troponin I, creatinine kinase, myoglobin & magnetic properties of sensors vary according to presence of biomarker &
\multicolumn{3}{c|}{N/A}
\tabularnewline \hline
Electrochemical dopamine sensor by Chan et al.\tnote{$\otimes$} & research & dopamine & Interdigitated micro electrodes measured electrochemical reaction& 50~Hz & -- & 10~pA
\tabularnewline \hline
\end{tabular}
\begin{tablenotes}[online]
\item[$\propto $] data rates are based on the frequency and the accuracy in bits
stated in the references; they do not include time stamps or message headers.
\item[$\Diamond$] values taken from the references as available.
\item[$\Join$] for each of the 3 axis.
\item [$\lhd$] sensor data is collected every 10~s by collector.
\item [$\rhd$] our assumption of 9 bits per measure (up to 512~mg/dL (\acs{USA} glucose units)).
\item[$\ddag$] prototypes and clinical trials exist for devices with 16 channels.
\item[$\aleph$] more channels are needed (e.g. for decoding arm joint angles); 100 channels arrays are being developed.
\item[$\star$] tested on mice; measurement using \ac{MRI}, but test values were extracted from explanted sensors.
\item [$\otimes$] the developed sensor was not made of nano tubes, which according to the authors, led to poor sensitivity.
\end{tablenotes}
}
\end{threeparttable}
\end{sidewaystable}
\section{Unidades}
Deve-se usar o sistema ISO de unidades e respeitar a capitalização das unidades de acordo com a norma. Ver \emph{Quantities and units} em \href{http://www.cl.cam.ac.uk/~mgk25/publ-tips/#typography}{Typographic conventions} (podem e devem dar uma olhada nas outras partes da página).
\section{Sugestões na escrita e uso do tex}
\begin{itemize}
\item usar o \texttt{\textasciitilde{}} para ligar as referências, evita a
possível mudança de página: ex.: \emph{como o Brandão
refere\textasciitilde\textbackslash{}cite\{bran99\}}
\begin{itemize}
\item também se deve seguir a mesma regra para
\texttt{\textbackslash{}ref}, unidades de dados, nº de standard (ex.:
\emph{IEEE\textasciitilde802.11}).
\end{itemize}
\end{itemize}
\section{Sugestões na escrita}
\begin{itemize}
\item em inglês não usar a \emph{short form}. Em textos formais deve-se manter a \emph{long form}. Incorreto: ``don't use''. \textbf{Correto:} ``do not use''
\item em inglês (e também português) não usar a forma reflexiva ou indireta, preferir sempre a forma direta (que é mais assertiva e torna as frases menos complexas).
\begin{itemize}
\item incorreto:``Python was used to program''; Correto: ``we used Python''
\end{itemize}
\item Em inglês deve-se ter em atenção quando usar \emph{that} ou
\emph{which}. O \emph{that} é usado quando se tem cláusulas restritivas, e
o \emph{which} quando são não-restritivas. Pela mesma razão o \emph{that}
não tem uma vírgula antes, enquanto o \emph{which} deve ter. Exemplos:
\begin{itemize}
\item \emph{The performance of the machine \textbf{that} hosted the
database was poor.}
\begin{itemize}
\item as outras máquinas (que não têm a BD) podem não ter tido má
performance.
\end{itemize}
\item \emph{The professors, \textbf{which} usually are knowledgeable,
made a huge blunder.}
\begin{itemize}
\item caracteriza-se os professores genericamente, não se
especifica que foram os que fizeram a asneira.
\end{itemize}
\item Nos casos em que se usa o which, pode-se qualificar, mas não se
restringe, i.e., não é considerada uma cláusula que restrinja, ex.:
\emph{The machine, which was noisy, had a poor performance}. Não é
relevante a máquina ser ruidosa.
\item Podem ver \url{Ver http://grammarist.com/usage/which-that/}{mais
alguma informação}.
\end{itemize}
\end{itemize}
\section{Referências para outras fontes de informação}
\subsection{Cost Model}
One essential part of our system is our cost model. We need to define a way to evaluate the cost of a program. We start by defining a map to store the cost of atomic operations. For instance, the cost of a sum ($C_{+}$) might be defined as 1, and the cost of multiplication ($C_{*}$) as 3. Besides our dictionary, we implement a semantic to define the cost of evaluating arithmetic expressions, boolean expressions, and statements. Our operational semantics uses the cost model to compute the real execution cost.
\subsection{Oracle}
Programs with loops require additional information to prove correctness, termination, and cost bounds. This information is provided through an oracle. This oracle will request user input when necessary to complete the VCG algorithm. Since information such as invariants is needed multiple times throughout the algorithm, we need to store this information to be easily accessible. To achieve this, we assign a unique identifier to each \textit{while-loop} and create an oracle hashtable to store the oracle information for each loop. The code for our oracle can be seen in listing~\ref{lis:oracle}.
\begin{lstlisting}[language=ml,label=lis:oracle,caption={Oracle Implementation.}]
let oracle_hashtbl = Hashtbl.create 43
let parse_info name parser_function =
Printf.printf name ;
let input = read_line () in
Lexing.from_string input |> parser_function Lexer.token
let oracle () =
try
let inv = parse_info "Invariant: " Parser.annot_start in
let f = parse_info "Variant: " Parser.aexp_start in
let n = parse_info "Number of iterations: " Parser.aexp_start in
let t = parse_info "Cost Function of While: " Parser.lambda_start in
(inv, f, n, t)
with _ -> failwith "Oracle Error\n"
let get_oracle id =
if Hashtbl.mem oracle_hashtbl id then Hashtbl.find oracle_hashtbl id
else
let inv, f, n, t = oracle () in
Hashtbl.add oracle_hashtbl id (inv, f, n, t) ;
(inv, f, n, t)
\end{lstlisting}
\subsection{VCG}
Let us look at our VCG implementation. We implemented this algorithm as similar to the theoretical definitions as possible to guarantee soundness, as per our proofs. We presented three theoretical definitions of our logic, one classic for upper bounds, one which uses amortized analysis to further refine our upper-bound estimation, and one that proves the exact costs of a restricted version of our language.
Consider the implementation of the wp function in listing~\ref{lis:wp}.
Note how in the while case we start by calling the $get\_oracle$ function to get our loop information.
\begin{lstlisting}[language=ml,label=lis:wp,float=htb,caption={Weakest Precondition Implementation.}]
let rec wpc s phi =
match s with
| Skip ->
(phi, Var "Skip")
| Assign (x, a) ->
(subst phi x a, Sum (Var "Assign", time_aexp a))
| ArrAssign (x, a1, a2) ->
let t' = Sum (time_aexp a1, time_aexp a2) in
(subst_arr phi x a1 a2, Sum (Var "Assign", t'))
| Seq (s1, s2) ->
let phi', t2 = wpc s2 phi in
let phi, t1 = wpc s1 phi' in
(phi, Sum (t1, t2))
| If (b, s1, s2) ->
let wp1, t1 = wpc s1 phi in
let wp2, t2 = wpc s2 phi in
let v_b = annot_of_bexp b in
let tb = time_bexp b in
(AAnd (AImpl (v_b, wp1), AImpl (ANeg v_b, wp2)), Sum (Sum (t1, t2), tb))
| While (id, b, _) ->
let inv, f, n, t = get_oracle id in
let time =
Sum
( Mul (Sum (n, Cons 1), time_bexp b)
, Sigma ("k", 0, Sub (n, Cons 1), lambda_app t (Var "k")) )
in
(AAnd (inv, AGe (f, Cons 0)), time)
\end{lstlisting}
Similarly, we can see the VC function implementation in listing~\ref{lis:vc}. The \textit{while} case calls the $get\_oracle$ function again to retrieve the information about loop invariant, variant, and cost.
\begin{lstlisting}[language=ml,label=lis:vc,float=htb,caption={VC Implementation.}]
let rec vc s phi: annot list =
match s with
| Skip | Assign (_, _) | ArrDef (_, _) | ArrAssign (_, _, _) ->
[]
| Seq (s1, s2) ->
vc s1 (wp s2 phi) @ vc s2 phi
| If (_, s1, s2) ->
vc s1 phi @ vc s2 phi
| While (id, b, s') ->
let inv, f, n, t = get_oracle id in
let b = annot_of_bexp b in
let wp, t' = wpc s' (AAnd (inv, AGt (f, Var "k"))) false in
AForall ("k", AImpl (AAnd (inv, AAnd (b, AEq (f, Var "k"))), wp))
:: AImpl (AAnd (inv, AAnd (b, AEq (f, Var "k"))), AGe (lambda_app t (Var "k"), t'))
:: AImpl (AAnd (inv, ANeg b), phi)
:: AImpl (AAnd (inv, b), ALe (f, n))
:: vc s' (AAnd (inv, ALe (f, Var "k")))
\end{lstlisting}
Finally, we show the entry point function VCG in listing~\ref{lis:vcg}. This function calls both $wpc$ and $VC$ and combines all the VCs together.
\begin{lstlisting}[language=ml,label=lis:vcg,float=htb,caption={VCG Implementation.}]
let vcg pre s t pos =
let wp, ts = wpc s pos true in
AImpl (pre, ALe (ts, t)) :: AImpl (pre, wp) :: vc s pos
\end{lstlisting}
The VCG implementation for amortized costs differs from the previous one, only for the \textit{while} case. The oracle will also request new information in this version, an amortized cost and a potential function instead of a function of cost.
The exact cost version of our logic requires additional changes. We start by extending our language with for-loops and implementing the required adaptations to our interpreter. A restriction will be added to \textit{if} statements to ensure equal run time for both branches. Our VCG algorithm will now be extended to deal with \textit{for}-loops and the inequality operator in the cost assertion will now be replaced with an equality operator to prove the exact-time bound.
\subsection{Insertion Sort}
Our first example is of a classic sorting algorithm, insertion sort. The implementation is presented in listing~\ref{lis:insertion-sort}.
\begin{lstlisting}[language=ml,label=lis:insertion-sort,float=htb,caption={Insertion Sort Implementation with Annotations.}]
{ n > 0 }
i = 1;
while i < n do
key = x[i];
j = i - 1;
while x[j] > key and j >= 0 do
x[j + 1] = x[j];
j = j - 1
end;
x[j + 1] = key;
i = i + 1
end
{ forall k. (0<=k and k<n) => x[k] >= x[k-1] | 9*n*n + 27*n + 13 }
\end{lstlisting}
The precondition simply states that $n$ is a positive number. The postcondition says that our final array is in ascending order.
Since the implementation has two {\em while} loops, we will have two calls to the oracle.
For the external loop, we define the maximum number of iterations as $n$. The variant is the $i$ variable since it always increases until it reaches the value of $n$. The invariant states that the array is always ordered from the first position until the $(i-1)$-th position: $\forall k. (0< k \land k < i) \rightarrow (x[k-1] \le x[k])$.
The cost of the body of the external {\em while} is not the same for all iterations, since we have a nested while. We define this cost with the function: $t(i) = 9 \times i + 15$.
For the internal loop, it will iterate $i$ times. The variant is the increasing expression $i-j$.
The invariant is that all elements between positions $i$ and $j$ are greater than the key and that from the first position until $i-1$ the array is sorted, excluding the element on position j: $(\forall k. (j < k \land k < i) \rightarrow x[k] > key) \rightarrow \forall k1, k2. 0 \le k1 \land k1 \le k2 \land k2 < i \land \neg (k1 = j) \land \neg (k2 = j) \rightarrow x[k1] \le x[k2]$.
Note that one can define a cost function $t(k)$ that would allow us to derive an exact cost for the {\em while} rule, however, our logic does not allow proving that this bound is tight.
Given this information, our VCG generates the conditions needed to prove the termination, correctness, and cost bound of our program.
\subsection{Binary Search}
Our next example is another classic algorithm, Binary Search. Here we want to prove that, not only our implementation is correct and terminates, but also that the algorithm runs in logarithmic time in the size of the array. The specification can be seen in listing~\ref{lis:binary-search}
\begin{lstlisting}[language=ml,label=lis:binary-search,float=htb,caption={Binary Search Implementation with Annotations.}]
{(forall i. (0 <= i and i < n) => a[i] < a[i+1]) and (exists j. a[j] = v)}
l = 0;
u = n - 1;
while l <= u do
m = l + ((u - l) / 2);
if a[m] < v then
l = m + 1
else
if a[m] > v then
u = m - 1
else
result = m;
l = u + 1
end
end
end
{0 <= result and result < n and a[result] = v | 43 * log(n) + 10}
\end{lstlisting}
The precondition states that the array $a$ is sorted, and that value $v$ is in the array. The postcondition says that $result$ is a valid position in $a$ and it corresponds to the position of $v$ in $a$, $a[result]=v$.
To prove the execution time bound, we provide to the oracle the maximum number of iterations as being $log(n)$ and a constant value as the cost of each loop body iteration.
To prove termination we must also provide $n-u+l$ as a variant. Our invariant says that the position we are looking for is between $l$ and $u$, $0 \le l \land u < n \land (\forall i. (0 \le i \land i < n \land a[i] = v) \rightarrow l \le i \land i \le u)$ as invariant.
\subsection{Binary Counter}
In the binary counter algorithm, we represent a binary number as an array of zeros and ones. We start with an array with every value at zero, and with each iteration, we increase the number by one until we reach the desired value. Our implementation can be seen in listing~\ref{lis:binary-counter}.
\begin{lstlisting}[language=ml,label=lis:binary-counter,float=htb,caption={Binary Counter Implementation with Annotations.}]
{n >= 0 and size = log(n)}
i = 0;
while i < n do
j = 0;
while B[j] = 1 do
B[j] = 0;
j = j + 1
end;
B[j] = 1;
i = i + 1
end
{n = sum(i,0,log(n) - 1, B[i]*2^i) | 20*c*n + 3*n + 30}
\end{lstlisting}
Unlike in previous examples, we applied our amortized logic to prove the bound of the binary counter algorithm.
Our precondition says that $n$ is a positive value, size is $log(n)$ and that all elements in array $B$ from 0 to $size$ start at zero. Our postcondition says that at the end of the program, array $B$ is a binary representation of decimal number $n$.
In order to prove this assertion, we must provide the oracle with the amortized cost ($2c$) and a potential function denoting the number of ones in the array at each iteration.
We must also specify the invariant $i = sum(k,0,size, B[k] * 2^k)$, the variant $i$, and the maximum number of iterations $size$, to prove correctness and termination respectively.
If we were to use a worst-case analysis on this implementation, we would get that this algorithm is $\mathcal{O}(n\ log n)$, meaning we would flip every bit ($log n$) a total of $n$ times. However, this is not the case. While the first bit ($B[0]$) does flip every iteration, the second bit($B[1]$) flips every other iteration, the third ($B[2]$) every 4th iteration, and so on. We can see a pattern where each bit $B[i]$ flips every $2^i$th iteration. This will mean that, at most, we have $2n$ bit flips, meaning our algorithm is actually $\mathcal{O}(n)$, as we successfully proved with our algorithm.
we can define a potential function as:
\subsection{Range Filter}
In our last example, we implement a simple filter where, given an array ($a$) and a range [l..u], we use an auxiliary array ($b$) to filter if the elements in $a$ are within the range, listing~\ref{lis:range-filter}.
\begin{lstlisting}[language=ml,label=lis:range-filter,float=htb,caption={Range Filter Implementation with Annotations.}]
{ 0 <= l and l < u and n >= 0}
j = 0;
for i=0 to n do
if (l <= a[i] and a[i] <= u)
then
b[j] = a[i];
j = j + 1
else
b[j] = b[j];
j = j + 0
end
end
{ forall i. (0<=i and i<n) => (l <= a[i] and a[i] <= u => exists j. B[j] = a[i] )
and (l > a[i] and a[i] > u => not (exists j. b[j] = a[i]) ) | 13*n + 10 }
\end{lstlisting}
Our pre-condition states that $l$ and $n$ are positive values, and $u$ is greater than $l$.
Our postcondition says that for every $i$ element in $a$ in the range [l..u], $i$ will also be in $B$. And for every $i$ element in $a$ not in the range [l..u], $i$ will not be in $B$.
We provide the invariant $\forall k. (0 \le k \land k<i) \rightarrow (l \le a[k] \land a[k] \le u \rightarrow \exists k. b[j] = a[i] ) \land (l > a[k] \land a[k] > u \rightarrow \neg (\exists j. b[k] = a[i]))$ in order to prove correctness.
Using our VC generator and EasyCrypt we prove that not only is this algorithm correct, the cost we provide of $13n + 10$ is the exact cost of this program. This result allows us to conclude the time it takes to run depends only on the size of the array, and not on its values.
\begin{table}[htpb]
\centering
\begin{tabular}{|c|c|c|}
\hline
Algorithm & Logic & Number of VCs Generated \\
\hline
Insertion Sort & Classic & 10 \\
Binary Search & Classic & 6 \\
Binary Counter & Amortized & 17 \\
Range Filter & Exact & 5\\
\hline
\end{tabular}
\caption{Logic Used and Number of VCs generated by each example.}
\label{tab:examples-info}
\end{table}
\subsection*{Soundness}
We need to prove theorem~\ref{theor:soundness-vcg}, which states that the VCG algorithm is sound if the VC generated implies the Hoare triple we wish to prove,
$$\models VCG(\{P\}Q\{R\})\ \Rightarrow\ \vdash \{P\}Q\{R\}$$
\begin{proof}
We prove $\Rightarrow$ by induction on the structure of Q and $\Leftarrow$ by induction in the derivation of $\vdash \{P\}Q\{R\}$.
\textit{Case if:}
Induction Hypothesis:
$$ \models VCG(\{P\} S_1 \{Q | t_1 \} \rightarrow \vdash \{P\} S_1 \{Q|t_1\})$$
$$ \models VCG(\{P\} S_2 \{Q | t_2 \} \rightarrow \vdash \{P\} S_2 \{Q|t_2\})$$
Let us consider
\begin{itemize}
\item $wp_1, t_1 = wpc(S_1,Q)$
\item $wp_2, t_2 = wpc(S_2,Q)$
\item $wpc(\mathrm{if}\ b\ \mathrm{then}\ S_1\ \mathrm{else}\ S_2, Q) = (\mathcal{B} \llbracket b \rrbracket \rightarrow wp_1 \land \neg \mathcal{B} \llbracket b \rrbracket \rightarrow wp_2 \land t_1=t_2, t_1+\mathcal{TB}\llbracket b \rrbracket)$
\end{itemize}
\begin{align*}
VCG(\{P\}\mathrm{if}\ b\ \mathrm{then}\ S_1\ \mathrm{else} S_2\{Q|T\}) = &\{P \rightarrow (b \rightarrow wp_1 \land \neg b \rightarrow wp_2 \land t_1=t_2)\} \cup \\
&\{T = t_1 + \mathcal{TB}\llbracket b \rrbracket\} \cup \\
&VC(\mathrm{if}\ b\ \mathrm{then}\ S_1\ \mathrm{else} S_2, Q)
\end{align*}
Where $VC(\mathrm{if}\ b\ \mathrm{then}\ S_1\ \mathrm{else} S_2, Q) = VC(S_1, Q) \cup VC(S_2,Q)$.
Assuming $\models VCG(\{P\}\mathrm{if}\ b\ \mathrm{then}\ S_1\ \mathrm{else}\ S_2\{Q|T\})$.
\begin{itemize}
\item Since $P \land \mathcal{B} \llbracket b \rrbracket \rightarrow wp_1$, $t_1 = t_1$, and $VC(S_1, Q)$, $$\models VCG(\{P\}S_1\{Q|t_1\})$$
\item Since $P \land \neg \mathcal{B} \llbracket b \rrbracket \rightarrow wp_2$, $t_2 = t_2$, and $VC(S_2, Q)$, $$\models VCG(\{P\}S_2\{Q|t_2\})$$
\end{itemize}
From our Induction Hypothesis, we have $\vdash \{P \land \mathcal{B} \llbracket b \rrbracket\} S_1 \{Q|t_1\}$, and $\vdash \{P \land \neg \mathcal{B} \llbracket b \rrbracket\} S_2 \{Q|t_2\}$.
By the \textit{if rule}, we get
$$\{P\}\mathrm{if}\ b\ \mathrm{then}\ S_1\ \mathrm{else}\ S_2 \{Q|t_1 + \mathcal{TB} \llbracket b \rrbracket\}$$
Since $T = t_1 + \mathcal{TB} \llbracket b \rrbracket$, by the \textit{weak rule}
$$\vdash \{P\}\mathrm{if}\ b\ \mathrm{then}\ S_1\ \mathrm{else}\ S_2\{Q|T\}$$
\textit{Case for:}
Induction Hypothesis:
$$ \models VCG(\{P\} S \{Q | T \} \rightarrow\ \vdash \{P\} S \{Q|T\})$$
Let us consider
$$wpc(\mathrm{for}\ i = a\ \mathrm{to}\ b\ \mathrm{do}\ S, Q) = (I[a/i], (b-a) \times (\mathcal{TA} \llbracket a \rrbracket + C_{ASSIGN\_V} + t)+ (b-a+1) \times \mathcal{TB}\llbracket a \le b \rrbracket)$$
where $wp_S, t_S = wpc(S, I[i+1/i])$
\begin{align}
VC(\mathrm{for}\ i = a\ \mathrm{to}\ b\ \mathrm{do}\ S,Q) =
&\{I[b/i] \rightarrow Q \}\ \cup \label{chap6:eq-1}\\
&\{(I \land \neg(a < b)) \rightarrow Q\}\ \cup \label{chap6:eq-2}\\
&\{(I \land a \le i < b) \rightarrow wp\}\ \cup \label{chap6:eq-3}\\
&VC(S,I[i+1/i]) \label{chap6:eq-4}
\end{align}
\begin{align}
VCG(\{P\}\mathrm{for}\ i = a\ &\mathrm{to}\ b\ \mathrm{do}\ S\{Q|T\}) = \{P \rightarrow I[a/i]\}\ \cup \label{chap6:eq-5}\\
&\{T = (b-a) \times t+ (b-a+1) \times \mathcal{TB}\llbracket a \le b \rrbracket\}\ \cup \label{chap6:eq-6}\\
&VC(\mathrm{for}\ i = a\ \mathrm{to}\ b\ \mathrm{do}\ S, Q) \label{chap6:eq-7}
\end{align}
Assuming $\models VCG(\{P\}\mathrm{for}\ i = a\ \mathrm{to}\ b\ \mathrm{do}\ S\{Q|T\})$.
Given \ref{chap6:eq-3}, , $t_S = t_S$, and VC(S,I[i+1/i]) then $\models VCG[\{I\land a \le i < b\}S\{I[i+1/i]|t\}]$
From our Induction Hypothesis
$$\vdash \{I\land a \le i < b\}S\{I[i+1/i]|t\}$$
By the \textit{for} rule
$$\vdash \{I[a/i]\}\mathrm{for}\ i = a\ \mathrm{to}\ b\ \mathrm{do}\ S\{I[b/i] | (b-a) \times (\mathcal{TA} \llbracket a \rrbracket + C_{ASSIGN\_V} + t)+ (b-a+1) \times \mathcal{TB}\llbracket a \le b \rrbracket\}$$
Given \ref{chap6:eq-5}, \ref{chap6:eq-1} and \ref{chap6:eq-6} we get
$$\vdash \{P\}\mathrm{for}\ i = a\ \mathrm{to}\ b\ \mathrm{do}\ S\{Q|T\}$$
\end{proof}
\subsection*{Example: Range Filter}
We now apply the VCG algorithm to the range filter example~\ref{fig:range-filter}.
We will use the same notation as in section~\ref{chap6:sec:axiomatic} and refer to our precondition as $P$, our program as $S$, our postcondition as $Q$, and our tight cost as $T$. We will also refer to the \textit{for} loop body statement as $S_{if}$. The algorithm starts with a call to the VCG function.
\begin{align}
VCG(\{P\}S\{Q|T\}) = &\{P \rightarrow wp)\ \cup \label{chap6:eq-8}\\
&\{T = t\}\ \cup \label{chap6:eq-9}\\
&VC(S,Q) \nonumber
\end{align}
Where
$$wp, t = wpc(S,Q) = (I[0/i][0/j] , n \times t_{if} + (n+1) \times \mathcal{TB}\llbracket 0 < n \rrbracket + 2)$$
$$wp_{if}, t_{if} = wpc(S_{if}, I[i+1/i])$$
Then the VCG function calls the VC function for $S$. Since only for loops generate extra VCs, we will omit other calls to the VC function for simplicity.
\begin{align}
VC(S,Q) =
&\{(I \land 0 \le i < n) \rightarrow wp_{if}\}\ \cup \label{chap6:eq-10}\\
&\{(I[n+1/i]) \rightarrow Q\}\ \cup \label{chap6:eq-11}\\
&\{(I[0/i] \land n < 0) \rightarrow Q\}\ \cup \label{chap6:eq-12}\\
&VC(S_f, I[i+1/i]) \nonumber
\end{align}
To prove our triple, we now simply need to prove all of the VCs generated by the algorithm (\ref{chap6:eq-8} to \ref{chap6:eq-12}), this can easily be done for all the conditions manually, or with the assistance of a theorem prover.
As would be expected, proving a Hoare triple by applying the VCG algorithm is simpler and more mechanic than proving it directly by applying our rules and deriving the inference tree.
\subsection*{Soundness}
We need to ensure that our Hoare logic is sound with respect to our operational semantic. For this version, our soundness theorem will be slightly different than the one we have previously presented.
\begin{theorem}[Soundness]
We have that $\models \{ P \} S \{ Q | t \}$ if and only if, forall state $\sigma$ such that $\sigma \models P$ and $\langle S, \sigma \rangle \rightarrow^{t'} \sigma'$, we have $\sigma' \models Q$ and $\mathcal{A}\llbracket t \rrbracket\sigma = t'.$
\end{theorem}
Even though our theorem changed, the proof for \textit{skip}, \textit{assign}, \textit{array} and \textit{seq} will look exactly the same since the upper bound calculated by our previous logic was already identical to the real cost of execution. Therefore we will only show the proof for \textit{if} and \textit{for}.
\begin{proof}
Case if:
Assume $\models \{ P \land b \} S_1 \{ Q | t \}$ and $\models \{P \land \neg b\} S_2 \{ Q | t\}$.
Suppose $\sigma \models P$.
If $\sigma \models b$, then $\sigma \models P \land b$ so, assuming $\langle S_1, \sigma \rangle \rightarrow^{t_1} \sigma_1$, we have that $\sigma_1 \models Q$
If $\sigma \models \neg b$, then $\sigma \models P \land \neg b$ e so, assuming $\langle S_2, \sigma \rangle \rightarrow^{t_2} \sigma_2$, we have that $\sigma_2 \models Q$.
Given our assumptions we know that $t_1 = t_2 = t$. We then have that the exact cost for the \textit{if} statement is $t + \mathcal{T}\mathcal{B}\llbracket b \rrbracket$. The rule for \textit{if} is sound.
$$\models \{\ P\ \}\ \mathrm{if}\ e\ \mathrm{then}\ S_1\ \mathrm{else}\ S_2\ \{\ t_1 + \mathcal{T} \mathcal{B} \llbracket b \rrbracket\ |\ Q\ \}$$
Case for:
Assume $\models \{\ P \land (a \le i) \land i < b\ \}\ S\ \{\ P[i+1/i]\ | t_S\}$ \\
Suppose $\sigma \models P[a/i]$ and $\langle \mathrm{for}\ i=a\ \mathrm{to}\ b\ \mathrm{do}\ S, \sigma\rangle \rightarrow^{t} \sigma_1$.
If $\sigma \models \neg (a \le b)$ then $\sigma = \sigma_1$. From our lemma~\ref{lemma:subst} we get that $\sigma \models P$. In this case, the execution time is $\mathcal{T}\mathcal{B} \llbracket a \le b \rrbracket$. When $a > b$, $t = \mathcal{T}\mathcal{B} \llbracket a < b \rrbracket$, then the axiom is sound.
If $\sigma \models a< b$ then $\sigma \models P[a/i] \land a < b$.
Let us consider a state $\sigma_2$ such that $\langle i=a, \sigma \rangle \rightarrow^{t_1} \sigma_2$. By the \textit{assign rule}, we know $\sigma_2 \models P$ and $t_1 = \mathcal{TA} \llbracket a \rrbracket + C_{ASSIGN\_V}$. We also have that $\sigma_2 \models a \le i < b$, therefore if we execute $S$ from state $\sigma_2$, we will get a state $\sigma_3$, $\langle S, \sigma_2 \rangle \rightarrow^{t_2} \sigma_3$, such that $\sigma_3 \models P[i+1/i]$, and $t_S = t_2$. By our induction hypothesis: $\langle \mathrm{for}\ i=a+1\ \mathrm{to}\ b\ \mathrm{do}\ S, \sigma_3 \rangle \rightarrow^{t_3} \sigma_1$, where $\sigma_1 \models P[b/i]$, and $t_3 = (b - a) \times \mathcal{TB} \llbracket a < b \rrbracket + (b-a-1) \times (t_S + \mathcal{TA}\llbracket a \rrbracket + C_{ASSIGN\_V)}$.
The real cost of executing statement $for$ from state $\sigma$ is $t = t_1 + t_2 + t_3 + \mathcal{TB} \llbracket a < b \rrbracket$. Knowing $t_1 = \mathcal{TA} \llbracket a \rrbracket + C_{ASSIGN\_V}$, $t_2 = t_S$, and $t_3 = (b - a) \times \mathcal{TB} \llbracket a < b \rrbracket + (b-a-1) \times (t_S + \mathcal{TA}\llbracket a \rrbracket + C_{ASSIGN\_V)}$, gives us
$$t = \mathcal{TA} \llbracket a \rrbracket + C_{ASSIGN\_V} + t_S + (b - a) \times \mathcal{TB} \llbracket a < b \rrbracket + (b-a-1) \times (t_S + \mathcal{TA}\llbracket a \rrbracket + C_{ASSIGN\_V)} + \mathcal{TB} \llbracket a<b \rrbracket$$
$$t = (b - a + 1) \times \mathcal{TB} \llbracket a < b \rrbracket + (b-a) \times (t_S + \mathcal{TA}\llbracket a \rrbracket + C_{ASSIGN\_V)} + \mathcal{TB} \llbracket a<b \rrbracket$$
Therefore the rule for \textit{for} is sound.
\end{proof}
\subsection*{Example: Range Filter}
To illustrate our logic, we will apply our rules to the range filter algorithm, as presented in figure~\ref{fig:range-filter}.
This algorithm consists of a simple filter where, given an array ($a$) and a range [l..u], we use an auxiliary array ($b$) to filter the elements in $a$ that are within the range.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{figures/range-filter.pdf}
\caption{Array filtering algorithm with annotations for exact cost.}
\label{fig:range-filter}
\end{figure}
We provide the invariant $I \equiv \forall k. (0 \le k \land k<i) \rightarrow (l \le a[k] \land a[k] \le u \rightarrow \exists k. b[j] = a[i] ) \land (l > a[k] \land a[k] > u \rightarrow \neg (\exists j. b[k] = a[i]))$ in order to prove correctness.
Let us refer to the \textit{if} statement as $S_{if}$ and to the \textit{for} statement as $S_{for}$.
By the \textit{assign rule} we have
$$\{I[i+1/i][j+1/j]\}j=j+1\{I[i+1/i]|\mathcal{TA} \llbracket j+1 \rrbracket + 1\}$$
By the \textit{assign rule} we also have
$$\{I[i+1/i][j+1/j][a[i]/b[j]]\}b[j]=a[i]\{I[i+1/i][j+1/j]|\mathcal{TA} \llbracket i \rrbracket + \mathcal{TA} \llbracket j \rrbracket + 1\}$$
By the \textit{seq rule} we get
$$\{I[i+1/i][j+1/j][a[i]/b[j]]\}b[j]=a[i]; j=j+1\{I[i+1/i]|4 + 3 + 1\}$$
By the \textit{assign rule} we have
$$\{I[i+1/i][j+0/j]\}j=j+0\{I[i+1/i]|\mathcal{TA} \llbracket j+0 \rrbracket + 1\}$$
By the \textit{assign rule} we also have
$$\{I[i+1/i][j+0/j][b[j]/b[j]]\}b[j]=b[j]\{I[i+1/i][j+0/j]|\mathcal{TA} \llbracket j \rrbracket + \mathcal{TA} \llbracket j \rrbracket + 1\}$$
By the \textit{seq rule} we get
$$\{I[i+1/i][j+0/j][b[j]/b[j]]\}b[j]=b[j]; j=j+0\{I[i+1/i]|4 + 3 + 1\}$$
Since $I[0/i] \land l \le a[i] \le u \rightarrow I[i+1/i][j+1/j][a[i]/b[j]]$, $I[0/i] \land \neg (l \le a[i] \le u) \rightarrow I[i+1/i][j+0/j][b[j]/b[j]]$, then by the \textit{if rule}
$$\{I[0/i]\} S_{if} \{ I[i+1/i] | 8 + \mathcal{TB} \llbracket l \le a[i] \land a[i] \le u \rrbracket \}$$
By the \textit{for rule} we get
$$\{I[0/i]\} S_{for} \{I[b/i] | n \times 17 + (n+1) \times \mathcal{TB} \llbracket i < n \rrbracket\}$$
By the \textit{assign rule}
$$\{I[0/i][0/j]\}j = 0\{\ I[b/i] | 2\}$$
By the \textit{seq rule}
$$\{I[0/i][0/j]\} S \{ I[i+1/i] | 8 + n \times 9 + (n+1) \times \mathcal{TB} \llbracket i < n \rrbracket + 2\}$$
Since $P \rightarrow I[0/i][0/j]$, $I[i+1/i] \rightarrow Q$, and $T \ge n \times 9 + (n+1) \times \mathcal{TB} \llbracket i < n \rrbracket + 2$, then by the \textit{weak rule}
$$\vdash \{P\} S \{Q | T\}$$
\section{Annotated While Language}\label{chap4:sec:language}
\input{sections/chapter 4/4.1-language.tex}
\section{Operational Semantics}\label{chap4:sec:formal-operational}
\input{sections/chapter 4/4.2-operational.tex}
\section{Axiomatic Semantics}\label{chap4:sec:axiomatic}
\input{sections/chapter 4/4.3-axiomatic.tex}
\section{Verification Conditions Generation}\label{chap4:sec:vcg}
\input{sections/chapter 4/4.4-vcg.tex}
\section{Goals and Contributions}\label{sec:intro-obj}
Our main goal is to define an axiomatic semantics-based proof system for reasoning about execution time bounds for a core imperative programming language. Such a system would be useful not only to understand the resource necessities of a program but also it could be applied to cryptographic implementations to prove the independence of resource usage from certain program variables.
This high-level goal translates into the following concrete objectives:
\begin{enumerate}
\item Study axiomatic systems. This will further our knowledge in the field and allow us to understand how to define our logic for resource analysis.
\item Study amortized analysis so we can understand how to apply amortization to a proof system to refine cost-bound estimation of while loops.
\item Analyze the state-of-the-art to understand what has already been developed to analyze resource consumption and the main limitations found.
\item Develop a sound logic capable of verifying correction, terminations, and bounds on resource consumption using a simple imperative programming language.
\item Create a tool based on our logic, capable of verifying time bounds, correction, and terminations, for example-problems.
\item Apply this logic to analyze the time complexity of classic algorithms.
\end{enumerate}
The main contribution of this paper is a proof system that can verify resource assumptions in three different scenarios:
\begin{enumerate}
\item Upper bounds on the required execution time. This is mostly an adaptation of previous work in \cite{barbosa21}.
\item Amortized costs denoting less pessimistic bounds on the execution time.
\item Exact costs for a fragment of the initial language with bounded recursion and a constrained form of conditional statements.
\end{enumerate}
The two last scenarios are a novel contribution of our system, and we treat them in a unified way to enable their integrated use.
Assertions on program behavior that establish upper bounds on execution time may be useful for general programming, where one wants to prove safety conditions concerning the worse case program complexity.
As in prior approaches, the tightness of the bound is not captured by the logic, and there is often a trade-off between the tightness of the proved bound and the required proof effort.
Proofs that leverage amortized costs may be used when trivially composing worst-case run-time bounds results in overly pessimistic analyses. This is particularly useful for algorithms where some components imply a significant cost in resources, whereas other components are not as costly. With amortized costs, we may prove assertions about the aggregate use of costly and less costly operations over the whole algorithm execution.
Finally, the third class of assertions denoting exact costs are useful in scenarios where the approximation of execution time is not enough to guarantee safety, as it happens for critical systems and real-time programming.
Moreover, proving that the exact execution time of a program is an expression that does not depend on confidential data provides a direct way to prove the absence of timing leakage, which is relevant in cryptographic implementations.
We must restrict the programming language to guarantee the ability to prove exact costs. Thus, in this third scenario, programs have bound recursion, and conditional statement branches have to have the same cost.
Before defining our proof system, we defined an operational semantics capable of computing the execution time for expressions and statements during program execution. This cost-aware operational semantics is another contribution of our work, and it is used to prove the soundness of our inference system.
A third contribution of this work, which shows the practicality of our proof system, is an implementation in OCaml of the different modules needed to apply it to example programs. We then present several application examples motivated by real-time and cryptographic software.
\section{Document Structure}\label{sec:intro-structure}
This document is organized as follows:
\begin{itemize}
\item The first chapter - \textbf{Introduction} - gives a context of our work in the field, the motivation for this project, our main goals, our contributions, and how the document is organized.
\item The second chapter - \textbf{Background} - elaborates on the theoretical results used in the basis of our work and needed to understand our definitions and results.
\item The third chapter - \textbf{Related Work} - presents an analysis of the literature on static resource analysis, from type-based systems to axiomatic semantics systems.
\item The fourth chapter - \textbf{Cost Aware Program Logic} - presents our language definition, our original logic for upper bound estimation, the respective VCG, and some illustrative examples.
\item The fifth chapter - \textbf{Amortized Costs} - briefly introduces the field of amortized analysis and presents an extension to the logic and VCG from chapter 4, with the use of amortized analysis to improve the upper-bound estimation.
\item The sixth chapter - \textbf{Exact Logic} - presents a variation to our language and an extension to our logic that allows for the derivation of the exact cost of a program.
\item The seventh chapter - \textbf{Implementation and Experimental Results} - shows the architecture of the tool developed, as well as some implementation details and practical results.
\item The eighth chapter - \textbf{Conclusion and Future Work} - reflects on the main conclusions from our research and developed work and presents some goals to further extend and improve our project.
\end{itemize}
\subsection{Semantic of Expressions}
In order to evaluate arithmetic expressions, we define a semantic function $\mathcal{A}$ which will receive two arguments, an arithmetic expression and a state.
$$\mathcal{A}: aexp \rightarrow state \rightarrow int$$
Writing $\mathcal{A} \llbracket a \rrbracket \sigma$ will return the value of evaluating expression $a$ in state $\sigma$. The function is defined in figure~\ref{fig:arith}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.85\linewidth]{figures/arith.pdf}
\caption{Semantics of arithmetic expressions.}
\label{fig:arith}
\end{figure}
Similarly, we define a semantic function $\mathcal{B}$ that, given a state, will convert a boolean expression to truth values.
$$\mathcal{B}: bexp \rightarrow state \rightarrow bool$$
In figure~\ref{fig:bool} we define $\mathcal{B}$ using the previous definition of $\mathcal{A}$.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{figures/bool.pdf}
\caption{Semantics of boolean expressions.}
\label{fig:bool}
\end{figure}
\subsection{Cost of expressions}
Our semantics will not only evaluate the meaning of a program but also compute the exact cost of executing it. To achieve this, we start by defining semantics for the cost of evaluating arithmetic and boolean expressions, as shown in Figure~\ref{fig:cost}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{figures/time.pdf}
\caption{Cost of Arithmetic and Boolean Expressions.}
\label{fig:cost}
\end{figure}
To evaluate the cost of an expression, we must establish the cost of atomic operations in our language, such as reading from memory or performing basic arithmetic (addition, multiplication, etc) and logic operations (disjunction, negation, etc).
For example, $C_{CST}$ corresponds to the cost of evaluating a constant, and $C_{VAR}$ is the cost of evaluating a variable. The cost of evaluating a multiplication $\mathcal{T}\mathcal{A}\llbracket a_1 * a_2\rrbracket$ is defined as a sum of the cost of evaluating each of the arithmetic expressions, $a_1$ and $a_2$, plus the cost of the multiplication operation $C_{*}$.
The cost of evaluating a sum $\sum_{i=b}^{e}a$ is the cost of evaluating $a$ multiplied by the number of times we evaluate it $e - b$.
These rules are simultaneously used by our operational semantics when executing our program, and by our axiomatic semantics, when proving time restrictions statically using our VCG.
For simplicity in the rest of the document, we will consider all the atomic costs as $1$, except in logic definitions and soundness proofs.
\subsection{Free Variables and Substitution}
Before we can define the semantics of a statement we need to first look at two important definitions: Free Variables, and Substitution.
\begin{definition}[Free Variables]
The Free Variables of an arithmetic expression can be defined as the set of variables occurring in an expression that are not bounded by any variable binding operator, such as $\sum$.\\
If we define this as a function $FV: a \rightarrow \{x\}$ we get the definition in figure~\ref{fig:free}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{figures/freevar.pdf}
\caption{Free Variables of Arithmetic Expressions.}
\label{fig:free}
\end{figure}
\end{definition}
For example, the free variables of $\sum_{x=0}^{10}x + 2^y + z$ are $\{y,z\}$.
\begin{definition}[Substitution]
A substitution consists of replacing every occurrence of a variable ($x_1$) in an arithmetic expression ($a$) with another arithmetic expression ($a_0$). This is written as $a[a_0/x_1]$ and the substitutions rules are as described in figure~\ref{fig:subst}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{figures/substitution.pdf}
\caption{Substitution algorithm for arithmetic expressions.}
\label{fig:subst}
\end{figure}
\end{definition}
As an example, let us look at the following substitution
$$(x+4y+3)[z+4/y] = x + 4(z+4) + 3$$
Substitutions might also be applied to states. For example, $\sigma[n/x]$ represents a state that is identical to $\sigma$, with the exception that $x$ takes the value of $n$. Note that, since a state is a mapping from variable to an integer value, $n$ has to always be an integer and never an expression.
\subsection{Evaluating statements}
In order to prove assertions on the execution time of a program, we need to define a cost-aware semantics. We define a natural operational semantics, where transitions are of the form $$\langle S, \sigma \rangle \rightarrow^t \sigma'$$
meaning that after executing statement $S$ from state $\sigma$ the final state is $\sigma'$ and the execution time was $t$.
The cost-instrumented operational semantics is defined in Figure~\ref{fig:operational}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{figures/operational.pdf}
\caption{Operational Semantics.}
\label{fig:operational}
\end{figure}
The skip axiom $[skip]$ says that \textit{skip} does not change the state of the program, and we associate a constant cost for its execution of $C_{SKIP}$.
The assignment axiom $[assign]$ says that executing the assignment $x = a$ in state $\sigma$ will lead to a state $\sigma[\mathcal{A}\llbracket a \rrbracket \sigma / x]$, which means state $\sigma$ where x takes the value of $\mathcal{A}\llbracket a \rrbracket \sigma$. The cost of this expression is defined as the cost of evaluating $a$, $\mathcal{T}\mathcal{A} \llbracket a \rrbracket$, plus the constant cost of an assignment, $C_{ASSIGN\_V}$.
Similarly, the array assignment axiom $[array]$ says that executing $x[a_1] = a_2$ from state $\sigma$ will lead to state $\sigma[\mathcal{A} \llbracket a_2 \rrbracket \sigma / x[\mathcal{A} \llbracket a_2 \rrbracket s\sigma]$, which is a similar state to $\sigma$, except $x[\mathcal{A} \llbracket a_1 \rrbracket \sigma]$ takes the value of $\mathcal{A} \llbracket a_2 \rrbracket \sigma$. This execution cost will be the cost of evaluating $a_1$, $\mathcal{T}\mathcal{A} \llbracket a_1 \rrbracket$, plus the cost of evaluating $a_2$, $\mathcal{T}\mathcal{A} \llbracket a_2 \rrbracket$, plus the constant cost of assigning a value to a position in an array, $C_{ASSIGN\_A}$.
The sequence rule $[seq]$ says that if we want to execute a sequence $S_1;S_2$ from state $\sigma$ we will first execute statement $S_1$ from state $\sigma$, this execution will lead to a certain state $\sigma'$ in $t_1$ time. If we execute $S_2$ from this state $\sigma'$ we will reach a final state $\sigma''$ in $t_2$ time. Therefore executing $S_1;S_2$ from state $\sigma$ will lead to state $\sigma''$ in $t_1+t_2$ time.
We have two conditional rules, $[if^{true}]$ and $[if^{false}]$. In order to decide which of the rules to apply, we must first evaluate $\mathcal{B}\llbracket b \rrbracket \sigma$. If this evaluates to true we apply rule $[if^{true}]$, which means we simply execute statement $S_1$, otherwise we apple rule $[if^{false}]$, which means we execute statement $S_2$. For both rules, the cost of executing the if statement is the cost of executing $S_1$ when true or $S_2$ when false, plus the cost of evaluating $b$, $\mathcal{T}\mathcal{B}\llbracket b \rrbracket$.
We have one rule and one axiom for while, $[while^{true}]$ and $[while^{false}]$.
If $\mathcal{B} \llbracket b \rrbracket \sigma$ is false, we apply the axiom $[while^{false}]$, that says we will remain in the same state $\sigma$ and the cost is simply the cost of the evaluation of b, $\mathcal{T}\mathcal{B} \llbracket b \rrbracket$.
If $\mathcal{B} \llbracket b \rrbracket \sigma$ is true, we apply rule $[while^{true}]$, which means we will execute the loop body, $S$, once from state $\sigma$ and this will lead to a state $\sigma''$. Finally, we execute the while loop again, but this time from state $\sigma''$. The cost of the while loop, in this case, is the cost of evaluating b, plus the cost of executing the body, plus the cost of executing the while loop from state $\sigma''$.
\subsection*{Example}
Let us consider the following program that swaps the values of x and y:
\begin{lstlisting}[language=ml]
z = x; x = y; y = z
\end{lstlisting}
Let the initial state $\sigma_0$ be a state such that $\sigma_0\ x = 3$, $\sigma_0\ y = 10$ and $\sigma_0\ z = 0$.
Then the derivation tree of this program will look like
\begin{prooftree}
\infer0{\langle z=x, \sigma_0 \rangle \rightarrow^{t_1} \sigma_1}
\infer0{\langle x=y, \sigma_1 \rangle \rightarrow^{t_2} \sigma_2}
\infer2{\langle z=x; x=y, \sigma \rangle \rightarrow^{t_1 + t_2} \sigma_2}
\infer0{\langle y=z , \sigma_2 \rangle \rightarrow^{t_3} \sigma_3}
\infer2{\langle z=x; x=y; y=z , \sigma_0 \rangle \rightarrow^{t_1 + t_2 + t_3} \sigma_3}
\end{prooftree}
Where $\sigma_1 = \sigma_0[3 / z]$, $\sigma_2 = \sigma_1[10/x]$ and $\sigma_3 = \sigma_2[3/y]$.
From the assign rule we get that $t_1 = \mathcal{TA} \llbracket x \rrbracket + C_{ASSIGN\_V} = C_{VAR} + C_{ASSIGN\_V}$, $t_2$ and $t_3$ will be the same. Therefore $t_1 + t_2 + t_3 = 3 \times (C_{VAR} + C_{ASSIGN\_V})$.
\subsection*{Example: Division Algorithm}
To illustrate our logic, let us apply our rules to the division algorithm, as presented in figure~\ref{fig:division}.
Since our example contains a while loop we will have to define, the invariant $I$ as $x = q \times y + r \land y \ge 0 \land r \ge 0$, the variant $f$ as $x-r$, the maximum number of iterations $N$ as $x$, and the function of cost $t(k)$ as $fun\ k \rightarrow 10$.
Let us consider $S_w \equiv r=r-y;q=q+1$.
By the \textit{assign rule} we know
\begin{equation}
\vdash \{ (I \land f > k)[q+1/q] \}q=q+1\{ I \land f > k | \mathcal{TA} \llbracket q+1 \rrbracket + 1 \} \label{chap4:eq-1}
\end{equation}
Where
$$\mathcal{TA} \llbracket q+1 \rrbracket = \mathcal{TA} \llbracket q \rrbracket + \mathcal{TA} \llbracket 1 \rrbracket + 1 = 3$$
$$(I \land f > k)[q+1/q] \equiv x = (q+1)\times y+r \land y \ge 0 \land r \ge 0 \land x-r>k$$
Again by the \textit{assign rule} we know
\begin{equation}
\vdash \{ (I \land f > k)[q+1/q][r-y/r] \}r=r-y\{ (I \land f > k)[q+1/q] | \mathcal{TA} \llbracket r-y \rrbracket + 1 \} \label{chap4:eq-2}
\end{equation}
Where
\begin{align*}
&\mathcal{TA} \llbracket r-y \rrbracket = \mathcal{TA} \llbracket r \rrbracket + \mathcal{TA} \llbracket y \rrbracket + 1 = 3\\
&(I \land f > k)[q+1/q][r-y/r] \equiv x = (q+1) \times y+r-y \land y \ge 0 \land r-y \ge 0 \land x-r + y>k
\end{align*}
Since \ref{chap4:eq-1}, and \ref{chap4:eq-2}, by the \textit{seq rule}, we have
\begin{equation}
\vdash \{ (I \land f > k)[q+1/q][r-y/r] \}S_w\{ I \land f > k | 8 \}
\end{equation}
\begin{align*}
I \land y \le r \land f=k \quad \equiv& \quad x = q \times y + r \land y \ge 0 \land r \ge 0 \land y \ge r \land x-r=k\\
&\rightarrow \quad x = (q+1) \times y + r - y \land y \ge 0 \land r \ge y \land x-r+y \ge k\\
&\equiv \quad (I \land f>k)[q+1/q][r-y/r]
\end{align*}
Given this result, and since $10 \ge 8$, by the \textit{weak rule} we get
\begin{equation}
\vdash \{I \land y \le r \land f = k \} r=r-y;q=q+1 \{I \land f>k | 10 \}
\end{equation}
It also apparent that
$$I \land y \le r \rightarrow r \ge 0 \rightarrow x- r \le x$$
We are now ready to apply the \textit{while rule}, and we get
$$\vdash \{I \land f \ge 0 \} S \{ I \land \neg (y \le r) | \sum_{i=0}^{x-1}10 + (x+1) \times \mathcal{TB} \llbracket y \le r \rrbracket \}$$
Where
$$\sum_{i=0}^{x-1}10 + \mathcal{TB} \llbracket y \le r \rrbracket = 10 \times x + (x+1) \times 3 = 13x + 3$$
Since $P \rightarrow I \land f \ge 0$, $I \land \neg (y \le r)$, and $20x+5 \ge 13x +3$, by the \textit{weak rule} we get
$$\vdash \{P\} S \{Q | T \}$$
\subsection*{Example}\label{chap4:subsec:vcg-example}
Let us apply the VCG algorithm to the division example~\ref{fig:division}. We will use the same notation as in section~\ref{chap4:sec:axiomatic} and refer to our precondition as $P$, our program as $S$, our postcondition as $Q$ and our cost upperbound as $T$. The algorithm starts with a call to the $VCG$ function.
\begin{equation}
VCG(\{P\}S\{Q|T\}) = \{P \rightarrow wp)\ \cup \\
\{P \rightarrow T \ge t\}\ \cup \\
VC(S,Q) \nonumber
\end{equation}
Where
$$wp, t = wpc(S,Q) = (I \land x-r \ge 0, \sum_{i=0}^{x}10 + (x+1)\times 3)$$
Then the $VCG$ function calls the VC function for $S$. Since only \textit{while} loops generate extra VCs, we will omit other calls to the $VC$ function for simplicity.
\begin{align}
VC(S,Q) = &\{(I \land y \le r \land x-r=k) \rightarrow wp_S\}\ \cup \\
&\{(I \land \neg (y \le r)) \rightarrow Q\}\ \cup \\
&\{(I \land y \le r) \rightarrow x-r \le x\}\ \cup \\
&\{(I \land y \le r \land x-r=k) \rightarrow t(k) \ge t_S\}\ \cup \\
&VC(r=r-y;q=q+1,I \land x-r>k) \nonumber
\end{align}
Where
\begin{align*}
wp_S, t_S &= wpc(r=r-y;q=q+1, I \land x-r>k) \\
&= (wp_1, t_1 + t_2) \\\\
wp_1, t_1 &= wpc(r=r-y, wp_2) \\
&= (wp_2[r-y/r], \mathcal{TA} \llbracket r-y \rrbracket + 1) \\\\
wp_2, t_2 &= wpc(q=q+1, I \land x-r>k) \\
&= ((I \land x-r>k)[q+1/q],\mathcal{TA} \llbracket q + 1 \rrbracket + 1)
\end{align*}
To prove our triple, we now simply need to prove all of the VCs generated by the algorithm (4.3 to 4.8), this can easily be done for all the conditions manually, or with the assistance of a theorem prover.
As would be expected, proving a Hoare triple by applying the VCG algorithm is simpler and more mechanic than proving it directly by applying our rules and deriving the inference tree.
\subsection*{Soundness}
\begin{proof}
We start by defining our Induction Hypothesis.
$$\models VCG(\{P\}S\{Q|T\}) \rightarrow\ \vdash \{P\}S\{Q|T\}$$
We also calculate the result of wpc, VC and VCG for \textit{while}.
$$wpc(\mathrm{while}\ b\ \textit{do}\ S, Q) = (I \land f \ge 0 \land \Phi = 0, N \times a + (N+1) \times \mathcal{TB} \llbracket b \rrbracket)$$
$$wp_S, t_S = wpc(S,I \land f > k \land \phi = P_k)$$
\begin{align}
VC(\mathrm{while}\ b\ \textit{do}\ S, Q) = &\{ (I \land \mathcal{B} \llbracket b \rrbracket \land f = k) \rightarrow {wp}_S \} \cup \label{chap5:eq-1}\\
&\{I \land \mathcal{B} \llbracket b \rrbracket \land f = k \rightarrow a + \Phi - P_k \ge t_s \} \cup \label{chap5:eq-2}\\
&\{ I \land \neg \mathcal{B} \llbracket b \rrbracket \rightarrow Q \} \cup \label{chap5:eq-3}\\
&\{I \land \mathcal{B} \llbracket b \rrbracket \rightarrow f \le N \} \cup \label{chap5:eq-4}\\
&\{ I \rightarrow \Phi \ge 0 \} \cup \label{chap5:eq-5}\\
&VC(S,I \land f>k \land \Phi=P_k) \label{chap5:eq-6}
\end{align}
\begin{align}
VCG(\{P\}\mathrm{while}\ b\ \textit{do}\ S\{Q|T\}) = &\{P \rightarrow (I \land f \ge 0 \land \Phi=0)\} \cup \label{chap5:eq-7}\\
&\{P \rightarrow T \ge N \times a + (N+1) \times \mathcal{TB} \llbracket b \rrbracket \} \cup \label{chap5:eq-8} \\
&VC(\mathrm{while}\ b\ \textit{do}\ S, Q) \label{chap5:eq-9}
\end{align}
Assuming $\models VCG(\{P\}\mathrm{while}\ b\ \textit{do}\ S\{Q|T\})$.
Since we have \ref{chap5:eq-1}, \ref{chap5:eq-2}, and \ref{chap5:eq-6}, then
$$\models VCG(\{I \land \mathcal{B} \llbracket b \rrbracket \land f=k\} S \{I \land f>k \land \Phi = P_k | a + \Phi - P_k \})$$
By the induction hypothesis
\begin{equation}
\vdash \{I \land \mathcal{B} \llbracket b \rrbracket \land f=k\} S \{I \land f>k \land \Phi = P_k | a + \Phi - P_k \} \label{chap5:eq-10}
\end{equation}
Given \ref{chap5:eq-4}, \ref{chap5:eq-5}, and \ref{chap5:eq-10}, by the \textit{while rule}
$$\vdash \{I \land f \ge 0 \land \Phi = 0\} \mathrm{while}\ b\ \textit{do}\ S \{I \land \neg \mathcal{B} \llbracket b \rrbracket | N \times a + (N+1) \times \mathcal{TB} \llbracket b \rrbracket\}$$
Given \ref{chap5:eq-3}, \ref{chap5:eq-7}, and \ref{chap5:eq-8}, by the \textit{weak rule}
$$\vdash \{P\} \mathrm{while}\ b\ \textit{do}\ S \{Q | T\} $$
\end{proof}
\subsection*{Example: Binary Counter}
\input{sections/chapter 5/vcg-example.tex}
\subsection*{Soundness}
Assume $I \land \mathcal{B} \llbracket b \rrbracket \Rightarrow f \le N$, $I \Rightarrow \phi \ge 0$, and
$$\models \{\ I \land \mathcal{B} \llbracket b \rrbracket\ \land f=k\}\ S\ \{\ I\ \land \phi = P_{k} \land f > k\ |\ a + \phi - P_{k}\ \}$$
Considering a state $\sigma$ such that $\sigma \models I\ \land\ f \ge 0\ \land\ \phi = 0$ and $\langle \mathrm{while}\ b\ \mathrm{do}\ S, \sigma\rangle \rightarrow^{t_w} \sigma_1$.
Given our assumption, we know that $\sigma \models f \ge 0$ and that every time we enter the \textit{while} body, f increases. We also know that as long as $I \land B[b]$ are true, $f \le N$. Therefore we know that the program will eventually stop and at most iterate $N+1$ times.
If $\sigma \models \neg \mathcal{B} \llbracket b \rrbracket$ then $\sigma_1 = \sigma$, therefore $\sigma_1 \models \neg \mathcal{B} \llbracket b \rrbracket \land I$. In this case $t_w = \mathcal{T}\mathcal{B}\llbracket b \rrbracket$. Since $N \ge 0$, then $\mathcal{A} \llbracket \sum_{i=0}^{N}a + \sum_{i=0}^{N+1} \mathcal{TB} \llbracket b \rrbracket \rrbracket \sigma \ge t_w$.
If $\sigma \models \mathcal{B} \llbracket b \rrbracket$, we have that $\sigma \models \mathcal{B} \llbracket b \rrbracket \land I$.
Considering a state $\sigma_2$ such that $\langle S, \sigma \rangle \rightarrow^{t_1} \sigma_2$ and $\langle \mathrm{while}\ b\ \mathrm{do}\ S, \sigma_2\rangle \rightarrow^{t_2} \sigma_1$. Applying our initial assumption we get $\sigma_2 \models I \land \phi = P_{k} \land f > k$.
Finally by applying the induction hypothesis we have that $\sigma_1 \models \neg \mathcal{B} \llbracket b \rrbracket \land I$.
\\
By our assumption we also know that $\forall k. \mathcal{A} \llbracket(a + \phi - P_{k})\rrbracket \sigma \ge t_1$.
By induction, we know that
$\langle \mathrm{while b do S} , \sigma_2 \rangle \rightarrow ^{t_2} \sigma_1$
where
$$\mathcal{A} \llbracket (N-1) \times a + N \times \mathcal{T} \mathcal{B} \llbracket b \rrbracket \rrbracket \sigma_2 \ge t_2$$
The real cost of the \textit{while} is given by $t_1 + t_2 + \mathcal{T}\mathcal{B}\llbracket b \rrbracket$.
$$\mathcal{A} \llbracket(a + \phi - P_{k})\rrbracket \sigma \ge t_1$$
Since $\sigma \models \phi = 0 \rightarrow \mathcal{A} \llbracket \phi \rrbracket \sigma = 0$. And since $\sigma_2 \models I$ and $I \rightarrow \phi \ge 0$, then $\sigma \models \phi \ge 0$. $P_k = \mathcal{A} \llbracket \phi \rrbracket \sigma_2$, then $P_k \ge 0$. Then we have
$$\mathcal{A} \llbracket a \rrbracket \sigma \ge t_1$$
Since $a$, $N$, and $\mathcal{TB} \llbracket b \rrbracket$ are all constant, we get
\begin{equation}
\mathcal{A} \llbracket N \times a + N \times \mathcal{T} \mathcal{B} \llbracket b \rrbracket \rrbracket \sigma \ge t_1 + t_2 \\
\mathcal{A} \llbracket N \times a + (N+1) \times \mathcal{T} \mathcal{B} \llbracket b \rrbracket \rrbracket \sigma \ge t_1 + t_2 + \mathcal{T} \mathcal{B} \llbracket b \rrbracket
\end{equation}
Therefore the rule for \textit{while} is sound.
$$\models \{\ I\ \land f \ge 0\ \}\ \mathrm{while}\ b\ \mathrm{do}\ S\ \{\ I \land \neg \mathcal{B} \llbracket e \rrbracket\ |\ N \times a + (N+1) \times \mathcal{T}\mathcal{B}\llbracket b \rrbracket\}$$
\subsection*{Example: Binary Counter}
\input{sections/chapter 5/axiomatic-example.tex}
\section{Operational Semantics for Exact Costs}\label{chap6:sec:operational}
\input{sections/chapter 6/6.1-operational.tex}
\section{Proof Rules for Exact Costs}\label{chap6:sec:axiomatic}
\input{sections/chapter 6/6.2-axiomatic.tex}
\section{Verification Conditions Generation for Exact Costs}\label{chap6:sec:vcg}
\input{sections/chapter 6/6.3-vcg.tex}
\section{Background}
\input{sections/chapter 5/5.1-background.tex}
\section{Proof Rules with Amortized Costs}\label{chap5:sec:axiomatic}
\input{sections/chapter 5/5.2-axiomatic.tex}
\section{Verification Conditions Generation with Amortized Costs}
\input{sections/chapter 5/5.3-vcg.tex}
\section{Introduction}\label{sec:introduction}
\input{sections/1-intro.tex}
\section{Background}\label{sec:background}
\input{sections/2-background.tex}
\section{Related Work}\label{sec:related}
\input{sections/3-related.tex}
\section{Cost-Aware Program Logic}\label{sec:logic}
\input{sections/4-logics.tex}
\section{Amortized Costs}\label{sec:amortized}
\input{sections/5-amortized.tex}
\section{Exact Costs}\label{sec:exact}
\input{sections/6-exactcost.tex}
\section{Implementation}\label{sec:results}
\input{sections/7-results.tex}
\section{Conclusions}\label{sec:conclusion}
\input{sections/8-conclusion.tex}
\bibliographystyle{splncs04}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,760
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// <copyright file="OrderMap.g.cs" company="MIT">
// Copyright (c) 2019 MIT
// </copyright>
// Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"),
// to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense,
// and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
// The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
// IN THE SOFTWARE.
// *******************************************************************
// GENERATED CODE. DOT NOT MODIFY MANUALLY AS CHANGES CAN BE LOST!!!
// USE A PARTIAL CLASS INSTEAD
// *******************************************************************
using Microsoft.EntityFrameworkCore;
using Microsoft.EntityFrameworkCore.Metadata.Builders;
using System.Configuration;
using System.ComponentModel.DataAnnotations.Schema;
using DotNetScaffolder.Domain.Data.Entities.DefaultEntity.Entity;
using System.Data.Common;
namespace DotNetScaffolder.Domain.Data.Contexts.EFCore.MySql.Mappings.MySqlFullContext
{
public partial class FullContextOrderMap : IEntityTypeConfiguration<Order>
{
public virtual void Configure(EntityTypeBuilder<Order> builder)
{
builder.ToTable("Order");
#region Primary keys
builder.HasKey(t => t.OrderId);
builder.Property(t => t.OrderId).HasColumnName("OrderId").ValueGeneratedNever();
#endregion
#region Constraints
builder.Property(t => t.OrderId).HasColumnName("OrderId").IsRequired();
builder.Property(t => t.CustomerId).HasColumnName("CustomerId").IsRequired(false);
builder.Property(t => t.OrderDate).HasColumnName("OrderDate").IsRequired(false);
builder.Property(t => t.DeliveryDate).HasColumnName("DeliveryDate").IsRequired(false);
builder.Property(t => t.ShippingName).HasMaxLength(50);
builder.Property(t => t.ShippingName).HasColumnName("ShippingName").IsRequired(false);
builder.Property(t => t.ShippingAddress).HasMaxLength(50);
builder.Property(t => t.ShippingAddress).HasColumnName("ShippingAddress").IsRequired(false);
builder.Property(t => t.ShippingCity).HasMaxLength(50);
builder.Property(t => t.ShippingCity).HasColumnName("ShippingCity").IsRequired(false);
builder.Property(t => t.ShippingZip).HasMaxLength(50);
builder.Property(t => t.ShippingZip).HasColumnName("ShippingZip").IsRequired(false);
#endregion
#region Indexes
builder.HasIndex(i => new {i.OrderId}).HasName("UQ__Order__C3905BCE3C614174").IsUnique(true);
builder.HasIndex(i => new {i.CustomerId}).HasName("IX_CustomerId").IsUnique(false);
#endregion
#region Relationships
builder.HasOne<Customer>(s => s.Customer).WithMany(s => s.Order).HasForeignKey(s => s.CustomerId).OnDelete(DeleteBehavior.Restrict);
builder.HasMany<OrderDetails>(s => s.OrderDetails).WithOne(s => s.Order).HasForeignKey(s => s.OrderId).OnDelete(DeleteBehavior.Restrict);
#endregion
#region Column Order
// Column ordering available in EF Core 2.1 - https://data.uservoice.com/forums/72025-entity-framework-core-feature-suggestions/suggestions/18936844-ef-core-migrations-column-ordering
// Waiting for that release before implementing
//TODO: builder.Property(t => t.OrderId).HasColumnOrder(1);
//TODO: builder.Property(t => t.CustomerId).HasColumnOrder(2);
//TODO: builder.Property(t => t.OrderDate).HasColumnOrder(3);
//TODO: builder.Property(t => t.DeliveryDate).HasColumnOrder(4);
//TODO: builder.Property(t => t.ShippingName).HasColumnOrder(5);
//TODO: builder.Property(t => t.ShippingAddress).HasColumnOrder(6);
//TODO: builder.Property(t => t.ShippingCity).HasColumnOrder(7);
//TODO: builder.Property(t => t.ShippingZip).HasColumnOrder(8);
#endregion
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 76
|
Q: Getting undesired cpu load while running a python script on cron I am trying to run a python script in my crontab which checks the CPU load of my system and writes the result into an html file. I am calculating the cpu load using the psutil module's cpu_percent() method.
When I execute the script in my jupyter notebook it runs perfectly OK.
But when I place the script in my crontab as below:-
*
*
*
*
*
*
*
*
*
*/Applications/anaconda/bin/python Html_python.py >Html_python.log 2>&1
The cron job does run and the output gets redirected in the log file but the html file does not show the correct cpu load and only shows 0.0
But after I run the script manually the correct cpu load is again reflected.
I think I am not using the psutil.cpu_percent() method correctly, by specifying the parameters like interval and percpu. Not able to understand how these two parameters work.
Can someone help me to understand these parameters clearly.
Thanks in advance.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,351
|
2023 Presidential Candidate: Northern Youth Accuses Wike Of…
2023 APC Presidential Ticket: I will Cage Tinubu,…
Experts Reveal Real Reasons Biden's Approval Rating Drops To 33%
by EDITOR January 13, 2022 028
President Joe Biden's approval rating dropped to 33% in a new Quinnipiac University poll, the lowest mark of any major public survey during his presidency, as he takes a beating over his handling of the economy and coronavirus.
The poll, conducted Jan. 7 through Jan. 10 and released Wednesday, found 33% of all 1,313 adults surveyed approve of Biden's job performance, while 53% disapprove. Ten per cent did not offer an opinion.
The poll has a margin of error of 2.7 percentage points. When isolating just registered voters, Biden's approval rating improves slightly to 35%.
Biden's dismal numbers come as inflation has soared to a 39-year high and COVID-19 cases, which were on the decline last summer, have spiked to an all-time high amid the rise of the highly contagious omicron variant. Biden, who is nearing his first full year in office, campaigned on getting the pandemic under control and reviving the economy.
The poll found wavering support among Biden's own party, with 75% of Democrats approving of Biden's job performance and 14% of Democrats disapproving. More than half of political independents, 57%, said they disapprove of Biden's job performance, while just 25% in that crucial voting bloc said they approve. Biden is getting virtually no crossover support from Republicans, with 95% saying they disapprove of his job and 2% approving.
"COVID," Tim Malloy, polling analyst for Quinnipiac, said of the main factor behind Biden's polling slide. "It's a looming cloud over the country right now and over the presidency."
For most of his presidency, Biden's handling of the COVID-19 pandemic had been his top polling strength. But the new poll found his support in this area has cratered, with 39% of Americans saying they approve of Biden's response to the coronavirus, compared to 55% who disapprove.
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The majority of Americans also gave Biden a thumbs down on the economy. Thirty-four per cent of poll respondents said they approve of Biden's handling of the economy, compared to 57% who disapprove, and 35% said they approve of his handling of foreign policy, while 54% disapprove.
In November, the Quinnipiac poll found Biden's approval rating at 36% among all American adults and 38% among registered voters. Other recent polls this year have found Biden's approval rating between 40% and 45%. The Real Clear Politics average of polls shows 42% of Americans approve of Biden's job performance and 53% disapprove.
After riding high during his first six months in office, Biden's job approval dipped below 50% in several polls for the first time in August, coinciding with his administration's chaotic withdrawal in Afghanistan and the rise of the delta COVID-19 variant. It picked up slightly in December but has dropped again in polling taken so far in 2022.
"The president is at a steady, slow, undramatic slide, and it continued in this poll," Malloy said. He called 33% "not a devastatingly low number," noting that other presidents, including President Donald Trump, polled at similar clips. "But this is not a great one-year anniversary for him. That's for sure."
Other findings in the poll include:
► In the generic congressional ballot, 43% of American adults said they want to see Republicans in control of Congress compared to 42% who said they prefer Democrats
►58% of Americans said democracy is in danger of collapse, compared to 37% who said it is not
►53% of American adults said it is either likely or somewhat likely there is another attack in the U.S. similar to what happened at the Capitol on Jan. 6, 2021
► 61% of Americans said they support the House committee investigating the Jan. 6 attack, while 33% oppose it
►57% of Americans said they believe there was no widespread fraud in the 2020 election, compared to 34% who believe there was
► 59% of those polled said they would not like to see Trump run for president in 2024
Reach Joey Garrison on Twitter @joeygarrison.
This article originally appeared on USA TODAY: Biden's approval rating drops to a new low of 33% on concerns about his handling of COVID, economy: poll.
US Today.
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Who's Afraid of
Postmodernism?
THE CHURCH
AND POSTMODERN
CULTURE
_James K. A. Smith, series editor
_www.churchandpomo.org
The Church and Postmodern
Culture series features high-profile
theorists in continental philosophy
and contemporary theology
writing for a broad, nonspecialist
audience interested in the impact of
postmodern theory on the faith and
practice of the church.
**Forthcoming authors and topics**
**Bruce Ellis Benson,** associate professor of philosophy at Wheaton College, writes on improvisation as a paradigm for thinking about worship and the arts.
**John D. Caputo,** David C. Cook Emeritus Professor of Philosophy at Villanova University and Watson Professor of Religious Studies at Syracuse University, asks "What would Jesus deconstruct?"
**Graham Ward,** professor of contextual theology and ethics at the University of Manchester, writes on political discipleship.
**Merold Westphal,** distinguished professor of philosophy at Fordham University, writes on transcendence, community, and interpretation in conversation with Kierkegaard and Levinas.
Who's Afraid of
**Postmodernism?**
Taking Derrida, Lyotard,
and Foucault to Church
James K. A. Smith
**Baker Academic**
Grand Rapids, Michigan
© 2006 by James K. A. Smith
Published by Baker Academic
a division of Baker Publishing Group
P.O. Box 6287, Grand Rapids, MI 49516-6287
www.bakeracademic.com
Printed in the United States of America
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—for example, electronic, photocopy, recording—without the prior written permission of the publisher. The only exception is brief quotations in printed reviews.
Library of Congress Cataloging-in-Publication Data
Smith, James K. A., 1970–
Who's afraid of postmodernism? : taking Derrida, Lyotard, and Foucault to church / James K. A. Smith.
p. cm. — (The church and postmodern culture)
Includes bibliographical references (p. ) and index.
ISBN 10: 0-8010-2918-X (pbk.)
ISBN 978-0-8010-2918-9 (pbk.)
1. Postmodernism—Religious aspects—Christianity. 2. Christianity— Philosophy. 3. Derrida, Jacques. 4. Lyotard, Jean François. 5. Foucault, Michel. I. Title. II. Series.
BR115.P74S66 2006
261.51—dc22 2005033865
Scripture quotations are from the Holy Bible, New International Version®. NIV®. Copyright © 1973, 1978, 1984 by International Bible Society. Used by permission of Zondervan. All rights reserved.
For Coleson,
whose dreams and visions of a fantastic world
encourage me to hope for a re-imagined church
* * *
Contents
Series Preface
Preface
1. Is the Devil from Paris? Postmodernism and the Church
2. Nothing outside the Text? Derrida, Deconstruction, and Scripture
3. Where Have All the Metanarratives Gone? Lyotard, Postmodernism, and the Christian Story
4. Power/Knowledge/Discipline: Foucault and the Possibilities of a Postmodern Church
5. Applied Radical Orthodoxy: A Proposal for the Emerging Church
[Annotated Bibliography: Further Reading on
Postmodernism and Christian Faith ](Smit_ISBN9781441200396_epub_c14_r1.html#d7e22238147)
Online Resources
* * *
Series Preface
Current discussions in the church—from emergent "postmodern" congregations to mainline "missional" congregations—are increasingly grappling with philosophical and theoretical questions related to postmodernity. In fact, it could be argued that developments in postmodern theory (especially questions of "post-foundationalist" epistemologies) have contributed to the breakdown of former barriers between evangelical, mainline, and Catholic faith communities. Postliberalism—a related "effect" of postmodernism—has engendered a new, confessional ecumenism wherein we find nondenominational evangelical congregations, mainline Protestant churches, and Catholic parishes all wrestling with the challenges of postmodernism and drawing on the culture of postmodernity as an _opportunity_ for rethinking the shape of our churches.
This context presents an exciting opportunity for contemporary philosophy and critical theory to "hit the ground," so to speak, by allowing high-level work in postmodern theory to serve the church's practice—including all the kinds of congregations and communions noted above. The goal of this series is to bring together high-profile theorists in continental philosophy and contemporary theology to write for a broad, nonspecialist audience interested in the impact of postmodern theory on the faith and practice of the church. Each book in the series will, from different angles and with different questions, undertake to answer questions such as What does postmodern theory have to say about the shape of the church? How should concrete, inthe-pew and on-the-ground religious practices be impacted by postmodernism? What should the church look like in postmodernity? What has Paris to do with Jerusalem?
The series is ecumenical not only with respect to its ecclesial destinations but also with respect to the facets of continental philosophy and theory that are represented. A wide variety of theoretical commitments will be included, ranging from deconstruction to Radical Orthodoxy, including voices from Badiou to Žižek and the usual suspects in between (Nietzsche, Heidegger, Levinas, Derrida, Foucault, Irigaray, Rorty, and others). Insofar as postmodernism occasions a retrieval of ancient sources, these contemporary sources will be brought into dialogue with Augustine, Irenaeus, Aquinas, and other resources. Drawing on the wisdom of established scholars in the field, the series will provide accessible introductions to postmodern thought with the specific aim of exploring its impact on ecclesial practice. The books are offered, one might say, as French lessons for the church.
* * *
Preface
**To the Reader**
A word about my intended audience and modus operandi: Although I wish seriously to engage philosophical currents and ideas (in the way Francis Schaeffer faced ideas head-on) and am a philosopher by both profession and training (though also engaged in ministry for the past dozen years), in this book I am writing not primarily to philosophers or theologians (though they might find it of interest) but rather to students and practitioners— to those trying to orient themselves to the issues and those on the front lines of cultural engagement in the postmodern world: pastors and youth pastors, campus ministers and worship leaders, clergy and laypersons, and those training for such vocations. Given its original context at L'Abri, the book also speaks to spiritual seekers trying to navigate the postmodern terrain while searching for meaning. To them I want to suggest that, quite unlike the anti-institutional mentality of postmodern "spirituality," it is actually a robust, vibrant, liturgical church that speaks meaning in and to a postmodern world. With these audiences in mind, I have not assumed any philosophical knowledge and have avoided becoming bogged down in textual exposition. Although it is important to trace ideas to their source and listen to theorists firsthand, I have minimized footnotes and citations and concentrated on the ideas and claims being made. Of course, sources and engagement with other literature could be multiplied almost ad infinitum. But rather than distract the reader with extensive footnotes, I have included a bibliography for further reading, which will also point interested readers to my more scholarly discussions of these matters.
**Acknowledgments**
This book came to me, its author, as a gift—as a wonderful effulgence at the intersection of two very deep, rich streams: the heritage of the Calvin College department of philosophy and the tradition of L'Abri Fellowship in Switzerland. The core of the book was initially presented as a series of lectures in the summer of 2003 at L'Abri, the study center founded by Francis and Edith Schaeffer. The opportunity to give the lectures arose from a working relationship between Calvin's philosophy department and L'Abri that regularly takes Calvin faculty and students to Switzerland. I am very grateful to the philosophy department, particularly Lee Hardy, for the opportunity to go to L'Abri, and to the generous hospitality of Jim and Gail Ingram and the folks at L'Abri for welcoming my wife, Deanna, and me. Both the faculty and the students at L'Abri provided thoughtful questions and insights that have helped me to clarify my thinking on the matters examined in this book.
Throughout my preparation for those lectures, I had a keen sense of being a dwarf standing on the shoulders of giants. On the one hand, the opportunity to present those lectures was a chance to repay a debt. I find myself on the path of Christian philosophy today because of an encounter with the work of Francis Schaeffer when I was a sophomore in college. To be able to carry on that tradition of philosophical reflection, cultural analysis, and biblical discernment in the original lectures and now in this book is an attempt to render thanks and honor to Schaeffer's legacy—even if I might take that legacy in directions that Schaeffer would not. On the other hand, I count myself blessed to be part of the legacy of philosophy at Calvin College, drawing on the stream of tradition handed down from figures like Jellema, Runner, Mouw, Plantinga, and Wolterstorff. And I feel a special sense of gratitude to my contemporary colleagues in the department, who have welcomed me and encouraged me in my labors.
The transformation of the original lectures into a book has also profited from the generosity of several friends who took the time to read the manuscript and offer their comments and criticisms, which have undoubtedly improved it, except in those places where I was too stubborn to listen. My thanks to John Franke, Brian McLaren, Bill VanGroningen, and Geoff Holsclaw for this very tangible expression of friendship.
Finally, I owe much to my family for their unflagging support and for keeping me rooted in the realities of faith and practice. This book brings back special memories of an adventure with Deanna in Switzerland and France that I shall never forget and shall always cherish. The challenge of raising children in a postmodern culture is what spurs many of my concerns in this book, particularly the reflections on the formation of desire in chapter 4. I dedicate this book to my second son, Coleson, whose passion for Tolkien, King Arthur, and things medieval spurs my own ancient-future sensibilities. My prayer is that he will keep seeking the Grail.
Some of the material in chapters 2 and 3 appeared in an earlier form in _Christianity and the Postmodern Turn: Six Views_ , ed. Myron B. Penner (Grand Rapids: Brazos, 2005) and is included here with the kind permission of the editor and publisher.
****
* * *
Is the Devil from Paris?
Postmodernism and the Church
Postmodernism tends to be something of a chameleon, portrayed as either monster or savior—either the new form of the enemy or the next best thing to come along. This chapter introduces the questions that the phenomenon of postmodernism poses for the church and suggests a strategy for engagement that avoids simple dichotomies of either demonizing or baptizing postmodernism.
**Raising the Curtain:** **The Matrix**
"Welcome to the _real_ world." With these words, Neo is welcomed by Morpheus after emerging from his imprisonment in the matrix, a "neural-active simulation complex" designed by machines to control human beings. Although Neo escapes from the trappings of a postapocalyptic world, the scene in fact replays one of the most ancient of philosophical images: emerging from Plato's cave. In Plato's _Republic_ , Socrates recounts the way in which the masses are enslaved to a world of images and shadows, as though chained in the depths of a cave, their heads locked in position. Because of these restraints, all they have ever seen are the shadows dancing on the cave wall, cast by a small fire that sends light across the puppets and artifacts carried along a wall behind them. Never knowing anything different, each of the cave's prisoners considers the shadows real, until one of them is released. Turning from the wall of shadows, this liberated prisoner begins to make his way out of the cave, steadily proceeding toward the world above. Immediately upon turning around, he realizes that the shadows are only images cast by the puppets and cutouts behind him. Moving past these and past the cave's interior fire, the prisoner slowly emerges from his subterranean confinement into the light of day and the world above.
At first the experience is bewildering and bedazzling; the sun's light blinds eyes accustomed only to darkness. Indeed, the light of the real world is painful to eyes that have not seen it. Unable to look up at first, the liberated prisoner must navigate his way around the world by looking at shadows on the ground and images in the water. But these images are cast not by copies and cutouts but by the things themselves. Indeed, the experience of emerging from the cave has slowly revealed that what the prisoner had thought was real was in fact but a shadow of reality, a copy of a copy. In the world above he could behold not just the shadow of a tree, nor even the cutout image of a tree, but the tree itself. What would have seemed ludicrous to him before his liberation—that the world of his birth, his entire environment was not real—was now clear. Clearer still is what the prisoner must now do: return to the cave, liberate his companions, and proclaim the truth of what is real.
The figure of Neo in the Wachowski brothers' _Matrix_ is a postmodern Platonic prisoner. He has spent his entire life in a prison of sorts, a pod of quasi-uterine liquid where he is nourished by hoses, and dancing before his consciousness is not a dim, darkened world of shadows but a Technicolor reality of highrises and coffee shops, computers and nightclubs. The "wall" on which all of this is played out is within Neo's own mind, where a "neural-active simulation" program feeds a world of images directly into his consciousness. Thus, though these human pris-oners are actually trapped in pods where they are "harvested"for energy to run AI, they believe themselves to be someone and somewhere else. While Neo's body is hooked up to a system of cords and hoses, he thinks that he is one Thomas Anderson, a mediocre employee of a growing technology firm.
Morpheus comes to Neo as a liberator, someone who knows the truth and can thus descend into the depths of the cave's darkness in order to liberate others. Although Morpheus meets Neo in the matrix, he and his crew are able to effect the release of his body—his real self—from the pod. When his questioning mind no longer receives the neural-active simulation that is the matrix (something like a body rejecting a transplanted organ), the system flushes him out of the pod as waste. Morpheus and his crew seize Neo, lifting him from the dark dungeon toward the luminous light of reality, dramatized by his body being hoisted from the dark caverns up toward the tunnel of light shining from the ship _Nebuchadnezzar_. When Neo awakes, Morpheus greets him: "Welcome to the real world." Slipping in and out of consciousness, Neo asks: "Why do my eyes hurt?" "Because you've never used them before" is the reply. This lack of use requires a rigorous regimen of rehabilitation. Spitting out a stream of questions and receiving a barrage of dizzying answers, Neo experiences a kind of vertigo and vomits out of disorientation. It's not easy getting used to the real world.
Our contemporary culture, including the church, has experienced a similar dis- and reorientation. This book focuses on a transition not unlike that experienced by Neo: an emerging from one place to another, from one construction of reality to another, from modernity to postmodernity. While we might not name it as such, our experience of cultural shifts and changes can be traced to the advent of postmodernity and the trickle-down effect of postmodernism on our popular culture. The transition calls into question almost all our previously held sureties and rattles a faith that has been too easily equated with such Cartesian "certainties,"sometimes issuing in a kind of vertigo. Like Neo's experience, our emergence into this new situation engenders a host of questions and a confused sense of being lost. As Morpheus puts it to Neo, whose mind is swirling in this new reality: "I imagine right now you must feel a bit like Alice, tumbling down the rabbit hole." Or as W. H. Auden once described this kind of cultural upheaval, "It's as if we had left our house for five minutes to mail a letter, and during that time the living room had changed places with the room behind the mirror over the fireplace." If the shadows we thought were real have been unveiled as mere shadows, doesn't it sometimes feel as if the whole world were dissolving? Even if we have a sense that this is "the real world," as Morpheus announces, we're not sure how to make our way in it.
While I don't want to claim the mantle of Morpheus, my hope is to offer a kind of therapy and rehabilitation, an orientation to the world of postmodernism, which is simply to say, the world in which we now find ourselves.
**What Is Postmodernism?**
The notion of postmodernism is invoked as both poison and cure within the contemporary church. To some, postmodernity is the bane of Christian faith, the new enemy taking over the role of secular humanism as object of fear and primary target of demonization. Others see postmodernism as a fresh wind of the Spirit sent to revitalize the dry bones of the church. This is particularly true of the "emerging church" movement (associated with Brian McLaren, Leonard Sweet, Robert Webber, and others), which castigates the modernity of pragmatic evangelicalism and seeks to retool the church's witness for a postmodern world. In both cases, however, postmodernism remains a nebulous concept—a slippery beast eluding our understanding. Or perhaps better, postmodernism tends to be a chameleon taking on whatever characteristics we want it to: if it is seen as enemy, postmodernism will be defined as monstrous; if it is seen as savior, postmodernism will be defined as redemptive. This ambiguity tends to make us—Christian scholars, pastors and ministers, laypersons engaged in ministry—skeptical about just what we're talking about. _What is postmodernism?_
The answer to this question is sometimes offered as a historical thesis: postmodernism has been variously described as a kind of _post_ \- (after-) modern condition and is sometimes even linked to particular historical events such as student riots in 1968, the abandonment of the gold standard, the fall of the Berlin Wall, or, to be specific, 3:32 p.m. on July 15, 1972 Each candidate for the advent of postmodernism relies on an account of the supposed collapse of modernity. Trying to pinpoint the advent of the postmodern condition by linking it to a historical epoch, particular event, or even a particular cultural sphere (architecture, literature, music, visual arts) seems counterproductive, given the widespread disagreement about such historical claims. Further, it seems naïve to think that a _Zeitgeist_ like postmodernism could be spawned by a single event.
Instead of trying to pinpoint its historical origin or essence, I want to unpack an assumption that most commentators on postmodernism seem to share in common: postmodernism, whether monster or savior, is something that has come slouching out of Paris. In particular, postmodernism owes its impetus to French philosophical influences. While most commentators from various disciplines (architecture, art, literature, theology) concede this point, few have facility with philosophy or French philosophy in particular. In other words, we tend to give French philosophy a nod as crucial for understanding postmodernism but then do not engage the philosophical underpinnings. Brian McLaren, for instance, regularly tips his hat to the philosophical but then pushes it aside as "too far removed from everyday life" or not nec-essary for understanding "postmodernity" as distinguished from "philosophical postmodernism." But I want to follow Francis Schaeffer's footsteps by taking philosophy very seriously precisely because it does impact everyday life. "Ideas have legs," and even in a culture of amusement, there is thought that shapes it.
As Schaeffer remarks in the foreword to _Escape from Reason_ , "If we are to understand present-day trends in thought, we must see how the situation has come about historically and also look in some detail at the development of philosophic thought-forms." In _The God Who Is There_ , Schaeffer analyzes the shifts of modernity as beginning with philosophy (the "first step" in the "line of despair"); thus cultural phenomena, for Schaeffer, are symptoms of philosophical shifts, not vice versa. In his critical cultural analyses as found in _The God Who Is There_ or _Escape from Reason_ , Schaeffer offers what we might call a trickle-down theory of philosophical influence: cultural phenomena tend to eventually reflect philosophical movements. Perhaps my analyses of philosophical postmodernism can be understood as a necessary supplement (or better, prerequisite) to McLaren's analyses of postmodernity.
Thus in this book I want to employ a Schaefferian strategy in considering postmodernism. As such, I consider it a sequel to Schaeffer's own engagements with humanism and existentialism; postmodernism (a term rarely used in France, by the way) is, in some sense, the heir to existentialism. By a Schaefferian strategy, I mean at least two things: first, we need to return to the philosophy itself to understand postmodernity. While postmodernity as a cultural phenomenon is often distinguished from postmodernism as a philosophical movement, I agree with Schaeffer that cultural phenomena tend to be a product of philosophical movements. We take culture seriously by taking ideas seriously. Second, my strategy is "Schaefferian" in the sense that my primary audience is not just philosophers but practitioners—more specifically, Christians engaged in ministry in a postmodern world, as well as searching inhabitants of this postmodern world. As such, these essays are not an academic project per se. Instead, their purpose is to introduce philosophical currents to people who don't usually travel that stream. Thus I avoid philosophical jargon as much as possible. Where special terminology is necessary, it is couched in a context of explanation and clarification. I see this as an _incarnational_ strategy, attempting to accommodate thought to language that is accessible to an audience, just as Calvin so often emphasized that God in Christ accommodates his thought to our language, appearing as a Word that we can understand.
With the humble goal of trying to unpack the primary philosophical impulses behind postmodernism, my strategy is to engage something of an unholy trinity of postmodern thinkers: Jacques Derrida, Jean-François Lyotard, and Michel Foucault. While their names might not be familiar to everyone, key aspects of their thought have now become commonplace not just in the academy but in the media as well. In particular, I carefully consider three slogans of postmodernism associated with these philosophers:
• "There is nothing outside the text" (Derrida).
• Postmodernity is "incredulity toward metanarratives" (Lyotard).
• "Power is knowledge" (Foucault).
Generally, these three slogans are invoked as being mutually exclusive to confessional Christian faith. How could someone who takes the sweeping narrative of the Scriptures as the Word of God reject metanarratives? How could someone who believes in the existence of a transcendent God and his creation deny that there is reality outside texts? How could someone who worships the God who is Love participate in a Nietzschean celebration of the will to power as the basis of reality?
The problem is that all these questions are rooted in a misunderstanding of the claims being made. In other words, these slogans (which were never intended as slogans by their authors) are treated like bumper stickers: claims made without a context. Once we appreciate the context of these claims, however, we see two things: First, they mean something different than what the "bumper-sticker" reading suggests. The bumper-sticker readings that turn these claims into slogans tend to perpetuate a number of myths about postmodernism. My goal is to demythologize postmodernism by showing that what we commonly think so-called postmodernists are saying is usually not the case. Second, and perhaps more provocatively, I will demonstrate that, in fact, all these claims have a deep affinity with central Christian claims.
As such, the studies that make up chapters 2–4 are intended to function as a two-edged sword. On the one hand, I critically introduce Christians to currents in contemporary thought, often described as postmodernism. This requires subjecting these ideas to criticism from an integrally Christian worldview. But they are also meant to cut the other way as well; that is, I also critique common Christian misunderstandings of postmodernism and suggest ways in which postmodernity is a condition that Christians should, in some sense, welcome. Something good _can_ come out of Paris. In this way, I'm simply replaying a Hebrew strategy, later adopted by Augustine and utilized by the likes of John Calvin and Abraham Kuyper: making off with Egyptian loot. As Augustine put it in his _Teaching Christianity_ ( _De doctrina chris-tiana_ ), just as the Hebrews left Egypt with Egyptian gold to be put to use in the worship of Yahweh (even if they misdirected its use at times), so Christians can find resources in non-Christian thought—whether that of Plato or of Derrida—that can be put to work for the glory of God and the furtherance of the kingdom. This book is an attempt to make off with postmodern loot for the sake of the kingdom.
In particular, I suggest that this unholy trinity of Derrida, Lyotard, and Foucault might in fact push us to recapture some truths about the nature of the church that have been overshadowed by modernity and especially by Christian appropriations of modernism. One of the reasons postmodernism has been the bogeyman for the Christian church is that we have become so thoroughly modern. But while postmodernism may be the enemy of our modernity, it can be an ally of our ancient heritage. In short, it might just be these Parisians who can help us _be_ the church. Specifically, each of the analyses made by these postmodern theorists entails a twofold effect for the church:
• _Derrida_. Deconstruction's claim that there is "nothing outside the text" [ _il n'y a pas de hors-texte_ ] can be considered a radical translation of the Reformation principle _sola scrip-tura_. In particular, Derrida's insight should push us to recover two key emphases of the church: (a) the centrality of Scripture for mediating our understanding of the world as a whole and (b) the role of community in the interpretation of Scripture.
• _Lyotard_. The assertion that postmodernity is "incredulity toward metanarratives" is ultimately a claim to be affirmed by the church, pushing us to recover (a) the narrative character of Christian faith, rather than understanding it as a collection of ideas, and (b) the confessional nature of our narrative and the way in which we find ourselves in a world of competing narratives.
• _Foucault_. The seemingly disturbing, even Nietzschean claim that "power is knowledge" should push us to realize what MTV learned long ago: (a) the cultural power of formation and discipline, and hence (b) the necessity of the church to enact counterformation by counterdisciplines. In other words, we need to think about discipline as a creational structure that needs proper direction. Foucault has something to tell us about what it means to be a disciple.
Of course, since none of these theorists are Christians, we should also expect some points of fundamental disagreement as well as the necessity to critique some of their conclusions.
Each chapter employs a common strategy to open up these postmodern thinkers and theories and begins with a brief discussion of a recent film as a way of illustrating some of the questions and issues at stake. Film is the new lingua franca of not just American culture but, increasingly, global culture. Further, it is a powerful "incarnational" medium that can reveal truth about our world, opening up our experience in a way that propositions and textbooks cannot. The film discussions, then, are intended to activate our curiosity—to get us asking the kinds of questions that these postmodern theorists are asking. Then, I introduce the central claim made by each figure, explaining what is being said by considering the context of the claim in the author's work. This will then require some clarification, particularly in the face of common misunderstandings of the claim, especially by Christian theologians and practitioners. We'll then explore the implications of this claim for the church, in both its theology and its practice (the two should never be separated), leading up to a constructive conclusion regarding the shape of the postmodern church.
Thus, each chapter ends with a "tour" of a postmodern church. After considering the thought of Derrida, Lyotard, and Foucault, each chapter concludes with a case study, considering recent developments in the church that have attempted to respond to the postmodern condition, such as the emerging church movement. In the work of Brian McLaren, Leonard Sweet, and others, we find Christian thinkers and practitioners who have specifically tried to engage the cultural shift to postmodernity. These bold explorers have been some of the first to map the postmodern terrain for the church. Given our engagement with the key theorists of postmodernity, we'll critically consider both their reading of postmodernism as well as their proposals for the shape of the postmodern church. As will become clear, although I share a deep sympathy with their concerns, I also think that at times their proposals remain captive to some modernist strategies. I will argue that the postmodern church could do nothing better than be ancient, that the most powerful way to reach a postmodern world is by recovering tradition, and that the most effective means of discipleship is found in liturgy. Each tour of a postmodern church will give a concrete picture of what Derrida, Lyotard, and Foucault might mean for how we worship.
A thoughtful engagement with postmodernism will encourage us to look backward. We will see that much that goes under the banner of postmodern philosophy has one eye on ancient and medieval sources and constitutes a significant recovery of premodern ways of knowing, being, and doing. Ancient and medieval sources provide a useful countervoice to modernity.Thus Derrida constantly engages Plato (and later, Augustine), Lyotard looks at tribal cultures, and Foucault considers ancient practices of discipline. Without being conservative or trying to recover a (mythical) pristine tradition in the name of "paleo-orthodoxy,"postmodernism does stage a certain creative recovery of ancient themes and figures.The three studies that make up chapters 2–4 set the stage for a concluding chapter that makes the case that the most persistent postmodernism should issue in a thickly confessional church that draws on the very particular (yet catholic) and ancient practices of the church's worship and discipleship. In other words, a "radical orthodoxy" is the only proper outcome of the postmodern critique, and insofar as the emerging church shrinks from an unapologetic dogmatics (which isn't a rabid fundamentalism), it remains captive to the dreams, ambitions, and skepticism of modernity. Thus chapter 5 considers why the best way to be postmodern is to be ancient, and the best way to proclaim Christian faith in the postmodern world is not quietly, with a chastised timidity, but unapologetically, with an embodied commitment to justice in community.
**Apologetics and Witness in a Postmodern World**
Talking about "postmodernism" can give the impression that we are describing a discrete, specific phenomenon, like a table or a cup. Further, one is tempted to suggest that with the advent of postmodernity, everything has changed. But neither of these is true: Postmodernism is an admittedly pluriform and variegated phenomenon. And postmodernism does not make a clean break from modernism. There are both continuities and discontinuities between modernity and postmodernity. The most significant continuity is that both deny grace; in other words, both modernity and postmodernity are characterized by an idolatrous notion of self-sufficiency and a deep naturalism.Noting this theological continuity, one also recognizes philosophical and cultural continuities, such that postmodernity is often an intensification of modernity, particularly with respect to notions of freedom, the use of technology, and so on.
These continuities influence the discussion that follows in two ways: First, I tend to emphasize the discontinuities between modernity and postmodernity. I recognize—and have argued elsewhere—that there are deep continuities between modern and postmodern thought, particularly in the work of Derrida and Foucault, both of whom confess that they are, in an important sense, Enlightenment thinkers.However, there is also an important sense in which they are critics of modernity, and thus postmodernism does in some way break with modernism. In the analyses presented in this book, I am particularly interested in the opportunities that this rupture represents for recapturing a more robust—and less modern—Christian faith.
This leads to my second emphasis on the continuities between postmodernism and orthodox Christian faith.Much in the work of Derrida, Lyotard, and Foucault merits criticism, especially from a Christian perspective. Here, and in the context of the original lectures that make up this book, however, it seemed necessary to emphasize points of overlap between postmodernism and historic, orthodox Christian faith, especially because I am, in some sense, carrying on the Schaefferian legacy. To do so, it is necessary to confront a latent modernism in Schaeffer's own construal of Christian faith as a "system of truth."Rather than explicitly critique Schaeffer on this score, I want to demonstrate that, perhaps to Schaeffer's surprise (and chagrin), the claims of postmodernists such as Derrida and Foucault have something in common with his own account of knowledge and truth (insofar as Schaeffer recognized the role of presuppositions).While Schaeffer generally focused on pointing out the discontinuities between, say, existentialism and Christian faith, there is a virtue to finding the "point of tension" by also finding the point of contact between Christian and non-Christian thought. And recognizing such a continuity may require that we jettison some of our own modern presuppositions. Our Christian faith—and correlatively, our account of apologetics—is tainted by modernism when we fail to appreciate the effects of sin on reason. When this is ignored, we adopt an Enlightenment optimism about the role of a supposedly neutral reason in the recognition of truth.(We also end up committed to "Constantinian" strategies that, under the banner of natural law, seek to build a "Christian America.")
To put this in more familiar terms, classical apologetics operates with a very modern notion of reason; "presuppositional" apologetics, on the other hand, is postmodern (and Augustinian!) insofar as it recognizes the role of presuppositions in both what counts as truth and what is recognized as true. For this reason, postmodernism can be a catalyst for the church to reclaim its faith not as a system of truth dictated by a neutral reason but rather as a story that requires "eyes to see and ears to hear." The primary responsibility of the church as witness, then, is not demonstration but rather proclamation—the kerygmatic vocation of proclaiming the Word made flesh rather than the thin realities of theism that a supposedly neutral reason yields.
To put it another way, unless our apologetic proclamation begins from revelation, we have conceded the game to modernity. On this score, I side with an even earlier Parisian philosopher and proto-postmodernist, Blaise Pascal, who adamantly protested that the God revealed in the incarnation and the Scriptures—the God of Abraham, Isaac, and Jesus Christ—is to be distinguished from the (modern) god of philosophical theism. But even more importantly, this new apologetic—which is, in fact, ancient—is one that is proclaimed by a community's way of life.As Peter Leithart has remarked, "The first and chief defense of the gospel, the first 'letter of commendation' not only for Paul but for Jesus, is not an argument but the life of the church conformed to Christ by the Spirit in service and suffering."The church doesn't _have_ an apologetic; it _is_ an apologetic.
**From Modern Christianity to a Postmodern Church**
If I am opposed to the epistemology, or theory of knowledge, that plagues modern Christianity, then I am also opposed to the ecclesiology (or lack thereof ) that accompanies this modernist version of the faith. Within the matrix of a modern Christianity, the base "ingredient" is the individual; the church, then, is simply a collection of individuals. Conceiving of Christian faith as a private affair between the individual and God—a matter of my asking Jesus to "come into my heart"—modern evangelicalism finds it hard to articulate just how or why the church has any role to play other than providing a place to fellowship with other individuals who have a private relationship with God. With this model in place, what matters is Christianity as a system of truth or ideas, not the church as a living community embodying its head. Modern Christianity tends to think of the church either as a place where individuals come to find answers to their questions or as one more stop where individuals can try to satisfy their consumerist desires. As such, Christianity becomes intellectualized rather than incarnate, commodified rather than the site of genuine community.
In discussing Christian faith emerging from modernity to postmodernity, however, I rarely speak of Christianity, and I even resist talking about Christians as individuals; rather, I tend to speak of the church—indeed, with a capital _C_. I want to advocate a shift from modern Christianity to a postmodern church, one akin to the paradigm shift experienced by Neo. My point here is confessional: as attested in the Apostles' Creed, I believe in the holy catholic church, and I believe that the very notion of the holy catholic church undoes the modern individualism that plagues contemporary evangelicalism.Indeed, we would do well to recover a much-maligned formula: "There is no salvation outside the church." This doesn't mean that a particular ecclesial body is the dispenser of grace or the arbiter of salvation; rather, there simply is no Christianity apart from the body of Christ, which is the church. The body is the New Testament's organic model of community that counters the modernist emphasis on the individual.
The church does not exist _for_ me; my salvation is not primarily a matter of intellectual mastery or emotional satisfaction. The church is the site where God renews and transforms us—a place where the practices of being the body of Christ form us into the image of the Son. What I, a sinner saved by grace, need is not so much answers as reformation of my will and heart. What I describe as the practices of the church include the traditional sacramental practices of baptism and Eucharist but also the practices of Christian marriage and child-rearing, even the simple but radical practices of friendship and being called to get along with those one doesn't like! The church, for instance, is a place to learn patience by practice. The fruit of the Spirit emerges in our lives from the seeds planted by the practices of being the church; and when the church begins to exhibit the fruit of the Spirit, it becomes a witness to a postmodern world (John 17). Nothing is more countercultural than a community serving the Suffering Servant in a world devoted to consumption and violence. But the church will have this countercultural, prophetic witness only when it jettisons its own modernity; in that respect postmodernism can be another catalyst for the church to _be_ the church.
1. _The Matrix,_ DVD, written and directed by Andy and Larry Wachowski (Burbank, CA: Warner Home Video, 1999).
2. W. H. Auden, "If, on account of the Political Situation," from _The Complete_ _Works of W. H. Auden_ (Princeton, NJ: Princeton University Press, 1988).
3. See, for instance, Charles Colson, "The Postmodern Crackup: From Soccer Moms to College Campuses, Signs of the End," _Christianity Today,_ December 2003, 72. See also the work of Millard Erickson, Douglas Groothuis, and D. A. Carson ( _The Gagging of God: Christianity Confronts Pluralism_ [Grand Rapids: Zondervan, 1996] and _Becoming Conversant with the Emerging Church: Understanding a_ _Movement and Its Implications_ [Grand Rapids: Zondervan, 2005]).
4. See, for instance, Brian D. McLaren, _A New Kind of Christian: A Tale of_ _Two Friends on a Spiritual Journey_ (San Francisco: Jossey-Bass, 2001); idem, _The Church on the Other Side: Doing Ministry in the Postmodern Matrix_ (Grand Rapids: Zondervan, 2000); Leonard Sweet, _SoulTsunami: Sink or Swim in New_ _Millennium Culture_ (Grand Rapids: Zondervan, 1999); Robert E. Webber, _The_ _Younger Evangelicals: Facing the Challenges of the New World_ (Grand Rapids: Baker, 2002); and Carl Raschke, _The Next Reformation: Why Evangelicals Must_ _Embrace Postmodernity_ (Grand Rapids: Baker, 2004).
5. This is when the Pruitt-Ingoe housing development (a prize-winning version of Le Corbusier's "machine for modern living") in St. Louis was dynamited as an uninhabitable environment for the low-income people it housed. See Charles Jencks, _Le Corbusier and the Continual Revolution in Architecture_ (New York: Monacelli, 2000).
6. McLaren, _Church on the Other Side_ , 160; and idem, _New Kind of Christian_ , 19.
7. Schaeffer, _Escape from Reason_ , in _The Francis A. Schaeffer Trilogy_ (Westchester, IL: Crossway, 1990), 208.
8. While I can't pursue this in detail, we should at least note a common heuristic distinction between postmodernism as an intellectual movement and postmodernity as a constellation of cultural phenomena. Derrida's deconstruction and Foucault's genealogy of power are examples of postmodern _ism_ ; adolescent absorption in virtual reality and the triumph of the mall as temple are examples of postmodern _ity_. Although there is a trickle-down effect between philosophical currents of postmodernism and cultural phenomena related to postmodernity, much that is associated with cultural postmodernity is, in fact, the fruit of modernity. In other words, cultural phenomena tend to not (yet?) reflect the radical implications of postmodernism. This might be because postmodernism itself has shrunk back from its own implications in both intellectual and cultural spheres. The individualism and consumerism that characterize contemporary culture are fruit nourished by deeply modern roots. So also relativism owes much more to modernism than to postmodernism. In this book, I focus primarily on the intellectual currents of postmodernism, but there remain important questions about cultural phenomena that could be rightly described as postmodern.
9. McLaren also looks to Schaeffer as a prototype in this regard. He cites Schaeffer's exhortation to the church a generation ago: "One of the greatest injustices we do to our young people is to ask them to be conservative. . . . If we want to be fair, we must teach the young to be revolutionaries, revolutionaries against the status quo" (McLaren, _Church on the Other Side_ , 16).
. I have explored the notion of incarnational language more fully in my _Speech and Theology: Language and the Logic of Incarnation_ , Radical Orthodoxy Series (London: Routledge, 2002).
. For example, "nondenominationalism" might be seen as a deeply modern phenomenon. I discuss this further in chapter 4.
. This resonance between ancient and postmodern thought has been suggested by medieval scholar David Burrell, who notes the affinity between postmodernism and medieval theology. "Postmodern," he suggests, "could be translated as 'anti-antimedieval' " (in _Faith and Freedom: An Interfaith Perspective_ [Oxford: Blackwell, 2004], 141).
. Many, I suspect, would be surprised to know that Parisian philosophical circles are currently abuzz with discussions of Saint Augustine, led by the work of Jean-Luc Marion at the Sorbonne.
. My thinking along these lines owes much to Robert Webber, _Ancient-_ _Future Faith: Rethinking Evangelicalism for a Postmodern World_ (Grand Rapids: Baker, 1999).
. Or as Graham Hughes suggests, both modernity and postmodernity are characterized by a trenchant "disenchantment of the world," to use Max Weber's phrase (Hughes, _Worship as Meaning: A Liturgical Theology for Late Modernity_ [Cambridge: Cambridge University Press, 2003], 2).
. See, for example, James K. A. Smith, _Jacques Derrida: Live Theory_ (London: Continuum, 2005), 3.3.2.
. I generally use this term as a shorthand to refer to Christian faith as rooted in the Scriptures and attested in the historic creeds and confessions, which I take to be amplified by later Reformed thought. While my project here is quite ecumenical, it will become clear that I think the Reformed confessions are an important extension of historic, orthodox Christian faith. I don't wish to engage in polemics about that here, except insofar as it impacts epistemology (the theory of knowledge) and hence apologetics. And in the end, what is at stake here is an Augustinian catholic theology.
. See the appendix on "Apologetics" in _Trilogy_.
. This question is debated, and it seems that even when he explicitly addressed it, Schaeffer remained ambiguous on this score, at times leaning toward a more classical approach that does not take seriously the role of presuppositions in knowledge. Schaeffer's own Reformed theology undercuts classical apologetics insofar as it is committed to the "noetic effects of sin"—that is, the effects of sin on the mind, distorting both what counts as true and what can be recognized as true for the unbeliever (Rom. 1:18–22; 1 Cor. 2). During my stay at L'Abri, I had occasion to look carefully at the classic seventeenth-century work of John Owen on the Holy Spirit. I could recommend no better statement on the noetic effects of sin than his account of "Corruption of the depravity of the mind by sin" (III.iii; _The Holy Spirit_ [repr., Grand Rapids: Kregel, 1954], 144–69). One finds the same point articulated in Kierkegaard's _Philosophical Fragments_ : in order to come to know the truth, the learner (disciple) must receive from the Teacher (God) not only the content of the truth but also the very condition for receiving it. The dispensation of the condition is an act of grace by the work of the Holy Spirit.
. In the contemporary context, what I have elsewhere called the "Biola school" of apologetics and Christian philosophy embodies this modernism. For a brief account, see James K. A. Smith, "Who's Afraid of Postmodernism? A Response to the 'Biola School,' " in _Christianity and the Postmodern Turn,_ ed. Myron Penner (Grand Rapids: Brazos, 2005).
. For further discussion of this new apologetic (following Robert Webber), see James K. A. Smith, _Introducing Radical Orthodoxy: Mapping a Post-secular_ _Theology_ (Grand Rapids: Baker, 2004), 179–82.
. Peter J. Leithart, _Against Christianity_ (Moscow, ID: Canon, 2003), 99.
. I remain concerned that, despite all of the talk about community in the emerging church, we have not yet explored the radical implications of it. The next task for the emerging church is to articulate an ecclesiology.
. Here we do well to return to the rich, sacramental theology of John Calvin as opposed to the thin, Zwinglian theologies that seem to have won the day in Reformed evangelical circles.
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* * *
Nothing outside the Text?
Derrida, Deconstruction, and Scripture
If postmodernism has anything close to a brand name, it is deconstruction, and if it has anything like a celebrity face, it is the dark face of a Parisian transplanted from North Africa: the face of Jacques Derrida. In this chapter we will consider the central themes of deconstruction through one of Derrida's most (in)famous claims: "There is nothing outside the text."
**Raising the Curtain:** **Memento**
Lenny has a problem. Well, he has lots of problems—believe me!—but one stands out: he can't remember what he did five minutes ago. Since a tragic incident involving the death of his wife, Lenny has not been able to make new memories. He can remember everything from _before_ the accident and thus can remember where he's from and how to navigate his way through day-to-day life: how to eat, how to drive, and very importantly, how to _write_. But while he's driving, he can't remember why he got in the car. Or when he enters a restaurant, he can't remember why he came. When he goes to meet a recent acquaintance, he can't remember what she looks like. And as the motel manager quickly figures out, Lenny can't remember how many rooms he's checked into—providing a lucrative business for the motel.
So how does someone without short-term memory make his or her way in the world? How can I drive to work if I can't remember why I got in the car? How can I ever make new friends if I can't remember a face? How can I ever read a book if I forget the first chapter five minutes after reading it? In the face of these challenges, Leonard in his resourcefulness comes up with a "system"; others with Leonard's condition couldn't survive because they didn't have one. The system is simple: _writing_. Leonard's navigation through existence is governed by writing, by a collection of texts and notes—coupled with Polaroid photographs —that substitute for memory. His pockets are filled with little texts, some written on napkins, others written on Polaroids, all providing the framework for him to understand his world. In his pocket is a snapshot of his Jaguar with the text "My Car" to remind him which vehicle in the parking lot belongs to him. All his acquaintances are noted in a similar way.
This system of texts and writing, however, works on the basis of two principles or beliefs: first, only trust your own handwriting; be suspicious of any writing you can't recognize. Second, really important information shouldn't be trusted to notes on napkins: vital information should be written on the body. Thus Leonard is himself a walking text, his body riddled with tattooed reminders: of historical events (like the murder of his wife), of basic principles (like "Consider the source" and "Memory is treachery"), and of "facts" about the case he is investigating, the murder of his wife (Leonard literally lives for revenge). Because his body is the source of so much knowledge, Leonard needs to spend a lot of time in front of the mirror to remind himself about the reality of his world: what's history ("John G raped and murdered my wife"), what he's doing (seeking to avenge his wife), who he's looking for (an elusive "John G"), what the facts are, and what his basic beliefs are ("Learn by repetition," "Don't trust your weakness," "Camera doesn't lie," etc.). Leonard's entire relationship to the world is mediated by texts—some on his body, more scribbled on notes—all of which function as the framework through which he sees the world.
Without these texts, Leonard literally would not be able to have an experience of the world. Just having an experience requires a certain ability to integrate one's sensations over time. If I'm listening to a U2 song, in order for me to "get it," I need to be able to listen from beginning to end and then integrate what I've heard. Leonard's condition makes this impossible: by the end of the song, Lenny would have forgotten the beginning! The only way Leonard can keep his experience together—and hence have anything like a world of experience—is by a series of notes/texts that put together the world for him. Without texts, Leonard lacks a world. And without a pen, Leonard lacks a text. So when someone like Natalie wants to manipulate Leonard by transforming his world, all she has to do is put all the pens in her purse! When a disturbing event happens—one that Leonard wants to remember but Natalie would rather forget—the absence of any writing utensils means that the event is obliterated from Leonard's world.
Of course, this kind of system has its problems. As Natalie points out: "It must be tough living your life according to a couple of scraps of paper. Mix up your laundry list with your grocery list, and you'll end up eating your underwear for breakfast!" Living on texts in a world composed of notes entails both doubt and anxiety: how does Leonard know his texts really represent the world outside his mind? In fact, this is one of the nagging doubts that requires a constant faith and reassertion of belief on his part. One of Leonard's fundamental beliefs—though he has to keep reminding himself—is that there _is_ a world outside his mind. As he confesses at the end of the film (which is the beginning of the story): "I have to believe in a world outside my own mind. I have to believe that my actions still have meaning, even if I can't remember it. I have to believe that when my eyes are closed, the world's still there. Do I believe . . . ?" Ultimately, the question he puts to himself is not whether the world exists outside his mind but whether he believes it.
**Derrida's Claim: There Is Nothing outside the Text**
According to many, Jacques Derrida is a kind of philosophical Leonard, or, conversely, _Memento_ is a "deconstructive" film. "Deconstruction"— a term coined by Derrida in 1967—has entered even the most colloquial American vocabulary and is used to describe everything from architecture and music videos to key lime pie. Often it is used simply as a synonym for destruction or criticism; hence, to "deconstruct" something is to take it apart, to knock it down, to pull it apart piece by piece. But when Derrida introduced the term in the late 1960s, he did not intend it as a primarily negative notion, even if he did intend it as a kind of criticism. For Derrida deconstruction is ultimately positive and constructive. We'll return to this later.
What is the link between Leonard and Derrida, between _Memento_ and deconstruction? It is the central role of texts or writing for mediating or putting together our experience of the world. For both Leonard and Derrida, language is the necessary filter through which the world comes to us. Just as Leonard depends on the writing of notes to give his world some coherence and order, so Derrida argues that all of us interpret our world on the basis of language (broadly understood). _Memento_ ends with Leonard claiming that he's really no different than anyone else. In a way, that is the heart of Derrida's claim: like Lenny, we all need crib notes and cheat sheets to make our way in the world. In one of his first books, published in 1967 (in French), Derrida famously puts it this way: "There is nothing outside the text" [ _Il n'y a pas de hors-texte_ ].
Now, immediately, we need to consider the ways in which this claim has been misunderstood by the "bumper-sticker" approach. When someone—especially a philosopher—claims that there is nothing outside the text, it sounds as if he is claiming that the whole world is a kind of book—that there are no cups or tables or spouses. If there is nothing outside the text, then all we have are texts; and if all we have are texts, then we don't have things. In other words, Derrida sounds like a metaphysical idealist (like Berkeley) who claims that there are no material things, only ideas in the mind of God. As such, many have understood Derrida as a linguistic idealist who thinks there is only language, not things—only texts, not cups or tables. This is how he is commonly understood by Christians, especially Christian theologians.
Of course, a Christian could not be a linguistic idealist (someone who thinks there are only words, not things) for at least two reasons: First, if there is nothing outside the text, then a transcendent Creator who is distinct from and prior to the world could not exist. In this sense, linguistic idealism would have to entail atheism. If Derrida is a linguistic idealist, then deconstruction and Christian faith are mutually exclusive. Second, if there is nothing outside the text, then it would seem that what the Bible (admittedly a text) talks about—what it refers to—is not real. When the Bible speaks about the incarnation, or the effects of the work of Christ, or a spiritual warfare in the heavenly realms, all these references must not be real. But if these claims are not real—if it is not the case that Jesus really was God in the flesh (John 1:14), or if his death on the cross did not effect a cosmic transformation (Col. 1:20)—then Christianity is at best a fiction and at worst a waste of time. Thus the common conclusion is that Derrida's claim that there is nothing outside the text is antithetical to authentic Christian confession.
I suggest, however, that there is a problem with this conclusion. In particular, the conclusion rests on a faulty premise, namely, a serious misunderstanding of what Derrida means when he claims that there is nothing outside the text. The following discussion carefully considers the bigger picture surrounding Derrida's claim and suggests that, in fact, Derrida offers insight into the structure of creation.
**Leonard's Condition as the Human Condition: Reading,** **Writing, and Interpretation**
Derrida's provocative claim that there is nothing outside the text arises in the context of a discussion about reading and interpretation. Let us consider the passage in some detail in order to appreciate the nuances of what Derrida is saying. If we are going to do justice to postmodernism, our engagement with it needs to be characterized by charity—and charity requires time.
In _Of Grammatology_ , the book in which Derrida makes this claim, he is engaged in an extended analysis of an essay by early modern thinker (and Genevan) Jean-Jacques Rousseau, "On the Origin of Language," which addresses a topic that deeply interests Derrida. In answering the question about the origins of language, Rousseau tends to think that language is an obstacle to the world, that language gets in the way of just experiencing the world itself. Language is a lens through which we see the world, albeit with some distortion, simply because this lens stands between us and the world. As soon as there is a lens, there is distortion. We can buff this lens for days or grind it as thin as possible, but this lens is mediation, and as soon as there is mediation, for Rousseau, there is distortion. Thus, Rousseau suggests that language is something that befalls us as a contingent evil, in a way corrupting what was a pure, unmediated experience of the world simply "as it is." Like Leonard in _Memento_ , we have a condition (a disease, an illness) that requires us to use language to make our way in the world. Rousseau longs for the good old days (what he calls "the state of Nature") when we weren't afflicted with this condition and could simply experience the world the way it is without mediation—without anything between us and the world. In other words, for Rousseau, as soon as the lens of language is inserted, we have to interpret the world. As soon as there is mediation, there is interpretation. The "state of Nature" is a state of immediacy where we don't have to "interpret" things; we simply "know" what they are. That's a cup. That's my wife. This is a computer. It's clear and simple. For Rousseau, Leonard— with his condition—is a freak, literally _un_ -natural.
But was there ever a time without interpretation? Will there ever be a time when we don't interpret? Does one ever just simply see a cup "as it is"? Enter Derrida. Although Rousseau offered his theory in the sixteenth century—at the center of the birth of modernity—Derrida thinks that most of us twenty-first-century inhabitants are Rousseaueans at heart. This becomes most clear in our ideas of what it means to read.
Often when we read—and biblical commentaries tend to be a great case study for this—we imagine that the text or the language of the book is something we have to get _through_ in order to recover the author's original intention. In other words, the text becomes a hurdle that we have to jump over—or a curtain we need to pass through—in order to get to what is behind the text, such as the author's idea or the referent (the thing to which the text points). Sometimes we concede that such a process requires that bothersome thing called interpretation—as when we're reading a poem or C. S. Lewis's more allegorical works. Then we concede that there is a kind of code that needs to be broken in order to understand the text. But most of the time, we don't think we interpret; we simply read. In these cases we assume that the text under consideration is clear and therefore doesn't require interpretation. We might need some background or context, but once those pieces are in place, we don't need to interpret. Instead, the text takes on a kind of transparency so that we can simply see what it means. So unlike Leonard, who needs notes and texts to help explain his world, we can move around without such supplements. When I read the newspaper, I don't need to "interpret"; I simply need to read. And most of us think that when we read the Bible, the same is true: yes, some passages are difficult, or the poetry of Song of Solomon might throw us for a loop, but if we're reading Paul's Epistle to the Romans, things are pretty clear. We simply need to provide a commentary that gives us the background and context. Such a commentary is like a cloth that cleans the text to grant it the transparency that makes interpretation unnecessary.
Derrida recognizes this kind of reading (he calls it "doubling commentary") and even concedes that there is a time and place for this kind of project. However, he worries that it assumes a kind of Rousseauean naiveté precisely because it assumes there can be a reading (or even experience) that does not involve interpretation. In other words, it assumes that we (who are either "normal," "healed," or "redeemed"—not beset by a "condition")are different from Leonard. Leonard is a freak; we are normal. Leonard needs notes and texts; we can simply look at the world and see it "as it is." Even if we are reading a text, we can get past it to what is behind it or the thing it is pointing to.
For Derrida, this is a naive assumption because it fails to recognize that we never really get "behind" or "past" texts; we never get beyond the realm of interpretation to some kind of kingdom of pure reading. We are never able to step out of our skins. Texts and language are not something that we get through to a world without language or a state of nature where interpretation is not necessary. If the text is construed as an Alice-in-Wonderland-like looking glass, on the other side is not a world without language or interpretations but simply more texts and interpretation. Down the rabbit hole of our experience, it is language all the way down. Thus, just before making his famous claim that "there is nothing outside the text," Derrida says that a reading or interpretation "cannot legitimately transgress the text toward something other than it, toward a referent . . . or toward a signified outside the text whose content could take place, could have taken place outside of language, that is to say, in the sense that we give here to that word, outside of writing in general" ( _OG_ , 158). In other words, if a line of text says, "The blue cup sat on Pilgrim's table," and I understand what it means (I can picture a blue cup sitting on a table), I have not, according to Derrida, stepped out of the realm of interpretation. Interpretation is not a series of hoops we jump through to eventually reach a realm of unmediated experience where we don't have to interpret anymore. Rather, interpretation is an inescapable part of being human and experiencing the world. So even this blue cup sitting on my table, from which I am drinking my coffee "firsthand," as it were, is still a matter of interpretation.
When Derrida says that we can't get beyond or behind the text to a referent (or signified) that is outside language, he means this in a radical way. There are a couple of less radical ways we could understand this, which he notes but does not emphasize. First, when he claims that there is nothing outside the text, this isn't simply because "Jean-Jacques' life, or the existence of Mama or Thérèse _themselves_ is not of prime interest to us" ( _OG_ , 158). In other words, he doesn't mean that we can just choose to act _as if_ Mama doesn't exist and play with the text without caring about what it really refers to. That there is nothing outside the text is not a voluntary condition that we can choose to effect. Second, when he claims that there is nothing outside the text, this is not simply "because we have access to their so-called 'real' existence only in the text and we have neither any means of altering this, nor any right to neglect this limitation" ( _OG_ , 158). For instance, one might claim that there is no Socrates outside the text because the only access one has to Socrates is now through the texts of Plato or Aristophanes. In that sense there would be no Socrates outside the text.
While both these reasons would be sufficient grounds to proclaim that there is nothing outside the text, Derrida says "there are more radical reasons" ( _OG_ , 158). He goes on to note: "In what one calls the real life of these existences 'of flesh and bone,' beyond and behind what one believes can be circumscribed as Rousseau's text, there has never been anything but writing" ( _OG_ , 159). It is not just that writing or texts are the portal through which we must pass in order to get to things or the gates that provide access to an uninterpreted reality; rather, when Derrida claims that there is nothing outside the text, he means there is no reality that is not always already interpreted through the mediating lens of language. Textuality, for Derrida, is linked to interpretation. To claim that there is nothing outside the text is to say that everything is a text, which means not that everything is a book, or that we live within a giant, all-encompassing book, but rather that everything must be interpreted in order to be experienced. Thus he is _not_ a linguistic idealist who denies the material existence of cups and tables; rather, in the line of Martin Heidegger (of _Being and Time_ ), he is what we might call—for lack of a better term—a comprehensive hermeneuticist who asserts the ubiquity of interpretation: all our experience is always already an interpretation.
Texts that require interpretation are not things that are inserted between me and the world; rather, the world _is_ a kind of text requiring interpretation. Even experiencing a cup "in person" or "in the flesh" demands that I interpret the thing as a cup, and this interpretation is informed by a number of different things: the context in which I encounter the thing, my own history and background, the set of presuppositions that I bring to the experience, and more. Given all these conditions, the things I experience are subject to interpretation—and as such, they are subject to _different_ interpretations. Let's consider another example from film, not the heady world of _Memento_ but the Disney world of _The Little Mermaid_. While I think the main character, Ariel, is corrupted by a consumerist desire, I'll bracket my extended social critique of the movie and here consider just one scene that illustrates the point that our firsthand experience of things themselves requires interpretation.
Ariel, though she is the daughter of King Triton and has all of the undersea world at her fingertips, still wants more. Indeed, the anthem of the film is her _cri de coeur_ in which she expresses her heartfelt desire for more—to be part of the human world. Still confined to life under the ocean's surface, every once in a while Ariel ascends to the surface, where she has befriended Scuttle, a seagull who, obviously, lives above the sea and has contact with the human world of sailors and landlubbers. Scuttle is thus a mediator between the undersea world of Ariel and the human culture she desires to join. One of the ways she harbors this desire for human culture is by collecting artifacts from the human world and storing them in a kind of _Wunderzimmer_ , or "room of wonders," her own undersea museum of humanity. Scuttle is one of her primary resources, not only acquiring the items for her but also naming and explaining them. When he gives her what we recognize as a smoking pipe, Scuttle tells Ariel it is a "snorflap," which is used for making music (and bubbles) by blowing into the mouthpiece. When he adds a fork to her collection, he names it a "dinglehopper" and explains that it is used for styling one's hair, like a comb or a brush. Each of the items in Ariel's collection—and her understanding of what they are—is situated by Scuttle's explanation.
Through a number of machinations, Ariel finally gets a chance to make a foray into the human world. In particular, she is afforded the opportunity to try to win the love of a prince with whom she has been infatuated ever since she glimpsed him during one of her trips to the surface. One of the prices for her adventure into human culture was selling her melodious voice to an evil witch. So when she encounters the prince, she can't talk or explain herself. Although mute, Ariel is nonetheless charming and beautiful, and the prince invites her to join him for dinner at his castle. Ariel excitedly accepts and, before we know it, is seated at the prince's table—a site for communion with this human world. This world, of course, is not her own, and she has difficulty navigating her way through the experience (on top of learning to walk). So when she is seated to dine and finally recognizes something familiar in this strange world, she is eager to demonstrate her facility with this cultural artifact. What does she spy there but—can it be?—a _dinglehopper_! Immediately she seizes the item and begins brushing her hair with the flair of a longtime user. The prince, as you can imagine, is puzzled by such a strange employment of a fork!
This thing—this strangely shaped piece of metal—even when we find it sitting on the table right in front of us, is subject to interpretation. Given our horizons of experience, our past history, what we've been told, and thus a whole host of presuppositions that we bring to the experience, we immediately see the object as a fork (and find it difficult to really see it as anything _but_ a fork). But for Ariel—with her different history, different experience, and thus different presuppositions—the item is interpreted as a dinglehopper. While it might seem as though we don't even interpret the object, we actually go through the interpretive process so quickly, without even thinking about it, that it seems as if we're not engaged in interpretation. But the speed with which the object is construed as a fork does not negate the fact of interpretation or the interpretive process involved. So we never get past texts and interpretations to things "simply as they are" in any kind of unmediated fashion (as Rousseau supposed); rather, we move from interpretation to interpretation. All the world is a text. Thus, "there is nothing outside the text."
Derrida does not deny that the material phenomena we bump into—like forks and cups—have existence outside books and our minds. As he has repeatedly emphasized, he is not advocating a kind of linguistic idealism. As such, the earlier criticisms of Derrida by Christian thinkers miss the mark. But does this mean there is nothing about Derrida's claim that might make Christians nervous? With an eye to appreciating the implications of Derrida's claim, we could loosely translate "There is nothing outside the text" simply with the axiom "Everything is interpretation." Or, in other words, "It is interpretation all the way down." For many Christians, this isn't much of an improvement. For some reason, at this point many Christians become nervous and assume that the claim that "everything is interpretation" is antithetical to Christian faith. Even if we understand Derrida's claim not as linguistic idealism but rather as ubiquitous interpretation, it would seem that we have a third reason why Derrida's claim is antithetic to Christian faith. If everything is interpretation, then even the gospel is only an interpretation and not objectively true.
**Derrida at the Foot of the Cross**
Let's consider this criticism of Derrida more carefully. If the claim that there is nothing outside the text means that everything is interpretation, then the gospel would be only an interpretation. If it is only an interpretation, then that means there might be other interpretations. And if the gospel is only an interpretation and there could be other interpretations, we can't know if the gospel is true. A version of this criticism can be found in D. A. Carson's criticisms of the emerging church. Carson is clearly worried that because folks like Stanley Grenz, Brian McLaren, and other "hard postmodernists" (as he calls them) reject modern notions of absolute or "objective" truth, they are giving up on truth altogether. But in his criticisms, it becomes clear that Carson simply conflates truth with objectivity: for Carson, one can only be said to know "truly" if one knows "objectively."While Carson rightly notes that human knowledge can never pretend to omniscience, this doesn't mean we can't claim to know in a finite but real manner. But his affirmation of finite knowledge always elides into an affirmation of objective knowledge. Although he does not define objectivity (quite an oversight, given his project), Carson clearly means this to carry some connotation of selfevident givenness: if a truth is objective, then it is not a matter of interpretation. Thus, if Derrida is not a linguistic idealist but nevertheless asserts that everything is interpretation, then according to folks like Carson, such a claim is antithetical to the (supposedly biblical!) requirement that what is true be objective. If the gospel is an interpretation, and therefore not "objective,"then it would seem that it cannot be true.
Again, allow me to put this into slow motion in order to do justice to the issues. On the one hand, this criticism is right. I would agree that the gospel is an interpretation and that we can't _know_ the gospel is true, _if_ by knowledge we mean unmediated objectivity or pure access to "the way things are" (a Rousseauean dream).On the other hand, it is wrong to conclude that this is antithetical to orthodox Christian faith. This third kind of criticism is loaded with unjustified assumptions about the nature of interpretation and the question of truth because it assumes that if something is an interpretation, it can't be true; or, conversely, it assumes that if something is true, it must be objective. As such, it harbors something of the Rousseauean notion that assumes interpretation is a disease—like Leonard's condition—that pollutes and corrupts our relationship to the world. But the fact that something is a matter of interpretation does not mean that an interpretation cannot be true or a good interpretation. When I construe this thing in front of me as a cup and use it to drink my coffee, although I am interpreting the cup, I am also interpreting it _well_. True, the cup does not exist as some brute fact, but that doesn't mean that my interpreted understanding of the cup is not good or true.
Let's get right to the heart of the matter by means of a thought experiment. Our central question here is whether Derrida's (paraphrased) claim that everything is interpretation is antithetical to orthodox Christian faith. I suggest that it is not. To do so, imagine with me two inhabitants of Jerusalem in the early first century. Here is their account of a particular day's event, not unfamiliar to us:
It was the Passover, and like most others—both natives and visitors for the feast—we'd heard about the events transpiring, first in the governor's palace and then just outside the city on Golgotha. So out of curiosity—we're too busy for silliness like religion—we made our way outside the city gates to catch a glimpse of the spectacle [CNN not yet being there to deceive them into thinking they've seen events unfold from the comfort of their living rooms]. There was a great deal of commotion—more than usual for Passover. Around Golgotha we heard both sneers and weeping, some people taunting and others sobbing. We were there to see some of the Roman guard gambling for a few scraps of clothes (Matt. 27:35), as well as some of the religious authorities gloating over their power. The center of attention, of course, was the crosses on this clump of a hill—particularly the one in the center, where a Nazarene hung. We've said it for years: What good can come out of Nazareth! Nazarenes end up as either servants, prisoners, or unfortunates punished for capital crimes. This one just confirmed the point again. On the central cross where this Nazarene was hanging—pathetic and beaten—a sign was posted proclaiming that he was "King of the Jews" (with something written in other languages, too, but we didn't know what they said). Are all these women crying because they actually believed that? we wondered. We're not sure who was more pathetic: the one hanging there on the tree or those who had put their hope in a Nazarene. King of the Jews. What is this? A joke? Judging from the smug laughter of the chief priests, that must be it.
After lunch, things did get a little strange: it was dark until about three o'clock in the afternoon—but authorities later reported it had been a solar eclipse. As the light began to return, the pathetic Nazarene started blurting out something about Elijah—but in doing so he seemed to expend his last gasp of energy and hung lifeless on the cross. Another cross, another Nazarene, another criminal—one less to worry about. The coincidence of an earth tremor at the same moment contributed to the commotion, but soon enough the crowds began to disperse. Sure, the women remained there weeping, still clinging to their naive hope that a Nazarene could be some kind of king—but we had been cured of such illusions long ago. So, like most of the other curiosity seekers, we joined the caravan heading back into the city, hoping to find some supper left at home.
Now consider an alternative account, that of a centurion posted as a guard on Golgotha that same afternoon (cf. Matt. 27:54):
I'm not sure why my number was called, except for experience and a certain acquaintance with the events leading up to that day. My responsibility—along with the rest of my cohort—was to keep surveillance at the scene of a crucifixion just outside Jerusalem. In many ways, this was like a hundred other crucifixions I had witnessed, but with a special, almost laughable twist: that day we would execute a Nazarene—a Nazarene!—who had claimed to be a king, "the King of the Jews." Everyone, both we Romans as well as Jerusalem's own religious leaders, got the joke, so the sign that we posted on the Nazarene's cross was intended just as a way of letting others in on it.
The background experience that familiarized me with the situation was somewhat secondhand. An old friend, Antony—a fellow with whom I'd gone through the academy—at one time had a direct encounter with this Nazarene [cf. Matt. 8:5–13]. When he was stationed just outside Capernaum, not far from Nazareth, news and rumors of the healings performed by a Jesus—a carpenter from Nazareth—often circulated throughout the area. Antony's servant, who had ministered to Antony in his own illness, was lying paralyzed at home, writhing in agony from a pain that Antony couldn't see and an illness that Roman doctors could not diagnose. In the throes of both desperation and what must have been madness, Antony made his way to Capernaum to make one last effort on behalf of his suffering servant. As Antony later recounted the story to me, what appeared mad desperation when he left his house took on both hope and a kind of reasonableness when he found himself near this Jesus. Though he had left his house with slim hope and vast doubts, in the presence of this Nazarene he found himself energized by deep faith and great hope. With a confidence that almost startled him, Antony told this Jesus: "Just say the word, and I know that my servant will be healed." By the time Antony reached home, his servant was already fixing dinner! But Antony wasn't surprised.
Not having been there, though being infected somewhat by the elation in Antony's narration, I found it easy in the months that followed to either forget the story or discount it. True, there was a certain irony when I was later given the assignment of overseeing the Nazarene's crucifixion, but life is full of such funny coincidences. A couple of times Antony's tale slipped out from the caverns of my memory into consciousness, but for the most part this was just another criminal, just another Nazarene. Another day, another crucifixion.
Of course, this particular crucifixion was unlike others: the chief priests and religious leaders weren't usually milling about outside the city gates like this. And they usually didn't take such delight in the crucifixion of a Jew. And while we could usually expect to find the mothers of criminals wailing at the foot of the cross, the cohort of women weeping for this Jesus was certainly out of the ordinary. And aside from continuing to protest their innocence and casting insults at those of us just doing our jobs, these defeated beings hanging on the cross usually didn't say much. And they certainly didn't say the kind of things this Jesus said.
At first I found it pitiable and a bit irritating, listening to this Nazarene making these pronouncements. But admittedly, I'd never heard a crucified one pray for us to be forgiven or promise other criminals that they would join him in paradise. While others in our cohort mocked and gambled for his clothes, slapping the backs of the chief priests, who usually detested them, I found myself retreating from the experience, as it were—finding myself a foreigner in a strange land, going through the motions of my job but with Antony's story ringing in my ear, reverberating with what the Nazarene himself had just said: "It is finished."
After that, things became eerie as the sky darkened for three hours. We had to remain on guard, not sure what was happening, but the darkness seemed to discourage not only commotion but even speech. And so there I found myself, in the dark, effectually alone (though surrounded by hundreds), with a strange silence echoing in such a way that it seemed to be asking me a question: "Who is this man?" I can't replay for you the turmoil of that questioning. I can simply witness to this: when the darkness lifted, and an earthquake seemed to shake the very foundations of the earth itself, those for me were but tiny cosmic ripples of a dawning within my own life and a shaking of my own foundations. For at that point, after wrestling with God for over three hours—struggling with the testimonies of Antony and my own experience on Golgotha—I could finally articulate an answer to my question: "Truly this was the Son of God!"
These two extended accounts of a series of events illustrate the way in which the gospel itself is an interpretation, helping us to appreciate Derrida's claim—and why Derrida's claim is not antithetical to Christian faith. Each of these accounts is an interpretation of events in first-century Jerusalem. Each of them is a response to Jesus of Nazareth. Each of them is a "reading" of what took place and the phenomena in front of each narrator. Each is a kind of textual rendering of what happened. But, of course, the renderings and interpretations are very different. If we, as Christians, agree with the interpretation of the centurion—and with him confess that this is the true account of what took place on that afternoon—our agreement does not mitigate the fact that this is an interpretation. If we appeal to God's special revelation about these events attested in the Scriptures, this does not change the fact that it remains an interpretation of what took place. In fact, the appeal to revelation only strengthens the claim that the centurion's reading is an interpretation: without that revelation we might be in the situation of the two natives of Jerusalem: all we see is another cross, another Nazarene.
Revelation informs our horizon. However, even the (objective) provision of a revelatory interpretation does not guarantee that everyone will read the event in this way. One must (subjectively) accept this revelatory interpretation, which requires faith—and such faith requires the regenerating work of the Holy Spirit. Again, see Owen's discussion in _The Holy Spirit_ (repr., Grand Rapids: Kregel, 1954). As he puts it, God's provision of objective light (revelation) does not resolve the problem of subjective darkness (148ff.). In other words, the objective provision of revelation in the Scriptures is ineffectual as revelation (i.e., to communicate) without the regeneration of the heart and mind in order to dispel blindness.
Christians who become skittish about the claim that everything is interpretation are usually hanging on to a very modern notion of knowledge, one that claims something is true only insofar as it is objective—insofar as it can be universally known by all people, at all times, in all places. On this account, the truth of the gospel—that God was in Christ reconciling the world to himself— is taken to be objectively true and thus capable of rational demonstration. (Classical apologetics buys this epistemology, or theory of knowledge.) If we say that the gospel is an interpretation, then it is not objectively true in the traditional or modern sense of being self-evident or universally demonstrable.
The problem with this very modern construal of the gospel is that it doesn't match up with the witness of the New Testament. It is clear from the Gospel narratives, for instance, that not everyone sees what the centurion sees. Of course, they all see and encounter the same material realities—crosses, bodies, and eventually corpses—but these material phenomena are texts that need to be interpreted. Thus the very fact that both the centurion and the chief priests are confronted by the same phenomena and yet see something very different seems to demonstrate Derrida's point: the very experience of the things themselves is a matter of interpretation. Even if we are confronted with the physical and historical evidence of the resurrection—even if we witnessed the resurrection firsthand—what exactly this meant would require interpretation. Only by interpreting the resurrection of Jesus does one see that it confirms that he is the Son of God (Rom. 1:4). As John Owen observes: "That Jesus Christ was crucified, is a proposition that any natural [i.e., unregenerate] man may understand and assent to, and be said to receive: and all the doctrines of the gospel may be taught in propositions and discourses, the sense and meaning of which a natural man may understand; but it is denied that he can receive the things themselves. For there is a wide difference between the mind's receiving doctrines notionally, and receiving the things taught in them really."
Moreover, in the epistles we get the same kind of claim, namely, that not everyone can see what the believer sees. While God's invisible attributes are, on the one hand, "clearly seen" (Rom. 1:20), Paul goes on to emphasize the way in which this is not seen by those whose "foolish hearts were darkened" (1:21), who thus construe or interpret the world as something other than God's creation. While I agree that interpreting the world as creation is the true interpretation, this does not negate its status as an interpretation. What is required to interpret the world well is the necessary conditions of interpretation—the right horizons of expectation and the right presuppositions. But as Paul repeatedly emphasizes, these conditions are themselves a gift; in other words, the presuppositions and horizons that make it possible to properly "read" creation are grace gifts that attend redemption and regeneration (Rom. 1:18–31; 1 Cor. 1:18–2:15; Eph. 4:17–18). This is precisely why we shouldn't be surprised that not everyone we encounter immediately grasps the rationality of the gospel. In fact, we should expect that someone will not be able to properly "see" creation or the crucifixion without the grace of redemption. Or, to put it another way, presuppositional apologetics—such as that developed by Francis Schaeffer, but also by Cornelius Van Til and, to a degree, Herman Dooyeweerd—rejects classical apologetics precisely because presuppositionalism recognizes the truth of Derrida's claim that everything is interpretation (though I am admittedly radicalizing their intuitions).
To embrace this (creational!) reality of ubiquitous interpretation requires that we embrace the corresponding reality of pluralism. Wherever there is interpretation, there will be conflict of interpretation or at least differences of interpretation. However, it is important to consider two levels, or modes, of this hermeneutic pluralism. On the one hand, a kind of pluralism and interpretive difference is inscribed into the very fabric of created finitude, such that we all see the same things but from different angles and locations. We all bump into the same stuff; it's just that some see it as a dinglehopper, others as a fork. In both Eden and the eschaton, we find interpretive pluralism that is rooted in this plurality of perspectives. As a factor of the conditions of a good creation, this kind of pluralism is something we must embrace as good (Gen. 1:31).And such interpretive pluralism remains a reality within the church. On the other hand, a kind of deep "directional"pluralism is endemic to our postlapsarian (postfall) condition; that is, there is a level of interpretive difference that concerns fundamental issues such as what it means to be authentically human and how we fit into the cosmos. In this respect, for instance, Christianity and Buddhism have very different interpretations about the nature of reality. However, we need to consider these as deep differences in interpretation rather than glibly supposing that the Christian account is objectively true and then castigating the Buddhist account for being merely an interpretation. In fact, both are interpretations; neither is _objectively_ true. And so, to a certain extent, we must also embrace this postlapsarian or directional pluralism as the given situation in which we find ourselves. To assert that our interpretation is not an interpretation but objectively true often translates into the worst kinds of imperial and colonial agendas, even within a pluralist culture. Acknowledging the interpreted status of the gospel should translate into a certain humility in our public theology. It should not, however, translate into skepticism about the truth of the Christian confession. If the interpretive status of the gospel rattles our confidence in its truth, this indicates that we remain haunted by the modern desire for objective certainty. But our confidence rests not on objectivity but rather on the convictional power of the Holy Spirit (which isn't exactly objective); the loss of objectivity, then, does not entail a loss of kerygmatic boldness about the truth of the gospel.
Deconstruction's recognition that everything is interpretation opens a space of questioning—a space to call into question the received and dominant interpretations that often claim not to be interpretations at all. As such, deconstruction is interested in interpretations that have been marginalized and sidelined, activating voices that have been silenced. This is the constructive, yea prophetic, aspect of Derrida's deconstruction: a concern for justice by being concerned about dominant, status quo interpretations that silence those who see differently. Thus, from its inception, deconstruction has been, at root, ethical—concerned for the paradigmatic marginalized described by the Old Testament as "the widow, the orphan, and the stranger." To put it differently: Wall Street and Washington both want us to think that their rendering of the world is "just the way things are." Deconstruction, by showing the way in which everything is interpretation, empowers us to question the interpretations of trigger-happy presidents and greedy CEOs—in a way not unlike the prophets' questioning of the dominant interpretations of the world. As such, we are free to interpret the world differently. When we reflect on the implications of deconstruction for the church later, we'll consider this prophetic conclusion of deconstruction's claim in more detail.
**Texts in Community**
Derrida's claim that there is nothing outside the text was often misunderstood, and not just by Christian theologians. Later, when presented with the opportunity, Derrida tried to clarify his claim: "The phrase that for some has become a sort of slogan of deconstruction, in general so badly understood ('there is nothing outside the text'), means nothing other than: _there is nothing_ _outside context_."In a way, Derrida is repeating the axiom of real estate as a central condition of interpretation: location, location, location! The context of both the phenomenon (whether a book, a cup, or an event) and the interpreter function as conditions or frameworks that determine just how a thing is seen or understood. Just as he claims that there is nothing outside the text, elsewhere Derrida claims that "there are only contexts." Context, then, determines the meaning of a text, the construal of a thing, or the "reading" of an event. For instance, part of the context of the centurion's "reading" of the crucifixion was his compatriot's earlier experience with the gentle healer from Nazareth—a context that the two natives of Jerusalem lacked. (I would also argue that grace formed part of the centurion's context.)
When Derrida talks about how contexts are "determined" or "filled in," we find a very important (though largely ignored) emphasis in his work: the role of community in interpretation. As he explains in his afterword to _Limited Inc_ , contexts are flexible and dynamic: contexts change as time and place changes, generating different meanings and interpretations. Derrida describes this as the possibility of _recontextualization_ : a phrase can mean one thing in one context and something different in another—just as the metal item is a fork in one context and a dinglehopper in another. Contexts change, and therefore meanings are given to change: if I shout "Duck!" in a field while we're hunting, you will look upward for a target; if we're golfing and I shout "Duck!" you should assume a fetal position to avoid an incoming projectile. The same word, _duck_ , is recontextualized. And insofar as a context can never be completely "filled in," any text, thing, or event is susceptible to different construals and interpretations. Because Derrida has emphasized this play and flexibility of contexts, many have concluded that he thinks we can just interpret things any way we want—that texts and events can be played with and we can simply make up the meaning as we go. For instance, they think Derrida's claim means that you can make the Bible say anything you want.
Of course, on the one hand, this is completely true—and once again, if we look at our experience, we see that Derrida is right:people and groups do interpret the Bible in all kinds of ways, and they do make the Bible say whatever they want it to say. We all know the truism that you can prove just about anything by quoting from the Bible, whether it be a justification of slavery or why Christians shouldn't have mortgages. Obviously, the Bible is subject to all kinds of interpretations. But this play of interpretations does not mean that all these interpretations are good or true. Deconstruction does not entail that one can say just anything at all about a text; it is not a celebration of sheer indeterminacy. "Otherwise," Derrida protests, "one could indeed say just anything at all and I have never accepted saying, or encouraging others to say, just anything at all."Instead, Derrida emphasizes that there are important, legitimate determinations of context; in particular, the context for understanding a text, thing, or event is established by a community of interpreters who come to an agreement about what constitutes the true interpretation of a text, thing, or event. Given the goals and purpose of a given community, it establishes a consensus regarding the rules that will govern good interpretation. So, within the human community, the metal object with pointy ends is to be understood as a fork and employed as an eating utensil, not as a comb. Although the object is susceptible to interpretation as a comb, such a construal is ruled a bad interpretation by the community. Without the rules established by a community, there would be no criteria to govern interpretation. And Derrida is not opposed to rules as such. In fact, he speaks positively about a community having a kind of "interpretive police" to govern interpretation for that community. Thus communities fix contexts, and contexts determine meanings. This role of community will become central as we think about what it means to interpret the Scriptures.
**Taking Derrida to Church**
We've now come quite a way from the bumper-sticker misunderstandings of Derrida to a clearer understanding of Derrida's claim that there is nothing outside the text. As we've seen, he means roughly that everything is interpretation; interpretation is governed by context and the role of the interpretive community. This entails abandoning the modern notion of objectivity and embracing a central theme of postmodernism: interpretation goes "all the way down."
But we still haven't fully answered our third objector, who suggested that if interpretation goes all the way down, then the gospel can't be known to be true. We've begun to answer that objection by showing the way in which the gospel must be an interpretation. But what are the implications of accepting Derrida's claim that there is nothing outside the text? What would that mean for our understanding of the gospel, Scripture, and church?
_Seeing the World through the Word_
Derrida has suggested that all the world is a text. As a text, it is subject to interpretation, and interpretation brings in the role of our horizons of perception and our presuppositions. These horizons or presuppositions are informed by our fundamental beliefs about the world as well as our past experiences and encounters with the world. There is no uninterpreted reality, no brute facts passively sitting there to be simply and purely seen. Rather, we see the world always already through the lens of an interpretive framework governed by ultimate beliefs. We could say that we always already see the world through a worldview. And part of Derrida's claim, much like the claim of presuppositional apologists like Schaeffer and Van Til, is that this is the case for everyone.We all—whether naturalists, atheists, Buddhists, or Christians—see the world through the grid of an interpretive framework—and ultimately this interpretive framework is reli-gious in nature, even if not allied with a particular institutional religion.
This insight should help us appreciate two things: First, if one of the crucial insights of postmodernism is that everyone comes to his or her experience of the world with a set of ultimate presuppositions, then Christians should not be afraid to lay their specifically Christian presuppositions on the table and allow their account to be tested in the marketplace of ideas (we'll talk more about this in chapter 2). In a way, Derrida has brought the broader culture to appreciate what Christian thinkers like Abraham Kuyper, Herman Dooyeweerd, Cornelius Van Til, and Francis Schaeffer have been saying for a long time: that our ultimately religious presuppositions govern our understanding of the world. Second, and more constructively, this should push us to ask ourselves whether the biblical text is what truly governs our seeing of the world. If all the world is a text to be interpreted, then for the church the narrative of the Scriptures is what should govern our very perception of the world. We should see the world through the Word. In this sense, then, Derrida's claim could be resonant with the Reformers claim of _sola scriptura,_ which simply emphasizes the priority of God's special revelation for our understanding of the world and making our way in it. There is nothing outside the Text, we might say. And to say that there is nothing outside the Text, then, is to emphasize that there is not a single square inch of our experience of the world that should not be governed by the revelation of God in the Scriptures. To say that there is nothing outside the Text is to say that there is no aspect of creation to which God's revelation does not speak. But do we really let the Text govern our seeing of the world? Or have we become more captivated by the stories and texts of a consumerist culture? Is our worldview shaped by the narratives of a hip-hop culture more than the stories of God's covenantal relationship with his people? One of the challenges of Christian discipleship is to make the text of the Scripture the Text outside which nothing stands. As U2's song "When You Look at the World" attests, this is not always easy; sometimes I "can't see what You see, when I look at the world." But the sanctification of the Spirit is aimed at enabling us to see the world through this lens.
_Interpreting as if We Believe the Apostles' Creed_
In many ways, modernity is characterized by a deep individualism that isolates us from one another, sealed up in our little egos or private spheres. It is not unimportant that Descartes' _Meditations_ —in many ways a manifesto of modernity—was the result of Descartes shutting himself up in his room alone for a number of days just to think by himself. This modern isolationist understanding of the human self has often crept into the church, which has too often valorized a notion of private interpretation (by wrongly appealing to the Reformation principle of the perspicuity of Scripture), suggesting that the meaning of the Scriptures is simply and objectively there—available for the taking.
Such an individualistic notion, however, has nothing to do with the Reformers, let alone the ancient church. As we confess in the Apostles' Creed, we believe in both "the holy, catholic church" and "the communion of the saints." And Derrida's critique of modernity, along with his emphasis on community, helps us appreciate the way in which postmodernity pushes us to recapture the central role of community not only for biblical interpretation but also for teaching us how to make our way in the world. One of the things that Leonard lacks in the film _Memento_ is a community of friends he can trust. (In fact, one of his rules—tattooed on his body—is to trust no one.) But as Derrida demonstrates, we can't interpret a text, thing, or event without the conventions and rules of an interpretive community; indeed, language itself is inherently communal and intersubjective. (And eventually even Leonard has to trust someone else, as when he trusts the note written in Natalie's handwriting.) For instance, to interpret the Scriptures, and interpret them _well_ , I cannot shut myself off from the community that is the church; rather, I need to be formed and informed by the breadth of this community, both geographically (the global church) and temporally (history of the church's witness). While the church is governed by the Scriptures, the Scriptures are only properly opened and active within the believing community. To say that there is nothing out-side the Text also entails that there is no proper understanding of the Text—and hence the world—apart from the Spirit-governed community of the church. The same Spirit is both author of the text and illuminator of the reading community.
**A Deconstructive Church**
If, as I have claimed, Derrida has something to say to the contemporary church, what would a deconstructive church look like? Here I would like to offer a tour of a church that has engaged Derrida and see what it might look like in action.
First, one is struck by the difference between a deconstructive church that affirms there is nothing outside the Text and the kind of postmodern church sketched by others. As we participate in the worship of a deconstructive church, we find that the Text is central for shaping our interpretation of the world. In order to take the totality of the text seriously, the deconstructive church employs the revered tradition of the lectionary, which, over the course of a few years, guides us through the entirety of the Text's narrative, rather than leaving us to the private canons and pet texts of the pastor. This use of the lectionary is part of a general impression that tradition is valued in the deconstructive church. We recite the ecumenical and historic creeds because these are the witness of our community past—the way for us to hear the interpretations of the ancient community, which was indwelt by the same Spirit that indwells us and grants illumination today. The pastor's preaching indicates a serious engagement with the early fathers and the Reformers as co-interpreters. All of this helps us understand that the church is a community, a "holy, catholic church," which has endured through millennia.
The voices of the community are not only ancient, however; they are also global. The singing and prayers are drawn from Christian communities in southern France and South Korea, from Scotland and Zimbabwe. These other voices—so often marginalized by the Western church—are received as voices of the Spirit at work in our global brothers and sisters, illuminating us by illuminating them. Thus the deconstructive church, while having a sense of being traditional, is nevertheless characterized by a diversity and global concern that disrupts the status quo. The deconstructive church embraces tradition but not the traditional _ism_ of the status quo. It is a community of interpretation that values marginalized readings—largely because the "foolishness" of the gospel itself is an interpretation of the human condition sidelined by secular modernity. To proclaim the gospel is actually already to speak from the margins.
Finally, we note that this church, while recognizing that the gospel is an interpretation of the world and the human condition— perhaps _because_ it recognizes this is an interpretation—focuses on the proclamation and witness of revelation. It does not focus on an apologetics of demonstration or on a "culture wars" agenda that, using logic as a weapon, seems to think that all Americans should simply see that Christianity is true. In fact, we can't help but be impressed by the prophetic stance of the deconstructive church with respect to its culture. One of the primary goals of the worship experience—embracing Word, sacrament, prayer, and singing—is to equip and empower the saints to see through the interpretations of the world and the human prospect offered by the cultural forces of capitalism, consumerism, and hedonism. In other words, the worship in a deconstructive church is aimed at forming believers who can recognize Wall Street's construal of happiness _as_ an interpretation as well as articulate the countercultural gospel's interpretation of human flourishing. The deconstructive church, in other words, is deeply prophetic—reflecting the voice not so much of Derrida as of Amos.
1. _Memento,_ DVD, written and directed by Christopher Nolan (Culver City, CA: Columbia TriStar Home Entertainment, 2001).
2. Leonard repeatedly claims, "I know who I am; I know all about myself." In fact, this is far from the truth, as Teddy constantly reminds him: "You don't know who you are." He knows only who he was.
3. The visual images are ultimately subject to the word, since pictures without writing are useless for Lenny. He writes on the photos to remind himself of what or who they are. He needs the text to interpret the image.
4. Jacques Derrida, _Of Grammatology_ , trans. G. Spivak (Baltimore: Johns Hopkins University Press, 1976), 158; henceforth referred to in the text as _OG_.
5. I have analyzed the hermeneutics of immediacy in much more detail in the first chapter of my _Fall of Interpretation: Philosophical Foundations for a_ _Creational Hermeneutic_ (Downers Grove, IL: InterVarsity, 2000).
6. Thus, I concede that Leonard in _Memento_ is not entirely a Derridean. Leonard still operates with a notion that there are facts that are not a matter of interpretation. Lenny thinks "memories are an interpretation" but that facts written on his body are not. But the film itself undermines Leonard's naive distinction, because one of the crucial facts that he writes down (the license number on Teddy's picture) is a pure fabrication just to give him someone to hunt down.
7. I have analyzed the process and conditions of interpretation in much more detail in my _Fall of Interpretation_ , especially chapter 5 (chapter 4 focuses on Derrida).
8. Because children's horizons are fluid and unsedimented, they are able to see the world in ways that adults do not: a pair of underpants can be construed as a space helmet, or for just about every boy (in North American culture, that is), any object is susceptible to being seen as a gun. As we gain experience, our horizons and presuppositions begin to solidify; as a result, our seeing becomes more quickly determined and habitual. But at the same time, it seems that we become less open to seeing differently. Much of twentieth-century art (e.g., Picasso) worked at trying to make our horizons of expectation fluid once again, inviting us to see differently, "with the eyes of a child," as it were. Thus Picasso once remarked that the key to his work was unlearning what he had learned as an adult and recovering the way he saw the world as a child (see E. H. Gombrich, _The Story of Art_ , 16th ed. [London: Phaidon, 1995], 573, 575). Given that we are invited to have faith "as a child," perhaps that entails keeping our horizons of expectation fluid rather than sedimented so that we are open to seeing God's miraculous work in the world.
9. Derrida later emphasizes that "there is nothing outside the text" does "not mean that all referents are suspended, denied, or enclosed in a book, as people have claimed, or have been naïve enough to believe and to have accused [me] of believing" (afterword to _Limited Inc_ , trans. Samuel Weber [Evanston: Northwestern University Press, 1988], 148). For further discussion, see my "Limited Inc/arnation: The Searle/Derrida Debate Revisited in Christian Context," in _Hermeneutics at the Crossroads: Interpretation in Christian Perspective_ , ed. Kevin Vanhoozer, James K. A. Smith, and Bruce Ellis Benson (Bloomington: Indiana University Press, forthcoming).
. See D. A. Carson, _Becoming Conversant with the Emerging Church: Understanding_ _a Movement and Its Implications_ (Grand Rapids: Zondervan, 2005), 105, 130–31, 143n46. The simplicity of this unwarranted conflation then underwrites the long concordance of passages he lists in chapter 7 (188–200), which he takes to be clear evidence that the Bible, because it speaks of "truth" and "knowing," therefore also advocates the modern epistemological notion of objectivity. As he puts it, "objective truth" is a "category . . . that both historic Christianity and the Bible itself have always insisted on" (126)! But this simply is not the case; in fact, as we will see below, the Scriptures give us good reasons to reject the very notion of objectivity, while at the same time affirming the reality of truth and knowledge.
. Here I agree with Carson that we should reject a common postmodern move that equates knowledge with omniscience, and then, since it is clear that such is not possible for finite beings, we must end up as skeptics with respect to knowledge (see ibid., 104–7). Such versions of postmodernism still adhere to the Cartesian criterion for knowledge; they just consider it impossible. We will consider the question of objectivity further in chapter 3, when we discuss why Christianity is not a metanarrative.
. For a helpful discussion of this point, see Philip Kenneson's "There's No Such Thing as Objective Truth, and It's a Good Thing, Too," in _Christian_ _Apologetics in a Postmodern World_ , ed. Timothy R. Phillips and Dennis Okholm (Downers Grove, IL: InterVarsity, 1995), 155–70.
. Owen, _Holy Spirit_ , 155.
. I have explained this in more detail in _The Fall of Interpretation_ , chapter 5.
. I am drawing here on a suggestive and underappreciated analysis of pluralism in Richard Mouw and Sander Griffioen, _Pluralisms and Horizons: An_ _Essay in Christian Public Philosophy_ (Grand Rapids: Eerdmans, 1993).
. Derrida, afterword to _Limited Inc_ , 136, emphasis added, translation modified.
. Jacques Derrida, "Signature Event Context," in _Margins of Philosophy_ , trans. Alan Bass (Chicago: University of Chicago Press, 1982), 320. For a more detailed discussion of this point, see my "Limited Inc/arnation."
. Derrida, afterword to _Limited Inc_ , 144–45.
. Ibid., 131, 146.
. This claim should be compared to Thomas Kuhn's analysis of the paradigms that govern scientific observation of the world in his landmark book _The_ _Structure of Scientific Revolutions,_ 2nd ed. (Chicago: University of Chicago Press, 1970). I have discussed this in my _Fall of Interpretation_ , 154–55.
. On this final point, see Roy A. Clouser, _The Myth of Religious Neutrality:_ _An Essay on the Hidden Role of Religious Belief in Theories,_ rev. ed. (Notre Dame: Notre Dame University Press, 2005); and idem., _Knowing with the Heart: Religious_ _Experience and Belief in God_ (Downers Grove, IL: InterVarsity, 1999).
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* * *
[Where Have All the
Metanarratives Gone?](Smit_ISBN9781441200396_epub_toc_r1.html#d7e3557)
[Lyotard, Postmodernism,
and the Christian Story](Smit_ISBN9781441200396_epub_toc_r1.html#d7e3557)
Perhaps no definition of postmodernity has seemed so opposed to Christian faith as Jean-François Lyotard's claim that postmodernism is "incredulity toward metanarratives." Isn't the Bible a metanarrative par excellence? Wouldn't being postmodern, then, require rejecting the Bible? That question is the central focus of this chapter.
**Raising the Curtain:** ** _O Brother, Where Art Thou?_**
We never tire of stories, especially grand epic tales of adventure and adversity. The Coen brothers' film _O Brother, Where Art_ _Thou?_ tells just such a grand story, replaying another epic— Homer's _Odyssey_ —in the context of Depression-era Mississippi. Like Homer's narrative, this story stars a Ulysses: the scheming and loquacious Ulysses Everett McGill, who goes by the more humble name Everett. Everett's odyssey begins on a work farm (he was sent up for practicing law without a license), where he is chained to two fellow convicts, Pete and Delmar. Needing to return home to prevent the marriage of his wife to another, Everett convinces Pete and Delmar to join him in his escape by telling them of a million-dollar treasure he buried after "knockin' off " an armored truck. The problem is that the site of the treasure is about to become the bottom of a lake: a river is being dammed in order to flood the valley, so there is some urgency to their adventure.
Like Homer's Ulysses, Everett and his crew encounter the usual suspects on their journey, including various versions of prophets, the Sirens, and a Cyclops. But what I find most intriguing in the film is an interesting—dare I say?—postmodern tension experienced by Ulysses Everett McGill. Everett is in many respects a devotee of modernity committed to a scientistic worldview. His trust is in reason, and like his Enlightenment forebears, he sees the traditions and "superstitions" of religion as an obstacle to true knowledge. When Pete and Delmar enter the waters of baptism and "get saved," Everett is incredulous and mockingly offers a quasi-Marxist account of the whole affair: "Well, I guess in hard times, flush the chumps! Everybody's lookin' for answers." After Delmar's "been saved," he exhorts Everett: "You should've joined us, Everett. It couldn'ta hurt none." But Everett will have none of it: "Join you two ignorant fools in a ridiculous superstition?! [Mocking and smug:] Baptism! You two're just dumber than a bag o' hammers."
While Everett always concedes that "everyone's lookin' for answers," he is confident that those answers are found in reason and "abstract thought," not in divination. The tales and fables of religion are to be overcome by the facts and propositions of science. However, Everett's modernist rationalism is challenged from beginning to end by a persistent religious and prophetic voice; his commitment to reason and science as the source of the answers everybody is looking for is called into question by the interruption of the divine, particularly by the tales and proclamations of a blind prophet. At the beginning of their journey they first encounter the blind prophet, who utters this revelation:
I work for no man. I have no name. You seek a great fortune. You will find a fortune, though it will not be the fortune you seek. But first you must travel a long and difficult road—a road fraught with peril. You shall _see_ things, wonderful to tell. You shall see a cow on the roof of a cotton house. I cannot tell you how long this road shall be. . . . Though the road may wind, yea your heart grow weary, still shall ye follow the way—even until your salvation.
Everett, as you would expect, responds with skepticism. "What the hell does he know? He's an ignorant old man." Throughout the story, Everett continues to resist the interruptions of the divine, seeing them as superstitions and silly stories in contrast to the facts and discoveries of science and rationality. But in the final chapters of the narrative, Everett's rationalism seems to meet its match.
After being pardoned by the governor of Mississippi and recapturing the love of his beloved Penelope, Everett has just one task remaining: to retrieve the wedding ring from their cabin. But when he arrives at the cabin, waiting for him and his companions is the diabolical warden, who has been pursuing them since their escape—the "devil" of the film who is interested not in proclamations of pardon but only in finishing his task: the execution of Everett, Delmar, and Pete. "End of the road, boys,"he hisses. "It's had its twists and turns. . . . You have eluded Satan, you have eluded me for the last time. . . . Perhaps you should start making your prayers," he mocks, tossing three nooses over the tree branch.
Pete launches into fervent prayers, beseeching God for mercy and forgiveness. And then, slowly, Everett descends to his knees and begins to pray. Everett, the devotee of modernity and master of suspicion, prays: "Please look down on us poor sinners. I just want to see my daughters again, Lord. I'm sorry. . . . Help us, Lord. Let me see my daughters again, Lord." At the conclusion of the prayer, water begins trickling through the dust surrounding their feet, and a rumbling can be heard in the distance, growing louder and louder. Suddenly, an Exodus-like wall of water comes crashing over the cabin, first sweeping away the devil/warden and then carrying out Everett, Delmar, and Pete. As they emerge to the surface, Delmar exclaims: "A miracle! That was a miracle!"
"Delmar, don't be ignorant," Everett replies. "I told you they was floodin' this valley."
"No, that ain't it," Delmar responds.
"We prayed to God, and he pitied us," Pete adds.
"Well, it never fails," Everett remarks. "Once again you two hayseeds are showin' how much you want for intellect. There's a perfectly scientific explanation for what just happened."
"That ain't the tune you were singing back there at the gallows!" Pete reminds him, to which Everett responds with a modernist soliloquy: "Well, any human being will cast about in a moment of distress. No, the fact is they're floodin' this valley so they can hydro-electric up the whole darn state. Yessir, the South is going to change—everything is going to be put on electricity and put on a payin' basis. Out with the spiritual mumbo jumbo, the superstitions, and the backward ways. We're gonna see a brave new world where they run everybody a wire and hook us up to a grid. Yessir, a veritable age of reason, like the one they had in France. Not a moment too soon." Just then, Everett sees a disconcerting sight: a cow on the roof of a cotton house.
Everett Ulysses McGill is never quite disabused of his (religious) commitment to modernity. Despite the persistent challenges to his scientistic faith, he clings to the religion of the Enlightenment. The film leaves the viewer with the responsibility for decision, as it closes with the blind prophet making his way down the tracks.
**Lyotard's Claim: Postmodernism Is Incredulity toward** **Metanarratives**
Postmodernism can be understood as the erosion of confidence in the rational as sole guarantor and deliverer of truth, coupled with a deep suspicion of science—particularly modern science's pretentious claims to an ultimate theory of everything. As such, Ulysses Everett McGill embodies modernity, but a modernity now haunted, from beginning to end, by the otherwise-than-rational—the blind prophet making his way on the railroad. In a way, then, we see a tension between a modern, scientistic worldview, on the one hand, and an ancient-postmodern, mythic worldview, on the other. This is true to our own experience:we have not emerged into a radically new postmodern world; rather, our modern world is disrupted and haunted by postmodern suspicions and critique. Our time is a bit like downtown Los Angeles, whose architecture reflects both epochs. It is not that the postmodern has come in and flattened the modern; rather, the curvaceous lines and eclectic ensembles of Frank Gehry's postmodern architecture assert themselves alongside the modernist glass boxes and crumbling "projects" inspired by Le Corbusier.
It is just such a tension and conflict between science and narrative that situates Jean-François Lyotard's account of postmodernism. Lyotard was one of the first to have the courage to attempt a definition of the new creature postmodernism. Writing a "report on knowledge" commissioned by the government of Quebec, Lyotard opened his analysis with the claim: "Simplifying to the extreme, I define _postmodern_ as incredulity toward metanarratives." The French term (curiously) translated by "metanarratives" is _grand reçits_ , big stories. Postmodernism, then, is the suspicion of and disbelief in "big stories." Now, if ever there was a big story, it is the grand narrative offered in Scripture, spinning a tale from before creation until the consummation of time (and beyond). Thus, if postmodernism is incredulity toward metanarratives, and Christian faith as informed by the Scriptures is just such a metanarrative, then postmodernism and Christian faith must be antithetical: postmoderns could never believe the Christian metanarrative, and Christians should not participate in postmodernism's incredulity. As with Derrida, Lyotard's claim and orthodox Christian faith are often understood to be mutually exclusive. And we find such a reading suggested by even the most nuanced Christian commentators on postmodernity.
However, this judgment is a bit hasty—another myth that needs to be demythologized. It is a bumper-sticker reading of Lyotard that is not informed by a careful understanding of just what Lyotard means by a metanarrative. As such, it also misunderstands what it would mean no longer to believe in metanarratives— to be incredulous with respect to metanarratives. This chapter more carefully engages Lyotard's critique and demonstrates that Christians should find in Lyotard not an enemy but an ally: orthodox Christian faith actually requires that we, too, stop believing in metanarratives.
The first thing we need to do, then, is define the term "metanarrative"— or more specifically, get a clear handle on Lyotard's definition of the term. Only then can we understand his claim that postmodernism is incredulity toward metanarratives. Generally, it is thought that the term refers simply to big stories— grand, epic narratives ( _grand reçits_ ) that tell an overarching tale about the world. In other words, many assume that metanarratives are the target of postmodern disbelief because of their scope, because they make grand, totalizing claims about reality and have universal pretensions. In other words, as Merold Westphal suggests, metanarratives would simply be meganarratives. If that were the case—that any grand story with a global scope were a metanarrative—then, indeed, the biblical narrative of creation, fall, redemption, and eschatological consummation would be a legitimate object of postmodern suspicion and incredulity.
But this is not what Lyotard means by a metanarrative. What is at stake for Lyotard is not the scope of these narratives but the nature of the claims they make. Put another way, the problem isn't the stories they tell but the way they tell them (and, to a degree, why they tell them). For Lyotard, metanarratives are a distinctly modern phenomenon: they are stories that not only tell a grand story (since even premodern and tribal stories do this) but also claim to be able to legitimate or prove the story's claim by an appeal to universal reason. Thus for Lyotard, the purveyor of metanarratives in _O Brother, Where Art Thou?_ would be not the religious believers or prophets but rather the enlightened man of science, Ulysses Everett McGill. It is the supposed rationality of modern scientistic stories about the world that makes them a metanarrative. On Lyotard's account, Homer's _Odyssey_ —though telling a grand story and making universal claims about human nature—is not a metanarrative because it does not claim to legitimate itself by an appeal to a supposed universal, scientific reason; rather, it is a matter of proclamation, or kerygma, which demands the response of faith. On the other hand, the scientific stories told by modern rationalism (Kant), scientific naturalism, or sociobiology are metanarratives insofar as they claim to be demonstrable by reason alone.
The central tension for Lyotard is not between big stories and little stories or global narratives versus local narratives. Instead, he formulates the tension as a conflict between science and narratives: when judged by the criteria of modern science, stories and narratives are little more than fables. When pushed, however, science must legitimate itself: it must produce a discourse of legitimation, which Lyotard simply calls philosophy. Thus, before determining what _postmodern_ means, he first defines what he means by _modern_ : "I will use the term _modern_ to designate any science that legitimates itself with reference to a metadiscourse of this kind making an explicit appeal to some grand narrative, such as the dialectics of Spirit [Hegel], the hermeneutics of meaning [Schleiermacher?], the emancipation of the rational [Kant] or working subject [Marx], or the creation of wealth [Adam Smith]" ( _PC_ , xxiii). The question of the relation between modernity and postmodernity revolves around this issue of "legitimation." Modernity, then, appeals to science to legitimate its claim—and by "science" we simply mean the notion of a universal, autonomous reason. Science, then, is opposed to narrative, which attempts not to prove its claims but rather to proclaim them within a story.
**The Myth of Truth and the Truth of Myth**
But postmodernism, according to Lyotard, has suggested that the emperor of modernity has no clothes! At the heart of the postmodern critique of modernity is an unveiling of the way that science—which is so critical of the "fables" of narrative—is itself grounded in a narrative. What modernity did not recognize about itself was the way in which narrative infiltrated science. Lyotard makes a distinction between "narrative knowledge" and "scientific knowledge"—the latter being distinctively modern, the former being both premodern and postmodern. He makes the same distinction by talking about the difference between science and myth (or "traditional knowledge"). Narrative knowledge is grounded in the custom of a culture and, as such, does not require legitimation. Lyotard links this to a tribal paradigm in which the homogeneity of a people ( _Volk_ ), coupled with the authority of a narrator, produces a kind of immediate autolegitimation. "The narratives themselves have this authority," he notes. In a sense, "the people are only that which actualizes the narratives" ( _PC_ , 23). Legitimation in terms of demonstration is not demanded but rather is implicit in the narrative itself as a story of the people.
In contrast to this auto-legitimation, modern scientific culture externalizes the problem of legitimation. Lyotard explains this in terms of the pragmatics of communication, where a speaker who makes a truth claim is a "sender," and the receiver or hearer of this claim is an "addressee." The two pragmatic poles of sender and addressee are distinguished, and the addressee demands of the sender justification for messages sent her way. I, as sender, must now provide "proof " ( _PC_ , 23–24). However, because the homogeneity of the premodern _Volk_ has dissolved, we have no immediate or previously agreed-on consensus. In Lyotard's terms, we do not all share the same language game. As such, modern legitimation has recourse to a universal criterion: reason—a (supposedly) universal stamp of legitimation. This move generates what Lyotard famously describes as metanarratives: appeals to criteria of legitimation that are understood as standing outside any particular language game and thus guarantee universal truth.
And it is precisely here that we locate postmodernity's incredulity toward metanarratives: they are just another language game, albeit masquerading as the game above all games. Or as Lyotard puts it, scientific knowledge, which considered itself to be a triumph over narrative knowledge, covertly grounds itself in a narrative (i.e., an originary myth).
In particular, Lyotard analyzes two modern narratives of legitimation: first, the humanistic metanarrative of emancipation (as found in Kant and Marx), and second, the metanarrative concerning the life of the Spirit in German Idealism. One could perform similar analyses of rationalist market economics or the steady rise of sociobiology from Darwin onward. But we can already see this infusing of myth in knowledge as far back as Plato, where "the new language game of science posed the problem of its own legitimation at the very beginning" ( _PC_ , 28). In books 6 and 7 of the _Republic_ , for instance, the answer to the question of legitimation (here both epistemological and sociopolitical) "comes in the form of a narrative—the allegory of the cave, which recounts how and why men yearn for narratives and fail to recognize knowledge. Knowledge is thus founded on the narrative of its own martyrdom" ( _PC_ , 28–29). In a similar way, Lyotard argues, modern scientific knowledge, when called on (by itself ) to legitimate itself, cannot help but appeal to narrative—this "return of the narrative in the non-narrative"is "inevitable" ( _PC_ , 27–28). Like the loquacious Ulysses Everett McGill, modernity and its science can't stop telling stories (is there a bigger story than _On the Origin of Species_?)—all the while claiming that they are opposed to such "fables." Scientists and modern philosophers still tell stories; as Lyotard comments, "the state spends large amounts of money to enable science to pass itself off as epic" ( _PC_ , 28). Whenever science attempts to legitimate itself, it is no longer scientific but narrative, appealing to an orienting myth that is not susceptible to scientific legitimation. Modernity's science demands of itself the impossible: "The language game of science desires its statements to be true but does not have the resources to legitimate their truth on its own" ( _PC_ , 28). The appeal to reason as the criterion for what constitutes knowledge is but one more language game among many, shaped by founding beliefs or commitments that determine what constitutes knowledge within the game; reason is grounded in myth. "Metanarratives," then, is the term Lyotard ascribes to these false appeals to universal, rational, scientific criteria—as though they were divorced from any particular myth or narrative. For the postmodernist, every scientist is a believer.
Here we must return to the question posed earlier: If postmodernity is incredulity toward metanarratives, then does postmodernism signal a rejection of Christian faith insofar as it is based on the grand story of the Scriptures? The answer is clearly negative, since the biblical narrative and Christian faith claim to be legitimated not by an appeal to a universal, autonomous reason but rather by an appeal to faith (or, to translate, myth or narrative). That said, some might argue that the Christian faith can be legitimated by reason. In evangelical apologetic discussions, for instance, classical or evidential apologists (versus presuppositionalists) might argue that Christian faith is grounded in reason and thus constitutes a metanarrative. Without rehearsing the history of debates regarding apologetic method, I would argue that classical or evidentialist apologetics would fall prey to Lyotard's critique of metanarratives (since it consorts with a notion of universal reason), and that such a critique would be welcomed by presuppositionalists. One of the constructive engagements with Lyotard would be to consider his discussion of language games and critique of metanarratives and its correlation with presuppositional discourses on worldviews and the critique of autonomous reason.To put it another way: classical apologetics is quite distinctly modern in its understanding of knowledge and truth.
Lyotard very specifically defines metanarratives as universal discourses of legitimation that mask their own particularity; that is, metanarratives deny their narrative ground even as they proceed on it as a basis. In particular, we must note that the postmodern critique is not aimed at metanarratives because they are really grounded in narratives; on the contrary, the problem with metanarratives is that they do not own up to their own mythic ground. Postmodernism is not incredulity toward narrative or myth; on the contrary, it unveils that all knowledge is grounded in such. Once we appreciate this, the (false) dichotomy that Middleton and Walsh, Grenz, Ingraffia, and others propose is dissolved insofar as the biblical narrative is not properly a metanarrative. As a result, new space is opened for a Christian appropriation of the postmodern critique of Enlightenment rationality.
What characterizes the postmodern condition, then, is not a rejection of grand stories in terms of scope or in the sense of epic claims, but rather an unveiling of the fact that all knowledge is rooted in _some_ narrative or myth—an insight earlier made by Schaeffer and Van Til. The result, however (and here I note one of the genuine problems of postmodernity), is what Lyotard describes as a "problem of legitimation" ( _PC_ , 8) (or what Habermas describes as a "legitimation crisis") since what we thought were universal criteria have been unveiled as just one game among many. If we consider, for instance, the reality of deep moral diversity and competing visions of the good, postmodern society is at a loss to adjudicate the competing claims. There can be no appeal to a higher court that would transcend a historical context or a language game, no neutral observer or "God's-eye view" that can legitimate or justify one paradigm or moral language game above another. If all moral claims are conditioned by paradigms of historical commitment, then they cannot transcend those conditions; thus every moral claim operates within a "logic" that is conditioned by the paradigm. In other words, every language game has its own set of rules. As a result, criteria that determine what constitutes evidence or proof must be game relative: they will function as rules only for those who share the same paradigm or participate in the same language game. The incommensurability of language games means that there is a plurality of logics that precludes any demonstrative appeal to a common reason. Recognition of the incommensurability of language games and the plurality of competing myths means that there is no consensus, no _sensus communis_. Many—especially Christians—lament this state of affairs (hence the renaissance of natural-law theories that purport to find a common ground for all). But is this situation as bad as we think? Are we lamenting the loss of what was a very modern hegemony of America, for instance? Is our situation really all that different from the situation of the apostle Paul or Augustine? Should we be trying to establish a common myth for an entire nation—a Constantinian strategy—or should the church simply be a witness amid this plurality of competing myths? (We'll return to these questions below.)
In the face of this problem, we must not lose sight of the fact that what constitutes the postmodern condition is precisely a plurality of language games—a condition in which no one story can claim either universal auto-legitimation (because of the plurality of "the people") nor appeal to a phantom universal reason (because reason is just one myth among others, which is itself rooted in a narrative). And this plurality is based on the fact that each game is grounded in different narratives or myths (i.e., founding beliefs). Whether we understand this as a new Babel or a new Pentecost, this situation—though posing a challenge—also presents a unique opportunity for Christian witness in postmodernity. The Ulysses Everett McGills will persist in our day, but postmodernism also opens the space for the faith narratives of a Delmar.
**Taking Lyotard to Church**
Having corrected the bumper-sticker understanding of postmodernism as incredulity toward metanarratives and seeing the way in which postmodernism, according to Lyotard, calls into question rationalist understandings of knowledge, we've begun to recognize the way in which postmodernism opens the space for Christian witness to be bold in its proclamation, its narration of the story. While in modernity science was the emperor who set the rules for what counted as truth and castigated faith as fable, postmodernity has shown us the emperor's nudity. As such, we no longer need to apologize for faith—we can be unapologetic in our kerygmatic proclamation of the gospel narrative. Thus the postmodern critique of metanarratives echoes Schaeffer and Dooyeweerd's earlier criticisms of the autonomy of reason. As such, Lyotard's analysis and critique of metanarratives have two important implications for Christian faith and the life of the church.
_The Faith-full-ness of Our Knowledge_
At root, what is at stake in postmodernism is the relationship between faith and reason. When Lyotard describes postmodernism as incredulity toward metanarratives, he indicates a suspicion and critique of the very idea of an autonomous reason, a universal rationality without ultimate commitments. Modernity's metanarratives cannot disengage themselves from narratives as their ultimate ground and thus cannot divorce themselves from myth, orienting beliefs that themselves are not subject to rational legitimation. In this light, consider, for instance, Thomas Kuhn's analysis concerning the role of paradigms in scientific research. Dominated by the language of faith,Kuhn's _Structure of Scientific_ _Revolutions_ points out the role of paradigms as "constellations of belief "that orient how we perceive our world and determine what we consider knowledge and truth. In other words, science finds itself grounded in prior beliefs that do not admit of legitimation but rather function as the basis for further legitimation. The paradigm itself is a belief, a matter of faith. It is also at this level that Wittgenstein notes: "If I have exhausted the justifications I have reached bedrock, and my spade is turned. Then I am inclined to say: 'This is simply what I do.' "To this list we could add Ga-damer, Polanyi, Derrida, and others; common to all of them is a delimitation of rationality, particularly Enlightenment ideals of scientific, objective rationality.
In this sense, the postmodern critique described by Lyotard as incredulity toward metanarratives represents a displacement of the notion of autonomous reason as itself a myth. And that is a project with which Christians ought to ally themselves, particularly once we have clarified that such an alliance does not require jettisoning the biblical narrative. By calling into question the idea of an autonomous, objective, neutral rationality, I have argued that postmodernity represents the retrieval of a fundamentally Augustinian epistemology that is attentive to the structural necessity of faith preceding reason, believing in order to understand—trusting in order to interpret.While this Augustinian structure is formalized— in the sense that there is a plurality of faiths, as many as there are language games—the structure (of faith preceding reason) remains in place, in contrast to modern (and perhaps even Thomistic)epistemologies (theories of knowledge).
The incredulity of postmodernity toward metanarratives derives from the fact that modernity denies its own commitments, renounces its faith, while at the same time never escaping it. Postmodernism refuses to believe the Enlightenment is without a creed. But note: the postmodern critique demands not that modern thought relinquish its faith (a modern gesture, to be sure) but that it own up to it—openly confess its credo. Thus we might consider the postmodern critique as a revaluing of myth, of orienting faith, providing new spaces for religious discourse—and in particular, an integrally Christian philosophy—in a climate where it has been demonstrated that everyone's "got religion."
How will this insight be helpful to Christian scholars and thinkers? My point is not to suggest that Lyotard's analysis concretely helps us understand Christian faith; in other words, I am not arguing that we look to Lyotard for assistance in understanding Christian faith commitments. Rather, Christian thinkers should find in Lyotard's critique of metanarratives and autonomous reason an ally that opens up the space for a radically Christian witness in the postmodern world—both in thought and in practice. By calling into question the very ideal of a universal, autonomous reason (which was, in the Enlightenment, the basis for rejecting religious thought) and further demonstrating that all knowledge is grounded in narrative or myth, Lyotard relativizes (secular) philosophy's claim to autonomy and so grants the legitimacy of a philosophy that grounds itself in Christian faith. Previously such a distinctly Christian philosophy would have been exiled from the "pure" arena of philosophy because of its "infection" with bias and prejudice. Lyotard's critique, however, demonstrates that no philosophy—indeed, no knowledge—is untainted by prejudice or faith commitments. In this way the playing field is leveled, and new opportunities to voice a Christian philosophy are created. Thus Lyotard's postmodern critique of metanarratives, rather than being a formidable foe of Christian faith and thought, can in fact be enlisted as an ally in the construction of a Christian philosophy.
But beyond the implications for Christian scholarship, this postmodern critique of metanarratives has important effects for Christian ministry and public witness. To the extent that the postmodern critique is effective, the modern notions of a neutral public space and secular sphere must be abandoned.The exclusion of faith from the public square is a modern agenda; postmodernity should signal new openings and opportunities for Christian witness in the broad marketplace of ideas. We must be careful, however, not to continue to propagate that witness in modernist ways: by attempting our own rationalist demonstrations of the truth of Christian faith and then imposing such on a pluralist culture (what is often described as a Constantinian agenda). The new apologetic of postmodernity will echo the patient presuppositionalist apologetic of Schaeffer—getting everyone's presuppositions on the table and then narrating the story of Christian faith, allowing others to see the way in which it makes sense of our experience and our world. While the new apologetics will be an _un_ apologetics, it will at the same time be characterized by faithful storytelling, not demonstration. It must be kerygmatic and charismatic: proclaiming the story of the gospel in the power of the Spirit.
_The Narrative Character of Our Faith_
Too many Christians are just pious versions of Ulysses Everett McGill; that is, too many Christians have bought into the modernist valorization of scientific facts and end up reducing Christianity to just another collection of propositions. Our beliefs are encapsulated in "statements of faith" that simply catalog a collection of statements about God, Jesus, the Spirit, sin, redemption, and so on. Knowledge is reduced to biblical information that can be encapsulated and encoded.And so, in more ways than one, our construal of the Christian faith has capitulated to modernity and what Lyotard calls its "computerization" of knowledge, indicating a condition wherein any knowledge that cannot be translated into a simple "code" or reduced to "data" is abandoned ( _PC_ , 4).But isn't it curious that God's revelation to humanity is given not as a collection of propositions or facts but rather within a narrative—a grand, sweeping story from Genesis to Revelation? Is there not a sense in which we've forgotten that God's primary vehicle for revelation is a story unfolded within the biblical canon?
By pointing to the fundamentally narrative basis of all knowledge, Lyotard reminds us about the ultimately narrative character of Christian faith. This resonates with postliberal theology (as found in the work of Stanley Hauerwas, for example), which emphasizes the narrative character of revelation.Why is narrative important, and how does it differ from propositional knowledge? First, narrative is a more fully orbed means of communication (and hence revelation), activating the imagination and involving the whole person in a concrete world where God's story unfolds. Second, Christian faith—unlike almost any other world religion (with the exception of Judaism)—is not a religion simply of ideas that have been collected. The faith is inextricably linked to the events and story of God's redemptive action in the world: Christian faith rests on the work of the Word, who "suffered under Pontius Pilate," and that work can only be properly proclaimed by being narrated, by telling a story. The notion of reducing Christian faith to four spiritual laws signals a deep capitulation to scientific knowledge, whereas postmodernism signals the recovery of narrative knowledge and should entail a more robust, unapologetic proclamation of the story of God in Christ. This is why the Scriptures must remain central for the postmodern church, for it is precisely the story of the canon of Scripture that narrates our faith.
The narrative character of our faith should affect not only our proclamation and witness but also our worship and formation. Although I focus on the question of formation in chapter 3, here I want to emphasize the way in which Christian worship should reenact the narrative of the gospel week by week in order to teach us how to find ourselves in the story. Crucial for our discipleship and formation is being able to write ourselves into the story of God's redeeming action in the world—being able to find our role in the play, our character in the story. To do that, we need to know the story, and that story should be communicated when we gather as the people of God, that is, in worship. This is why the most postmodern congregations will be those that learn to be ancient, reenacting the biblical narrative _._Just as Lyotard's account of narrative knowledge shows a link between premodern and postmodern, so worship in postmodernity (which appreciates the role of narrative) should signal a recovery of liturgical tales—the narrating of creation, fall, redemption (as well as crucifixion, burial, and resurrection) in the very manner in which we worship.
**A Storytelling Church**
How should the church look different after this encounter with Lyotard and postmodernism? What implications does this have for practice? Building on our initial tour of a deconstructive church in chapter 2, as well as suggestions in this chapter, let's continue our tour of the postmodern church by visiting a "storytelling" church. What's happening?
Again, we find that the role of Scripture is central, not just as the Text that mediates our understanding of the world but also as the Story that narrates our role in it. Each week the worshipping community is confronted by the narrative of a God who makes a covenant with his people, who is faithful to his promises, and who acts in history to effect a relationship with his people.The story of Yahweh unfolds as the drama of salvation—a drama in which Yahweh is the "star"but also in which each of us finds a role. In order to appreciate the breadth of this story, each week the postmodern church narrates "scenes" from the different "acts" of the drama: a scene from the Old Testament, a scene from the Gospel narratives, and a scene from the Epistles.
But in addition to the narration of the story in Word, this postmodern church narrates the story in its practices. Like the early church (Acts 2:42; 20:7), this postmodern community celebrates communion each week, for in the celebration of the Eucharist it narrates the gospel: the death, burial, and resurrection of the Lord Jesus Christ, which enacted a new covenant between God and his people. In this act of eating and drinking, the community proclaims the Lord's death "until he comes" (1 Cor. 11:26). While the postmodern church is a storied community centered on the narrative of Scripture, it is also a eucharistic community that replays the narrative in deed. Further, the symbols and signs of the Lord's Supper embody the gospel for us. Because the postmodern church values narrative, it values story and as such values the aesthetic experience engendered by material signs and symbols. Put another way, because of the renewed role of story as a kind of literature activating the imagination, the postmodern church values the arts in general as an incarnational medium that embodies the story of God's faithfulness. Seeing iconoclasm as a symptom of a kind of modern Platonism, the postmodern church affirms the role of the aesthetic (what plays on the senses) in telling the story. Just as God communicates to humanity through the incarnation of the Word as flesh—the image of the invisible God (Col. 1:15)—so God continues to speak to the church through the material symbols of bread and wine but also through images and dance.
The postmodern church resists the tendency of pragmatic evangelicalism, which tries to "dumb down" the story to make it accessible or attractive to the culture. Instead, the postmodern church affirms the timelessness (and timeliness) of the biblical narrative as it is told. Rather than trying to translate the biblical story into a contemporary, more "acceptable" narrative (which usually ends up compromising the narrative to culture), the postmodern church seeks to initiate listeners into the narrative. Authentic Christian worship both invites outsiders into the gospel story and provides a significant means for the formation of disciples of Jesus Christ. In other words, authentic worship does not have to choose between reaching seekers and building up the saints. Incarnational worship does both. As the Church of Scotland's _Common Order_ of worship puts it, public worship is both a "converting ordinance" and "an edifying ordinance"; that is, worship can be both a way of inviting the lost into the body of Christ and a way of building up the saints, forming them into the kinds of people that pursue the kingdom with heart, soul, mind, and strength.
Worship, then, needs to be characterized by hospitality; it needs to be inviting. But at the same time, it should be inviting seekers into the church and its unique story and language.Worship should be an occasion of cross-cultural hospitality. Consider an analogy: When I travel to France, I hope to be made to feel welcome. However, I don't expect my French hosts to become Americans in order to make me feel at home. I don't expect them to start speaking English, ordering pizza, talking about the New York Yankees, and so on. Indeed, if I wanted that, I would have just stayed home! Instead, what I'm hoping for is to be welcomed into their unique French culture; that's why I've come to France in the first place. And I know that this will take some work on my part. I'm expecting things to be different; indeed, I'm looking for just this difference. So also, I think, with hospitable worship: seekers are looking for something our culture can't provide. Many don't want a religious version of what they can already get at the mall. And this is especially true of postmodern or Gen X seekers: they are looking for elements of transcendence and challenge that MTV could never give them. Rather than an MTVized version of the gospel, they are searching for the mysterious practices of the ancient gospel.
Quinn Fox recently captured this point with a brilliant analogy between Starbucks and the church. The church, he suggests, "might learn about corporate worship language from the language of coffee. Starbucks realizes, it seems, that a distinctive menu that people need to learn is not a bad thing." In his testimony, he confesses: "I did not inherit my parent's commitment to coffee. Rather, 'I found it' in my late 30s." His coming to faith in coffee required learning a new language, the Italian parlance of Starbucks. But this wasn't off-putting to him; he didn't complain that it wasn't user friendly. Rather, he felt he was being invited into a world:
At busy times an orderly (if slow) procession of the faithful crowd toward the counter. An order may be something like "I'd like a _grande_ , non-fat, triple shot, 2 pump peppermint _latte_ with extra whip cream." The money changer loudly relays the request. And one should not worry if the strangeness of the terms causes a stumble. The temple assistant mediates these early morning "sighs that are too deep for words" by translating them into flawless coffee Italian. The Barista (it even sounds a little like "priest") who feverishly prepares coffee drinks behind the espresso bar repeats the petition verbatim, as if by uttering the words s/he speaks them into being. At the more relational franchises, the customer's name will be attached to the order. When the brew is ready, complete in all of its uniqueness, the Barista chants the request once again, just to indicate that the unction is complete.
Postmodern worship does something of the same: orienting itself by ancient, strange practices but in a way that invites not only the faithful but also the searching into the story's rhythms and cadences.
Finally, the postmodern church recognizes that its primary responsibility is to live the story for the world. The church is the stage where God's drama is played out; as such, we Christians have a responsibility to "act well," we might say, to faithfully play out the love of God in the church as a community of love and justice. Our storytelling should be supported by our story living.
1. _O Brother, Where Art Thou?_ DVD, directed by Joel and Ethan Coen (Burbank, CA: Touchstone Home Video, 2001).
2. The Cyclopean Bible salesman, Big Dan, shares this notion, taking advantage of the Depression to sell the Bible as an opiate for the impoverished masses. As he puts it, there's a big market for Bibles "in this time of woe and want. People are lookin' for answers."
3. Or women! As Everett commands Delmar: "Never trust a female. . . . Truth means nothing to a woman, Delmar. Triumph of the subjective!"
4. Jean-François Lyotard, _The Postmodern Condition: A Report on Knowledge_ , trans. G. Bennington and B. Massumi (French original, 1979; Minneapolis: University of Minnesota Press, 1984), xxiv; henceforth abbreviated in the text as _PC_. For a more detailed, scholarly account of Lyotard's argument, see my "Little Story about Metanarratives: Lyotard, Religion, and Postmodernism Revisited," _Faith and Philosophy_ 18 (2001): 353–68.
5. I am thinking of Richard Middleton and Brian Walsh's otherwise excellent book, _Truth Is Stranger Than It Used to Be_ (Downers Grove, IL: InterVarsity, 1995). A similar understanding of the antithesis between postmodernism and Christianity is offered by Stanley Grenz, Henry H. Knight III, and Brian Ingraffia.
6. See, for instance, Middleton and Walsh, _Truth Is Stranger_ , 70–71.
7. Merold Westphal, _Overcoming Onto-Theology: Toward a Postmodern Christian_ _Faith_ (Bronx, NY: Fordham University Press, 2001), xiii.
8. It is hard to resist comparing this to John Calvin's account of the _autopistie,_ or self-authentication, of Scripture (see _Institutes_ , I.vii.5). This is why Michael S. Horton, in _Covenant and Eschatology: The Divine Drama_ (Louisville: Westminster John Knox, 2002), has suggested that Calvin and the post-Reformation scholastics were, in a sense, "postfoundationalist" theologians. For a historical account, see Richard A. Muller, "Sources of Reformed Orthodoxy: The Symmetrical Unity of Exegesis and Synthesis," in _A Confessing Theology for Postmodern Times_ , ed. Michael Horton (Wheaton: Crossway, 2000), 43–62.
9. "Myth" here should not be understood in the modern, scientistic sense as a "fable" opposed to truth; instead, it indicates the religious, confessional status of a truth. We should understand myth here in the sense also suggested by C. S. Lewis. According to Lewis, the imagination is a truth-bearing faculty that communicates not via propositions but via myths. Myths allow us to experience "as a concrete what can otherwise be understood only as an abstraction." See C. S. Lewis, "Myth Became Fact," in _God in the Dock: Essays on Theology and Ethics_ , ed. Walter Hooper (Grand Rapids: Eerdmans, 1970), 67. My thanks to Kevin Vanhoozer for pointing me to this Lewis source.
. For a nuanced discussion of the latter, see Herman Dooyeweerd, _In the_ _Twilight of Western Thought: Studies in the Pretended Autonomy of Theoretical_ _Thought_ , ed. James K. A. Smith, Collected Works, B/4 (Lewiston, NY: Edwin Mellen Press, 1999).
. For just a selective example of such passages, see Thomas Kuhn, _The_ _Structure of Scientific Revolutions_ , 2nd ed. (Chicago: University of Chicago Press, 1970), on belief, see 2, 4, 17, 43, 113; on commitments, see 4–5, 7, 11, 40–43; on tradition, see 6, 10, 39, 43.
. Ibid., 175.
. Ludwig Wittgenstein, _Philosophical Investigations_ , trans. G. E. M. Anscombe (New York: Macmillan, 1959), §217.
. See my "The Art of Christian Atheism: Faith and Philosophy in Early Heidegger," _Faith and Philosophy_ 14 (1997): 71–81; and idem, "Is Deconstruction an Augustinian Science? Augustine, Derrida, and Caputo on the Commitments of Philosophy," in _Religion with/out Religion: The Prayers and Tears of John D._ _Caputo,_ ed. James H. Olthuis (London: Routledge, 2002), 50–61.
. As Francis Schaeffer suggested in _Escape from Reason_ , the notion of an autonomous reason is not unique to the Enlightenment but can already be located in Aquinas's understanding of natural reason. Aquinas and Augustine disagree on this point, as seen in Aquinas's commentary on Boethius's _De trinitate_ , Q. 1, art. 1. My goal here is not to mediate that debate but to raise a question that demands further consideration. For an argument to the contrary, see John Milbank and Catherine Pickstock, _Truth in Aquinas_ , Radical Orthodoxy Series (New York: Routledge, 2001).
. For an insightful analysis of these secular "bully" strategies, see Stephen L. Carter, _The Culture of Disbelief: How American Law and Politics Trivialize_ _Religious Devotion_ (New York: BasicBooks, 1993).
. This can been seen in quite a remarkable way in D. A. Carson, _Becoming_ _Conversant with the Emerging Church: Understanding a Movement and Its Implications_ (Grand Rapids: Zondervan, 2005), chap. 7: "Some Biblical Passages for Evaluation." The chapter is a collection of lists of proof texts that are supposed to have the self-evident force of criticizing "hard postmodernism" just by documenting the texts—a sort of miniconcordance of Bible verses that use the words "true" or "truth." Carson's critique of McLaren on this score, particularly on questions of narrative (ibid., 163–66), is an epic adventure in missing the point.
. This "propositionalization" of the gospel is also anti-missional. If all the "facts" or "data" of the gospel are "out there," so to speak, then there is nothing more to be done. However, if narration and enactment are central, then mission is central and ongoing. My thanks to Bill VanGroningen for a conversation on this point.
. See also the recent call to recover this theme in Michael S. Horton, _Covenant_ _and Eschatology: The Divine Drama_ (Louisville: Westminster John Knox, 2002) and Kevin J. Vanhoozer, _The Drama of Doctrine: A Canonical-Linguistic_ _Approach to Christian Theology_ (Louisville: Westminster John Knox, 2005).
. Robert Webber has suggested something similar in _Ancient-Future_ _Faith: Rethinking Evangelicalism for a Postmodern World_ (Grand Rapids: Baker, 1999).
. I would note the work of Marva Dawn as crucial for reflection on the theology of worship.
. For a rich account of the story of the God who acts, see Michael Horton, _Covenant and Eschatology._
. That is, worship is about the glory of God, not the satisfaction of one's needs. For a discussion that recovers the centrality of God as the focus of worship, see Michael S. Horton, _A Better Way: Recovering the Drama of God-Centered_ _Worship_ (Grand Rapids: Baker, 2002).
. Building on the use of the lectionary in chapter 2, which would help the community cover the entirety of the story over time, not just our favorite scenes.
. We will return to a fuller consideration of aesthetics and liturgy in chapter 5.
. On this point, see William Storrar, "From _Braveheart_ to Faint-Heart: Worship and Culture in Postmodern Scotland," in _To Glorify God: Essays on_ _Modern Reformed Liturgy_ , ed. Bryan Spinks and Iain Torrance (Grand Rapids: Eerdmans, 1999), 70–71.
. On these matters, I am deeply indebted to Marva Dawn, _Reaching Out_ _without Dumbing Down: A Theology of Worship for This Urgent Time_ (Grand Rapids: Eerdmans, 1995).
. See Quinn Fox, "Liturgy and Starbucks," _Perspectives_ (February 2003). Online: http://www.perspectivesjournal.org/perspectives/2003/02/seeit-starbucks .php.
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* * *
Power/Knowledge/Discipline
[Foucault and the Possibilities
of a Postmodern Church](Smit_ISBN9781441200396_epub_toc_r1.html#d7e3637)
Michel Foucault's attention is drawn to institutions of power: prisons and schools, hospitals and factories, sex and money. What could he have to say to a postmodern church? In this chapter, we'll explore Foucault's claim that "power is knowledge"in order to see its insight into Christian formation and discipleship.
**Raising the Curtain:** **One Flew over the Cuckoo's Nest**
The hospital is a machine—that is Chief 's thesis in _One Flew_ _over the Cuckoo's Nest_. The hospital is part of the Combine that "works over" individuals by means of its power, its control. The hospital's walls hum with the sound of its machinery of surveillance and repression—"hum of black machinery, humming hate and death and other hospital secrets." On the mental ward of this Oregon hospital, Chief—a longtime resident—is able, in a way, to see through walls and can thus observe the hospital's machinations of power firsthand. Even the strategies aimed toward cure are in fact systems of control and domination: from the orderlies to the medication, from the strict schedule and work regimen to the "therapeutic circle" that serves only to humiliate— all performed to the tune of banal background music that functions as the soundtrack for repression.
At the center of the machine—but also an effect of it—is the watchful eye of Nurse Ratched. From the glass-enclosed nurse's station, she keeps watch over the ward like the warden from a prison watchtower. Indeed, the mental ward is a panopticon of sorts, a structure where a central hub of power is able to see all of those subjected to it, monitoring their actions sheerly by surveillance—by the threat of being seen. Nurse Ratched oversees this environment and employs all the tools of her trade to carry out her surveillance and punishment. Robotlike herself, Nurse Ratched is the eyes and ears of the machine, the agent of its disciplines. Every once in a while, Chief glimpses her real self, her inner workings, when "she really lets herself go and her painted smile twists, stretches to an open snarl, and she blows up bigger than a tractor, so big I can smell the machinery inside the way you smell a motor pulling too big a load." But as Chief realizes, while Nurse Ratched is the face of the system, she is but a part of it: "It's not just the Big Nurse by herself," he concludes, "but the whole Combine, the nation-wide Combine that's the really big force, and the nurse is just a high-ranking official for them."
Of course, Nurse Ratched does not speak or think of her work as surveillance and certainly not as punishment. The stated goal of the institution is healing and cure. Or, to stick with Chief 's mechanistic metaphor, the hospital is a shop for repair—not quite manufacturing, since the specimens are brought into the shop, but rather remanufacturing, setting broken objects into a mold that knocks off the edges and makes them conform to the shape of societal expectations. The physicians are merely technicians whose procedures are "installations" that are engineered for cure. Sometimes the patient's wiring is so malfunctioning that the hospital needs to reset the circuitry with a dose of high voltage (electroshock therapy). Although some patients have an initial concern, patient Daniel Harding explains that the goal of the procedure is simply a microcosm of the goal of the institution: "In this country, when something is out of order, then the quickest way to get it fixed is the best way." Responding to the suggestion that this is similar to "electrocuting a guy for murder," Harding continues: "Both activities are much more closely related than you think; they are both cures."
What really concerns Chief, however, is not the regimen of medicines and timetables, restraints and electroshock therapy. What is of greater concern are the Combine's more covert operations, its stealth control and manipulation—the way it "works on you" without your knowing it, the way it worked on Chief 's father. "They work on you ways you can't fight!" Chief warns. "They _install_ things. They start as quick as they see you're gonna be big and go to working and installing their filthy machinery when you're little, and keep on and on and on till you're _fixed_!"
Wandering into the panopticon one day—into the clutches of the Combine and Nurse Ratched—is Randle P. McMurphy, shipped to the mental ward from a work farm at Pendleton. What stands most in need of repair in McMurphy is the fact that he doesn't think he needs repair. (McMurphy is committed, whereas most of the other patients are in the ward voluntarily, submitting to the repair work of therapy by choice.) McMurphy becomes something of a project for Nurse Ratched, who sees that he is a "manipulator" planning to "take over." But the Combine and its emissaries will have none of that. The narrative of _One_ _Flew over the Cuckoo's Nest_ recounts the struggle of the system in its attempts to repair R. P. McMurphy and his resistance to its power, which inspires others, like Chief, to resist its clutches. But ultimately the story ends with the triumph of the system, culminating in McMurphy's lobotomy, which leaves him a near lifeless shell on the Acutes ward. (To prevent McMurphy's my-thology from being tarnished, Chief smothers the now-docile McMurphy and, because of McMurphy's heroic resistance, finally musters the strength and drive to break through the glass of the mental ward and escape into the surrounding hills.)
The film and novel paint a vivid and disturbing picture of institutional power and its attempts to whitewash its mechanisms with paternalistic claims about "cure" and the "good of the patient." Although McMurphy is the hero because of his libido-driven resistance to the system, even his empowerment of the other patients cannot stop the driving cogs of the Combine: McMurphy, though mythologized, is crushed by the machine. Thus _One Flew over the Cuckoo's Nest_ leaves us just where we would expect from a work dating from the 1960s: with a deep suspicion of institutions, institutional power, and the control they exert over us. We are left simply with the return of Nurse Ratched, the continued machinations of the hospital, and the remaining hum of the Combine in its walls. If we are to escape control and the trappings of institutional power, the only way out is to follow Chief through the shattered window, wandering alone, institutionless but "free."
**Foucault's Claim: Power Is Knowledge**
There is a sense in which Michel Foucault is the Randle P. McMurphy of our unholy trinity—a somewhat libido-driven rebel, protesting control and resisting systems by documenting their covert domination in modern culture. If Forman's _One_ _Flew over the Cuckoo's Nest_ became something of a visual anthem for a generation, Michel Foucault's _Discipline and Punish_ has become an analogous anthem for postmodernity across the disciplines, from criminal justice to education. What is fascinating is the way in which Foucault's history of prisons says so much about other institutions such as schools, factories, and hospitals. While we might expect the hospital of _One Flew over_ _the Cuckoo's Nest_ to mirror Foucault's specific analysis of hospitals in his earlier work _The Birth of the Clinic_ , the mental ward under Nurse Ratched's surveillance more resembles the carceral systems described in _Discipline and Punish_. Indeed, we need to appreciate that Foucault's account of the "birth of the prison" is not really about penitentiaries; it is about the way in which society as a whole reflects the prison. The prison is but a microcosm of society itself.
For Foucault, at the root of our most cherished and central institutions—hospitals, schools, businesses, and, yes, prisons—is a network of power relations. The same is true of our most celebrated ideals; at root, Foucault claims, knowledge and justice reduce to power. While we moderns—especially we moderns who grew up on "Schoolhouse Rock"—were shaped by the maxim of Francis Bacon proclaiming that "knowledge is power," Foucault's postmodern axiom is that "power is knowledge." However, Foucault himself resists any bumper-stickerization of this notion. As he clarifies, he does not mean that knowledge and power are identical;instead, he means to emphasize the inextricable relationship between knowledge and power. Knowledge, or what counts as knowledge, is not neutrally determined.Instead, what counts as knowledge is constituted within networks of power— social, political, and economic. As he states near the beginning of _Discipline and Punish_ , we should give up the notion that power leads to madness; quite to the contrary, "we should admit rather that power produces knowledge (and not simply by encouraging it because it serves power or by applying it because it is useful); that power and knowledge directly imply one another; that there is no power relation without the correlative constitution of a field of knowledge, nor any knowledge that does not presuppose and constitute at the same time power relations" ( _DP_ , 28). Thus Foucault regularly speaks of "power-knowledge relations" or the "nexus" of power/knowledge.
Like Chief in _One Flew over the Cuckoo's Nest_ , postmodernism is characterized by a deep hermeneutic of suspicion.This is why Foucault, following Nietzsche, describes his method in intellectual history as "genealogy" or "archaeology" whose task is to uncover the secret, submerged biases and prejudices that go into shaping what is called the truth.There is no claim to truth that is innocent; there is no knowledge that simply falls into our minds from the sky, pristine and untainted. What might be claimed as obvious or self-evident is, in fact, covertly motivated by other interests—the interest of power. If someone says, "What do you mean? This is just the way things are. Can't you see that?" Foucault the genealogist traces the lineage of such thinking to the beliefs that really motivate it. Or to use his archaeological metaphor, he digs beneath the surface of what goes around as objective truth to show the machinations of power at work below the surface. Like Chief, who can see through the pristine white walls of the hospital to its more monstrous workings, Foucault sees through the neat and tidy claims to objective truth, seeing them as only masks of power. The genealogist, then, shares Chief 's critical X-ray vision, and "finds that there is 'something altogether different' behind things: not a timeless essential secret, but the secret that they have no essence or that their essence was fabricated in a piecemeal fashion from alien forms."Like a genealogist whose patient documentation of a family tree shows the family's complicity in the evils of slavery, so Foucault's genealogy intends to show that modernity's claims to scientific objectivity or moral truth are fruits of a poisoned tree of power relations. Or to use an architectural metaphor, Foucault's archaeology sets out to show that what we thought were sure foundations are more like collections of fragments piled in the bottom of the hole.Foucault is not out to lament this situation, as though we had lost our foundations, but rather to get us to own up to what has always been the case.
To claim that power is knowledge, then, is to make a claim about the power relations that stand behind both institutions and ideals. As Nietzsche earlier claimed in his _Genealogy of Morals_ , good and evil are just names that we give to the power interests of the strong versus those of the weak. Thus "in a sense," Foucault concludes, "only a single drama is ever staged in this 'non-place,' _the endlessly repeated play of dominations_."The story of humanity is not the Enlightenment fiction of perpetual progress or the constant progression of the race, as Kant (and Richard Rorty) suggest, but rather simply the shift from one combat to another, from one form of domination to another.
Foucault's claim, however, is not a proclamation made from on high, as if it were just the kind of heaven-sent axiom he protests against. His claim about the relation between power and knowledge is not an a priori or abstract claim; rather, it is a claim that bubbles up from his analyses of concrete institutions and ideals such as hospitals and prisons, notions of madness versus reason, or the history of sexuality. Thus Foucault's claim is always made on the basis of case studies—where the axiom is not applied to a case but rather arises from it. If Foucault thinks that power is knowledge, it is because the history of modern institutions bears that out. Let us consider one of his case studies in more detail both to see how Foucault operates and to understand the context of his more general claim. I have chosen to focus on his most influential study—that of prisons offered in _Discipline_ _and Punish_.
**The Subject of Discipline**
Foucault's account of the modern prison begins with a ghastly scene from 1757, recounting the punishment and execution of a regicide named Damiens. Wearing nothing but a shirt, he was conveyed in an oxcart "to the Place de Grève, where, on a scaffold that will be erected there, the flesh will be torn from his breasts, arms, thighs and calves with red-hot pincers, his right hand, holding the knife with which he committed the said parricide, burnt with sulfur . . . then his body drawn and quartered by four horses and his limbs and body consumed by fire, reduced to ashes and his ashes thrown to the winds" ( _DP_ , 3). As it turned out, six horses were eventually needed for this final operation, and "when that did not suffice, they were forced, in order to cut off the wretch's thighs, to sever the sinews and hack at the joints" ( _DP_ , 3).
Though we are horrified by the description of such an event orchestrated in the name of justice—and surely Foucault opens with just such an intention—we will properly understand Foucault's analysis only if we appreciate that, for him, modern society is in some sense worse than the one that tortured and executed Damiens. In other words, the thesis of Foucault's _Discipline and_ _Punish_ is that the society that tortured Damiens is less dangerous than the society that locked up Chief in that Oregon mental institution—that what Damiens suffered was, in some sense, less evil than what Randle P. McMurphy suffered. By the end of _Discipline_ _and Punish_ , Foucault wants us to be equally horrified by the mechanisms of domination that suffuse modern society.
This is why _Discipline and Punish_ —like almost all of Foucault's case studies—is not ultimately about prisons but rather concerns modern society as a whole. If he documents a change in the strategies of punishment and discipline within the penal system, this is only a microcosm of broader movements in modern Western culture. Although the book is divided into three parts that trace the historical development of penal theory from torture (roughly, sixteenth and seventeenth centuries) to punishment (eighteenth century) and finally discipline (nineteenth and twentieth centuries), the underlying force of Foucault's descriptive argument is to show that there is no qualitative difference between these epochs—that, if anything, the later developments are somehow more brutal. The story of change in punishment is not a narrative of progress, let alone a story of the triumph of the humane, but rather the substitution of one form of domination for another (more insidious) mode of domination. Foucault is interested in narrating not just the subplot of the history of prisons but rather the story of how we got to where we are: a modern "disciplinary"society where all of us, like Chief and McMurphy, are subject to mechanisms of control and repression.
What interests me about Foucault is his analysis of the way in which society is put together by sinews of power—and that it could not be otherwise. A different configuration of society is just a different (and not necessarily better) constellation of forces of power. All three parts of the book are trying to lay out the "mechanisms of power" at work in the different epochs of penal history. In the epoch of torture, this power "works on" the criminal in order to produce a confession, because a confession produces the truth—inscribes the truth in and on the body of the condemned ( _DP_ , 37, 38, 39–40, 41). This was brought about by both public and ritual means ( _DP_ , 43). The confession meant the condemned agreed with his sentence, which justified his punishment. But what was the end of such a "production of truth"? To solidify the power of the sovereign ( _DP_ , 47–49, 50). This is why he concludes that truth is always a function of power and vice versa: "The truth-power relation remains at the heart of all mechanisms of punishment," and that is precisely what _"is still to be found in contemporary penal practice_ " ( _DP_ , 55). At the core of Foucault's historical analysis is really a genealogy of the present.
A change did take place in how punishment is administered:eighty years after the torture and execution of Damiens, the model of punishment was no longer the rack but the regimen of discipline with its specification of rules and schedules (see _DP_ , 6–7). According to Foucault, the changes were not motivated by a desire to be more humane (as is commonly supposed) but rather were a means of dealing with political (even revolutionary) problems that attended the public display of torture. What began to happen was the opposite of what was intended: instead of enforcing new allegiance to the king, the spectacles of torture tended to make the people identify with the criminal! So punishment gradually became less violent and more secret. A "new age for penal justice" dawned ( _DP_ , 7). But the result of this shift was a social cost: "Crimes seemed to lose their violence, while punishments, reciprocally, lost some of their intensity, _but at_ _the cost of greater intervention_ " ( _DP_ , 75, emphasis added). Crime was constructed differently, based on new economic structures (focusing on property rather than violence). This created a certain "class justice" ( _DP_ , 75). But what interests Foucault is the way in which this shift in crime also entailed new emphasis on control and prevention and "stricter methods of surveillance" ( _DP_ , 77).
Was penal reform really a change in attitude, the result of a new "humanism"? Foucault is suspicious (see _DP_ , 78): "More certainly and more immediately, it was an effort to adjust mechanisms of power that frame the everyday lives of individuals; an adaptation and a refinement of the machinery that assumes responsibility for and places under surveillance their everyday behavior, their identity, their activity, their apparently unimportant gestures" ( _DP_ , 77). Such surveillance becomes synonymous with society itself, entailing "a closer penal mapping of the social body." The result—and this is decisive—was to "make punishment and the repression of illegalities a regular function, _coextensive with society_ " ( _DP_ , 82). Foucault goes on to speak of this as the generalization of punishment—the extension of controlling mechanisms to society itself—and the rest of _Discipline_ _and Punish_ traces this spreading and suffusion of discipline in modern society.
Through the course of his analysis, Foucault documents the formation of what he calls a "disciplinary society"—the primary goal of which is the creation of the individual—a "reality fabricated by this specific technology of power that [he has] called 'discipline' " ( _DP_ , 194). So the goal of a disciplinary society, and the institutions within that society, is the formation of individuals by mechanisms of power. Society makes individuals in its own image, and the tools for such manufacturing are the disciplines of power. Here Foucault adds an important proviso: "We must cease once and for all to describe the effects of power in nega-tive terms: it 'excludes,' it 'represses,' it 'censors,' it 'abstracts,' it 'masks,' it 'conceals.' In fact, power produces; it produces reality" ( _DP_ , 194). In Foucault's descriptive analyses, he does not attempt to offer any kind of evaluation of power as either positive or negative. But this does at least prohibit describing its effects as negative. For Foucault—and this is close to the heart of the theory of society offered in _Discipline and Punish_ —power is necessary and constitutive of society. All that changes are the mechanisms and technologies of power. One could not have a society that is not fundamentally characterized by power relations.
As a case study of the disciplined society, Foucault asks us to compare the difference between the way medieval cities responded to disease and the way the early modern city responded to the plague. The "political dream" of a disciplinary society is found, in fact, in the historical organization of the plague-stricken town. Why? Because the plague-stricken town is the paradigmatic example of a regulated, disciplined organization of individuals subjected to constant surveillance and registration ("the gaze"): "a compact model of the disciplinary mechanism" ( _DP_ , 197). While the medieval case of the leper produced procedures of exclusion and isolation, "the [modern] plague gave rise to disciplinary projects" ( _DP_ , 198). "The first is that of a pure community, the second that of a disciplined society. Two ways of exercising power over men, of controlling their relations, of separating out their dangerous mixtures. The plague-stricken town . . . is the utopia of the perfectly governed city" ( _DP_ , 198). Note that here Foucault's conception of the nature of power comes to the surface: the two different modes (exclusion, discipline), insofar as they are manifestations of power, always involve power over others; power is always some form of control. But couldn't one conceive of power differently? We'll return to this question.
The organization of society around discipline (as a new mode of power) culminates in an "architecture" concerned with much more than buildings; it is an architecture for society itself that generalizes "the gaze." The architectural ideal is the panopticon envisioned by Bentham's model for a penitentiary. "The Panopticon must not be understood as a dream building: it is the diagram of a mechanism of power reduced to its ideal form; its functioning, abstracted from any obstacle, resistance, or friction, must be represented as a pure architectural and optical system: it is in fact a figure of political technology that may and must be detached from any specific use" ( _DP_ , 205). That is why, whether in Bentham's prison or in Nurse Ratched's mental ward, "whenever one is dealing with a multiplicity of individuals [i.e., society] on whom a task or a particular form of behavior must be imposed, the panoptic schema may be used" ( _DP_ , 205). Thus the panopticon gives rise to panopticism, the generalized mode of asymmetrical surveillance, of being seen without seeing: "The panoptic schema . . . was destined to spread throughout the social body; its vocation was to become a generalized function" ( _DP_ , 207). The panopticon, then, far from being just an architectural ideal for the prison, was the utopian dream of a disciplinary society, "a generalizable mode of functioning; a way of defining power relations in terms of the everyday life of men" ( _DP_ , 205). Panopticism accomplishes the generalization of discipline throughout the social body.
Hence, "our society is one not of spectacle, but of surveillance" ( _DP_ , 217); that is, the generalization of discipline as constitutive of society itself ("coextensive with the entire social body," _DP_ , 213) weaves observation and documentation into the warp and woof of society in order to create "docile" and "useful" subjects—of the state, of capitalism, and so on ( _DP_ , 216–17, 220–21). The disciplinary society forms individuals into what it wants them to be: docile, productive consumers who are obedient to the state. Like McMurphy in the hands of Nurse Ratched, we are "projects" in need of reengineering and repair. We are all like the residents of the mental ward, in the place of being supervised and controlled, watched over and dominated by structures of surveillance and discipline that have become suffused through society.
In the final section of _Discipline and Punish_ , a very important shift in the analysis and argument of the book takes place: in a sense, we come back to the prison here, since the predominant focus of part 3 is not the prison but society as a whole. In other words, part 3 traces the development of a disciplinary society. In part 4, Foucault suggests that the modern penitentiary is a product _of_ a disciplinary society, rather than a disciplinary society being the reflection of penal practice. Thus, when we look at the modern penitentiary, we are looking at ourselves, seeing our society reflected in a mirror, as it were. The modern prison only codifies and localizes the generalization of discipline that has already been effected throughout the social body. It is not that the prison came to be a model for society; rather, the disciplinary mechanisms of society "colonized the legal institution" ( _DP_ , 231). If Foucault is documenting the birth of the prison, it is delivered from the matrix of a disciplinary society already in place. As an "apparatus for transforming individuals," the penitentiary "merely reproduces, with a little more emphasis, all the mechanisms that are to be found in the social body" ( _DP_ , 233)—in the military, schools, hospitals, and factories.As he comments, "This prison came from elsewhere" ( _DP_ , 256). And later he concludes that the prison "continues, on those who are entrusted to it, a work begun elsewhere, which the whole of society pursues on each individual through innumerable mechanisms of discipline" ( _DP_ , 302–3). It is not that society has come to mimic prisons but rather that prisons are microcosmic crystallizations of what is characteristic of society itself. In modern society discipline is ubiquitous.
This condition is analyzed through three particular regimens:isolation, compulsory labor, and treatment. Isolation is meant to bring the offender into confrontation with himself; it is a moral discipline. Work creates a laboring subject (proletarian) fit for a capitalist society, reformed to meet the requirements of the machines of production ( _DP_ , 242); it is an economic discipline. Treatment or reform is a means of transforming the abnormal into a normal individual, to "cure" his abnormality; it is a kind of medical discipline.As such, the modern penitentiary is about much more than mere detention or deprivation of liberty. These are necessary "supplements" to detention ( _DP_ , 248). They inscribe the structures and disciplines of society onto the offender. And it is this that gives birth to the delinquent as the target of reform—no longer a criminal or a monster, the offender is an "abnormality" ( _DP_ , 251–52). If the penitentiary is intended for the delinquent—the abnormal one—it is because the broader society has determined what counts as normal. In fact, this is almost precisely the theory of the "therapeutic community" on the ward in _One Flew over the Cuckoo's Nest_. As Chief recounts the theory, "A guy has to learn to get along in a group before he'll be able to function in a normal society; how the group can help the guy by showing him where he's out of place; how society is what decides who's sane and who isn't, so you got to measure up." The creation of the delinquent gets to the heart of what Foucault sees at work, not only in modern penal practices but also in modern society as a whole: disciplinary power aimed at normalization. In fact, at the conclusion of the chapter (and the book) we see what function _Discipline and Punish_ is meant to serve: it is a book intended to "serve as a historical background to various studies of the power of normalization and the formation of knowledge in modern society" ( _DP_ , 308). Hence, the prison is just one of many (all?) modern institutions that tend "to exercise a power of normalization" ( _DP_ , 308). What is this power? And how do we arrive at this conclusion?
The historic penitentiary at Mettray represents the culmination of this "carceral" ideal, incorporating the omni-disciplinarity of "cloister, prison, school, regiment" ( _DP_ , 293; cf. III.1). It included all the elements of the panoptic ideal, to the extent that even the observers were subjected to discipline when they were "taught the art of power relations" ( _DP_ , 295). "In the normalization of the power of normalization, in the arrangement of a power-knowledge over individuals, Mettray and its school marked a new era" ( _DP_ , 296). This new era involved the widening of carceral circles to society itself: "The carceral archipelago transported this technique from the penal institution to the entire social body" ( _DP_ , 298). This in fact erased any kind of qualitative distinction between the "least irregularity" and the "greatest crime," since both, on this register, were to be considered "a de-parture from the norm" ( _DP_ , 299). Rather than being an enemy of the sovereign or the social contract, "the social enemy was transformed into a deviant" ( _DP_ , 299)—hence the necessity for discipline and normalization from cradle to grave ( _DP_ , 300). The result is that nothing falls outside this carceral network: "There is no outside" ( _DP_ , 301). "In this panoptic society," Foucault observes, "of which incarceration is the omnipresent armature, the delinquent is not outside the law [there is no 'outlaw']; he is, from the very outset, in the law, at the very heart of the law, or at least in the midst of those mechanisms that transfer the individual imperceptibly from discipline to the law, from deviation to offence" ( _DP_ , 301). In fact, he seems to suggest that the deviant is a product of society.
This widening of the carceral circle means that society is characterized not by a disjunction between the social and the carceral but rather by a "carceral continuum" ( _DP_ , 303) that operates on the basis of a new law, the norm. Normalizing power thus spreads: "Borne along by the omnipresence of the mechanisms of discipline, basing itself on all the carceral apparatuses, it has become one of the major functions of our society. The judges of normality are present everywhere" ( _DP_ , 304). So the political issue at stake in the question of the prison is not about whether it is corrective or not: "The problem lies rather in the steep rise in the use of these mechanisms of normalization and the wide-ranging powers which, through the proliferation of new disciplines, they bring with them" ( _DP_ , 306). And it is precisely these powers of normalization that are the object of Foucault's concern.
**Will the Real Foucault Please Stand Up?**
So far I have summarized Foucault's case study in which he tries to concretely demonstrate the way in which power is knowledge or, more specifically, the necessary and ubiquitous role of power in society and social institutions. However, the next question is: Just what are we supposed to do with this analysis? Just why is Foucault painting this picture? What does he want us to conclude from this? What is Foucault trying to convince us of? After Foucault has powerfully described the development of modern, disciplinary society, what does he want us to do with this? Is this intended simply as a neutral, objective description of the way things are? As we've seen, such a notion runs counter to Foucault's own notion of genealogy. Is there, then, behind this description a latent prescription? Or, to put it otherwise, is Foucault painting this intricate picture for us to show us what's _wrong_ with modern society? Is his work a protest—a call for liberation from such repressive structures?
Here we hit on a difficult matter of interpretation, not just for outsiders but even for Foucault scholars: Who is the real Michel Foucault?Is he some kind of modern—an ultimately Enlightenment thinker committed to the autonomy and freedom of the individual? Is he a Marxist protesting the abuse of power and the oppressive structures of society? Is he a closet classic liberal who is lashing out against anything that would restrict the freedom and autonomy of the individual? These questions also place another question on the table: If Foucault is a postmodern thinker, just how modern is postmodernism? (And later we face another set of questions concerning just how far Christians can go in appropriating the Enlightenment project of freedom.) There are two basic ways that Foucault can be read in light of these questions:
1. _The Nietzschean Foucault._ On this reading, Foucault's analyses are not intended to convey any kind of moralizing stance. In other words, the Nietzschean Foucault is not painting this picture of power to show that power is bad and should thus be undone. If Foucault is a Nietzschean, his project is purely descriptive and not intended to harbor any kind of prescription: he is just showing us the way things are, not how they're supposed to be. If one started to talk about power as bad or to think about one organization of society as better than another, then one would be invoking a system of values. But as Foucault's own exposition of Nietzsche demonstrates, for him all such values only reduce to power.Beyond Foucault's own confessed Nietzscheanism, one can cite other evidence for this reading. It is certainly the case that Foucault does not think that power is bad; unlike Lord Byron, he does not think that power necessarily corrupts. As we've already seen, Foucault thinks we should stop talking about power in negative terms of repression and exclusion and instead think of it positively in terms of production. The Nietzschean Foucault would be trying not to change the world but only to describe it, perhaps even to celebrate it.
2. _The Liberal or Enlightenment Foucault._ Marx famously remarked that whereas philosophers usually just interpret the world, the point is to change it. The other way to read Foucault is to see him working in this broadly modern or Enlightenment tradition that includes both Kant and Marx. And, in fact, Foucault himself later acknowledged that he saw himself as a kind of Enlightenment thinker working in the tradition of critical theory stemming from Kant, through Marx, up to the Frankfurt School of Jürgen Habermas and others.In that case, the way to read Foucault is to see him giving us this disturbing picture of control and domination in order to motivate us to change things. The evidence for this reading of Foucault (which I think is the better one) is twofold: On the one hand, we have external evidence, such as Foucault's own activist involvement in prison reform movements in France. On the other hand, the very language of _Discipline and Punish_ already seems to communicate a negative evaluation of the way things are, eliciting a call for reform and revolution. When he describes particular configurations of power relations as networks of domination, such a descriptor already seems to entail an evaluation. A critical theory needs criteria. Indeed, the very notion of a neutral description of things runs counter to the core of Foucault's thought (and, as we've already seen, Derrida's too).
While there is certainly ambiguity on this score, the best reading of Foucault is to read him as a kind of closeted Enlightenment thinker; in fact, later in his work Foucault "comes out" on just this point.Moreover, this is certainly the way that Foucault has been _used_ —as a protest thinker, co-opted by various versions of the political left in order to resist the continued vestiges of control and domination in modern culture. As such, Foucault has been adopted by various movements—from gay rights to educational reform—that protest any form of control over the individual.
If Foucault is a kind of covert Enlightenment liberal, what does that mean? First, we need to specify what we mean by "liberal" here: a classical political liberal who places priority on the individual as a sovereign, autonomous agent—a being who is lord of his or her domain and thus resists any mode of external control. The watchword of liberalism is freedom: the free agent should not be controlled—by a king, by tradition, by religion, or by institutions. So the liberal's slogan is varying versions of "Hands off! Don't try to control what I think; don't try to control what I believe; don't try to control what I do."Any institution that tries to control beliefs or behavior is inherently dominating and repressive. And since institutions tend to be erected for just these reasons, there is a deep sense that institutions per se are structures of domination. Hence, liberalism in this sense is deeply anti-institutional; while leftists—whether politicos or filmmakers—speak about being radicals, it is usually a radicalizing of this Enlightenment notion of freedom. Thus there is a deeply libertarian streak to liberalism that eschews control and discipline. The very goal of Enlightenment is liberation, which is why both Kant and Marx are Enlightenment thinkers. And insofar as Foucault's work feeds into just such impulses, it is hard not to see a libertarian streak in his descriptions.
Just as it is difficult not to side with McMurphy in the face of the grinding repression of the Combine and Big Nurse, so it is difficult not to be sympathetic to Foucault's suspicion about institutions of discipline and formation. But I would argue that Christians should resist the temptation to side with either Mc-Murphy or Foucault (but neither should they side with Nurse Ratched or Mettray). On the other hand, Foucault provides some critical insights into the nature of discipline and its role in the formation of individuals. Thus my engagement with Foucault is complicated, as is my criticism. The critical point is that Foucault is absolutely right in his analysis of the way in which mechanisms of discipline serve to form individuals, but he is wrong to cast all such discipline and formation in a negative light. In other words, Christians should understand discipline positively, precisely because Christians should not be liberals in the classical sense described above. Christians should eschew the very notion of an autonomous agent who resists any form of control. By rejecting Foucault's liberal Enlightenment commitments, but appropriating his analyses of the role of discipline in formation, we can almost turn Foucault's project on its head.
**Is Power All Bad?**
If I have so far argued that Foucault is a kind of closet liberal and thus deeply modern, I need to be equally critical of evangelical (and especially American) Christianity's modernity and its appropriation of Enlightenment notions of the autonomous self. Indeed, many otherwise orthodox Christians, who recoil at the notion of theological liberalism, have unwittingly adopted notions of freedom and autonomy that are liberal to the core. Averse to hierarchies and control, contemporary evangelicalism thrives on autonomy: the autonomy of the nondenominational church, at a macrocosmic level, and the autonomy of the individual Christian, at a microcosmic level. And it does not seem to me that the emerging church has changed much on this score; indeed, some elements of emergent spirituality are intensifications of this affirmation of autonomy and a laissez-faire attitude with respect to institutions.We don't want denominations to tell us how to run our churches, and we don't want churches to tell us how to run our lives. If either of these institutions threatens our autonomous sphere with control—let alone discipline—we jump ship: the church splits from the denomination to become an independent congregation, or the individual leaves the church and hops to another. So when we see the experience of Chief and McMurphy, because of our own (American) liberal suspicions about institutions and institutional control, we identify with these anti-institutional figures—all in the name of freedom (even if we also talk in terms of "law and order").
But it is crucial to distinguish truly biblical conceptions of positive freedom and empowerment from liberal Enlightenment conceptions of negative freedom as a kind of hands-off stance. To put it a little more staunchly: freedom is an idol of the contemporary church, and we will only properly resist Foucault's liberalism if we give up our own.
Let me anticipate an initial response: Why should Christians resist these liberal conceptions of freedom? How can I be against freedom? Am I going to offer a defense of domination? If we're opposed to liberal conceptions of freedom, doesn't that mean we're _for_ control and domination?
Well, yes. But in order for me to show why this isn't a revival of fascism, let's return to Foucault's conception of power. As Foucault describes it, social institutions and relationships are necessarily constructed on the basis of power relations; power is ubiquitous. Moreover, power is understood as power over others—some kind of domination (even if it isn't a simple bifurcation of haves and have-nots, those with power and those without).This power is channeled through mechanisms of discipline—various practices and regimens—that form the individual by conforming him to what society wants—a good worker and consumer. And while he cautions that we should not think of this negatively, the overwhelming impression of his work is that this situation is both repressive and oppressive.
But should we accept this negative view of power? Is power all bad? Specifically, can Christians share in this devaluation of power and discipline as inherently evil? Can we who claim to be disciples—who are called and predestined to be conformed to the likeness of the Son (Rom. 8:29)—be opposed to discipline and formation as such? Can we who are called to be subject to the Lord of life really agree with the liberal Enlightenment notion of the autonomous self? Are we not above all called to subject ourselves to our _Domine_ and conform to his image? Of course, we are called not to conform to the patterns of "this world" (Rom. 12:2) or to our previous evil desires (1 Peter 1:14), but that is a call not to nonconformity as such but rather to an alternative conformity through a counterformation in Christ, a transformation and renewal directed toward conformity to his image. By appropriating the liberal Enlightenment notion of negative freedom and participating in its nonconformist resistance to discipline (and hence a resistance to the classical spiritual disciplines),Christians are in fact being conformed to the patterns of this world (contra Rom. 12:2).
Once we reject the liberal conception of the autonomous agent who resists control and discipline, Foucault's analyses of the mechanisms of discipline take on a very different light. Admittedly, Foucault seems to suggest that modern society simply took over religious disciplines and rituals and generalized or altered them. Thus he suggests that modern factories resemble medieval monasteries ( _DP_ , 149), that modern prisons bear the marks of earlier convents ( _DP_ , 243), and that the general structures of a disciplinary society mimic monastic communities ( _DP_ , 149). And insofar as the prisons and factories of a disciplinary society are seen as repressive and dominating, the charge flows back against these earlier communities of religious discipline.
How can we respond to this charge? Of course, on the one hand, this simply flows from Foucault's liberalism, his opposition to domination and control as such. Insofar as we don't accept such a conception of the autonomous self, the criticism doesn't hold. But more importantly, on the other hand, this raises a crucial point: while formally or structurally speaking, there are mechanisms of discipline operative in both the convent and the prison, in both the factory and the monastery, more specifically these disciplines and practices are aimed at very different ends. And here we must make an important distinction: we can distinguish good discipline from bad discipline by its _telos_ , its goal or end. So the difference between the disciplines that form us into disciples of Christ and the disciplines of contemporary culture that produce consumers is precisely the goal they are aiming at. Discipline and formation are good insofar as they are directed toward the end, or telos, that is proper to human beings: to glorify God and enjoy him forever (Westminster Catechism, question 1). Or, to put it otherwise, a disciplinary form is proper when it corresponds with the proper end of humanity, which is to be (renewed) image bearers of God. So other forms of disciplinary formation are bad and wrong insofar as they try to mold human beings into something other than what they are called to be. Almost universally these other modes of discipline are reductionistic because they reduce human beings to something less than they are called to be. Some modes of discipline reduce us to economic animals whose primary end is production and consumption; other modes of discipline reduce us to sexual animals whose primary end is instinctual satisfaction; still other modes of disciplinary society try to mold us into violent creatures whose primary end is destruction. What is wrong with all these disciplinary structures is not that they are bent on forming or molding human beings into something, but rather _what_ they are aiming for in that process. Thus it is helpful to distinguish the formal _structure_ of disciplinary formation as such from the specific _direction_ discipline takes.
Admittedly, as we've learned in previous chapters, what constitutes the proper end, or telos, of human formation depends on the ultimate story we tell of what human beings are and what humans are called to be. The Christian story specifies that human beings are creatures whose ultimate telos is to image their Creator and be conformed to the image of his Son. Different stories obviously envision different ends for humanity. So what constitutes good or proper formation must be determined in relation to the particular founding narrative that we confess tells the truth about the world and the human condition. As such, we can draw an important link between Lyotard's emphasis on the role of narrative and Foucault's emphasis on the role of formation: discipline is aimed at formation for a specific end, and that end is determined by our founding narrative.
**Taking Foucault to Church**
Unlike our engagements with Lyotard and Derrida, there is a temptation for Christians to side with Foucault because much of modern Christianity has unwittingly bought into the Enlightenment notion of autonomy. As such, if we're going to put Foucault's analysis to work in shaping a postmodern church, we have to stand him on his head a bit: we need to see what he describes but reject what he thinks about disciplinary society as such. If we do that, what exactly does Foucault give us? In what way can he be a catalyst for thinking about a postmodern church?
_The Cultural Power of Discipline Formation_
Keep in mind that Foucault offers studies of all kinds of practices (from bells ringing to get us to move according to timetables to the use of negative stimuli to get us to stop doing something) that shape and mold human beings to act in a certain way—to be certain kinds of persons. Foucault is absolutely right about the fact that this works! Disciplinary mechanisms in our society _do_ make humans into certain kinds of people who are aimed at particular goals. For instance, many Americans are defined by the primary goal of consumption. They stake their identity on their material possessions—on labels, objects of luxury, and the never-ending process of keeping up with trends. If we look at the way upper-middle-class Americans spend their time and money, we have to conclude that their ultimate goal is to be faithful consumers. How did they get to be that way? How did they become that kind of person? The answer isn't simple, but we can easily identify several disciplinary practices that form human beings into these consuming animals.
First, the very success of capitalism depends on a consuming culture as market, and particularly a culture that wants ever-new products (otherwise the market becomes quickly saturated and the possibilities of profits quickly diminish). Thus we have a culture—or at least a class within the culture—that has a vested interest in seeing a society of consumers. How does it create this population of consumers? One of the primary ways has been the advent of mass media, which, from its inception, has been aimed at marketing. We must understand, for instance, that television programs were basically invented to gain an audience for commercials. Thus the majority of mass media is undertaken as a means for creating an audience for advertising that will eventually become a market of consumers. Marketing, then, is driven by investing products with social, sexual, and even religious value, which makes them something much more than they are.In other words, marketing capitalizes on fundamental structural human desires for meaning and transcendence and presents products and services as ways to satisfy these human longings. It then utilizes the tools of disciplinary practice to inject these values into the very character of human beings—internalizing the values so that they become part of the person. By using repetition, images, and other strategies—all of which communicate truths in ways that are not cognitive or propositional—marketing forms us into the kind of persons who want to buy beer to have meaningful relationships, or buy a car to be respected, or buy the latest thing to come along simply to satisfy the desire that has been formed and implanted in us. It is important to appreciate that these disciplinary mechanisms transmit values and truth claims, but not via propositions or cognitive means; rather, the values are transmitted more covertly, as Chief recognized. They are communicated by a world of images and through a range of practices that teach the body, as it were. This covertness of the operation is also what makes it so powerful: the truths are inscribed in us through the powerful instruments of imagination and ritual.
It is absolutely crucial that the church recognize this process. In other words, the first thing we need to learn from Foucault is how pervasive disciplinary formation is within our culture—from public education to MTV. Anyone who raises children will with some degree of reflection recognize that this is the case. Nothing frustrates me more than the "label idolatry" already evident in my children. Indeed, raising children in American culture has made me see and appreciate the forces of disciplinary formation, and the globalization of American values makes this a reality around the world. All of us certainly find ourselves in multiple webs of power relations and subject to multiple disciplinary mechanisms bent on forming us into certain kinds of people. In the world of late-modern capitalism, many of these disciplinary interests coalesce. So those disciplinary mechanisms that would form us into primarily sexual animals have been co-opted by capitalist interests that want to form us into consumers. Everything from beer and deodorant to shampoo and rice is sold on the basis of sex. By unveiling the cultural power of disciplinary formation, Foucault can be a catalyst for the scales falling from our eyes so that we see what is happening.
_The Necessity of Counterformation by Counterdisciplines_
But beyond simply recognizing that such cultural formation is pervasive, we also need to recognize that the telos, or goal, at which these disciplines aim is fundamentally inconsistent with (and even competing with) the message of the gospel and what it specifies as the proper end of humanity. We need to recognize the inconsistency between how late-modern capitalism defines human beings and how Christian faith defines us. Because of the covertness of this formation, Christians are often not alert to what they are becoming. To use a metaphor that George Barna employed for quite different ends: Christians are sometimes like frogs in a kettle. Reportedly, if you place a frog in a pot of room-temperature water and gradually increase the temperature of the water, even to the boiling point, the frog will not jump out of the kettle, even if it means death. This is either because the frog doesn't sense the change, or because the change is so gradual it lulls the frog into accepting the environment. So also with the church: because the disciplinary mechanisms of Disney, MTV, and the Gap are so insidious and covert, we don't recognize the way in which their message—and their vision of the human telos—is shaping our own identity. Christians need first to recognize that disciplinary formation takes place in culture, then second, to recognize the antithesis between the dominant culture's understanding of the human calling and the biblical understanding of our ultimate vocation.
But the church must also do a third thing: enact countermeasures, counterdisciplines that will form us into the kinds of people that God calls us to be. Too often we imagine that the goal of Christian discipleship is to train us to think the right way, to believe the right things. But the ultimate goal of sanctification and discipleship is to shape us into a certain kind of person: one who is like Jesus, exhibiting the fruit of the Spirit (Gal. 5:22–23), loving God and neighbor, caring for the orphan, the widow, and the stranger (Jer. 22:3; James 1:27). He has shown us what is good and what the Lord requires of us: to do justice, to love mercy, and to walk humbly with God (Mic. 6:8). These are all just translations of the broader human vocation, which is to bear the image of Christ as renewed image bearers of God. The primary aim of discipleship is to create a certain kind of person who acts in a certain way, not someone who simply thinks in a certain way. According to the Scriptures, knowing the truth is only instrumental to ultimately doing the truth (Jer. 22:16).
But how do we become that kind of people? How do I become the kind of person who "does" the truth? It takes practice. First, it requires grace. Because no one is good (no, not one!), being properly directed to our proper telos requires a regeneration and redirection of the heart by the Holy Spirit. That is why they are fruits of the Spirit. Insofar as the Spirit indwells believers, they are being formed into the image of Christ to the extent that they learn to walk in the Spirit and in the Spirit's power. However, while regeneration is a necessary condition for becoming this kind of person, it is not a sufficient condition. This must be cultivated by practices of sanctification.
Second, recognizing the structural goodness of disciplinary formation, the church must utilize disciplines that will form us into these kinds of people—disciplines that will counteract the formation of MTV and television commercials. We would do well to recover the tradition of spiritual disciplines such as prayer and fasting, meditation, simplicity, and so on as a means of shaping our souls through the rituals of the body. Further, as I've already suggested, our corporate worship should be aimed at constituting us as disciples who are countercultural agents of redemption. Communion and confession, foot washing and economic redistribution are ways of practicing what it means to be citizens of the kingdom. And such practices inscribe this telos of the kingdom into our character.Christian worship is one of the primary arenas in which we participate in the practices that shape who we are. If our worship simply mimics the disciplinary practices and goals of a consumer culture, we will not be formed otherwise. Conceiving of the church as a disciplinary society aimed at forming human beings to reflect the image of Christ, we will offer an alternative society to the hollow formations of late-modern culture.
1. I will refer to the 1975 film directed by Milos Forman but also to the original novel by Ken Kesey (1962; repr., New York: Penguin, 1976). The novel is narrated in the first person from Chief 's perspective.
Kesey, _One Flew over the Cuckoo's Nest,_ 1. One could also compare the role of ducts in the film _Brazil._
3. Ibid., 5.
4. Ibid., 181.
5. Ibid., 179.
6. Ibid., 209.
7. This is the (somewhat controversial) interpretation of Foucault offered by James Miller in his _Passion of Michel Foucault_ (New York: Simon & Schuster, 1993). Miller argues that Foucault's theories reflect an outgrowth of his own experiential and experimental sexual escapades.
8. Michel Foucault, _Discipline and Punish: The Birth of the Prison_ , trans. Alan Sheridan (French original, 1975; repr., New York: Vintage, 1977); henceforth abbreviated in the text as _DP_.
9. Foucault, _The Birth of the Clinic: An Archaeology of Medical Perception_ , trans. A. M. Sheridan Smith (French original, 1963; New York: Vintage, 1973). A second French edition appeared in 1972 and was published in English in 1994.
. As Foucault puts it: "It has been said but you have to understand that when I read—and I know it has been attributed to me—the thesis, 'Knowledge is power,' or 'Power is knowledge,' I begin to laugh, since studying their _relation_ is precisely my problem. If they were identical, I would not have to study them, and I would be spared a lot of fatigue as a result. The very fact that I pose the question of their relation proves clearly that I do not _identify_ them" ("Critical Theory/Intellectual History," an interview reprinted in _Critique and Power_ , ed. Michael Kelly [Cambridge, MA: MIT Press, 1994], 133).
. This is why I suggest below that Foucault's "genealogy" shares something in common with presuppositionalist approaches to epistemology, which emphasize the role of "control beliefs" [Wolterstorff] in the constitution of knowledge.
. Christians, aware of the deep structural effects of sin, should also operate with a hermeneutics of suspicion, even if suspicion doesn't get the last (or first) word. For discussions along these lines, with an eye on practice, see Merold Westphal, _Suspicion and Faith_ (Bronx, NY: Fordham University Press, 1998).
. Foucault unpacks this method most carefully in his essay "Nietzsche, Genealogy, History," in _Language, Counter-Memory, Practice_ , ed. Donald F. Bouchard (Ithaca, NY: Cornell University Press, 1977).
. Ibid., 142.
. "The search for descent is not the erecting of foundations: on the contrary, it disturbs what was previously considered immobile; it fragments what was thought unified; it shows the heterogeneity of what was imagined consistent with itself " (ibid., 147).
. Ibid., 150, emphasis added.
. One of the most important aspects of the panopticon was this asymmetry, which allowed the "subjects" to be seen but not the observers. The subject "is seen, but he does not see" ( _DP_ , 200). This invisibility of the observers also means that they need not always be observing: "for what matters is that he knows himself to be observed," that he might be the subject of observation at any moment ( _DP_ , 201). Cf. the words written on the cell walls at Mettray: "God sees you." The panopticon is the substitution for an omniscient deity (or Santa!).
. Foucault analyzes these other social institutions, especially in the chapter titled "Docile Bodies" (III.1).
. This is powerfully illustrated in another film, _Brazil_.
. One can see all of these at work in _One Flew over the Cuckoo's Nest_.
. Kesey, _One Flew over the Cuckoo's Nest_ , 47. All of this is undertaken under the banner of "democracy" (ibid.).
. It is not accidental that David Macey's landmark biography is titled _The_ _Lives of Michel Foucault_ , in the plural.
. See Foucault, "Nietzsche, Genealogy, History."
. Foucault articulates this very clearly in lectures and interviews collected in _The Politics of Truth_ , ed. Sylvère Lotringer (New York: Semiotext[e], 1997).
. See especially the lectures collected in _The Politics of Truth_.
. The term _liberal_ is used in a more restrictive sense in American political parlance. But it is important to recognize that, with respect to the tradition of political liberalism I am describing here, both Democrats and Republicans are liberals. It is just that their "Hands off!" stances apply to different things: a Democrat is more likely to assert, "Hands off my body—I can do what I want with it!"; a Republican is more likely to assert, "Get your grubby tax-collecting hands off my money—it's mine to do what I want with it!" These are two points on the same liberal continuum.
. Although I am sympathetic to his critique, I worry, for instance, that Spencer Burke's critique of the church has absorbed some version of this penchant for liberal autonomy. See Spencer Burke, "From the Third Floor to the Garage," in _Stories of Emergence: Moving from Absolute to Authentic_ (Grand Rapids: Zondervan, 2003), 27–39. I worry that what is being offered is a spiritual version of Chief 's escape from the institution into the "freedom" of the wilderness.
. That said, we should also honor the complexity of situations that give rise to nondenominational churches. Some, no doubt, emerge from denominations that have become so static and modern that the denomination is no longer a link to the catholic tradition but rather a reified modernist institution. Not all denominations represent a link to the great catholic tradition; this is especially true of anti-creedal Protestant denominations that emerged in modernity. Vis-à-vis these, nondenominational congregations could actually provide the opportunity to be more catholic. However, nondenominational churches must grapple with how to connect with the _normativity_ of the catholic tradition. My thanks to Brian McLaren for helping me to complexify these matters.
. This is why Foucault tends to resist being described as a Marxist; he thinks the Marxist conception of power in society is too simplistic, boiling down to a structure of haves and have-nots. But for Foucault, even the oppressors are effects of power, just as Nurse Ratched is an effect of the system, a product of the Combine.
. The very notion of spiritual disciplines remains foreign and even anathema to many evangelicals. However, there are signs of a shift, engendered in large part by Richard J. Foster's classic, _Celebration of Discipline: The Path to_ _Spiritual Growth_ (San Francisco: Harper & Row, 1978).
. For a nuanced, brilliant account of these matters, drawing on Foucault, see Daniel M. Bell Jr., _Liberation Theology after the End of History: The Refusal to_ _Cease Suffering_ , Radical Orthodoxy Series (London: Routledge, 2001). I discuss this in some detail in _Introducing Radical Orthodoxy: Mapping a Post-secular_ _Theology_ (Grand Rapids: Baker, 2004), 243–54.
. For a very helpful and insightful discussion of the religious nature of advertising, see Charles Colson and Nancy Pearcey, _How Now Shall We Live?_ (Wheaton: Tyndale House, 1999), chap. 23. See also Jean Kilbourne, _Can't Buy_ _My Love_ (New York: Free Press, 2000); and her video series _Still Killing Us Softly_ ; and James B. Twitchell, _Adcult U.S.A.: The Triumph of Advertising in American_ _Culture_ (New York: Columbia University Press, 1996).
. For further discussion of worship as a means of forming character, see Marva Dawn, _Reaching Out without Dumbing Down: A Theology of Worship for_ _This Urgent Time_ (Grand Rapids: Eerdmans, 1995), chap. 6; Stanley Hauerwas, _The Peaceable Kingdom: A Primer in Christian Ethics_ (Notre Dame: University of Notre Dame Press, 1983), 107–10; and Smith, _Introducing Radical Orthodoxy_ , 235–39.
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* * *
Applied Radical Orthodoxy
A Proposal for the Emerging Church
We have seen that postmodern thought can be an occasion for a recovery of ancient Christian themes and sources because the critique of modernity reopens a significant role for tradition. In this chapter we will consider the unique connections between tradition and postmodernism by examining the voice of Radical Orthodoxy.
**Raising the Curtain:** **Whale Rider**
If there is one thing that postmodernism is opposed to, it is the traditional. The very notion of the postmodern has become synonymous with the new, the novel, the _avant-garde_ , and at the very least, the contemporary. But is it possible to be faithful to tradition in the contemporary world? Is that even something we should want? Don't the advances of modernity—instantaneous global communication, the virtual connection of the four corners of the world, the steady march of technological mastery, the fluidity of trends and self-invention—don't these represent the overcoming of tradition and an escape from its static past? Who would want to go back to crawling when we've learned to fly? Or could it be that the price of flying is not worth the so-called freedom? Might the progressive, ahistorical detachment of our modern life be a denial of something that is part of the fabric of being human? Could it be that we are traditioned creatures, in which case unhooking ourselves from tradition would end only in self-alienation, even self-destruction?
These tensions between tradition and contemporary culture are powerfully illustrated in the film _Whale Rider_. The film opens with the words "In the old days . . . ," then immediately cuts to the very modern, technological scene of childbirth in a contemporary hospital. The film's opening is both ancient and future: appealing to "the ancestors" while documenting the birth of the next generation. This ancient-future tension is what drives the narrative, centered around the young girl Paikea.
_Whale Rider_ tells the story of a Maori tribe struggling to flourish in the Eastland region of New Zealand. The Maori have not weathered modernity well: the "opportunities" it has presented to the youth have produced something of a lost generation. Some, like Porourangi (the firstborn who should be chief-in-waiting), have welcomed the possibility of international travel as a way to escape the restrictions and expectations of a traditional culture. His younger brother, Rawiri, like other young men in the tribe (such as Hemi's father) has also escaped, but into a world of drug-induced immobilization and dilapidated squalor. In response to this rejection of Maori identity—particularly the rituals and "old ways" of the tribe—the chief, Koro, has responded with a retrenched commitment to recover the tradition in its most stringent form, further alienating his sons, Porourangi and Rawiri.
Koro has pinned all his hopes on Porourangi's firstborn, who, as the film opens, is being delivered into this world with much difficulty. This son, he believes, will be "the One" who will recover the power of the ancestors and redirect the Maori community— a prophet who will lead them. But the birth of this baby boy results in the death of his mother—only for the son to die moments later. Porourangi has lost his wife and son; Koro has lost his hope. And lost in this tumult of grief and shattered hopes, a twin has arrived: a baby girl, ignored by her grandfather and later abandoned by her father. But in his confrontation with his father—who in the presence of Porourangi's dead wife could only ask, "Where's the boy?"—Porourangi has the audacity (and the hope?) to tell Koro his daughter's name: "Her name is Paikea," he announces.
"What?" Koro responds.
"You heard me."
"No, not that name."
Why Koro's horror on this announcement? Because the name both retrieves and challenges the heart of Maori tradition. The narrative or myth that orients the tribe revolves around the story of an ancient Paikea who, when his canoe capsized in the South Pacific, rode on the back of a whale to the Eastland region of New Zealand. Since that time, the tribe's chiefs have been the firstborn, male descendants of Paikea. For Porourangi to announce that his daughter would be called Paikea, when his firstborn son and rightful heir had died, was, from Koro's perspective, an act of defiance and a patent rejection of the traditions of the ancestors. And indeed this was Porourangi's last act before abandoning the island—and his responsibilities as firstborn—altogether, leaving the baby Paikea in the care of a loving grandmother but a disdaining grandfather, or Paka. In the face of Koro's rigid understanding of the tradition, Porourangi's only option is rejection and abandonment: he leaves his daughter, he escapes the tribe, and in a powerful symbol throughout the film, he leaves on the shore an unfinished _te waka_ , or war canoe. Once the object of his artistic gifts and passion and a powerful expression of the tribe's tradition, the half-finished hulk of a vessel sits abandoned on the coastline—left to the elements but still having an intimidating presence that haunts the tribe. While they have sought to reject and forget their heritage, the massive, empty hull just won't go away.
When we next see the young Paikea, we find her in a tortured relationship with her Paka—tortured on both sides (for Paikea refuses to stop loving her grandfather). On the one hand, Paka can't seem to stop loving Paikea, playing with her on his bike, smiling gently into her face; on the other hand, his passion for the revival of the tribe through a new leader—firstborn and male—is frustrated every time he calls Paikea's name. She represents to him his own failure, and he worries that she is the sign that the tribe has been abandoned by the ancestors; indeed, he sees Paikea as bad luck. With the death of the firstborn male, the line of chiefs has returned to the ancestors, and Koro must find some way to get it back—some way to retrieve the charism of chief in someone. But he knows that can't be Paikea: the charism of chief could never be given to a girl.
So Koro undertakes measures to try to retrieve the tradition: he gathers all the firstborn males of the village and launches a sacred school meant to form them in the ancient ways. Paikea is systematically excluded, even told to stay off the grounds of the _marae_ , or temple, because it is, according to Koro, "the one place where our old ways are upheld." In this sacred school of learning, they are taught the old ways: the chants and songs of the tribe, its stories and myths, its dances and rituals. The young boys find these old ways strange vis-à-vis the modernity in which they feel most comfortable. This is seen, for instance, in a ritual war dance where the boys are taught to slap their chests "until they bleed" and to stick out their tongues in the face of the opponent. "When you stick out your tongue," Koro explains, "you're saying to your enemy, 'I'm going to eat you.' " The boys, bewildered, are tentative to enter the practice.
They do take to one of the practices: a ritual form of combat using long staffs. Their formation in modernity has fostered a certain interest in violence, even if ritualized. Paikea, excluded from the training, tries to mimic the actions from a distance, using a broom handle. When Koro finds out, he hastily banishes her: "Do you want me to fail?" he asks her. But her grandmother shares an interesting piece of family history: her uncle Rawiri—who is now mired in idleness and squalor by a drug addiction—was once a champion of this ritual fighting ("before he was fat and ugly," she comments). So she suggests that Paikea ask him for private lessons. Rawiri's friends and girlfriend are surprised to learn that as a young boy he was once a master of ancient traditions; and indeed, Rawiri has really forgotten this about himself—a microcosmic picture of the way he has generally forgotten who he was (and is). But when Paikea invites him to train her, as soon as he grabs the staff, there is a transformation. This artifact of the tradition has an almost sacramental character and seems to immediately recall him to himself, to remind him of not only who he was but who he is called to be. This reacquaintance with the tradition has a remarkably humanizing effect on one from the lost generation. Rawiri takes up the challenge with fervor and not only trains Paikea to be the best in the tribe but also recovers his own sense of identity and worth.
Koro's sacred school of firstborns is finally subjected to one last test to discern which is the One destined to bring the charism of chief back to the tribe. Koro takes the boys out into the bay in a rickety aluminum boat (which is a small, ugly artifact of modernity compared with the massive decorated hull of the _waka_ they can see on shore the whole time). Anchoring in a deep section of the bay, he removes from his neck a whale-tooth amulet (a _reiputa_ ) and throws it into the deep: "One of you will bring it back to me," he announces. The boys dive into the water, eager to retrieve the whale's tooth and seal their identity as chief-inwaiting. One by one they bob up to the surface, until Koro finally asks, "Well, which one of you has it?" None could retrieve the amulet. They return to shore in silence; Koro makes his way to his bed and remains there for days on end. He has failed; the ancestors have ignored his prayers; his people are doomed to the darkness of forgetting.
But Paikea has heard her grandfather's prayers and sympathizes with his struggles. "He was calling to the ancient ones," she tells us in a voice-over, "asking them to help him. But they weren't listening. . . . So I tried." She makes her way out to the empty, haunting hull of the _waka_. It is into the skeleton-like presence of the _waka_ that Paikea retreats when she feels her grandfather's rejection most intensely. It was also the site of an important conversation with her father, when he briefly returned to the tribe. At that time Paikea was intensely aware of Koro's rejection, and Porourangi had explained to her: "He's looking for something that doesn't exist anymore." He's looking for a "prophet," someone to "lead [their] people out of the darkness."
But at this time Paikea retreats to the _waka_ to pray for her grandfather—to call the ancestors in his stead. "And they heard me." This initiates the transformative sequence of the film: one night Paikea gives her award-winning speech in "love and respect" for her grandfather, in which she explains the story of her people and tearfully recognizes that she was not the leader that her grandfather expected; on this same night, Paka finally emerges from his room and is about to attend Paikea's speech but is turned back to the beach. Distant cries lure him to the shore, where he finds an entire pod of whales have beached themselves and are slowly dying. "Who's to blame?" he asks himself. The ancestors had heard Paikea's prayers and tears, and had come. But what did it mean? "It was a test," Paikea concludes.
The entire tribe comes together in the face of this tragedy and labors through the night to try to save the whales. They drape the creatures with dampened blankets and run to and from the sea with pails of water. When one of the whales dies in the hour before dawn, Rawiri comforts a grief-stricken woman who, just hours before, was one of the cynical old women gathered around the card table. The death of the whale has sparked in her a memory of a way of life she has forgotten. But their nocturnal labors for the whales is tinged with futility as the sun begins to rise: further down the beach they see the massive hulk of a whale that must be ancient, dwarfing those they have been tending. If it has seemed almost hopeless to get these smaller creatures back in the water, what is the hope for saving this mammoth?
Koro approaches the gentle, colossal creature with chants and prayers and quickly diagnoses the situation: they need to find some way to turn the animal around and get it headed back out to sea. Then the others will follow. He barks at Rawiri to gather everyone for this task, but Rawiri is less confident: they've been working all night, he replies. "They'll do it for you," Koro responds—a first indication that Koro sees in his second-born the charism of a leader. Rawiri gathers together the tribe to devote themselves to this Herculean, impossible task. The strategy seems simple: attaching a massive rope to the tail, which will be pulled with a tractor, both men and women will push simultane-ously on the head and attempt to redirect the whale. The tractor strains and the rope begins to fray, eventually snapping. What now? Is there any hope?
Paikea has been watching all this from the hull of the _waka_. After the tribe leaves the shore, she makes her way down to the water's edge and inches cautiously toward the whale. Mimicking the traditional Maori greeting she has known since she was a child, Paikea rubs noses with the whale, prayerfully, trying to discern what she ought to do. Slowly, but resolutely, she climbs up on the back of the animal and assumes the stance of her namesake: Paikea the ancient whale rider. The whale responds to her entreaties, beats its tail fin against the sand, and begins to maneuver its massive body toward the ocean's depths. Paikea remains on the whale's back, riding the creature as it leads the other beached whales back to the depths, where they can flourish. Paikea seems to refuse to let go and remains attached to the whale as it makes its descent into the water.
As this is happening, Paikea's grandmother begins to search for her: "Where is she? Where is she?" the grandmother cries. This turns the tribe's attention to the beach and to the ocean, where they see the young Paikea riding the whale back out to sea. With tears of sadness and anger, the grandmother puts something into Koro's hand: the whale's tooth none of the boys had been able to retrieve. "Which one?" he asks.
"What do you mean 'which one?' " his wife replies, indignantly. Koro knows.
The force of the dive pushes Paikea off the whale's back, and she is later retrieved from the waters, in serious condition. Koro cautiously enters her hospital room, humbles himself before young Paikea, kneeling beside her in submission: "Forgive me, O Wise Leader. I am just a fledgling new to flight." The scene then cuts to the underwater serenity of a young whale playfully dancing underneath its mother. Koro has found the charism of the ancestors where he had least expected it.
In the closing scene of the film, we see that the creative, unexpected retrieval of the tradition led by Paikea has transformed and renewed the community: cutting across a shot of blue sky is the bow of the _waka_ , now completed and shimmering in bright colors, its intricate carving completed by Porourangi, who has returned home. He and Rawiri lead the team of those from the lost generation as they roll the _waka_ out to sea and launch it on its maiden voyage. On the shore are groups of tribal dancers in traditional costume and paint, chanting and singing according to the old ways and led by some of the young men and women who had been mired in drugs and the underside of modernity. While the film does not present a simplistic rejection of modernity (modern medicine nurses Paikea back to health), it is the recovery of tradition _in_ modernity, and sometimes against modernity, that makes possible the renewal of the identity of Paikea's people. It is the strangeness of ancient ritual and the outlandish notion of whale riding that grants them a future.
**Redeeming Dogma: A More Persistent Postmodernism**
The church would do well by learning to ride whales. We need to be attentive and discerning about the way modernity has eroded our identity as the "peculiar people" who make up the body of Christ and seek to retrieve the strange ways and ancient practices of the communion of the saints in order to re-form who we are. In this concluding chapter, we will see that the outcome of the postmodernism sketched in earlier chapters should be a robust confessional theology and ecclesiology that unapologetically reclaims premodern practices in and for a postmodern culture. A more persistent postmodernism—one that really follows through on the implications of claims made by Derrida, Lyotard, and Foucault (or better, the meshing of their central claims with insights from the Christian theological tradition)—will issue not in a thinned-out, sanctified version of religious skepticism (a "religion without religion") offered in the name of humility and compassion but rather should be the ground for the proclama-tion and adoption of "thick" confessional identities. Much that we find in the name of postmodern spirituality, or even in the name of an "emerging" Christianity, is a timidity with respect to the particularities of the Christian confessional tradition. While this is almost certainly a corrective with respect to rabid forms of fundamentalism—whether Protestant or Catholic—a retreat into a thinly "ecumenical" Christianity that reduces confession to bland concerns with justice or love still remains a latent version of a very modern project. In this respect much of the dominant discussion in postmodern theology or philosophy of religion actually shrinks back from the more radical implications of the postmodern critique.
The most persistent postmodernism will issue in a postmodern dogmatics—or what we might call a postcritical dogmatics of second naiveté. And on the level of practice, a more persistent postmodernism will engender not quite a postmodern church but rather a postmodern catalyst for the church to _be_ the church.
In this respect a recent movement or sensibility in Christian theology embodies this more persistent postmodernism. Radical Orthodoxy—a sensibility that seeks to articulate a robust confessional theology in postmodernity—represents a more persistent or thoroughgoing postmodernism insofar as it refuses the modern (and skeptical) equation of knowledge with omniscience. In other words, unlike much postmodern theology or Continental philosophy of religion, Radical Orthodoxy refuses to be haunted by Cartesian anxiety.
What do we mean by this? We must appreciate the sense in which many advocates of postmodern theology or religion are deeply critical of particular, determinate formulations of religious confession. Figures such as Derrida and John D. Caputo rightly point out (and many who are part of the emergent conversation are very sympathetic on this score) that the modern Cartesian dream of absolute certainty is just that: a dream, and, admittedly, one that has been a nightmare for those who have become victims to such rational confidence (colonized peoples, an exploited creation, etc.). And far too often, some version of Cartesian certainty has attached itself to particular religious expressions—the result is what we call fundamentalism—and engendered untold harm. The problem with such modern religion—whether in the form of post-Kantian liberal theology or the equally modernist versions of Protestant fundamentalism—is twofold: on the one hand, it rests on a mythical epistemology of immediate access and cognitive certainty; on the other hand, its fruit has included harm, violence, and suffering for communities, both to those within such communities and to those regarded as "other" by these religious communities.
If, then, we are going to be _post_ modern—if we are going to get rid of what is worst about modernity—then surely we need to abandon not only foundationalist epistemology but also the forms of religion that have hitched their wagon to this Cartesian train. But for Derrida, Caputo, and others, the rejection of modernist religion (and its attendant epistemology) takes the form of a critique that might be said to still accept the rules of the game laid down by Descartes. In particular, a common move in postmodern theology is to reject the Cartesian equation of knowledge with quasi-omniscient certainty, instead asserting a kind of radical skepticism that opposes faith to knowledge but thereby actually retains the Cartesian equation of knowledge and certainty. "I don't _know_ ," Derrida once said; "I must _believe._ " In other words, the postmodern theologian says, "We can't _know_ that God was in Christ reconciling the world to himself. The best we can do is _believe_." Why? Because to know would mean being certain. We know that such certainty is an impossible dream; therefore, we actually lack knowledge. We don't know; we can only believe, and such faith will always be mysterious and ambiguous. But this isn't a bad thing; quite to the contrary, it is liberating and just. It is precisely when we think we know something about God that we start erecting boundaries and instituting discipline. People who _know_ what God wants effect the worst sort of violence on those who don't know, even on those who are part of the "knowing" community. Not only infidels are harmed by such "believers" (who are really "knowers") but also those internal to the religious community, who are subjected to all kinds of legalistic rules, even if they are self-imposed. So postmodern religious faith eschews knowledge and therefore also eschews the particularity of dogma and doctrine. In other words, according to this line of thinking, postmodern faith sees any particular, determinate religious confession as still tainted by knowledge; instead, the postmodernist advocates a "religion without religion" that is not linked to any particular creed or denomination—a more transcendent, less determinate (or even indeterminate) commitment to justice or "love."
Much in this critique has been rightly affirmed by many who have tried to think through the shape of the emerging church in postmodernity. Those whose Christian experience has been shaped by American fundamentalism (like myself ) are particularly open and receptive to this critique of determinate modern religion since we have seen and experienced firsthand the kind of harm that is done—both to people and the gospel—by such practices and theological formulations. So it is understandable that the emergent church has flirted with a religion without religion, sympathetic to versions of postmodern spirituality that undercut the role of dogma and the institutional church.However, I suggest that such a religion without religion is not really postmodern:it is rather an extension of deeply modern sensibilities. Further, a more properly postmodern theology will reject the very terms of this critique and, in fact, be much more hospitable to both dogmatic theology and the institutional church.
First, this quasi-postmodern religion without religion does not upset the modern Cartesian formulation of the problem. Instead, it proceeds by accepting the Cartesian equation of knowledge with certainty; then, because such certainty is impossible, it must conclude that knowledge is impossible. But we need not accept this all-or-nothing logic. Indeed, before Descartes this would have seemed simply mistaken. From Augustine through Aquinas, medieval theologians were very attentive to the difference between "comprehending" God (which was impossible) and "knowing" God (which was possible, because God had given himself to us in terms that could be received).Why should we think that the criterion for knowledge is godlike certainty or omniscience? Why should we accept the clearly mistaken modern equation of the two? Quasi-postmodern religion without religion actually accepts and works from this Cartesian paradigm, whereas a more persistent or proper postmodernism rejects this paradigm as an aberration in the history of philosophy and theology.
It is precisely this refusal of the Cartesian paradigm that characterizes Radical Orthodoxy, which seeks to reanimate the account of knowledge offered by Augustine and Aquinas. On this ancient-medieval-properly-postmodern model, we rightly give up pretensions to absolute knowledge or certainty, but we do not thereby give up on knowledge altogether. Rather, we can properly confess that we know God was in Christ reconciling the world to himself, but such knowledge rests on the gift of (particular, special) revelation,is not universally objective or demonstrable, and remains a matter of interpretation and perspective (with a significant appreciation for the role of the Spirit's regeneration and illumination as a condition for knowledge). We confess knowledge without certainty, truth without objectivity.
Second, it is the acceptance of the modern Cartesian paradigm that undergirds Derrida's and Caputo's critique of dogma and determinate religious confession. To confess something determinate and embody this in dogma or doctrine would be to claim to know something about the transcendent, and the inverted Cartesian skepticism of this quasi-postmodernism can't have that. Postmodern religion without religion's resulting affirmation of "faith" seems both deeply fideistic and anti-institutional. The most significant problem with this, from a Christian perspective, is that it is deeply unincarnational. It operates with what I have elsewhere described as a "logic of determination" rather than a "logic of incarnation."According to this logic of determination, particularity itself is violent and leads to violence; therefore, in order to avoid violence we must have, for instance, a social hope that is indeterminate and hopes for a justice that is unspecified, or we must have a religious community without dogma or discipline. But Derrida's premise, which equates determination with violence, can and must be called into question. The determinate and finite would be construed as violent and exclusionary only if one assumes that finitude is somehow a failure—implying that we are somehow called to be infinite. In short, to accept Derrida's premise that all determination or finitude constitutes violence, one would have to adopt some version of a gnostic ontology, which construes finitude as a kind of fall, an original violation. But we are free to reject this premise, particularly on Christian grounds.
Instead of adopting a logic of determination that construes finitude or particularity as a violence, I advocate a logic of incarnation that honors finitude and particularity as a good. If one begins, instead, with an affirmation of embodiment as good, then the fact of finitude and particularity—for example, the confession that God became flesh at a particular time ("under Pontius Pilate") and in a particular place ("born of the Virgin Mary")—is not construed as injustice or violence, because with the rejection of Derrida's logic of determination one must also reject the very modern notion of an ahistorical, a-geographical, transcendental religion. Therefore, it follows that the particularity of religious confession is not violent per se. (It can even be argued that one can locate the seeds of such an incarnational logic in undeveloped aspects of Derrida's early work, such that we might be able to deconstruct Derrida on just this point.)
Christian confession begins from the scandalous reality that God became flesh, and became flesh in a particular person, at a particular time, and in a particular place. The affirmation of particularity is at the very heart of the incarnation, which is itself a reaffirmation of the goodness of particularity affirmed at creation. This affirmation of particularity is then extended in and by the body of Christ, which is the church. But such an incarnational affirmation of embodiment and particularity—including the particularities of dogmatic confession, institutional organization, historical unfolding of doctrine, and so on—is more properly postmodern than the lingering modernism of a religion without religion that, in Kantian fashion, reduces faith to a generic affirmation of love or justice. A more persistent postmodernism embraces the incarnational scandal of determinate confession and its institutions: dogmatic theology and a confessionally governed church.Perhaps in its most scandalous form, there is nothing more postmodern than hierarchy (And nothing more modern than autonomous, nondenominational anarchy.)
So far, I have been suggesting that a properly postmodern theology will be dogmatic, not skeptical. This is not to advocate a return to an uncritical fundamentalism or the triumphalist stance of the Religious Right. Rather, it is to affirm that our confession and practice must proceed unapologetically from the particularities of Christian confession as given in God's historical revelation in Christ and as unfolded in the history of the church's response to that revelation. To be dogmatic, then, is to be unapologetically confessional, which requires being unapologetic about the determinate character of our confession, contra the Cartesian anxiety exhibited by much postmodern theology. This should translate into a robust appropriation of the church's language as the paradigm for both thought and practice. While this affirmation of the primacy of revelation is a core tenet of Radical Orthodoxy, it is one shared with other movements in postmodern theology, including postliberalism.But this issue of the primacy of revelation raises another concern I'd like to briefly address before moving to a more specific consideration of what an incarnational affirmation of history entails for worship and discipleship.
One of the concerns I have about the shape of the postmodern or emerging church is what could technically be described as a correlationist model."Correlation" refers to a theological strategy whose pedigree is distinctly modern. It operates as follows: beginning with a certain confidence in the findings of a secular discipline—whether philosophy, psychology, history, or sociology—a correlationist theology adapts this neutral or scientific framework as a foundation and then correlates Christian theological claims with the facts disclosed by secular science. For instance, Bultmann accepted the neutral (supposed) facts of Heidegger's existential account of the human condition and then correlated Christian theology to fit this model. Or liberation theology took the findings of Marxist sociology as disclosing the scientific facts about human community and then correlated Christian theology with this "scientific" foundation. In every case, correlationist theology has a deeply _apologetic_ interest: ultimately, the goal is to make Christianity intelligible or rational to a given culture (even if it operates on the assumption of a transcultural, neutral, objective reason). In the process, however, primacy is given not to the particularity of Christian revelation or the confessional tradition but rather to the poles of science, experience, and so on, which are taken to be neutral "givens."
But such a correlational method is true not only in theology; we can also clearly see it in church practice. In fact, one of the most trenchant critiques of contemporary evangelicalism has charged the church with looking primarily to its surrounding culture for the norms of what it means to be, or better, to "do," church. Thus seeker-sensitive churches have sought to translate or correlate the gospel into terms of a given (usually white, upper-middle-class) culture, even giving a certain priority to this cultural pole. What Robert Webber helpfully describes as "pragmatic evangelicalism" operates on a deeply modernist level. And many in the emerging church have been critical of just this cultural assimilation, which has dominated the megachurch, church-growth paradigm. But I wonder whether, in the name of creating a postmodern church, the emerging church continues this correlation by other means. While this is by no means a monolithic phenomenon, there are certainly streams in emerging discussions that are simply looking to update the church and bring it into correlation with a postmodern rather than a modern culture. Those in the emergent conversation who are more reflective see this for what it is: more of the same and really just an extension of (modern) pragmatic evangelicalism. However, even among more reflective emergent thinkers, one can see hints of a retained correlational stance. There remains a certain notion that the church needs to "get with"postmodernity such that postmodern culture sets an agenda for the church, rather than postmodernity being a catalyst for the church to recover its own authentic mission.
If we hope to be properly postmodern, then we must intentionally resist this correlational model. And here Radical Orthodoxy is instructive in both its diagnosis and prescription. At the level of diagnosis, Milbank observes that "the pathos of modern theology is its false humility."Conceding its foundations to the conditions of modernity and accepting the notion of a neutral science that must position theological discourse, modern theology had to be apologetic. But "once theology surrenders its claim to be a metadiscourse, it cannot any longer articulate the word of the creator God, but is bound to turn into some oracular voice of some finite idol, such as historical scholarship, humanist psychology, or transcendental philosophy. If theology no longer seeks to position, qualify or criticize other discourses, then it is inevitable that these discourses will position theology."But it is precisely the postmodern critique of Derrida, Lyotard, and Foucault that has unmasked this myth of a neutral scientific discourse that could position the supposed irrationality of theological discourse. All discourses and disciplines proceed from commitments and beliefs that are ultimately religious in nature. No scientific discourse (whether natural science or social science) simply discloses to us the facts of reality to which theology must submit; rather, every discourse is, in some sense, religious. The playing field has been leveled. Theology is most persistently postmodern when it rejects a lingering correlational false humility and instead speaks unapologetically from the primacy of Christian revelation and the church's confessional language. Radical Orthodoxy, then, is more properly postmodern than Derridean religious skepticism precisely because it embraces this situation. Indeed, it is "intended to overcome the pathos of modern theology, and to restore in postmodern terms, the possibility of theology as a metadiscourse."
The rejection of correlation with respect to theology should also be true of our understanding of church practice, worship, and discipleship. If the pathos of modern theology is its false humility, it might be that the persistent pathos of postmodern Christian practice and the emerging church is a continued false humility. If Christian theology should proceed from the primacy of God's revelation in Christ and Scripture, then Christian practices of worship and discipleship should do the same. Our understanding of what it means to be the church must be shaped by the priority of revelation and the Christian tradition, not what (even) a postmodern culture needs or is looking for. A radically orthodox church practice will refuse the correlational idol of relevance without giving up the central impetus of hospitality. We see this modeled in the case of _Whale Rider_ : the community's capitulation to modernity only spelled disaster. Rejection of tradition in favor of modernity showed itself to be a failure. But the solution was not to broker a compromise with modernity either—to come up with a correlation between modernity's tenets and a thinned-out version of the faith tradition that was "suitable" for moderns. (Correlation always privileges the culture, whether of modernity or postmodernity.) Rather, healing and communal wholeness were found when the community risked putting its tradition first—when it granted primacy to its faith stories and let those stories position their response to and appropriation of modernity (or postmodernity).This was not a simple going backward either: it was a nonidentical repetition of the tradition in a postmodern context. It was not a nostalgic, romantic return to old paths. This was a creative retrieval of the tradition _for_ a postmodern culture. After all, the charism of chief was unexpectedly incarnated in a girl.
**Recovering Tradition: Taking History Seriously**
I have suggested that a more persistent postmodernism, articulated by Radical Orthodoxy, begins from a primary affirmation of the incarnation. In the preceding section I argued that if our theology and practice are going to be fundamentally incarnational, then they should be the catalysts for a reaffirmation of the particularities of Christian dogma, confession, and ecclesial practice. I want to extend this incarnational logic into two more spheres that are significant for Christian worship and discipleship in the postmodern world. First, in this section, I suggest that the incarnation should entail a deep affirmation of time and history, which should translate into church practice that is catholic and traditional (though in a postmodern mode). The following section explores the implications of the incarnational affirmation of space along two axes: an affirmation of liturgy and the arts and a commitment to place and local communities.
Let us first think about time. There is a significant sense in which modernity tried to transcend time in its quest for universal, ahistorical principles and truths that applied at all times, in all places, to all people. This universal penchant for ahistoricality resulted in the colonial imposition of one particular set of practices as rational and universal, when in fact they were the fruit of a very determinate history and geography. In this respect modernity represented a revival of traditional Platonism, which held that ideas—and it is ideas that modernity really cared about—trafficked in the eternal, unchanging, atemporal realm of the Forms.In other words, to grasp an idea was to transcend time, and the ideas that really mattered were not conditioned by time or change. In fact, it was the realm of bodies and matter— the realm of generation and decay—that was also the realm of time, history, and change. Thus it is no surprise that modernity, launched by the disembodied Cartesian "thinking thing," would come to have an ambivalent relationship to the world of bodies and time. History, rather than being affirmed as the arena for the material unfolding of latent possibilities, was something to be subdued and transcended. For Kant, for instance, what was properly ethical or good could not bear any relationship to the particular contingencies of time or place.
The church's theology bought into this ahistoricism in different ways: along a more liberal, post-Kantian trajectory, the historical particularities of Christian faith were reduced to atemporal moral teachings that were universal and unconditioned. Thus it turned out that what Jesus taught was something like Kant's categorical imperative—a universal ethics based on reason rather than a set of concrete practices related to a specific community. Liberal Christianity fostered ahistoricism by reducing Christianity to a universal, rational kernel of moral teaching. Along a more conservative, evangelical trajectory (and the Reformation is not wholly innocent here), it was recognized that Christians could not simply jettison the historical particularities of the Christian event: the birth, life, death, and resurrection of Jesus Christ. However, there was still a quasi-Platonic, quasi-gnostic rejection of material history such that evangelicalism, while not devolving to a pure ahistoricism, became dominated by a modified ahistoricism we can call primitivism. Primitivism retains the most minimal commitment to God's action in history (in the life of Christ and usually in the first century of apostolic activity) and then seeks to make only this first-century "New Testament church" normative for contemporary practice.This is usually articulated by a rigid distinction between Scripture and tradition (the latter then usually castigated as "the traditions of men" as opposed to the "God-given" realities of Scripture).Such primitivism is thus anticreedal and anticatholic, rejecting any sense that what was unfolded by the church between the first and the twenty-first centuries is at all normative for current faith and practice (the question of the canon's formation being an interesting exception here). Ecumencial creeds and confessions—such as the Apostles' Creed or the Nicene Creed—that unite the church across time and around the globe are not "live" in primitivist worship practices, which enforce a sense of autonomy or even isolation, while at the same time claiming a direct connection to first-century apostolic practices.
I'm suggesting that this anticatholic ahistoricism stems from the absorption of a modern aversion to the logic of incarnation and the affirmation of the goodness of creation—and the attendant affirmation of embodiment, change, time, history, and therefore tradition. To affirm the goodness of creation (Gen. 1:31) is to affirm the goodness of time, time's unfolding in history, and the fruit of this process in tradition. As John Milbank puts it, it is not just the material creation that "participates" in God; our own human _poiesis_ , or "making," is a kind of co-creation that also participates in God's transcendence. In other words, human cultural making—including the culture making of the church's institutions and practices over time—is an arena of the Spirit's continued activity and revelation.
While the emerging church rightly rejects the disincarnate theologies and practices of pragmatic evangelicalism, I wonder if it has retained something of modernity's ahistoricism or its evangelical version, primitivism. In the name of postmodern Christianity, we often hear sentiments about believing in Jesus but not Christianity, not letting tradition distort Jesus's radical message in the Gospels, or sorting out the nonnegotiable essentials of the gospel from traditions—where, ironically, the traditions in question are usually the machinations of pragmatic evangelicalism. Sometimes this is expressed in a regulative principle about theological concepts: if a theological concept isn't in the Bible, then it lacks any normativity.This also underwrites a persistent non- or even antidenominationalism in the emerging church, which rejects the normative confessional boundaries of any institutional hierarchies. This relates to our earlier observation of a lingering affirmation of autonomy in the emerging church; along the axis of time, we see a lingering, disincarnate rejection of time, history, and tradition.
Radical Orthodoxy articulates an incarnational affirmation of history that could help the emerging church think through its commitment to the incarnation by exorcising its latent primitivism. For, as Catherine Pickstock announces, "one of the most central aims of a radically orthodox perspective is to restore time and embodiment to our understanding of reality."On this more incarnational account, time is not "something to be lamented or circumvented by means of the instruments of nostalgia, but rather as our very condition of possibility _per se_."If we are created as finite, temporal creatures, then time is part of the good creational air that we breathe, so to speak. "Changefulness in time is actually what defines us."To be human is to be temporal; to be temporal is to be traditioned, which is simply to say that we are always and only temporal in a social or communal manner.
But this is not a traditional _ism_ ; an affirmation of time, history, and tradition rejects the notion of a reified, static past that feeds the nostalgia of traditionalism. Rather, Pickstock emphasizes an ancient-future affirmation of time. We are constituted "as much by the past as by the future. For no co-ordinate of time—past, present, or future—wields supreme sway."This "peculiar relationship to time," she concludes, "distances us from both liberals and conservatives, for both these latter tend to invoke theology or the notion of God to underwrite some pre-existing value—whether, for conservatives, some fetish of tradition, or for liberals, some timeless humanist value. Against such positions, [Radical Orthodoxy] would prefer to emphasize that there are no such pre-established givens, for everything is a never-finished work, which yet discloses what lies invisibly within the interstices of time."What is wrong with modernity is its suppression of time, and this suppression of time is seen in both liberal ahistoricism and the conservative evangelical version of ahistoricism: primitivism. In contrast to both, Radical Orthodoxy asserts an affirmation of time as the incarnate arena for the Spirit's unfolding and thus takes seriously the fruits of time as it becomes embodied in tradition. This is not to make a fetish of tradition but rather to recognize that time is a medium for God's continued revelation and to concede a certain authority and normativity to what precedes us.
The shorthand to describe this affirmation of time and tradition is simple: this is catholic faith. In order for the church to be postmodern, it should be catholic. This might seem counterintuitive at first. But what the emerging church is reacting against is a deep, hurtful experience of sectarianism; the antidote to this is a generous orthodoxy and healthy catholicity. To be emergent should entail being catholic.
It is no secret that the evangelical tradition can take shape in forms that are deeply sectarian, provincial, and polemical.But when we diagnose the cause of such instantiations of evangelical faith, we find one common cause: memory loss. In particular, such sectarian versions of evangelical identity tend to see themselves as relatively new inventions, or—following the logic of primitivism that we've already noted—new recoveries of the "true" faith and "New Testament church principles." The most polemical and schismatic permutations of evangelical faith and practice tend to exhibit a paradoxical blend of primitivism and temporal hubris: on the one hand, they tend to have an air of having just dropped from the sky, but, on the other hand, they claim to give us the only authentic version of Pauline Christianity. While trumpeting notions of recovering the truth, these polemical elements of the evangelical tradition seem to be characterized by a deep forgetting. We might suggest that these versions of Christianity are more interested in being "holy" and "apostolic" than in being "catholic"—as if these traits could be separated.
Much earlier, in the early fifth century, Augustine grappled with another version of sectarianism—Donatism—which also tended to suffer from memory loss. And thus, when pastorally addressing the challenge for his parishioners, Augustine advocated remembering. In particular, he charged them: "Remember, you are catholic" (Sermon 52). The emerging church could heed the same admonition today. With Augustine's admonition in mind, the emerging church might find a resource in an unlikely place:papal biographer George Weigel's _Letters to a Young Catholic_.
That an admonition to be catholic would give evangelicals pause is evidence of precisely why such an exhortation is so important. (I've been in evangelical congregations that, if they recite the creed, project it on a screen and expect us to confess "the holy universal church," just to keep things straight.) In an era when even confessional churches are being co-opted by a kind of generic evangelical pragmatism, American civic theology, or mainstream liberalism, Weigel's _Letters_ should be received as a reminder of the Augustinian challenge to remember our catholicity. Such a remembering of who we are—disciples of Jesus who are members of one, holy, catholic, apostolic church—is a powerful antidote to both the schismatic and the polemical elements of the evangelical tradition and should also revitalize a sense of _antithesis_ , or what Weigel describes as "the catholic difference."
It will be most difficult for evangelicals—particularly "emerging" evangelicals—to imagine themselves as recipients of these letters. Because of various shifts in identity and historical factors, we might not immediately think of ourselves as the audience for a book addressed to "catholics"; but insofar as our credo includes the confession of one holy catholic church, what Weigel articulates is the core of Christian faith and practice. While we often hear the term "catholic" as a way of marking off a body of Christians from other Christians, when Weigel speaks of "the catholic faith," he means the faith that distinguishes the people of God from the secular and pagan faiths of the contemporary world. If there is a polemics here, it is directed not against other Christians (Weigel is not out to demonize Protestants) but against the Christian faith's most seductive foes: secularism, naturalism, and liberalism. When Weigel articulates "the catholic difference," he is not marking off Roman Catholics from Presbyterians but rather describing what distinguishes the people of God as a peculiar people and a holy nation. Indeed, Weigel even wants to reinvigorate the notion of the ghetto, recalling the Catholic ghetto of his Baltimore youth: "The most ghettoized people of all," he concludes, "are those who don't know they grew up in a particular time and place and culture, and who think they can get to universal truths outside particular realities and communities." In the same way that some of my Dutch friends are drawn to the accounts of Hasidic communities in Chaim Potok's fiction, Weigel gives us a sense in which a community constituted by "the catholic difference" functions as an empowering ghetto—though with its own set of struggles and challenges. "The real question," he offers, "is not whether you grow up in a ghetto, but whether the ideas and customs and rhythms of your particular ghetto prepare you to engage other ideas and customs and life experiences without losing touch with your roots."
Of course, the same peculiar people that are marked by "the catholic difference" also comprise a transnational and enduring community. So Weigel's strategy for introducing his young interlocutor to the catholic faith is via a world tour of catholic places inhabited by exemplars of catholic faith. Beginning with the Baltimore ghetto of his youth, Weigel takes us on "an epistolary tour" of places such as Saint Peter's Rome, Chesterton's pub in London, Saint Catherine's Monastery on Mount Sinai, the Oratory that was home to Cardinal Newman in Birmingham, and the Basilica of the Holy Trinity in Kraków, one of several Polish sites in the book. (Curiously absent from the tour are any sites outside the Northern Hemisphere; indeed, the entire continents of South America and Africa are silent in this account.) The result is a rich sketch of the core themes and affirmations that constitute "the catholic difference," which is, "at bottom, a way of seeing the world."If you'll permit a Kuyperian indulgence, I take Weigel to be providing a lucid account of the Christian world- and life-view. And indeed, the reason I think this book is such a wonderful reminder of our catholicity—why I receive it as an Augustinian injunction to remember I am catholic—is because Weigel helps to locate key themes we traditionally re-gard as part of a Reformed worldview as ultimately _catholic_ Christianity. "While Catholicism is a body of beliefs and a way of life," he remarks, " _Catholicism is also an optic, a way of seeing_ _things, a distinctive perception of reality_."And one of the core features of the optic of Catholicism is its emphasis on tradition. Catholic faith constitutes a community of memory that resists both romanticism and the kind of temporal hubris that dismisses everything prior to 1968. "Christian thinking," Weigel suggests, "should adopt an _ecumenism of time_ , employing wisdom and insight from any historical era."In other words, Catholicism is what Chesterton called "a democracy of the dead" because it affirms tradition, which "means giving votes to the most obscure of all classes, our ancestors."It is precisely this ecumenism of time that makes catholic Christianity critical of what Newman described as liberalism in religion.In an era when what we are getting in the name of postmodern spirituality might be more akin to "liberalism," Newman's voice and critique could be an ally in recovering the more antithetical side of the confessional tradition—what Weigel has been calling "the catholic _difference_ ," or "countercultural Catholicism."
A more persistently postmodern church must be radically incarnational. And to affirm the incarnation is to affirm the scandal of particularity with respect to both space and time. This requires a healthy sense of being constituted by our traditions as we look forward to an eschatological hope in the future. The postmodern church will be a witness to its contemporary generation by being a peculiar people oriented to a coming kingdom through the practices and language of a living tradition.The postmodern church must take the risk of learning to ride whales.
**Renewing the Body: Space, Place, and Incarnation**
A radical affirmation of the incarnation means affirming not only time (and history and tradition) but also space; that is, it must entail an affirmation of the goodness of the stuff that Descartes described as extended and then wrote off so quickly: bodies, buildings, and bowls of soup. ("Thinking things" never get hungry.) The materiality of God's good creation, like time, is something that modernity sought to repress. And modernist, fundamentalist worship and spirituality reflected this: focused on a didactic sermon meant to convey the ideas that make up the "system" of Christian truth, evangelical worship services have fostered a talking-head Christianity that accords well with the "thinking things" of Cartesian modernity, but not with the robust, fleshy, communal beings that God called into being in Adam and Eve. The iconoclasm and ritual-phobia of evangelical worship bear direct affinity with the disenchanted world bequeathed to us by the immanentism of modern science.
Thus here again, it seems, if we want to be postmodern in some sense, we must recover elements of ancient ritual and practice, for it is liturgy that honors our fleshiness. But this is not a merely traditionalist fiat; it stems from the very way we think about the world and what it means to be human. In other words, an incarnational affirmation of liturgy and the aesthetics of worship is the fruit of an incarnational ontology (an account of the nature of reality) and a holistic anthropology (an account of what it means to be human).If we want to resist the reductionistic Cartesian picture of human persons as "thinking things" (and we should also resist other reductionistic accounts of the human person as merely consuming things or biological things), we must recover the holistic anthropology we find intimated in Scripture and unpacked in the Augustinian catholic tradition. An incarnational anthropology begins with the affirmation that human persons are material: that we don't just inhabit flesh and blood, but we _are_ flesh and blood. Being embodied is an essential feature of being a human creature. As such, we are not defined by thinking; rather, we are primarily affective: the center of the person is not the mind, but the heart. (That's not to say we are irrational, but only that rationality [mind] is relative to what Augustine calls "the right order of love"—the direction of our heart.) When Pascal famously stated that "the heart has reasons of which reason knows nothing," he meant to assert this primacy of affective, embodied being-in-the-world. This holistic anthropology (or account of the human person) is postmodern precisely because it rejects the reductionism of modernity, but it admittedly recovers key insights of a premodern, biblical worldview.
Because of this fundamental affirmation of embodiment, materiality, and affectivity, a radically orthodox worldview is fundamentally sacramental. It affirms not only the goodness of material bodies but also that the whole realm of the material has a revelational potential. And when this is coupled with the incarnational affirmation of time and tradition, a radically orthodox vision asserts a special mode of sacramentality for aspects of the church's tradition, ritual, and liturgy.
George Weigel gets at the same point when he says that the catholic optic is animated by a _sacramental imagination_. (Once again, I'm suggesting that the best way to be postmodern is to be premodern; to be emergent, one must be catholic.) Weigel sees this sacramental imagination unfolded for us in G. K. Chesterton's old haunt, the Cheshire Cheese pub in London, where we find the rotund apologist enjoying the very material blessings of food and ale. As Weigel puts it, Chesterton's delight in the material world illustrates "the bedrock Catholic conviction that _stuff_ _counts_." Indeed, Weigel makes the radically orthodox claim that only a catholic account of the world can really affirm materiality: "Catholicism takes the world, and the things of the world, far more seriously than those who like to think of themselves as worldly."Both fundamentalists and so-called materialists, he argues, subscribe to a _gnostic imagination_ ; only those who affirm the paradox of the incarnation can see the world with a sacramental imagination.
We find the same affirmation of stuff in the poetry of Gerald Manley Hopkins and in the almost sacramental (or theurgical) ladder of love in Evelyn Waugh's _Brideshead Revisited_.And it is ultimately this affirmation of creational stuff that makes us take history seriously, as illustrated in sites of pilgrimage and veneration. In commenting on the _scavi_ (excavations) beneath Saint Peter's, Weigel suggests that "the _scavi_ and the obelisk— Peter's remains and the last thing Peter may have seen in this life—confront us with the historical tangibility, the sheer grittiness, of Catholicism." The foundations of the Catholic faith are something we can touch.
Although Catholicism may seem to be an otherworldly ideology, Weigel contends that, paradoxically, only catholic faith can really affirm the world—and modern materialists and naturalists actually flatten the world and reduce it to nothingness (and an attendant nihilism). This point is illustrated in what Radical Orthodoxy describes as a "participatory ontology": the sense that we properly understand the nature of the world as creation only when we see that the world "participates" in God. Or, as proponents of this view put it, creation is "suspended" from the divine. The material stuff of the world is "suspended" from the immaterial, invisible God in whom "we live and move and have our being" (Acts 17:28). This suspension of the material is what gives matter its depth, as it were; it makes it more than material. The disenchanted, flattened matter of modern naturalism, on the other hand, actually dissolves matter into nothing. Thus only Christians can be proper materialists!
Because of this Christian materialism, a catholic postmodernism (or postmodern catholicity) affirms sacramentality on two levels. On the one hand, it affirms a general sacramentality: the whole world has potential to function as a window to God and a means of grace from God because God himself affirms materiality as a good thing. We see this not only in creation itself but also in the reaffirmation of it in the incarnation, in which God is happy to inhabit the goodness of flesh. Furthermore, materiality receives an eschatological affirmation in our hope for the resurrection of the body. Even the future kingdom will be a material environment of sacramentality. On the other hand, when an incarnational ontology and anthropology are linked with our earlier affirmation of time and tradition, a catholic postmodernism also affirms a special sacramentality—a special presence and means of grace in the sacraments of baptism and Eucharist. Thus a properly postmodern ecclesiology must overcome the triumph of a deeply modernist (and Zwinglian) notion of the ordinances of baptism and communion and recover a thicker, more sacramental practice of worship.
If it seems strange to suggest that only Christians (or what I'm now calling "postmodern catholics") can properly be materialists, a related theme offers a similar reversal of first impressions:despite assumptions that Catholicism is Victorian in its supposed repression of sexuality, in fact at the root of Catholicism is a rich, affirmative theology of the body. Building on the founding affirmation of the incarnation, Weigel provides a kind of exegesis of the Sistine Chapel to help us reach John Paul II's conclusion: that the Sistine Chapel is "the sanctuary of the theology of the human body."The sacramental imagination, which affirms the goodness of creation, animates an iconic imagination that affirms the presence of the invisible in the visible—that "lifts up" the messiness of bodies to be more than biological machines. "Human bodies," Weigel summarizes, "are icons." And if this is pictured in the Sistine Chapel (Letter 8), and undergirds the beauty of Chartres Cathedral as a kind of "antechamber" of heaven (Letter 12), it is articulated most forcefully by John Paul II. "In a move that takes the argument about the sexual revolution as far beyond prudishness as you can imagine, John Paul has proposed that sexual love within the bond of faithful and fruitful marriage is nothing less than an icon of the interior life of God himself." Contrary to the assumptions of the _New York Times_ reporter who thought the pope should have been embarrassed by the nudes in the Sistine Chapel, John Paul II steadily sketched a theology of the body in 129 addresses to general audiences between 1979 and 1984. Weigel does an excellent job of showing how countercultural this affirmation of embodiment and sexuality is in our contemporary context.
Taking the incarnation seriously means taking bodies seriously, which means affirming the space that they occupy as an arena of revelation and grace. The sacramental imagination begins from the assumption that our discipleship depends not only—not even primarily—on the conveyance of ideas into our minds, but on our immersion in embodied practices and rituals that form us into the kind of people God calls us to be. It is only Cartesian "thinking things" that can do without liturgy; for we embodied creatures, whether ancient or postmodern, the rhythms of ritual and liturgy are gracious practices that enable discipleship and formation. Thus postmodern worship stages a recovery of the aesthetic aspects of the Christian tradition as a crucial means for redirecting our imagination in community—a means for reordering our love.We were created for stories, not propositions; for drama, not bullet points. As someone has suggested, humanity cannot live on prose alone.The story of God-become-flesh is best rendered by the poetry and painting of affective worship rather than the narrowly cognitive didacticism of Power-Pointed "messages." Properly postmodern worship resists such reductionism by reclaiming the holistic, full-orbed materiality of liturgical worship that activates all the senses: hearing (not just "messages" but the poetry of the preached Word), sight (with a renewed appreciation for the visual arts, iconicity, and the architectural space of worship), touch (in communal engagement, but also touching the bread that is Christ's body), taste (the body and blood), and even smell (of wine in the cup of the new covenant but also the fragrance of worship in candles and incense). God's taking on a human body also takes up our bodies into worship and participation in the divine.
Finally, if a radically orthodox, incarnational vision takes time and tradition seriously and affirms the goodness of bodies and space, it should also think carefully about _place_. A radically or-thodox vision entails not only a distinct liturgy and aesthetics but also a distinct geography. If modernity fosters an ahistorical penchant for timelessness and a disembodied notion of persons as merely thinking things, it also fosters a disconnection from space and locality. David Matzko McCarthy relates this to the increasing hegemony of the (modern, capitalist) market for which, Marx famously noted, all that is solid melts into air: "Our modern growth economy," he observes, "requires that our attachments to people and things be superficial. We must be on the move in order to follow the market."Not only are we increasingly mobile across national and international territories; we also find that the modern market makes us the kind of people who can't be satisfied in one place for very long. Smaller urban homes can't meet the desire for bigger and better, so we make the market-driven pilgrimage into the suburbs to secure the requisite square footage and adequate number of garage doors (three being the new standard)—even though this also means that we spend increased amounts of time commuting in the solitary space of our SUVs. (Descartes had to retreat into a private room to dream up a "thinking thing"; we have the long drive on I-95 to reinforce this solipsism.)
The suburbs, we might suggest, are quintessentially modern, and so it is not surprising that evangelical churches not only have followed the market but also thrive in "mega" forms in this suburban environment. For instance, the modern facilities of First Family Church in Kansas City include a vast lobby surrounded by plasma-screen TVs, buttressed by a food court, a sprawling Barnes-&-Noble-like bookstore, and entryway to the Christianized version of the magical kingdom for kids (where children are admitted to Sunday school by scanning their bar codes).But perhaps what best signals First Family's dislocation from place is the vast sea of vehicles that surround the church like a metallic moat. Because of this growing distance from the parking lot's edge to the entrance of the gymlike sanctuary (the architecture fosters the iconoclasm of pragmatic evangelicalism), visitors are greeted by a large golf cart to shuttle primly dressed families into worship. And at the conclusion of worship, parking attendants help direct the rush of SUVs as they head for the exit, dispersing across the suburbs.
The convoy of SUVs making its way to and from "worship" at this suburban congregation owes more to the disembodied, disincarnate worldview of Cartesian modernity than to the radically incarnational confession of the church catholic. Being a properly incarnational, more persistently postmodern church entails not only a sacramental, embodied mode of worship practice but also translates into considerations about the place where we worship. The Christian _ekklesia_ must be not only liturgical but also local; it must transform not only hearts but also neighborhoods; its worship must foster not only discipleship but also justice—indeed, disciples who are passionate about justice.
If the emerging church needs to be catholic, as I've suggested, then it must also recover the notion of parish ministry.The postmodern church must be willing to embrace, above all, those who have been crushed by the underside of modernity: those who inhabit the urban core of our cities. And to do so, it is important that the postmodern church stay put; that is, the church is properly postmodern not when it seeks to "plant" new congregations in the comfortable environs of American suburbia but when it struggles to revitalize existing congregations and communities in our inner cities. (Indeed, I wonder if church "planting" isn't a rather modernist phenomenon, given to an infatuation with the new and wanting to work from a clean slate rather than the messiness of given communities. New construction is always easier than renovation. But there is something about the givenness and grittiness of existing communities that challenges our autonomous dreams to create or plant the next best thing. And if we can run with the architectural metaphor, I would take a restored Arts and Crafts home from the 1900s over the blandness of new developments any day.)
As Eric Jacobsen suggests, in an era of disincarnate suburban dislocation, incarnational ministry might simply mean praying for sidewalks.Embodied worship must be symbiotically related to the place in which we worship: the neighbor sometimes actually means the one next door. If the Word became flesh and dwelt among us, this should translate into an incarnational geography for the church, countering the disembodied abstraction of modernity that has too often been adopted by pragmatic evangelicalism. We need to counter not only the ideas of modernity but also the practices of modernity, and one of its most insidious practices involves a flight from the messy realities of urban community. Sidewalks might represent a threat to Cartesian autonomy, but they can also be a means for the inbreaking of grace.
**Taking Radical Orthodoxy to Church**
The radically orthodox church, while perhaps not Roman, is nevertheless catholic. And it is precisely this catholicity that takes up the key elements of the churches we toured with Derrida, Lyotard, and Foucault. In other words, we can see that this unholy trinity of Parisians has unwittingly already pointed us to elements of a more incarnational, even radically orthodox understanding of the church and practice. This is a kind of whale riding: creatively retrieving the empowering core of traditional identity but enacting a rendition of this in and for postmodernity. Such a project is motivated not by nostalgic traditionalism or fear of modernity's eroding effects but rather by an incarnational logic that assumes we are by nature traditioned creatures who properly find our identity only by being traditioned well. Just as Paikea took risks to pray in the tongues of her ancestors, so we must consider that the way forward might run along ancient paths. What we have already affirmed in our earlier tours of postmodern churches—the centrality of the Word, the use of the lectionary, the engagement with the arts, practices as ritual discipline—can now be seen as undergirded by the incarnational affirmation of time (tradition) and space (embodiment).
As we enter the radically orthodox church, we enter a space that is organized by a certain "ergonomics" of community: an eclectic collection of chairs is arranged in concentric circles around a table bearing the sacraments, contained in pottery fashioned by a member of the local parish. This organization of space means that during each phase of worship, members of the congregation are faced by others: they see and are seen by others, which reminds them of the iconic gaze of God, who confronts us in the other (Matt. 25). The worship space is also organized by dynamics of light and darkness: surrealist stained glass casts a colored light over portions of the sanctuary, while candles flicker both light and shadow from chapel stations on the fringes of the sanctuary. Several screens display shifting digital images that function as a kind of digital glass of images drawing us into worship.Like traditional icons—which can be found in one of the side chapels— these digital images function as windows to transcendence. But it is not only the visual arts that draw us into participatory worship. Immediately upon our entering the sanctuary, the scent of burning candles conveys a difference from the concrete jungle we've just emerged from and also distinguishes this experience from the scentless passivity of MTV and film.There is also a curious ambience emitted by an unlikely ensemble playing from one of the chapel stations: a jazz combo with sax, double bass, lead guitar, harmonica, and musical saw.
We are signaled to more intentional worship by an a cappella call to worship in the form of a chant from Afghanistan. This draws together the families around the table for the recitation of a poem by one of the congregation's gifted poets. The eclectic ensemble then leads us in worship in song, drawing on hymns of the faith, choruses from around the globe, and U2's "40," based on Psalm 40. The Old Testament reading from the lectionary is staged as a drama and liturgical dance, while the reading from the Gospel is backed up by a soulful anthem from the sax. The homily focuses on the Epistle, challenging the congregation to reorient their desires to what really matters (Phil. 1:9–10). This ultimately points us to two important communal experiences of our identity and opportunities for formation. First, this week a young family has brought their daughter to be baptized. Utilizing a beautiful baptismal formula from the sixteenth-century Huguenots, the parents express their desire and passion to see their daughter formed in the faith; but we too, as the congregation, pledge to be the village that will raise her together in Christ. Second, baby Anthea, newly welcomed into the body of Christ, pulls up to the table with her family to participate in her first meal at Christ's table: the Eucharist. Anthea, with her siblings and parents, remain seated at the table. After the consecration of the meal (including a poem by Anne Sexton), the celebrant invites the congregation to share in Christ's body and blood by being seated at the table with Anthea, newest member of the church's family. Anthea's parents pour wine and break bread for each of us as we sit briefly at the table of fellowship and communion. As we proceed to and from the table, the ensemble has spread out around the sanctuary, and the sounds of the instruments bounce back and forth across the worship space. The digital glass has shifted to images of children from around our community— the local space that is our parish. We are reminded that our commitment to Anthea is both a communal commitment and a commitment to our community.
At the conclusion of worship, we are sent out into our neighborhood as ambassadors of the King-in-waiting, reminded of Monday's meeting about the neighborhood co-housing project, and reminded of our Sabbath commitment to abstain from the economic cycle for the day. The walk home with parishioners who are also neighbors solidifies the sense that we are a peculiar people.
The radically orthodox church, then, is not traditionalist, even if it is traditioned; it is not a rote system of repetition but a creative repetition of the core features of what constitutes us as the people of God; it is not a nostalgic retreat into "the way we used to do it" but a dynamic reappropriation of ancient practices as the very material means to be formed differently, as agents who will counter the practices of modernity's market and empire. The radically orthodox church is the space for the formation of postmodern catholics.
1. _Whale Rider,_ DVD, directed by Niki Caro (Culver City, CA: Columbia TriStar Home Entertainment, 2003).
2. This positive impact of the tradition is seen in another episode related to the sacred school. One of the sharpest young boys, Hemi (clearly one that Koro has his hopes set on to be the One), is the son of an absentee, wayward father who is also part of this lost generation, sucked in by the wiles of modern culture. But when he briefly visits the _marae_ to see Hemi's ritual performance, he swells with pride—even though he immediately leaves the _marae_ , as if to protect himself from this pull of the ancestors. But by the end of the film he cannot withstand this pull; he is one of the decorated tribesmen who launches the _waka_. Then Hemi beams in pride at his father.
3. This recalls an earlier incident in the film: Paikea is with her Paka as he works on the pull cord of a battered old outboard engine. Paikea's curiosity about the tribe's myth leads her to ask about their connection to the ancestors. Paka responds by using the analogy of the rope in his hands: Their heritage is like this rope, he suggests, which is made up of many tiny strands. So, too, they are part of a long line of chiefs that is made up of many ancestors. When Koro then uses the rope to start the motor, it snaps. As he goes to retrieve another cord, Paikea repairs the rope and starts the engine. "I don't want you to do that again," Paka scolds her. "It's dangerous."
4. For my criticism of Derrida on this point, see James K. A. Smith, "Re-Kanting Postmodernism: Derrida's Religion within the Limits of Reason Alone," _Faith and Philosophy_ 17 (2000): 558–71.
5. For a more extensive introduction, including a consideration of Radical Orthodoxy's relation to other like movements (e.g., postliberalism), see James K. A. Smith, _Introducing Radical Orthodoxy: Mapping a Post-secular Theology_ (Grand Rapids: Baker, 2004).
6. For a helpful discussion of the long shadow cast over theology by Descartes' doubt, see Nancey Murphy and Brad J. Kallenberg, "Anglo-American Postmodernity: A Theology of Communal Practice," in _The Cambridge Companion_ _to Postmodern Theology_ , ed. Kevin Vanhoozer (Cambridge: Cambridge University Press, 2003), 26–41. As they point out, a properly postmodern theology will refuse the terms of the debate set by Descartes at the origins of modernity.
7. "Theology" is not quite the right term here, and Caputo and Derrida would be somewhat uncomfortable with the term, since it seems to have too much a sense of being linked to a determinate confession. Instead, they would describe this as "postmodern philosophy of religion" or simply "religious studies." My employment of the term "theology" in this context is largely heuristic and a shorthand.
8. See Jacques Derrida, _Memoirs of the Blind_ , trans. Pascale-Anne Brault and Michel Naas (Chicago: University of Chicago Press, 1993), 155.
9. For a lucid, entertaining articulation of such a postmodern "religion without religion," see John D. Caputo, _On Religion_ (London: Routledge, 2001).
. It is this aspect of the emerging church that D. A. Carson criticizes so harshly regarding questions of truth and objective knowledge. But as I noted in chapter 2, I am trying to sketch a third way between radical, albeit religious, skepticism and Carson's confidence in objectivity. This third, Augustinian way affirms the possibility and reality of knowledge and truth but rejects the modern notion of objectivity. It is, we might say, a confessional realism.
. For more on this point, see James K. A. Smith, _Speech and Theology:_ _Language and the Logic of the Incarnation_ , Radical Orthodoxy Series (London: Routledge, 2002), chap. 5.
. Murphy and Kallenberg note Stephen Toulmin's suggestion that modernity "is a giant Ù-shaped detour" ("Anglo-American Postmodernity," 39, citing Toulmin, _Cosmopolis: The Hidden Agenda of Modernity_ [Chicago: University of Chicago Press, 1990], 167). The idea is that the turn made with Descartes has been shown to be a wrong turn, and as we get back on track, we find significant continuity with premodern forebears. This affirms Webber's "ancient-future" thesis and explains the invocation of the epistemologies of Augustine and Aquinas as more properly postmodern.
. It must be noted that Derrida and Caputo rule out a priori any possibility of a particular, determinate revelation. This is perhaps one of the most fundamental differences with Radical Orthodoxy, which, like Barth, begins from an affirmation of a given, particular revelation of God in Christ. The religion-without-religion paradigm seems to deny the very _possibility_ of revelation, whereas postliberalism and Radical Orthodoxy affirm the _primacy_ of a given revelation.
. See James K. A. Smith, "Determined Violence: Derrida's Structural Religion," _Journal of Religion_ 78, no. 2 (April 1998): 197–212; and _Speech and_ _Theology_ , chap. 5. I will unpack this further in a forthcoming book, the working title of which is _Holy Wars and Democratic Crusades: Deconstructing Myths of_ _Religious Violence and Secular Peace_.
. See James K. A. Smith, "A Principle of Incarnation in Derrida's ( _Theolo-gische?_ ) _Jugendschriften_ : Towards a Confessional Theology," _Modern Theology_ 18 (2002): 217–30.
. To reaffirm our point in chapter 4 in dialogue with Foucault, where we noted that the continued penchant for nondenominational spirituality could be seen as a lingering form of modern autonomy.
. For some background considerations, see John Milbank, _Theology and_ _Social Theory_ (Oxford: Blackwell, 1990), chap. 10.
. For a lucid introduction to postliberal theology, see George Hunsinger, "Postliberal Theology," in _Cambridge Companion to Postmodern Theology_ , 42–57. See also Brian D. McLaren, _A Generous Orthodoxy_ (El Cajon, CA: Emergent Youth Specialties; Grand Rapids: Zondervan, 2004), chap. 8.
. I unpack this model in more detail in _Introducing Radical Orthodoxy_ , 33–42.
. A classic representative of correlational theology is David Tracy. See his _Blessed Rage for Order: The New Pluralism in Theology_ (Chicago: University of Chicago Press, 1996); and idem, _The Analogical Imagination: Christian Theology_ _and the Culture of Pluralism_ (New York: Herder & Herder, 1998). The same correlational method lies, I would suggest, behind the Wesleyan quadrilateral (which appeals to Scripture, tradition, reason, and experience as "sources" of theology), which has been widely recovered as of late.
. Webber, _The Younger Evangelicals: Facing the Challenges of the New_ _World_ (Grand Rapids: Baker, 2002), 41.
. I suspect that some of this lingering correlationism in the emerging church is at least partly due to a lingering correlationism in one of its theological leaders, Stanley Grenz ( _requiescat in pace_ ). This could be seen in his sympathy for the Wesleyan quadrilateral in _Revisioning Evangelical Theology_ (Downers Grove, IL: InterVarsity, 1993), a book to which I owe a great deal. (Brian D. McLaren also appeals to the Wesleyan quadrilateral in _A New Kind of Christian: A Tale_ _of Two Friends on a Spiritual Journey_ [San Francisco: Jossey-Bass, 2001], 55.) Grenz was attentive to this concern. For instance, in a recent contribution on ecclesiology, he explicitly rejects what he calls a "sociological foundationalism of community," specifically referring to Milbank's critique (see Stanley Grenz, "Ecclesiology," in _Cambridge Companion to Postmodern Theology_ , 258). But his method in earlier works, and even in this same essay, seems to indicate a lingering foundationalism or correlational method. For a discussion along these lines, with a response from Grenz, see Archie Spencer, "Culture, Community and Commitments: Stanley J. Grenz on Theological Method," _Scottish Journal_ _of Theology_ 57 (2004): 338–60.
. John Milbank, _Theology and Social Theory: Beyond Secular Reason_ (Oxford: Blackwell, 1990), 1.
. Ibid.
. Ibid.
. This might be embodied in the way that modern technology and tools could be appropriated to complete and launch the _waka_ (war canoe). What mattered was the identity that was wrapped up in the _waka_.
. I should clarify that my reference to "Platonism" here is to a traditional understanding of Plato as a dualist. In fact, Radical Orthodoxy seeks to retrieve quite a different Plato in a positive, non-dualistic way. Though I criticize the dualism or "Platonism" of modern Christianity here, I don't mean to suggest that Radical Orthodoxy is critical of Platonism per se. For a discussion of the issues, see my chapter, "Will the Real Plato Please Stand Up? Participation versus Incarnation," in _Radical Orthodoxy and the Reformed Tradition_ , ed. James K. A. Smith and James H. Olthuis (Grand Rapids: Baker, 2005), 61–72. My thanks to Geoff Holsclaw for noting this tension.
. The fact that primitivists accept the shape of the biblical canon as determined several centuries later is a nasty little exception to this rule.
. This might be most radically played out in Plymouth Brethren traditions (the tradition of my own conversion to Christian faith), but this stance is quintessentially Baptist. However, Pentecostal Christianity tends to operate on the same principle. It is not a coincidence that all these streams emphasize the autonomy of the local congregation.
. For a lucid critique of this framework from a card-carrying evangelical, see F. F. Bruce, "Scripture and Tradition in the New Testament," in _Holy Book_ _and Holy Tradition_ , ed. F. F. Bruce and E. G. Rupp (Manchester: Manchester University Press, 1968), where Bruce argues that the New Testament itself constitutes an interpretive tradition.
. This primitivism opens up such evangelical traditions to any new wind of doctrine. The key is for such new doctrines to assert their first-century, primitive origins. So, for instance, the radically novel eschatology of dispensationalism could become the dominant orthodoxy in just half a century because it claimed biblical rather than traditional warrant. For a relevant discussion, see Larry V. Crutchfield, _The Origins of Dispensationalism: The Darby Factor_ (Lanham, MD: University Press of America, 1992).
. The incarnation is a reaffirmation of the goodness of both time and space.
. As Milbank stipulates, "I have always tried to suggest that participation can be extended also to language, history and culture: the whole realm of human _making_. Not only do being and knowledge participate in a God who is and who comprehends; also human making participates in a God who is infinite poetic utterance: the second person of the Trinity" (John Milbank, _Being Reconciled_ [London: Routledge, 2003], ix).
. For instance, despite McLaren's healthy affirmation of tradition in _A_ _Generous Orthodoxy_ (El Cajon, CA: Emergent Youth Specialties; Grand Rapids: Zondervan, 2004), 87, he often makes such observations as the following: "For much of Western Christianity, the doctrine of creation (a biblical term) has been eaten alive by the doctrine of the fall (not a biblical term)" (234). I understand the distinction, but a thoroughgoing incarnational theology will think the distinction a moot point.
. Pickstock, "Radical Orthodoxy and the Mediations of Time," in _Radical_ _Orthodoxy? A Catholic Enquiry_ , ed. Laurence Paul Hemming (Aldershot: Ashgate, 2000), 64.
. Ibid.
. Ibid.
. For more on traditionality as an essential aspect of creaturehood, see James K. A. Smith, _The Fall of Interpretation: Philosophical Foundations for a_ _Creational Hermeneutic_ (Downers Grove, IL: InterVarsity, 2000), 152–57.
. Pickstock, "Radical Orthodoxy and the Mediations of Time," 64.
. Ibid., 65.
. For a comprehensive, and disturbing, account of this in the twentiethcentury American Reformed tradition, see John M. Frame, "Machen's Warrior Children," in _Alister E. McGrath and Evangelical Theology: A Dynamic Engagement_ , ed. Sung Wook Chung (Carlisle: Paternoster; Grand Rapids: Baker, 2003), 113–46.
. George Weigel, _Letters to a Young Catholic_ (New York: Basic Books, 2004), 9.
. Ibid.
. Ibid.
. Ibid.
. Ibid., 10. This way of putting it suggests an overlap with Radical Orthodoxy. Indeed, when showing the way in which Flannery O'Connor's work countered the flatness of "debonair nihilism," Weigel notes: "If Mary McCarthy was right, and the Eucharist only represented Christ in some magical way, then Flannery O'Connor was being utterly, thoroughly, radically orthodox when she muttered, 'Well, if it's a symbol, to hell with it' " (16).
. Ibid., 80.
. Ibid., 92.
. Ibid., Letter 5.
. On the possibility of speaking ancient languages to a postmodern culture, see Marva Dawn, _Talking the Walk: Letting Christian Language Live Again_ (Grand Rapids: Brazos, 2005).
. I unpack this connection in more detail in my _Introducing Radical_ _Orthodoxy_ , 223–29.
. Weigel, _Letters to a Young Catholic_ , 86.
. Ibid., 87, 94.
. Ibid., 98–100, 101–14.
. Ibid., 26–27.
. Ibid., 130.
. Ibid., 131.
. For more on liturgy and sacramentality, including the ontology of participation that undergirds this, see my _Introducing Radical Orthodoxy_ , chap. 6.
. See _Alternative Worship: Resources from and for the Emerging Church_ , compiled by Jonny Baker and Doug Gay, with Jenny Brown (Grand Rapids: Baker, 2003), 63. The authors employ Les Murray's distinction between the "narrowspeak" of reductive modernity and the "wholespeak" of a more imaginative worldview, calling for a recovery of wholespeak as the church's language. This "poetic discourse" represents "the re-enchantment or re-mythologization of speech, where speech reflects the Christian imagination, recognizing the importance of symbols, images, 'myths,' and metaphors as well as sharing space and time with music and the visual arts" (ibid.).
. David Matzko McCarthy, _The Good Life: Genuine Christianity for the_ _Middle Class_ (Grand Rapids: Brazos, 2004), 42.
. For a survey of the megachurch phenomenon with a critical eye, see James B. Twitchell, _Branded Nation: The Marketing of Megachurch, College Inc.,_ _and Museumworld_ (New York: Simon & Schuster, 2004), 47–108.
. Admittedly, the same placelessness and mobility can be true of urban congregations that draw people from long distances—including people who want to worship in "diverse" communities but don't necessarily want to live there. So it is not only suburban churches that fail to enact a "parish" theology committed to place. Simply being located in the core city does not make a congregation a parish. And conversely, it is possible for a suburban church to be actually more properly "parish" oriented. My thanks to Brian McLaren for pushing me on these matters.
. For an argument along these lines, based on a concrete case study, see Mark Mulder, "A Dissonant Faith: The Exodus of Reformed Dutch Churches from the South Side of Chicago" (Ph.D. diss., University of Wisconsin–Milwaukee, 2003), esp. 141–208, on the interrelation of issues of place and church polity. Mulder advocates recovering a sense of parish as a necessary condition for justice.
. See Eric Jacobsen, _Sidewalks in the Kingdom: New Urbanism and the_ _Christian Faith_ (Grand Rapids: Brazos, 2003), 84. Jacobsen provides an excellent introduction to these issues, even if he does not situate the concerns vis-à-vis postmodernism. For more on an incarnational theology of place, see T. J. Gorringe, _A Theology of the Built Environment: Justice, Empowerment, Redemption_ (Cambridge: Cambridge University Press, 2002).
. For samples of this kind of visual material, visit sacramentis.com.
. We also catch a whiff of the scent of good Sumatran (fair-trade) coffee— the new wine of the postmodern church!
. See Tod Bolsinger, _It Takes a Church to Raise a Christian: How the Community_ _of God Transforms Lives_ (Grand Rapids: Brazos, 2004).
* * *
Annotated Bibliography
[Further Reading on Postmodernism and
Christian Faith](Smit_ISBN9781441200396_epub_toc_r1.html#d7e3797)
In this book we've only scratched the surface of thinking about the shape of Christian faith in postmodernity. For those interested in thinking about this further, here are some books that can help you along the way. Texts marked with an asterisk (*) are a little more technical.
Baker, Jonny, and Doug Gay, with Jenny Brown. _Alternative_ _Worship: Resources from and for the Emerging Church._ Grand Rapids: Baker, 2004. A wonderful, unique book that provides concrete worship and liturgical resources for alternative worship that has "one hand in the past and one in the future." Includes a CD-ROM. An excellent resource.
*Benson, Bruce Ellis. _Graven Ideologies: Nietzsche, Derrida, and_ _Marion on Modern Idolatry._ Downers Grove, IL: InterVarsity, 2002. A book that demonstrates the way postmodernism can be seen as a critique of idols and thus amenable to Christian faith.
Dawn, Marva. _Reaching Out without Dumbing Down: A Theology of_ _Worship for This Urgent Time._ Grand Rapids: Eerdmans, 1995. Dawn tends to be critical of postmodernity but largely because she simply identifies it with modernity. She sees "seeker-sensitive" models of the church as postmodern, whereas I would say they are thoroughly modern. As a result, her suggestion of a rich liturgy as an antidote is, I suggest, postmodern.
Hauerwas, Stanley. _A Better Hope: Resources for a Church Confronting_ _Capitalism, Democracy, and Postmodernity._ Grand Rapids: Brazos, 2000. I would recommend almost anything Hauerwas writes as relevant to the question of postmodernity, but here he addresses the question head-on. More accessible than you might think.
*Hughes, Graham. _Worship as Meaning: A Liturgical Theology_ _for Late Modernity._ Cambridge: Cambridge University Press, 2003. This book considers the way in which worship weaves a symbolic web that both draws on and challenges the meanings available to us in late modernity.
Kitchens, Jim. _The Postmodern Parish: New Ministry for a New_ _Era._ Herndon, VA: Alban Institute, 2003. An interesting book; sort of Brian McLaren meets Stanley Hauerwas.
McLaren, Brian D. _The Church on the Other Side: Doing Ministry_ _in the Postmodern Matrix._ Grand Rapids: Zondervan, 2000. Though a bit glitzy in the way it hails postmodernity as a radically new era, this book takes up very specific strategies for the church to rethink itself in the contemporary world. In general, I wish McLaren's vision of the church were more sacramental (like Webber's), but this is still a helpful book.
———. _A Generous Orthodoxy: Why I Am a Missional, Evangelical,_ _Post/Protestant, Liberal/Conservative, Mystical/Poetic, Biblical,_ _Charismatic/Contemplative, Fundamentalist/Calvinist, Anabaptist/_ _Anglican, Methodist, Catholic, Green, Incarnational, Depressed-_ _Yet-Hopeful, Emergent, Unfinished Christian._ El Cajon, CA: Emergent Youth Specialties; Grand Rapids: Zondervan, 2004. This book offers theology in a confessional mode, honestly tackling the hard questions. Chapter 5, on the church as missional, is itself worth the price of the book.
———. _A New Kind of Christian: A Tale of Two Friends on a Spiritual_ _Journey._ San Francisco: Jossey-Bass, 2001. This book, which generated some controversy, put the emerging church conversation on the map. It is written as a dialogue/novel and raises the grassroots questions of postmodernity, tackling doubt and anxiety head-on. Highly recommended, and a great book for small-group discussion. It has now been extended into a trilogy: _The Story We Find Ourselves In: Further Adventures_ _of a New Kind of Christian_ (San Francisco: Jossey-Bass, 2003) and _The Last Word and the Word after That: A Tale of_ _Faith, Doubt, and a New Kind of Christianity_ (San Francisco: Jossey-Bass, 2005).
Middleton, J. Richard, and Brian J. Walsh. _Truth Is Stranger_ _Than It Used to Be: Biblical Faith in a Postmodern Age._ Downers Grove, IL: InterVarsity, 1995. From the authors of the classic _Transforming Vision: Shaping a Christian World View_ (Downers Grove, IL: InterVarsity, 1984), this book is one of the first nuanced engagements with postmodernism. Very strong on biblical engagement.
Raschke, Carl. _The Next Reformation: Why Evangelicals Must Embrace_ _Postmodernity._ Grand Rapids: Baker, 2004. An introduction to postmodern thought from a professional philosopher with extensive engagement in real-world ministry. One of the few books that specifically tackles both theory and practice.
*Smith, James K. A. _The Fall of Interpretation: Philosophical_ _Foundations for a Creational Hermeneutic._ Downers Grove, IL: InterVarsity, 2000. In this book I explore Derrida (and others) in more detail and argue that interpretation is a constitutive (and therefore good) feature of being a creature.
*———. _Introducing Radical Orthodoxy: Mapping a Post-secular_ _Theology._ Grand Rapids: Baker, 2004. This book will introduce you to an important sensibility in contemporary theology and at least point toward how it should make a difference for worship and discipleship.
Walsh, Brian J., and Sylvia C. Keesmaat. _Colossians Remixed:_ _Subverting the Empire._ Downers Grove, IL: InterVarsity, 2004. An outstanding "anti-commentary" that shows the enduring relevance of the biblical narrative in the context of postmodernity. One of a kind.
Webber, Robert E. _Ancient-Future Faith: Rethinking Evangelicalism_ _for a Postmodern World._ Grand Rapids: Baker, 1999. In this very readable book, Webber makes the case that I have suggested: that a truly postmodern church will be deeply historical (recovering its ancient heritage) and liturgical (activating the imagination through symbol and sacrament).
———. _The Younger Evangelicals: Facing the Challenges of the_ _New World._ Grand Rapids: Baker, 2002. Here Webber provides a report on the state of the emerging church. He does so by showing the differences between traditional evangelicals, pragmatic evangelicals (seeker-sensitive), and younger evangelicals. Includes both diagnosis and prognosis with specific, concrete suggestions for worship, youth ministry, discipleship, the arts, and more. A great book.
* * *
Online Resources
As a complement to the annotated bibliography on postmodernism and Christian faith, here are some select Web-based resources for further exploring the shape of the church in postmodernity.
**The Ooze** (<http://www.theooze.com>). This is "the" site for thinking about the emerging church. Includes articles that are regularly updated and organized under the categories of culture, faith, and ministry. Also provides information on new books, upcoming events, and opportunities to connect with others via online forums. Excellent design to boot.
**Emergent** (<http://www.emergentvillage.com>). Recently redesigned to be more clear and intuitive, it contains very helpful resources, including articles, online forums, and information about upcoming Emerging Gatherings, conferences, and other events, including online conferences and lectures. You can also sign up for an Emergent Village e-newsletter.
**Ancient-Future Worship** (<http://www.an>cientfutureworship .com). A site hosted by Robert Webber's Institute for Worship Studies that provides some resources for churches trying to integrate the insights of _Ancient-Future Faith_ into their worship.
**Sacramentis.com** (<http://www.sacramentis.com>). Hosted by Sally Morgenthaler, a leader in helping churches rethink worship in ways that are both ancient and postmodern. Lots of wisdom and some beautiful images on this site.
**The Ekklesia Project** (<http://www.ekklesiaproject.org>). The Ekklesia Project is an ecumenical, cross-denominational movement that seeks to think about a more radical understanding of being disciples of Jesus, emphasizing the church's countercultural calling. Lots of resources, including articles and an e-zine.
**Journal for Cultural and Religious Theory** (<http://www.jcrt.org>). This online journal provides access to some of the best work at the intersection of Continental philosophy, theology, and religious studies. Excellent design and rich archives.
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{"url":"https:\/\/brilliant.org\/discussions\/thread\/retrograde-analysis-2\/","text":"# Retrograde analysis\n\nChess is probably one of the most well-known board games in the world, with millions of players playing it around the world. There are two major divisions of chess: those that play chess as a game and those that play chess as a puzzle. Chess puzzles are further divided into two, tactical puzzles that are relevant for chess players and chess compositions that are rarely relevant for players, appealing for puzzle enthusiasts instead.\n\nSomething not interesting\n\nAbove: Mate in 3, a tactical puzzle taken from some chess game\n\nSomething interesting\n\nAbove: Mate in 4, composed by Vincente Maria N. Portilla in 1873\n\nSomething unique\n\nAbove: Helpmate in 8 (in helpmate, the two players cooperate to mate Black), composed by Z. Maslar in 1981\n\nChess compositions are divided into a large number of divisions (directmates and helpmates are two examples, shown above), but here I will talk about a quirky one, called retrograde analysis.\n\nAll you need to know are the rules of chess; you don't even need to know the strategies for playing chess, but you have to completely understand the rules. If you don't know the rules, turn back now and read that article until you completely understand it.\n\nIn retrograde analysis, you're not going forward in time...you're looking back in time.\n\nMY COMPOSITION WHEE\n\nAbove: What was the last move? Composed by Ivan Koswara, 2014\n\nYou don't have to think about possible defenses like games and directmates; it's pure logic, much closer to a logic puzzle. Helpmates are similar.\n\nBased on that sample puzzle above, do you find it interesting? Let me know!\n\nNote by Ivan Koswara\n6\u00a0years, 2\u00a0months ago\n\nThis discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution \u2014 they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.\n\nWhen posting on Brilliant:\n\n\u2022 Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .\n\u2022 Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting \"I don't understand!\" doesn't help anyone.\n\u2022 Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.\n\u2022 Stay on topic \u2014 we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.\n\nMarkdownAppears as\n*italics* or _italics_ italics\n**bold** or __bold__ bold\n- bulleted- list\n\u2022 bulleted\n\u2022 list\n1. numbered2. list\n1. numbered\n2. list\nNote: you must add a full line of space before and after lists for them to show up correctly\nparagraph 1paragraph 2\n\nparagraph 1\n\nparagraph 2\n\n[example link](https:\/\/brilliant.org)example link\n> This is a quote\nThis is a quote\n # I indented these lines\n# 4 spaces, and now they show\n# up as a code block.\n\nprint \"hello world\"\n# I indented these lines\n# 4 spaces, and now they show\n# up as a code block.\n\nprint \"hello world\"\nMathAppears as\nRemember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.\n2 \\times 3 $2 \\times 3$\n2^{34} $2^{34}$\na_{i-1} $a_{i-1}$\n\\frac{2}{3} $\\frac{2}{3}$\n\\sqrt{2} $\\sqrt{2}$\n\\sum_{i=1}^3 $\\sum_{i=1}^3$\n\\sin \\theta $\\sin \\theta$\n\\boxed{123} $\\boxed{123}$\n\n## Comments\n\nSort by:\n\nTop Newest\n\naxb8=R+\n\n- 5\u00a0years, 12\u00a0months ago\n\nLog in to reply\n\nThank you for that thoughtful contribution to the discussion. :D\n\n- 5\u00a0years, 11\u00a0months ago\n\nLog in to reply\n\nWow, that's actually really cool!\n\n- 6\u00a0years, 2\u00a0months ago\n\nLog in to reply\n\nI love all things about chess! Thanks for posting this.\n\nI believe in the first problem, labeled \"mate in 3\", there is actually a mate in 2 moves for white.\n\nOf course, I could be mistaken. Would someone like to look for the mate in 2 and respond?\n\n- 4\u00a0years, 6\u00a0months ago\n\nLog in to reply\n\nCouldn't find it. Which move were you thinking about?\n\n- 4\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\nThe move I saw was: 1. Rxa8.\n\nThe threat is: 2. Qxa6#\n\nIf 1. .... Kxa8 2. Qxa6#\n\nIf the black knight moves, then: 2. Rb8#\n\nI've probably missed something. Let me know if you see something I may have missed please?\n\n- 4\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\nHmm how about sac'ing the queen with 2.Qxd4 . Gives black king the escape square with Kxc6\n\n- 4\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\nI believe you're right!\n\nThanks for finding that for me.\n\n- 4\u00a0years, 3\u00a0months ago\n\nLog in to reply\n\nWhile I was able to calculate that the last move played in the fourth problem above is aXb8=R, I am unable to determine which black piece was taken on b8. It could not have been the Queen or a Rook because then the white king would have been in check. But, that piece could either be a Knight or a Bishop. Which one was it?\n\n- 4\u00a0years, 1\u00a0month ago\n\nLog in to reply\n\nThe White king can be in check; after all, because the White king is in check, White plays axb8=R+ to remove the check. The reason it can't be queen, rook, or bishop is that before that, Black has no last move.\n\n- 4\u00a0years, 1\u00a0month ago\n\nLog in to reply\n\nThe answer to the first problem is 1.Nxa5+ Ka7 2.Qxa6+! Kxa6 3.Rxa8#\n\n- 3\u00a0years, 4\u00a0months ago\n\nLog in to reply\n\n\u00d7\n\nProblem Loading...\n\nNote Loading...\n\nSet Loading...","date":"2020-07-06 00:29:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 8, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8738722205162048, \"perplexity\": 4283.835499511164}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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-2020-29\/segments\/1593655889877.72\/warc\/CC-MAIN-20200705215728-20200706005728-00169.warc.gz\"}"}
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Q: Set custom UITabBar font color under iOS7 It seems that [[UITabBar appearance] setSelectedImageTintColor:[UIColor redColor]]; is no longer working propertly under iOS7. I can set the background image, but the selected text is no longer getting the red color. I'm calling this from my application delegate. Has anyone tried to set the UITabBar font color and style under iOS7?
A: It worked for me..
[[UITabBar appearance] setBarTintColor:[UIColor blackColor]];
[[UITabBar appearance] setTintColor:[UIColor whiteColor]];
A: It is tintColor in iOS7, try the below:
[[UITabBar appearance] setTintColor:[UIColor redColor]];
Edit:
To tint the non active buttons, put the below code in your VC's viewDidLoad:
[self.tabBarItem setFinishedSelectedImage:[UIImage imageNamed:@"item_seleted.png"] withFinishedUnselectedImage:[UIImage imageNamed:@"item_unselected.png"]];
A: To tint not active items I used this
UITabBarItem *item = [self.tabBar.items objectAtIndex:1];
// here you need to use the icon with the color you want, as it will be rendered as it is
item.image = [[UIImage imageNamed:@"unselected.png"] imageWithRenderingMode:UIImageRenderingModeAlwaysOriginal];
// this icon is used for selected tab and it will get tinted as defined in self.tabBar.tintColor
item.selectedImage = [UIImage imageNamed:@"selected.png"];
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 2,232
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Q: Finding if current time in between a specified time slot I am trying to write a utility method which, by taking current time into consideration, would return the time slot (Breakfast, Lunch, Snacks or Dinner).
Below is the step by step proceedings I took to crack this. But, I am always being retuned NextTimeslotNone by my nextTimeslot method. When I print breakfastBeginingDate and breakfastEndingDate - both prints 2016-02-22 18:30:00 +0000 on console.
Any idea what am I doing wrong here?
Step 1: Defined an enum for slot identification
typedef NS_ENUM(NSInteger, NextTimeslot) {
NextTimeslotBreakfast = 0,
NextTimeslotLunch,
NextTimeslotSnacks,
NextTimeslotDinner,
NextTimeslotNone
};
Step 2: Wrote a method to set some arbitrary date component (to ignore date while comparing time slot)
+ (NSDate *)dateByNeutralizingDateComponentsOfDate:(NSDate *)originalDate {
NSCalendar *gregorian = [[NSCalendar alloc] initWithCalendarIdentifier:NSCalendarIdentifierGregorian];
// Get the components for this date
NSDateComponents *components = [gregorian components: (NSCalendarUnitYear | NSCalendarUnitMonth | NSCalendarUnitDay | NSCalendarUnitHour | NSCalendarUnitMinute | NSCalendarUnitSecond) fromDate: originalDate];
// Set the year, month and day to same arbitrary values
[components setYear:2000];
[components setMonth:1];
[components setDay:1];
return [gregorian dateFromComponents:components];
}
Step 3: Wrote a method to set return BOOL after comparing if a date is in between 2 specified dates
+ (BOOL)isTimeOfDate:(NSDate *)targetDate betweenStartDate:(NSDate *)startDate andEndDate:(NSDate *)endDate {
if (!targetDate || !startDate || !endDate) {
return NO;
}
// Make sure all the dates have the same date component.
NSDate *newStartDate = [MyUtility dateByNeutralizingDateComponentsOfDate:startDate];
NSDate *newEndDate = [MyUtility dateByNeutralizingDateComponentsOfDate:endDate];
NSDate *newTargetDate = [MyUtility dateByNeutralizingDateComponentsOfDate:targetDate];
// Compare the target with the start and end dates
NSComparisonResult compareTargetToStart = [newTargetDate compare:newStartDate];
NSComparisonResult compareTargetToEnd = [newTargetDate compare:newEndDate];
return (compareTargetToStart == NSOrderedDescending && compareTargetToEnd == NSOrderedAscending);
}
Step 4: Wrote a method to return the correct time slot for current time.
+ (NextTimeslot)nextTimeslot {
NSDate *currentDate = [NSDate dateWithTimeIntervalSinceNow:3600];
NSDateFormatter *formatter = [[NSDateFormatter alloc] init];
[formatter setDateFormat:@"yyyy-MM-dd HH:mm a"];
// Breakfast slot (9:00 AM - 11:00 AM)
NSString *breakfastBeginingTime = @"2016-02-23 09:00 am";
NSString *breakfastEndingTime = @"2016-02-23 11:00 am";
NSDate *breakfastBeginingDate = [formatter dateFromString:breakfastBeginingTime];
NSDate *breakfastEndingDate = [formatter dateFromString:breakfastEndingTime];
if ([MyUtility isTimeOfDate:currentDate betweenStartDate:breakfastBeginingDate andEndDate:breakfastEndingDate]) {
return NextTimeslotBreakfast;
}
// Lunch slot (11:30 AM - 2:00 PM)
NSString *lunchBeginingTime = @"2016-02-23 11:30 am";
NSString *lunchEndingTime = @"2016-02-23 02:00 pm";
NSDate *lunchBeginingDate = [formatter dateFromString:lunchBeginingTime];
NSDate *lunchEndingDate = [formatter dateFromString:lunchEndingTime];
if ([MyUtility isTimeOfDate:currentDate betweenStartDate:lunchBeginingDate andEndDate:lunchEndingDate]) {
return NextTimeslotLunch;
}
// Snacks slot (3:30 PM - 6:30 PM)
NSString *snacksBeginingTime = @"2016-02-23 03:30 pm";
NSString *snacksEndingTime = @"2016-02-23 06:30 pm";
NSDate *snacksBeginingDate = [formatter dateFromString:snacksBeginingTime];
NSDate *snacksEndingDate = [formatter dateFromString:snacksEndingTime];
if ([MyUtility isTimeOfDate:currentDate betweenStartDate:snacksBeginingDate andEndDate:snacksEndingDate]) {
return NextTimeslotSnacks;
}
// Dinner slot (07:30 PM - 10:00 PM)
NSString *dinnerBeginingTime = @"2016-02-23 07:30 pm";
NSString *dinnerEndingTime = @"2016-02-23 10:00 pm";
NSDate *dinnerBeginingDate = [formatter dateFromString:dinnerBeginingTime];
NSDate *dinnerEndingDate = [formatter dateFromString:dinnerEndingTime];
if ([MyUtility isTimeOfDate:currentDate betweenStartDate:dinnerBeginingDate andEndDate:dinnerEndingDate]) {
return NextTimeslotDinner;
}
return NextTimeslotNone;
}
A: Ok, fixed it by adding another method to convert UTC to current system time zone. This worked perfectly fine. Sharing it in a hope to save someone else the valuable time.
+ (NSDate *)systemDateForDate:(NSDate *)iDate {
NSTimeZone *tz = [NSTimeZone defaultTimeZone];
NSInteger seconds = [tz secondsFromGMTForDate:iDate];
return [NSDate dateWithTimeInterval: seconds sinceDate:iDate];
}
And using this method in my time slot calculation.
+ (NextTimeslot)nextTimeslot {
NSDate *currentDate = [MyFCUtility systemDateForDate:[NSDate date]];
NSDateFormatter *formatter = [[NSDateFormatter alloc] init];
[formatter setTimeZone:[NSTimeZone systemTimeZone]];
[formatter setDateFormat:@"yyyy-MM-dd HH:mm:ss"];
// Breakfast slot (9:00 AM - 11:00 AM)
NSString *breakfastBeginingTime = @"2016-02-23 09:00:00";
NSString *breakfastEndingTime = @"2016-02-23 11:00:00";
NSDate *breakfastBeginingDate = [MyFCUtility systemDateForDate:[formatter dateFromString:breakfastBeginingTime]];
NSDate *breakfastEndingDate = [MyFCUtility systemDateForDate:[formatter dateFromString:breakfastEndingTime]];
if ([MyFCUtility isTimeOfDate:currentDate betweenStartDate:breakfastBeginingDate andEndDate:breakfastEndingDate]) {
return NextTimeslotBreakfast;
}
// Lunch slot (11:30 AM - 2:00 PM)
NSString *lunchBeginingTime = @"2016-02-23 11:30:00";
NSString *lunchEndingTime = @"2016-02-23 15:00:00";
NSDate *lunchBeginingDate = [MyFCUtility systemDateForDate:[formatter dateFromString:lunchBeginingTime]];
NSDate *lunchEndingDate = [MyFCUtility systemDateForDate:[formatter dateFromString:lunchEndingTime]];
if ([MyFCUtility isTimeOfDate:currentDate betweenStartDate:lunchBeginingDate andEndDate:lunchEndingDate]) {
return NextTimeslotLunch;
}
// Snacks slot (3:30 PM - 6:30 PM)
NSString *snacksBeginingTime = @"2016-02-23 15:30:00";
NSString *snacksEndingTime = @"2016-02-23 18:30:00";
NSDate *snacksBeginingDate = [MyFCUtility systemDateForDate:[formatter dateFromString:snacksBeginingTime]];
NSDate *snacksEndingDate = [MyFCUtility systemDateForDate:[formatter dateFromString:snacksEndingTime]];
if ([MyFCUtility isTimeOfDate:currentDate betweenStartDate:snacksBeginingDate andEndDate:snacksEndingDate]) {
return NextTimeslotSnacks;
}
// Dinner slot (07:30 PM - 10:00 PM)
NSString *dinnerBeginingTime = @"2016-02-23 19:30:00";
NSString *dinnerEndingTime = @"2016-02-23 22:00:00";
NSDate *dinnerBeginingDate = [MyFCUtility systemDateForDate:[formatter dateFromString:dinnerBeginingTime]];
NSDate *dinnerEndingDate = [MyFCUtility systemDateForDate:[formatter dateFromString:dinnerEndingTime]];
if ([MyFCUtility isTimeOfDate:currentDate betweenStartDate:dinnerBeginingDate andEndDate:dinnerEndingDate]) {
return NextTimeslotDinner;
}
return NextTimeslotNone;
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,118
|
Felipe Abdiel Baloy Ramírez (Cidade do Panamá, 24 de fevereiro de 1981) é um ex-futebolista panamenho que atuava como zagueiro ou volante. Atualmente está aposentado.
Entrou para a história do futebol por ter sido o autor do primeiro gol do Panamá em Copas do Mundo. Ele marcou o gol com 37 anos e 120 dias, o que fez dele, à época, o 3º jogador mais velho da história a balançar as redes em um Mundial.
Carreira
Ele foi o primeiro jogador do Panamá a disputar o Campeonato Brasileiro, passando por e entre os anos de 2003 e 2005.
Foi autor do primeiro gol da Seleção Panamenha de Futebol em Copas do Mundo, na derrota para a Inglaterra, em 2018.
Fez parte do elenco do seu país na Copa América de 2016.
Títulos
Atlético-PR
Campeonato Paranaense: 2005
Monterrey
Campeonato Mexicano: 2009-10 (Apertura)
Prêmios individuais
Seleção Panamenha
Seleção dos 11 melhores da Copa Ouro da CONCACAF: 2005 e 2007
Ligações externas
Página oficial do jogador
Estatísticas de Felipe Baloy na Futpédia
Estatísticas de Felipe Baloy na FIFA
Naturais da Cidade do Panamá
Futebolistas do Panamá
Futebolistas do Sporting San Miguelito
Futebolistas da Corporación Deportiva Envigado Fútbol Club
Futebolistas do Deportivo Independiente Medellín
Futebolistas do Grêmio Foot-Ball Porto Alegrense
Futebolistas do Club Athletico Paranaense
Futebolistas do Club de Fútbol Monterrey
Futebolistas do Club Santos Laguna
Futebolistas do Club Atlético Monarcas Morelia
Futebolistas do Club Atlas
Jogadores da Seleção Panamenha de Futebol
Jogadores da Copa Ouro da CONCACAF de 2007
Jogadores da Copa Ouro da CONCACAF de 2009
Jogadores da Copa Ouro da CONCACAF de 2011
Jogadores da Copa América de 2016
Jogadores da Copa do Mundo FIFA de 2018
|
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"redpajama_set_name": "RedPajamaWikipedia"
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| 5,357
|
When Hedi Slimane debuted his version of YSL, ahem, excuse me, Saint Laurent, there was an outcry and backlash for some, and a cheer of pure joy for many, many, others. And increasingly, many, many more. I admit, I was on the fence. I sided with style.com's review, that it was something we've seen before and not what we'd hope from a house that constantly produced incredible innovation for decades. But as I poured through Yves' older shows and Hedi's earliest work, and once I realized how well the separates from Hedi's collection fit into every fashionable woman's wardrobe, I was able to jump on board the chic train to Grungetown! Chic! Chiiiic!
Want to hop on board, too? No ticket required! Here are the basics to get you there safely and chicly, of course. First off?
What to Wear it With: Everything below.
What to Wear it With: Layer it with a skinny black pant and ivory silk blouse along with a pointy toed bootie for the full Hedi effect, or add it to a tee and boyfriend jeans and d'orsay flats.
What to Wear it With: Chunky knit sweaters and a plaid shirt around your waist with platforms of any kind or opt for a boatneck sweater and denim jacket with cool fall sneakers.
What to Wear it With: Layer over jeans and a tee with a jean jacket, tuck it into skirts, in lieu of a cardigan or wrapped around the waist of your dresses.
Tell me which look you'll be adopting into your wardrobe, stat? And don't forget: I know getting dressed isn't always easy and so many trends can be so confusing. I'm here for any q's you might have, so hit me up!
« Quick Tip Featured on RealSimple.com: Make Your Diamonds Sparkle Like Your Pearly Whites!
|
{
"redpajama_set_name": "RedPajamaC4"
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| 592
|
<div class="video">
<div ng-show="data.length == 0">
No results.
</div>
<div class="video-item" ng-repeat="item in data" ng-show="data.length > 0">
<video preload="{{data.length < 2 ? 'auto' : 'none'}}" ng-click="selectVideo(item)" poster="squebi/img/poster.jpg" src="{{item.src}}" ng-show="item.type == 'video'"></video>
<div class="video-overlay">
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<i ng-click="closeVideo()" class="fa fa-times" ng-show="singlevideo"></i>
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<div style="{{singlevideo.fragment.xywh.style}}" ng-show="singlevideo && singlevideo.fragment.xywh" class="region-overlay"></div>
<video preload="auto" poster="squebi/img/poster.jpg" class="video-single" ng-show="singlevideo" src="{{singlevideo.src}}" controls="" autoplay=""></video>
</div>
|
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"redpajama_set_name": "RedPajamaGithub"
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| 1,886
|
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Pass the needle through the following two small pearl units.
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| 3,235
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Need help navigating TIS earnings?
Orchids Paper Products Company Announces First Quarter 2018 Results
April 25, 2018 5:00pm Comments
BRENTWOOD, Tenn., April 25, 2018
BRENTWOOD, Tenn., April 25, 2018 /PRNewswire/ -- Orchids Paper Products Company (NYSE:TIS), a national supplier of high-quality consumer tissue products, today reported results for the quarter ended March 31, 2018. The following table provides selected financial results for first quarter 2018 compared to first quarter 2017 and fourth quarter 2017.
1Q 2018
(Dollars in thousands, except per share data)
Net sales:
Converted product
Parent rolls
Total net sales
Net (loss) income
$ (2,294)
Diluted net (loss) income per share
$ (0.21)
Other Selected Financial Data:
EBITDA margin
Jeff Schoen, President and Chief Executive Officer, stated, "Orchids continues to ramp production volume at its new Barnwell, SC facility, which drove sequential revenue growth of 10.9% in the first quarter of 2018. We expect sales at this facility to continue to increase in coming quarters as a result of continued ramp of ultra-premium retail and parent roll business. The mill is successfully producing ultra-premium quality paper and we are selling excess parent rolls."
"On April 17, 2018, we appointed Mindy Bartel as our new chief financial officer. Ms. Bartel has a strong manufacturing background, with over 25 years of experience in manufacturing and technology service industries. Ms. Bartel's solid background in manufacturing, finance and operations will be a significant asset to Orchids as we continue to execute on our long-term strategic objectives."
"In March 2018, we engaged Guggenheim Securities LLS as a financial advisor to, among other things, explore strategic and financial alternatives for the Company, including to explore possibilities for the Company to repay its existing credit facilities."
First Quarter Results
The company reported net sales of $48.2 million in the first quarter of 2018, up 36.5% compared to the year-ago period. Converted product sales were $43.6 million, a year over year increase of 32.7%, and parent roll sales were $4.6 million, up 86.8% over the first quarter of 2017. The increase in converted product sales was a result of the company ramping new customer volume at the Barnwell facility. Parent roll growth was driven by an increase in the volume of excess parent rolls sold from Barnwell.
Gross profit in the first quarter of 2018 was $2.9 million, an increase from $2.0 million in the first quarter of 2017, or growth of 46%. Gross profit margins increased from 5.6% in the first quarter of 2017 to 6.0% in the first quarter of 2018. The improvement in gross profit margins reflects the favorable impact of the increased sales volume combined with higher average selling prices, primarily due to changes in the mix of products sold. However, gross profit margins remained under pressure from challenging, industry-wide conditions, as input costs, including fiber and freight costs, continued to rise.
Operating Loss
Orchids had a first quarter operating loss of $1.0 million in 2018 and a loss of $0.9 million in the first quarter of 2017. The increase in the operating loss was driven by SG&A increasing to $3.6 million in the first quarter of 2018, up $1.0 million from $2.6 million in the year-ago period. SG&A increased due to professional and consulting fees associated with our previously announced initiatives to review strategic alternatives and our debt refinancing efforts.
The company reported a tax benefit of $1.9 million in the first quarter of 2018 and a tax benefit of $.4 million in the year ago period.
Net loss in the first quarter of 2018 was $2.3 million and diluted loss per share was ($0.21). Net loss and diluted loss per share in the first quarter of 2017 were $0.9 million and ($0.08), respectively.
Cash Flow Provided by (Used in):
Operating cash flow net of changes in working capital
Changes in working capital
$ (18,027)
Cash, including restricted cash, beginning
Cash, including restricted cash, ending
Cash (used in) provided by operations was ($2.5) million in the first quarter of 2018, $5.2 million in the fourth quarter of 2017, and $3.6 million in the first quarter of 2017. Changes in working capital (used) provided operating cash flows of ($2.2) million in the first quarter of 2018, $4.1 million in the fourth quarter of 2017, and ($3.8) million in the first quarter of 2017.
Conference Call/Webcast
The Company will hold a teleconference to discuss its first quarter results at 11:30 a.m. (ET) on Thursday, April 26, 2018. All interested parties may participate in the teleconference by calling 888-346-7791 and requesting the Orchids Paper Products teleconference. A question and answer session will be part of the teleconference's agenda. Those intending to access the teleconference should dial in fifteen minutes prior to the start. The call may also be accessed live via webcast through the Company's website at www.orchidspaper.com under "Investors." A replay of the teleconference will be available for 30 days on the Company's website.
This press release contains non-GAAP financial measures. A non-GAAP financial measure is a numerical measure of a company's financial performance that excludes or includes amounts so as to be different than the most directly comparable measure calculated and presented in accordance with Generally Accepted Accounting Principles ("GAAP") in the United States in the statement of income, balance sheet or statement of cash flows of a company. The non-GAAP financial measures used within this press release are: (1) EBITDA, (2) Adjusted EBITDA, (3) Operating Cash Flow excluding changes in working capital, (4) Changes in working capital, (5) Adjusted net (loss) income and (6) Adjusted net (loss) income per diluted share.
EBITDA, Adjusted EBITDA, Operating Cash Flow less changes in working capital, Changes in working capital, and Adjusted net (loss) income and Adjusted net (loss) income per diluted share are not measurements of financial performance under GAAP and should not be considered as an alternative to net income, operating income, diluted net income per share or any other performance measure derived in accordance with GAAP, or as an alternative to cash flow from operating activities or a measure of the Company's liquidity. EBITDA represents net (loss) income before net interest expense, income tax expense, depreciation and amortization. Adjusted represents EBITDA before specified items. Changes in working capital is the subtotal of changes in operating assets and liabilities shown on the Consolidated Statements of Cash Flows. Operating Cash Flow less changes in working capital is Net Cash provided by operating activities less Changes in working capital. Adjusted net (loss) income and Adjusted net (loss) income per diluted share exclude the impact of certain items that management does not believe are indicative of the Company's core operating performance. Management believes EBITDA and Adjusted EBITDA facilitate operating performance comparisons between periods and between companies by eliminating potential differences caused by variations in capital structures (affecting relative interest expense), tax positions (such as the impact on periods or companies of changes in effective tax rates or net operating losses), the age and book depreciation of facilities and equipment (affecting relative depreciation expense), sporadic expenses (including start-up costs, foreign exchange adjustments, failed refinancing costs, consulting and professional fees, and relocation), and non-cash compensation (affecting stock-based compensation expense). These measures are also commonly used in the industry and are used by the Company's lenders in monitoring adherence to covenants. Management believes that Changes in working capital provides an indication of the cash invested in or provided by changes in operating assets and liabilities and therefore may indicate trends in operating performance and may call out a significant source or use of cash during any period. Operating Cash Flow less changes in working capital is believed to provide an estimate of the cash generated from all operating activities, prior to investments in or liquidations of operating assets and liabilities and therefore may indicate trends in operating performance and may call out significant changes in the generation of cash through operating activities. Management provides Adjusted net (loss) income and Adjusted net (loss) income per diluted share because it believes these measures assist investors and analysts in comparing the Company's performance across reporting periods on a consistent basis by excluding items that the company does not believe are indicative of its core operating performance.
This release contains forward-looking statements that involve certain contingencies and uncertainties. The Company intends these forward-looking statements to be covered by the safe harbor provision for forward-looking statements contained in the Private Securities Litigation Reform Act of 1995. These statements relate to future events or future financial performance, and involve known and unknown risks, uncertainties and other factors that may cause its actual results, levels of activity, performance or achievements to be materially different from any future results, levels of activity, performance or achievements expressed or implied by such forward-looking statements. In some cases, forward-looking statements can be identified by terminology such as "may," "should," "could," "expects," "plans," "intends," "anticipates," "believes," "estimates," "predicts," "potential," "will" or "continue" or the negative of such terms or other comparable terminology. Such forward-looking statements include, without limitation, the Company's beliefs, expectations, focus and/or plans about future events, including those regarding any potential refinancing, and the terms, conditions, timing and costs of any such refinancing. Although the Company believes that the expectations reflected in the forward-looking statements are reasonable, it cannot guarantee future results, levels of activity, performance or achievements. These statements are only predictions.
Factors that could materially affect the Company's actual results, levels of activity, performance or achievements include, without limitation, those detailed under the caption "Risk Factors" in the Company's Annual Report on Form 10-K for the year ended December 31, 2017, as filed with the Securities and Exchange Commission on March 16, 2018.
The Company's actual results may be materially different from what it expects. The Company does not undertake any duty to update these forward-looking statements after the date hereof, even though the Company's situation may change in the future. All of the forward-looking statements herein are qualified by these cautionary statements.
About Orchids Paper Products Company
Orchids Paper Products Company is a customer-focused, national supplier of high quality consumer tissue products primarily serving the at home private label consumer market. The Company produces a full line of tissue products, including paper towels, bathroom tissue and paper napkins, to serve the value through ultra-premium quality market segments from its operations in northeast Oklahoma, Barnwell, South Carolina and Mexicali, Mexico. The Company provides these products primarily to retail chains throughout the United States. For more information on the Company and its products, visit the Company's website at http://www.orchidspaper.com.
Investor Relations Contact:
Water's Edge Investor Relations Consulting Group
Louie Toma
louie.toma@watersedgeir.com
Orchids Paper Products Company and Subsidiaries
Selected Income Statement Data
(Dollars in thousands, except per share data) (unaudited)
Converted product net sales
Parent roll net sales
Cost of sales less depreciation
Depreciation in cost of sales
Total cost of sales
Intangibles amortization
Other (income) expense, net
Loss before income taxes
Benefits from income taxes
Average number of shares outstanding, basic
Average number of shares outstanding, diluted
Net (loss) income per share:
Cash dividends paid
Cash dividends per share
Selected Balance Sheet Data
(Dollars in thousands)
Inventory, net
Property plant and equipment
Net property plant and equipment
Intangibles and goodwill, net
Accounts payable, inclusive of amounts due to related parties
Current portion of long-term debt
Deferred income taxes
Total stockholders' equity
Total liabilities and stockholders' equity
Debt, current and long term
Reconciliations of Non-GAAP and GAAP Measurements
(Dollars in thousands) (unaudited)
Federal tax rate change
Adjustments, after tax (1):
Barnwell start-up costs
Relocation costs
Stock compensation expense
Failed debt refinancing costs
Consulting and other professional fees
Amortization of intangible assets
Adjusted net loss (2)
Net (loss) income per diluted share
Adjusted net loss per diluted share (2)
Weighted average diluted shares
Tax effect was calculated using the estimated annual effective tax rate for the period presented.
Adjusted net loss and adjusted net loss per diluted share exclude the impact of the listed items that we do not believe are indicative of our core operating performance.
Reconciliations of Non-GAAP and GAAP Measurements (continued)
EBITDA Reconciliation:
Plus: Interest expense
Plus: Income tax benefit
Plus: Depreciation
Plus: Intangible amortization
Earnings Before Interest, Income Taxes, Depreciation, and Amortization (EBITDA)
Adjusted EBITDA Reconciliation:
Plus: Barnwell start-up costs
Plus: Foreign exchange (gain) loss
Plus: Relocation costs
Plus: Stock compensation expense
Plus: Failed debt refinancing costs
Plus: Consulting and other professional fees
Operating Cash Flow Reconciliation:
Adjustments to reconcile net (loss) income to net cash (used in) provided by operating activities:
Provision for doubtful accounts
Operating cash flow excluding changes in working capital
Changes in cash due to changes in operating assets and liabilities:
Income taxes receivable
Prepaid expenses
Accrued liabilities
Net cash (used in) provided by operating activities
View original content with multimedia:http://www.prnewswire.com/news-releases/orchids-paper-products-company-announces-first-quarter-2018-results-300636723.html
SOURCE Orchids Paper Products Company
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{"url":"http:\/\/aas.org\/archives\/BAAS\/v26n4\/aas185\/abs\/S7515.html","text":"CO Observations of the Dwarf Starburst Galaxy NGC 5253 and the $\\bf I_{\\bf CO}\/N_{\\bf H_2}$ Ratio\nSession 75 -- Interacting Galaxies and Starbursts\nDisplay presentation, Wednesday, 11, 1995, 9:20am - 6:30pm\n\n## [75.15] CO Observations of the Dwarf Starburst Galaxy NGC 5253 and the $\\bf I_{\\bf CO}\/N_{\\bf H_2}$ Ratio\n\nJ.L.Turner (UCLA), S.C.Beck (Tel Aviv)\n\nWe present observations of the 3 mm continuum and CO emission in the dwarf galaxy NGC 5253. NGC 5253 appears to be undergoing a recent burst of star formation which may be tidally induced by its companion, M83. These aperture synthesis maps were made with the Owens Valley Millimeter Array, and have a resolution of 12$^{\\prime\\prime}$ x 7$^{\\prime\\prime}$. There is strong thermal continuum emission at 3 mm at the location of the starburst. However the CO emission is extraordinarily weak. This indicates that CO is a poor tracer of the H$_2$ mass in this galaxy, since the high resolution of the maps rules out beam dilution as a cause of the low single dish CO temperatures. We present a value for the conversion factor from CO intensity to H$_2$ column density, $\\rm I_{CO}\/N_{H_2}$.\n\nThis work was supported in part by NSF grant AST90-22996.","date":"2013-12-09 22:39:53","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\": 0.8009579181671143, \"perplexity\": 2924.4612904314554}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-48\/segments\/1386164000853\/warc\/CC-MAIN-20131204133320-00079-ip-10-33-133-15.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/lara.epfl.ch\/w\/sav08\/deciding_quantifier-free_fol_over_ground_terms?rev=1429630213&do=diff","text":"# Differences\n\nThis shows you the differences between two versions of the page.\n\n sav08:deciding_quantifier-free_fol_over_ground_terms [2008\/05\/19 14:04]vkuncak sav08:deciding_quantifier-free_fol_over_ground_terms [2015\/04\/21 17:30] (current) Both sides previous revision Previous revision 2008\/05\/19 14:08 vkuncak 2008\/05\/19 14:04 vkuncak 2008\/05\/19 14:04 vkuncak 2008\/04\/27 17:26 damien 2008\/04\/25 14:06 vkuncak 2008\/04\/23 15:10 vkuncak 2008\/04\/23 11:07 vkuncak 2008\/04\/23 11:07 vkuncak 2008\/04\/23 09:10 vkuncak 2008\/04\/23 09:03 vkuncak 2008\/04\/23 08:37 vkuncak 2008\/04\/23 08:36 vkuncak 2008\/04\/23 08:33 vkuncak 2008\/04\/23 08:32 vkuncak 2008\/04\/17 13:48 vkuncak 2008\/04\/17 13:48 vkuncak 2008\/04\/17 13:08 vkuncak 2008\/04\/17 13:07 vkuncak 2008\/04\/17 13:06 vkuncak created Next revision Previous revision 2008\/05\/19 14:08 vkuncak 2008\/05\/19 14:04 vkuncak 2008\/05\/19 14:04 vkuncak 2008\/04\/27 17:26 damien 2008\/04\/25 14:06 vkuncak 2008\/04\/23 15:10 vkuncak 2008\/04\/23 11:07 vkuncak 2008\/04\/23 11:07 vkuncak 2008\/04\/23 09:10 vkuncak 2008\/04\/23 09:03 vkuncak 2008\/04\/23 08:37 vkuncak 2008\/04\/23 08:36 vkuncak 2008\/04\/23 08:33 vkuncak 2008\/04\/23 08:32 vkuncak 2008\/04\/17 13:48 vkuncak 2008\/04\/17 13:48 vkuncak 2008\/04\/17 13:08 vkuncak 2008\/04\/17 13:07 vkuncak 2008\/04\/17 13:06 vkuncak created Line 10: Line 10: Is the following formula satisfiable ? Is the following formula satisfiable ? - $+ \\begin{equation*} a=f(a) \\land a \\neq b a=f(a) \\land a \\neq b -$\u00a0++|No. When interpreting variables as constants, two syntactically different terms are different ($a \\neq f(a)$).++ + \\end{equation*}\u00a0\u200b++|No. When interpreting variables as constants, two syntactically different terms are different ($a \\neq f(a)$).++ And the following ? And the following ? - $+ \\begin{equation*} \u200b(f(a)=b \\lor (f(f(f(a))) = a \\land f(a) \\neq f(b))) \\land f(f(f(f(a)) \\neq b \\land b=f(b) \u200b(f(a)=b \\lor (f(f(f(a))) = a \\land f(a) \\neq f(b))) \\land f(f(f(f(a)) \\neq b \\land b=f(b) -$ + \\end{equation*} ===== Axioms for Term Algebras ===== ===== Axioms for Term Algebras ===== Term algebra axioms for function symbols (recall [[Unification]]):\u200b Term algebra axioms for function symbols (recall [[Unification]]):\u200b - $+ \\begin{equation*} \u200b\\forall x_1,\u200b\\ldots,\u200bx_n,\u200by_1,\u200b\\ldots,\u200by_n.\\ f(x_1,\u200b\\ldots,\u200bx_n)=f(y_1,\u200b\\ldots,\u200by_n) \\rightarrow \\bigwedge_{i=1}^n x_i=y_i \u200b\\forall x_1,\u200b\\ldots,\u200bx_n,\u200by_1,\u200b\\ldots,\u200by_n.\\ f(x_1,\u200b\\ldots,\u200bx_n)=f(y_1,\u200b\\ldots,\u200by_n) \\rightarrow \\bigwedge_{i=1}^n x_i=y_i -$ + \\end{equation*} and and - $+ \\begin{equation*} \u200b\\forall x_1,\u200b\\ldots,\u200bx_n,\u200by_1,\u200b\\ldots,\u200by_m.\\ f(x_1,\u200b\\ldots,\u200bx_n) \\neq g(y_1,\u200b\\ldots,\u200by_m) \u200b\\forall x_1,\u200b\\ldots,\u200bx_n,\u200by_1,\u200b\\ldots,\u200by_m.\\ f(x_1,\u200b\\ldots,\u200bx_n) \\neq g(y_1,\u200b\\ldots,\u200by_m) -$ + \\end{equation*} for $g$ and $f$ distinct symbols. for $g$ and $f$ distinct symbols. Unless the language is infinite, we have that each element is either a constant or is result of application of some function symbols: Unless the language is infinite, we have that each element is either a constant or is result of application of some function symbols: - $+ \\begin{equation*} \u200b\\forall x. \\bigvee_{f \\in {\\cal L}} \\exists y_1,\u200b\\ldots,\u200by_{ar(f)}.\\ x=f(y_1,\u200b\\ldots,\u200by_{ar(f)}) \u200b\\forall x. \\bigvee_{f \\in {\\cal L}} \\exists y_1,\u200b\\ldots,\u200by_{ar(f)}.\\ x=f(y_1,\u200b\\ldots,\u200by_{ar(f)}) -$ + \\end{equation*} Additional axiom schema for finite terms: Additional axiom schema for finite terms: - $+ \\begin{equation*} \\forall x. t(x) \\neq x \\forall x. t(x) \\neq x -$ + \\end{equation*} if $t(x)$ is a term containing $x$ but not identical to $x$. if $t(x)$ is a term containing $x$ but not identical to $x$. Line 46: Line 46: Represent each disequation $t_1 \\neq t_2$ as Represent each disequation $t_1 \\neq t_2$ as - $+ \\begin{equation*} x_1 = t_1 \\land x_2 = t_2 \\land x_1 \\neq x_2 x_1 = t_1 \\land x_2 = t_2 \\land x_1 \\neq x_2 -$ + \\end{equation*} where $x_1,x_2$ are fresh variables. where $x_1,x_2$ are fresh variables. Line 76: Line 76: Represent $cons(x,\u200by)=z$ by Represent $cons(x,\u200by)=z$ by - $+ \\begin{equation*} x=car(x) \\land y = cdr(z) \\land \\lnot atom(z) x=car(x) \\land y = cdr(z) \\land \\lnot atom(z) -$ + \\end{equation*} here here * $atom(x)$ is negation of $Is_{cons}$ * $atom(x)$ is negation of $Is_{cons}$ Line 98: Line 98: ===== References ===== ===== References ===== - * [[http:\/\/\u200blara.epfl.ch\/\u200b~kuncak\/\u200bpapers\/\u200bKuncakRinard03TheoryStructuralSubtyping.html|On the Theory of Structural Subtyping]],\u200b see Lemma 25 on page 13 + * [[http:\/\/\u200blara.epfl.ch\/\u200b~kuncak\/\u200bpapers\/\u200bKuncakRinard03TheoryStructuralSubtyping.html|On the Theory of Structural Subtyping]],\u200b see\u00a0proof of\u00a0Lemma 25 on page 13","date":"2019-11-18 15:17:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"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\": 0.7411731481552124, \"perplexity\": 10456.233086648623}, \"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-2019-47\/segments\/1573496669795.59\/warc\/CC-MAIN-20191118131311-20191118155311-00511.warc.gz\"}"}
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midlifelove 11:53 pm on March 8, 2010 Permalink | Reply
Tags: bored, Boredom, dangerous to health, depression ( 7 ), heart risk, raise blood pressure
My Grandma used to say boring people are bored people – but sometimes a touch of the old ennui is unavoidable.
Waiting for web pages to load, getting stuck in traffic, watching someone play video games – they come high on most people's Top Five Most Boring Things to do list.
They're all things that test your patience, but they're not terminal you say. Until now, that is, because according to new research, you really can be bored to death.
Boredom Bad For Heart
It seems the more bored you are, the more likely you are to die early, although scientists caution it's probably not the boredom alone that will kill you, but the risky behaviour – drinking, eating, inhaling the wrong things – that often accompanies it.
It doesn't hurt to be just occasionally bored – that happens to everyone. It's the chronic kind of boredom that's dangerous.
In the study 7500 London civil servants aged 35 – 55 were questioned about how bored they were at work the previous month, and then those who were still alive were followed up 20 years later.
University College London researchers Annie Britton and Martin Shipley found that those who reported they had been very bored were two and a half times more likely to die of a heart problem than those who hadn't reported being bored.
But when the authors made a statistical adjustment for other potential risk factors, like physical activity levels and employment grade, the effect was reduced.
Boredom As Dangerous As Stress
Researchers point out someone who is bored may not be motivated to eat well or exercise, and boredom is often linked to depression, which has long been recognized as a risk factor for heart disease.
Others said boredom was potentially as dangerous as stress.
"Boredom is not innocuous," said Sandi Mann, a senior lecturer in occupational psychology at the University of Central Lancashire who studies boredom.
She said boredom is linked to anger suppression, which can raise blood pressure and suppress the body's natural immunity. "People who are bored also tend to eat and drink more, and they're probably not eating carrots and celery sticks," she said.
midlifelove 4:30 am on March 6, 2010 Permalink | Reply
Tags: Ballina, Bass Coast, Best Places To Retire in Australia ( 2 ), Bribie Island, Copper Coast, Echuca, Geraldton, Inverloch, Kadina, Mandurah, Moonta, Murray River, New South Wales, Perth, Phillip Island, Port Macquarie, Queensland, South Australia, St Helens, Tamar Valley, Tasmania, Toowoomba, Torquay, Tweed Valley, Victor Harbor, victoria ( 2 ), Wallaroo, Western Australia, Wonthaggi
You're thinking of escaping from the big city for a more peaceful, less polluted life in a country town or beachside community. You want to avoid the known – and unknown hazards – like man-eating koalas and deranged gunmen. So before you sell up and move, it's a great idea to do investigate your options. And there's no better guide on how to avoid disaster than Jill and Owens Weeks's well-researched retirement advice.
For more than a decade, they've investigated Australia from north to south, east to west, for their best-selling publications.
Here's their pick of the best places to retire.
The state with "by far the greatest number of ideal retirement locations. North of Sydney, the entire coastline all the way up to Newcastle is popular, particularly as retirement trends change in favour of semi retirement."
Port Macquarie – "vibrant and dynamic area which offers many of the services and culture of a large city."
Ballina "good for those who want to get away from capital cities and can live without many of the facilities they offer."
Tweed Valley "offers most of the good features of the Gold Coast without many of the downsides . . and close to Coolangatta international airport just across the border in Queensland."
Torquay (near Geelong) "There's a buzz about the place . . . close to Geelong's facilities. Fastest growing non-metropolitan area in Victoria and a clear favourite with retirees."
Bass Coast (Inverloch, Wonthaggi, Phillip Island) or Echuca (on the Murray River) "Cost of housing reasonable and a compelling list of positive features with few if any down sides."
Mandurah (south of Perth) – "Special appeal because of the number of services focused on retirees," or Geraldton.
Toowoomba "affordable housing, good facilities, relaxed pace, and not over-crowded with tourists in holidays. Not as humid as the coast, and Brisbane is accessible."
Bribie Island "preferred destination for those who want to get away from the congestion of the Gold coast."
Victor Harbor "Outstanding geography, access to Adelaide, vineyards and fine food. Holiday destination for over 100 years, town is full of well-preserved buildings, museums," or the Copper Coast Yorke Peninsula (Kadina, Wallaroo, Moonta.)
St Helens "Ideal climate on the Sun Coast, an affordable and relaxed lifestyle that few places in Australia can match," or the Tamar Valley
Checklist for Retiring
Jill and Owen Weeks suggest:
Consider renting in the location of your choice. House-sit or even caravan-sit before you buy.
Subscribe to the local newspaper for at least 12 months before you move. This provides a good source of jobs, real estate and what's happening in the community.
Research the utilities and services. Does the mobile phone work? Is there internet connection and educational services? This is particularly important if you are running a home-based business. Also check local services such as the nearest mechanic or electrician.
Will your new home require major renovations? Is it 'older body friendly?' Will you eventually need to move because of stairs? What happens if your mobility is impaired?
See also Where to Retire in Australia for additional advice.
Tags: 70th birthday, All-Starr summer tour, Barbara Bach, George Harrison, John Lennon, Paul McCartney, Ringo Starr, Ringo turns 70, The Beatles, vegetarian, YNot 15th album
With a new album, an enduring and happy marriage, and a zest for life that's the envy of many 30-year-olds, Ringo Starr enters his 70th year with high spirits.
He's the eldest of the famous Beatles – he will hit his 70-year milestone on July 7 – and he's got the newest music.
He's not missing a beat as he embarks on a three-week promotional tour for his just-released Y Not album, his 15th solo outing on which fellow Beatle Paul McCartney plays bass on the song 'Peace Dream'.
And he plans to mark his 70th by flashing a two-fingered peace sign at noon and playing an evening gig at Radio City Music Hall as part of a summer tour with his latest All-Starr band.
Last year on the Larry King Show, Starr noted: "I work out. I have a trainer. And I watch what I eat. That's it really. And I'm in love with a beautiful girl, so it keeps me young."
Turning 40 was Harder
And as he told Randy Lewis of the Los Angeles Times, 70 is "not as big as 40 was. Forty was: 'Oh, God, 40!
"There's that damn song, 'Life Begins at 40.' No, it's not so big anymore. I am nearly 70, and I'd love to be nearly 40, but that's never going to happen.
"I feel the older I get, the more I'm learning to handle life,"
Notes Lewis: "His charming Liverpudlian accent is nearly as strong as ever, even though he's maintained a home in Los Angeles for the last 34 years — the majority of it with actress Barbara Bach, whom he married in 1981 — along with residences in England and Monte Carlo."
God Now 'My Life'
He's trim – like McCartney and his late pal Harrison, he's an avowed vegetarian – looks 15 years younger than his age, and as the years roll by spiritual issues have become more prominent, he says.
"Being on this quest for a long time, it's all about finding yourself," Starr says. "For me, God is in my life. I don't hide from that. … I think the search has been on since the '60s. … I stepped off the path there for many years and found my way (back) onto it, thank God."
There's no secret to his successful marriage to Barbara, Starr told USA Today. "I'm just blessed that she puts up with me. I love the woman. She loves me. There's less down days than up, and we get on really well. We do spend a lot of time together. That's the deal."
Many Artistic Projects
Starr says he also needs diverse creative outlets to keep him engaged when he's not making albums or touring with his All-Starr band. In the 1970s it was acting, now it's art — a selection of his photos appears inside the album.
"I am always painting," he says. "I love photography. It's easy to take shots. But if you have to choose, it's music. I love music, I love playing."
Ringo was three months older than John Lennon, who would have turned 70 this year. Lennon was 40 when he was shot dead in New York City.
Paul McCartney will be 68 this year. George Harrison would have been 67.
Tags: australia ( 2 ), Baby boomers, Best Places To Retire in Australia ( 2 ), Bette Davis, Jill and Owen Weeks, London Marathon, retire, Richard Branson
You might sympathise with Hollywood star Bette Davis when she said "I will not retire while I've still got my legs and my make-up box."
Many Baby Boomers see age as "just a number" and like Virgin boss Richard Branson (64) – who's added training for the 2010 London Marathon to his punishing work schedule – they're busy packing more life into their years.
But even for those to whom "retirement" is a foreign concept, changing lifestyle, moving home, or splitting their year between two places, can be tempting as a mid-life option.
Looking For A Change
The children are independent, you don't need to worry about climbing the corporate ladder any more, and you can take time out to think about what you want to do with the rest of your life.
If you're living in Australia or thinking of moving there, Jill and Owen Weeks have done a heap of research to help with your decisions.
Their book Where to Retire in Australia is a great resource for anyone in "the Moving Generation" looking for a change of lifestyle.
The Weeks' golden rule is: the most successful moves are usually less than 200km or no more than two hours from the old location. Why? So people remain in easy reach of old friends, family and social networks.
Key Issues To Consider
They've identified the key issues to decide on when making a change:
Taking care of yourself; Consider closeness to doctors, hospitals and specialist medicine if required; social life – how easy is for family and friends to visit, will you find your others who share your interests, do you want to spend part of your year in another location?
Weather; Check out the climate before you buy. If you've only holidayed at the location at certain times of the year rent for 18 months before making a permanent move while you decide if the climate is for you.
Keeping the Body Moving; Ensure you can pursue your interest, hobbies and sporting activities.
Keeping the Brain Active; Access to the Internet, opportunities for further study, or part time work, libraries.
Let's Go Shopping; Proximity to a reasonably large shopping centre will have impact on your cost of living. Can you use "Loyalty "programmes like FlyBuys and discount cards.
Can You Afford to Move; Consider real estate, tax and social security implications, any cost of living changes.
Social Compatibility; Moving away from close friends and family may be painful. They may not visit regularly even if they promise they will. Are you good at making new friends and does the place you are moving to welcome new residents.
Part 2: The Best Places in Australia to Retire.
midlifelove 1:29 am on February 21, 2010 Permalink | Reply
Tags: best places to retire, Country Brand ratings, Forbes magazine, medical care costs, retirement
Twenty years from now you will be more disappointed by the things you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover.
Author Twain's advice applies at any age, but especially for those in their middle years looking ahead to possible "retirement". You maybe won't ever want to "retire" like your parents did, but chances are you'll be changing locations, changing your means of earning income, and maybe even changing countries.
There are stacks of "Best Places to Retire Lists" around, all using different criteria. CBS' Moneywatch has done a good analysis of some of the more popular list sites get their ratings.
But now Forbes magazine has compiled its own list of the 10 best retirement havens for US citizens, based on a wide variety of criteria ranging from safety to retiree-friendly visa requirements to decent medical care.
Forbes' Top Ten Places to Retire
Austria – A "lower-cost Switzerland" with mountains, the great outdoors, elegant architecture and trams that run. Vienna offers the highest quality of life on the globe, according to Mercer, and medical insurers say its private clinics are world-class.
Thailand – Expect a warmly welcome, ideal for seniors on modest budgets. Considered Asia's best buy for quality health care at reasonable costs, but the good private hospitals are in Bangkok, so make sure your beach retreat is within three hours of the capital.
Italy – Warm Mediterranean sociability and one of the world's best health care systems. Great bargains in the South in Puglia or Sicily, or in cities like Sorrento on the Amalfi Coast.
Panama – Year-round sun, low taxes, massive discounts for seniors, first-world amenities, quality private hospitals, bird-filled rainforests, a dollar economy and easy flights from the U.S.
Ireland – Stunning countryside, cultural affinity, low taxes and many rebates for seniors. Overpriced Dublin reachable in a couple hours from anywhere in the country, so look for housing inland.
Australia – The world's best place to live, according to the Country Brand Index. The highest quality cities at the lowest cost, claims Mercer. The famously friendly Aussie has created a first-world country with low Asian costs.
France – offers perhaps the friendliest of policies toward American retirees of any European Union nation. Considered most affordable quality health care in the world, plus low taxes for American ex-pats. Paris for wealthy, but Brittany, Normandy and the Dordogne, a short train ride away, are more affordable.
Malaysia – Welcoming to retirees, low costs and spectacular coastline make it a strong contender for the budget-conscious, but also increasingly for the wealthy wanting an Asian tax haven. For health care reasons, avoid straying too far from Kuala Lumpur.
Spain – The 'Florida of Europe' has long been a magnet for sun-starved Brits, so retirement infrastructure all in place. Coast overdeveloped but great value in lesser-known interior cities like Salamanca or Burgos.
Canada – Cities from Vancouver to Montreal consistently score among the best in the world, both on quality-of-life and value-for-money benchmarks. Friendly policies toward retirees, affordable medical care and a natural cultural fit for Americans.
Nowhere Is Perfect
Says Forbes: "No place is perfect. Some countries rank high in one area but lower in others. Australia is by one well-regarded rating, the Country Brand Index, the most livable place in the world. But if you plan to return to the U.S. frequently, Australia makes for a long slog.
"Canada is No. 2 in the Country Brand ratings and certainly convenient for Americans, but its harsh winters are well-known.
"Italy scores high on quality of life, medical care, and cost of living and climate . . . But its complicated taxes and bureaucracy require patience.
"So, the key to any decision: Know yourself and do your homework."
Tags: cancer, Coffee good for you, Coffee health benefits, Harvard Medical School, heart disease ( 2 ), longevity, Parkinsons Disease ( 2 ), type 2 diabetes ( 2 )
Drink up, coffee lovers. Not only is coffee aromatic and delicious, it's good for you.
Who says? None other than Harvard Medical School.
Once considered questionable for your health, it turns out that the beloved beverage is actually healthful in moderation. That means a few cups a day.
At about 20 cents per 6-ounce cup, coffee is a good deal if you brew it yourself.
Harvard researchers say drinking coffee may help prevent diseases such as:
Cancer: Some studies have found coffee drinkers have lower rates of colon and rectal cancers, better survival rates with prostate cancer, and are 50 percent less likely to get liver cancer than coffee abstainers.
Type 2 diabetes: Coffee is thought to contain chemicals that lower blood sugar because heavy coffee drinkers may be half as likely to get diabetes as those who drink little or no coffee. Coffee also may increase your resting metabolism rate, which could help prevent diabetes.
Parkinson's disease: Coffee seems to help protect men from Parkinson's disease, but not women. The difference might be due to estrogen, researchers say.
Heart disease: Coffee is not linked to the development of heart disease. In the past few years, Harvard scientists say, coffee has been shown to be safe even for heart attack survivors. Scientists think antioxidants in coffee may reduce inflammation and protect blood vessel walls.
Life span: Recent studies suggest that drinking coffee decreases the risk of premature death, especially in women. Women who drank at least five to seven cups a week had a death rate 26 percent lower than non-consumers, a large investigation by researchers in Spain and at Harvard Medical School found.
Three Or More Cups Good
It's not only Harvard researchers who are touting the brew's benefits. As the Palm Beach Post reported, in September a study led by Neal Freedman of the National Cancer Institute showed that people with chronic hepatitis C and advanced liver disease who drank three or more cups of coffee a day cut their risk of the disease progressing by 53 percent.
Although caffeine might be considered the "active ingredient" in coffee, coffee is only 2 percent caffeine and 98 percent "other stuff," including more than 1,000 different compounds such as vitamins, minerals and amino acids.
It even contains fiber. Each cup contains from 1.1 to 1.8 grams of soluble dietary fiber, the kind that dissolves in water and helps prevent cholesterol from being absorbed by the intestines, according to researchers at the Spanish National Research Council in Madrid.
But Not If You Are Pregnant
Do researchers have any words of caution? Yes — although regular coffee drinking isn't harmful for most people, that might not hold true for pregnant women. Research has linked miscarriage to caffeine consumption of 200 milligrams or more per day. A typical cup of coffee has 100 to 150 milligrams, Harvard reports.
midlifelove 11:28 pm on February 5, 2010 Permalink | Reply
Tags: belly fat bad, big butt healthy, Dr Konstantinos Manolopoulos, fat stored on hips, Large thighs good, Oxford University, spare tire unhealthy
Long Live Big Butt Women
Those curves are making cardiologists smile.
A new study out of Oxford University in England reported in the NY Post found women with "pear-shaped" figures — ample extra padding around the hips, buttocks and thighs — are actually at a lower risk for heart and metabolic diseases.
"It is shape that matters and where the fat gathers," said Oxford's Dr. Konstantinos Manolopoulos, explaining that fat stored on the hips and in the rear absorb harmful fatty acids and further prevent arteries from clogging.
Belly Fat Is Bad
"Fat around the hips and thighs is good for you, but around the tummy is bad," he told BBC News, noting that the proverbial "spare tire" around the belly leads to higher incidences of heart disease.
The findings help explain why women traditionally have lower rates of heart disease then men until they reach menopause, when their rates rise as they begin to gather fat around their midsections.
"The only thing I can say is that women who have large thighs shouldn't be anxious about it," Manolopoulos said. "Their body shape is associated with health."
Scientists believe genetics play a large role in where a body stores fat.
midlifelove 2:31 am on January 30, 2010 Permalink | Reply
Tags: divorce ( 9 ), how to have a great marriage, Ian and Mary Grant, marriage ( 6 ), mens and womens brains different, tips for better marriage, unhappy marriage
How to Have a Great Marriage
Marriage is a two-storey house – that's her story and his story, quips Ian Grant*, parenting and relationship guru, who with Mary, his wife of 40 years runs the highly successful Hot Tips on Marriage seminars in New Zealand and Australia.
Now the couple have combined to write Growing Great Marriages, a book based on the Hot Tips seminars which offers hundreds of practical strategies for giving and getting the love you want in your marriage. Some of their ideas from the book:
Top Five Tips for A Better Marriage
1) Watch your language
Research shows the ratio of praise to blame is like spring rain – praise and your marriage will flourish, blame and it will wither. Couples who say five positive things for one negative should be OK. When the threshold drops to one to two they are in trouble.
Thousands of trials have shown men are much more affected by arguments than women are: "flooding" – when brain messages bypass the thinking and judging mechanisms and go straight to the "old reptile brain" – the section of the brain that generates negative emotions and knee jerk reactions – happens at much lower levels of criticism in men than women and they stay in that flooded state for longer.
2) Commit and invest in the relationship
A 2002 study by the Institute for American Values in New York showed it was wrong to assume that someone in a troubled marriage has two choices: stay married and be miserable, or get a divorce and be happy. A survey of so-called "unhappily married" couples showed two thirds of the couples who stayed married were actually happier five years later.
Other research shows if a partner invests into a relationship, he or she is happier. When you invest in something, you bond to it.
3) Remember men's and women's brains work differently:
Think of the brain like a house: In men brain activity focuses on different sections of the brain and they switch from section to section like someone moving from appliance to appliance in a house, switching them on when needed and then turning them off.
Women's brains function more globally – they've switched on everything in the house and left it on!
Bill and Pam Farrel's best seller Men are Like Waffles Women are Like Spaghetti (Harvest House) gives a great picture for the way gender affects the way men and women think.
Women are good at multi-tasking because like a plate of spaghetti , their brain wiring is made up of lots of different strands, touching and intertwining with each other. Women process life through interconnections. Every thought and issue is connected to every other thought and issue.
Men' brains are geared to work in a more compartmentalised way – like squares on a waffle. Don't engage Dad in meaningful conversation while he is cooking the barbecue because he is focused on one thing – getting the meat cooked. Men will work on one square at a time, and then move on to the next one.
4) Operate a love bank
When difficulties arise, focus on restoring love, not resolving conflicts.
Only one in four marriages is saved through counselling, which has traditionally focused on conflict resolution. But what most couples want is to maintain the feelings of romantic love, and if they can do that the conflict resolves itself. (Romantic love triggers the endorphins which give the pleasurable feeling of being in love)
The Love Bank idea was developed by Dr William Harley, author of Fall in Love, Stay in Love. Think about the "love currency" you can deposit for your partner.
Ask yourself "What can I do that makes him feel the best?" Men thrive when offered recreational companionship, sexual fulfilment and admiration from their wives. Women thrive on affection, openness and honesty. What makes people happiest is receiving attention from the people who matter most to us.
5) Plan dates
Ian and Mary Grant have dozens of great ideas of sharing positive time together. Amongst their suggestions:
A 48 hour retreat; book somewhere special, take candles, special lingerie, etc and create your own haven.
Work through the alphabet for date nights – some may be elaborate, and others as simples as a DVD you both want to see.
Stage regular "cloth napkin dinners" with a formal setting, best cutlery and candles.
On your wedding anniversary each year plan a special dinner and repeat your wedding vows to one another.
*Ian and Mary Grant are high profile parenting and relationship gurus who founded Parents Inc, an Auckland based centre which runs nationwide seminars and courses on family and relationships.
midlifelove 12:43 am on January 28, 2010 Permalink | Reply
Tags: incurable optimist, Michael J Fox ( 2 ), Parkinsons Disease ( 2 ), successful Hollywood marriage, Tracy Pollan
Incurable Optimist: Michael J Fox
Family Ties and Spin City star Michael J. Fox is the first to acknowledge his diagnosis with Parkinson's Disease at the age of 29 could have been the undoing of his life in many ways.
He'd been married to fellow actor Tracy Pollan for just two years. The first of their four children, Sam, was just a baby.
"It could have left us undone," the Back to the Future megastar says in his latest book Always Looking Up – The Adventures of an Incurable Optimist.
And if there was ever evidence needed that he is indeed, an optimist, you only need to look at how he has handled the challenge of living with illness.
He and Tracy went on to have three more children (twins Schulyer and Aquinnah, and six years later, Esme) and build one of the most enduring marriages in Hollywood.
An Amazing Life
Michael was faced with a choice, he says. "I could concentrate on the loss – or I could just get on with my life and see if maybe those holes started filling in themselves. Over the last ten years they have, in the most amazing ways.
"For everything the disease has taken something of greater value has been given. It may be one step forward two steps back but I've learned what is important is making that one step count.
"Parkinson's Disease has taken physical strength, spontaneity, physical balance, manual dexterity, the freedom to do the work I want to do when I want to do it, and the confidence that I can always be there for my family when they need me."
Happy Marriage
In a chapter on family life titled 'Why I'm still with Tracy and Shaky When I'm Not' Michael says "Some people ask me the secret of a long and happy marriage, just as they ask me about the key to raising children. My flip answer in the kid department is "love 'em, feed 'em and keep 'em out of traffic." As for marriage, I often reply with equal brevity "Keep the fights clean and the sex dirty."
"Parkinson's is always putting me in a box, and Tracy has become expert at folding back the flaps, tipping it over and easing me out.
"She'd tell you probably with a laugh, that the greatest challenge she faces isn't having a Parkinson's patient for a husband, it's having me for a husband. And by the way, I am a Parkinson's patient."
Gains Greater Than Losses
"The more complicated our marriage has got, the more it seems to bring out the best in us.
"I was a big believer in my own PR: a happy-go-lucky lottery winner who had it all, a great career, a beautiful wife, a healthy son. I was struggling, though, with figuring out how to keep it all going. I was working more than I needed to, worrying more than I liked to admit, and drinking more than anyone should. I was, to put it mildly, not well positioned to deal with what was coming.
"The change that Parkinson's has forced up me and Tracy and the family, pales in comparison with the changes we have brought upon ourselves. We give more to each other than Parkinson's could ever take away".
Tags: Bill Gates, Dr Hilary Tindle, fewer heart attacks, Lance Armstrong ( 2 ), live longer, lower heart risk, Michael J Fox ( 2 ), optimist, University of Pittsburgh research study.research study on optimists
Optimists Enjoy Better Health
It seems life's most public optimists – "Can Do" power houses like Lance Armstrong, Bill Gates and Michael J Fox – are onto something.
New research indicates being an optimist significantly cuts your chances of suffering a heart attack, and even helps you live longer.
In the largest study done on the effect of positive thinking on health, University of Pittsburgh researchers found that compared to pessimists, optimists had a nine per cent lower risk of developing heart disease and a 14 per cent lower risk of dying from any cause.
Bad Times Bring Good
Seven times Tour de France winner Lance Armstrong credits much of his own success on the bike to his battle with testicular cancer.
"I don't think it's a stretch to say that none of, none of my success on the bike would have been possible without that disease," he said. "Life wouldn't have been necessarily empty, but it would not have looked like this." Armstrong also said his optimism is inspired by his mother, who overcame her own set of challenges as a teen mom.
"I give all the credit to my mom," he said. "She's really a survivor. She's as strong and tough as they come and she never looks at anything in a negative light."
Incurable Optimist Michael J Fox
Armstrong's personal example, and the LiveStrong foundation he set up to support cancer survivors, inspired actor Michael J Fox to start his own Fox Team foundation for research into Parkinson's Disease.
In his most recent book Always Looking Up – the Adventures of an Incurable Optimist Fox says that "for everything the disease has taken something of greater value has been given. It may be one step forward two steps, back but I've learned what is important is making that one step count."
Bill Gates – Impatient Optimist
He's in good company. Microsoft founder Bill Gates has characterized himself as an "impatient optimist." And for those that know him, both terms describe him well.
Gates has focused on his philanthropic efforts–which focus on areas where there is great suffering as well as the means to alleviate that suffering through attention and increased resources. But, too often, he says change is not coming quickly enough.
The University of Pittsburgh study – on post-menopausal women – found the positive benefits of being optimistic were independent of income, education, or "health behaviors like [controlling] blood pressure and whether or not you are physically active, or whether or not you drink or smoke," says Dr. Hilary Tindle, lead author of the study. "I was surprised that the relationship was independent of all of these factors."
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What is Calligraphy & Its benefits – Does beautiful handwriting leaves a positive impression on you? Do you know the various benefits that Calligraphy has in store for you? Calligraphy is one of the most beneficial parts of the academics that also help you in the future. If you do not know much about this art of handwriting then do not worry. Here, we will discuss Calligraphy and its advantages.
Calligraphy is the beautiful art in itself. It is more like the drawing with letters than the handwriting. Most of the people actually hire calligraphers and spend a huge amount. But there is no need to do it when you can learn it yourself. There are plenty of advantages of learning this art of handwriting.
Calligraphy is basically the art or talent of forming beautiful symbols by hand and arranging them well. It is also known as the visual art of writing. It's a set of skills and techniques for positioning and inscribing words so they show integrity, harmony, some sort of ancestry, rhythm and leaves a charismatic impression on whoever sees it.
Modern calligraphy ranges from functional inscriptions and designs to fine-art pieces where the letters may or may not be readable. If you are wondering where can this art be used then you should not wonder more. Below mentioned are the places where you can put your skill to bring out the best.
Wedding or any kind of event invitations.
What are the benefits of learning Calligraphy?
Calligraphy in itself is beautiful and good for your soul in a number of ways. It has plenty of benefits to shower to the people who learn it. If you are having double thoughts about learning this beautiful art then do not worry an ounce. We have come up with the major reasons that will compel you to learn this visual art of writing.
We cannot deny the fact that the lifestyle we all have adopted has some major distractions. No one either wants or gets the alone time to relax their mind or brains. Calligraphy is therapeutic i.e. it can have a positive psychological effect on the writer in plenty ways.
After learning this art of handwriting you will become independent in your own ways. People never run out of the functions and events in their life. Therefore, they will always need the calligraphers to impress their guests. You can write your invitations which will eventually save your time & money.
It is an unquestionable fact that calligraphy impresses people to the most extent. Beautiful handwriting and a stunning envelope presentation send a message beyond what you actually write in the note. People will automatically be impressed by your talent and will help in their ways.
Unlike the older days, you can now make a great career in this field. By being a professional calligraphy artist you can make a lot of money and charm people with your art. You can forge a unique business identity or can easily incorporate this into your art to get immense benefits.
The art of calligraphy is different in every part of the world and every type is beautiful equally. You can learn any type depending on your interest and choice. Below listed are some of the types of calligraphy. Scroll down to have a look on them and choose any one of them to get the best results and peace of mind.
You can learn calligraphy at any age without any restrictions. This is one of the most beautiful kind of art that everyone should focus on learning. These benefits have definitely been helpful for you to enroll yourself in the calligraphy training classes. So, I hope this might have brushed up your knowledge about calligraphy and its various perks.
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\section{Introduction}
The photosynthetic conversion of physical energy of sunlight into its
chemical form suitable for cellular processes involves many physical
and chemical mechanisms~\cite{Bio1,Bio2}. Photosynthesis starts with
the absorption of a photon by a light-harvesting pigment forming an
exciton, followed by the transfer of the exciton to the reaction
center, where charge separation is initiated. It is the nature of this
transfer process which is presently in the focus of intense research.
Recent experiments \cite{BioEn2007,Bio5} provided evidence that an
incoherent hopping model seems not to be sufficient to
describe long-lasting beating signals in a two-dimensional Fourier
transform electronic spectrum \cite{Brixner}
recorded from green sulfur bacteria such as {\it Chlorobium tepidum}.
Here, the energy transfer between the main chlorosome antenna and the
reaction centers is mediated by the Fenna-Matthews-Olson (FMO)
protein~\cite{Bio3,Bio4,Rienk} which
contains bacteriochlorophyll (BChl) molecules. The FMO protein is a
trimer made of identical subunits, each of which contains seven BChl
molecules and no carotenoids. Due to its small size, it represents an
important model system for photosynthetic energy transfer and has been
extensively studied experimentally and theoretically. The observed
\cite{BioEn2007} long lived electronic coherence lasted up to time
scales comparable to the time scale of the energy transport. The
experiments were performed at low temperature $T=77$~K and clearly
suggest that the exciton moves coherently through the FMO complex
rather than by incoherent hopping. Similarly, Lee et al.~\cite{Bio5}
found coherent beating signals at low temperatures in a two-color
electronic coherence photon echo experiment. It allows to directly
probe electronic coherences by mixing of the bacteriopheophytin and
accessory bacteriochlorophyll excited states in the reaction center of
the purple bacterium Rhodobacter sphaeroides. These measurements were
performed at 77 K and at 180 K. The coherence beatings in these
measurements can only be explained \cite{Fleming09a} by a strong
correlation between protein-induced fluctuations in the transition
energies of neighboring chromophores, leading to the conclusion that
protein-correlated environments in fact preserve and support
electronic coherence in photosynthetic complexes.
Recently, an ultrafast polarization experiment \cite{Scholes} has
revealed quantum coherent intrachain (but not interchain) electronic
energy transfer in conjugated polymers with different chain
conformations as model multichromophoric systems at room temperature.
The data suggest that chemical donor-acceptor bonds help to correlate
dephasing perturbations. By introducing an angle-resolved coherent
optical wave mixing technique, the quantum beating signals between
coherently coupled electronic transitions in the light-harvesting
complex of purple bacteria are directly observed \cite{Cogdell09}.
Also in the light-harvesting proteins of cryptophyte marine algae,
quantum coherent couplings have been identified by exceptionally
long-lasting excitation oscillations even at ambient temperature
\cite{Scholes10}.
The consequences of these seminal experiments on long-lived
electronic coherences are immediate: the excitation can move rapidly
and reversibly in space, allowing for a very efficient search for an
energetic minimum in photosynthesis. The subsequent trapping of the
excitation, however, must be optimized by properly adjusted
environmental fluctuations. In view of potential applications, the
recent progress helps to understand the design principle of
photosynthetic complexes \cite{Fleming09c} and to exploit the
near-unity efficiency of energy transfer which is believed to result
from the constructive interplay of quantum coherence and slow,
spatially correlated environmental fluctuations. This could open the
door to efficient future artificial light-harvesting complexes finding
applications in optimized organic solar cells. The combination of
optimized exciton trapping \cite{Silbey} with powerful quantum
coherent adaptive control schemes \cite{Motzkus02,Motzkus05,Motzkus08}
could, in addition, allow to exploit quantum effects to direct the
outcome of photochemical processes. Effects beyond the lowest-order
F\"orster treatment, such as the failure of the point-dipole
approximation and the ensuing solvent screening and the
sharing of common bath modes have been considered in a generalized
F\"orster theory \cite{Silbey09}.
Theoretically, photosynthetic energy transfer processes in
light-harvesting complexes are often discussed in simplified
low-dimensional models describing a few individual chromophores
which mutually interact by dipolar couplings and which are
exposed to the fluctuations of the polar solvent molecules and the
protein host
\cite{Bio1,Bio2,SpiBoLe1987,SpiBoWeiss,BioGi2005,BioGi2006,
BioGi2007}.
Two limiting cases are commonly considered: (i) When the dipole
coupling between the chromophores is weak in comparison to the
coupling of the chromophore to environmental fluctuations, the
excitons are considered to be localized at the chromophore sites. The
weak
electronic coupling can then be treated perturbatively, resulting in
an incoherent hopping dynamics described by the standard F\"orster
theory~\cite{Bio6}. (ii) In the opposite limit of weak coupling to the
environmental fluctuations, standard perturbative quantum master
equations are used
\cite{Aspuru08,Aspuru09a,Aspuru09b,Alexandra08,Alexandra09a,
Alexandra09b,Plenio08,Plenio09,Plenio10a,Plenio10b,Plenio10c},
resulting
in a damped coherent dynamics for the exciton transfer. They naturally
are based on the Markovian approximation, which renders the time
evolution
memoryless and allows for a straightforward numerical solution.
These types of master equations are appropriate in the case of a clear
separation of time scales, i.e.,
when the time scale on which the environmental fluctuations occur is
much smaller than that on which the system dynamics evolves.
Formally, this is captured by the requirement that the bath
reorganization energy \cite{SpiBoWeiss} is much larger than the typical
system energy. This condition is typically not fulfilled for the
energy transfer dynamics in biomolecular light-harvesting complexes in
a protein-solvent environment
\cite{BioGi2005,BioGi2006,BioGi2007,Valkunas}. Here, in addition to
the rather slow polarization fluctuations of the polar solvent
molecules, the protein cage acts also as a frequency filter which
particularly shapes the frequency distribution of the environmental
modes. A
similar effect arises for electronic spin qubits in semiconductor
quantum dots \cite{Bio10a} and in donor-based charge qubit crystal
systems
\cite{Bio10b}. Here, the designed geometrical shape of the system
structures the acoustic phonon spectrum and gives rise to strong
non-Markovian effects. In the context of excitonic energy transfer, it
has recently been shown~\cite{Bio7,BioTh2008,Bio8}
that under realistic biomolecular circumstances, time-local master
equations become increasingly unreliable when both time scales become
comparable. In addition to the violation of the Markovian assumption,
the coupling between chromophores and environment cannot be
considered as weak enough to allow for a lowest order perturbative
treatment \cite{BioGi2005,BioGi2006,BioGi2007}. It was recently shown
that the latter condition renders any weak coupling approach
questionable~\cite{Bio8}. Non-Markovian approaches have been employed
beyond a lowest-order treatment
\cite{Bio7} and by coupling each chromophore to a single damped
harmonic mode \cite{Plenio10c}, whose existence, however, was not
further motivated. Alternatively, numerically exact simulations of the
real-time dynamics of quantum coherent energy transfer
under realistic conditions have been carried out
\cite{BioTh2008,Bio8,BioTh2009}
by employing the quasiadiabatic propagator path-integral (QUAPI)
\cite{QUAPI1,QUAPI2,Bio11}.
By this, it has been shown that the rather slow polarization
fluctuations are one possibility to enhance quantum coherence
in the transfer processes. The coupling of two chromophore pairs to
the common slowly fluctuating modes even allows to entangle two
excitonic pairs over surprisingly long times even at room temperature
\cite{BioTh2008}. However, when the fluctuations are fast, no
entanglement is created even when the two pairs couple to the same
modes. The possibility of entanglement and the role of non-Markovian
contributions in biomolecular complexes have also been re-addressed in
recent works \cite{Plenio10a,Plenio10b,Buch10}.
Slow fluctuations have also been treated \cite{Mukamel10} by
different variants of cumulant expansion techniques and by statistical
averaging approaches over static disorder in sum-over-eigenstates
approaches. Recently, the standard Redfield equations, which are valid
in the weak-coupling regime, have been extended by generalizing the
Redfield relaxation tensor on the basis of the Lindblad quantum master
equation \cite{Mukamel09}. This technique goes beyond the secular
approximation and thus can include effects of stronger coupling.
However, the approach is still memoryless and leads to time-local
evolution equations.
Despite the fact that the experimental coherence
beatings~\cite{Fleming09a} could only be explained including strong
correlations between protein-induced fluctuations in the transition
energies of neighboring chromophores, the influence of these
correlations received little attention in the theoretical
investigations. Nazir~\cite{Nazir09} investigated the influence of
correlated fluctuations on a donor-acceptor system for strong system
bath coupling. Correlations in a super-Ohmic bath are found to
suppress the crossover to incoherent dynamics at high temperatures,
which is in line with the common expectation \cite{SpiBoWeiss} that a
super-Ohmic bath naturally provides only weakened influence of the
fluctuations on the system. Similar effects would be expected for
correlations in the environment of the chromophores whose spectrum is
typically assumed to be Ohmic. Fassiolo et al.~\cite{Alexandra09a}
discussed the influence of correlations on the trapping probability in
a ring of chromophores within a Lindblad master equation approach.
They find that the correlations between environmental fluctuations
allows to tune the trapping probability.
From a condensed matter point of view, environmental fluctuations are
normal modes of the bulk material and thus are typically phonons which
propagate through the material. Accordingly they couple to all
chromophores with amplitude differences determined by the phase
differences due to finite times the modes need to propagate from one
chromophore site to the next. However, rattling of side chains of
macromolecules in these highly disordered protein environments might
well be viewed as a localized excitation. In most existing studies it
is assumed that each site of a multichromophoric array is coupled to
its local environment. Including in these approaches spatial
correlations could be achieved by assuming a finite localization
length of the excitations.
In the next section we discuss the influence of spatially correlated
environmental fluctuations on a donor-acceptor model using the
numerical exact quasi-adiabatic path integral propagator approach
which allows us to treat realistic strong couplings and slow
environments. Although being a clear oversimplification to realistic
exciton
transport the donor-acceptor model serves as a toy model to study the
influence of spatial correlations and the difference between
propagating and localized modes on quantum coherence in detail. In the
third section we will discuss how spatial correlations influence two
donor-acceptor pairs which are initially uncoupled. Depending on the
distance $r_{da}$ between donor and acceptor and the distance $r$
between the two donor-acceptor pairs spatially correlated fluctuations
increase or decrease the decay rates of the coherent dynamics. Finally
we discuss and summarize our results.
\section{Spatial environmental correlations in a single chromophore
pair}
\label{single}
\subsection{Model}
The simplest way to model a single chromophore (or pigment) is by
describing it as a quantum two-level system consisting of a ground and
an excited state, which are separated by the energy gap $\epsilon$.
When the electron is in the excited state, it is
localized by its attractive interaction with the hole it left. This
dipole electron-hole configuration forms an exciton. A formal
description can be given
in terms of the Pauli matrix $\tau_z$. Environmental
fluctuations will cause transitions between the ground and the excited
state and will add a fluctuating energy. Experimentally it is known
that the recombination time is of the order of nanoseconds, whereas the
complete energy transfer through the complex is of the order of
picoseconds. Thus the environmental fluctuations causing recombination
are negligible. Describing the fluctuations by harmonic oscillators,
which couple linearly to the chromophore, results in the independent
boson model for a single
chromophore
\begin{equation}
H \,=\, \epsilon\frac{\tau_z}{2} \,- |e\rangle\langle
e|\sum_{\bf k}\lambda_{\bf k}({\bf r}) q_{\bf k} \,+\frac{1}{2}\sum_{\bf k}
\left(p_{\bf k}^2+\omega_{\bf k}^2q_{\bf k}^2 \right)
\end{equation}
where we introduced the position and momentum operators, $q_{\bf k}$ and
$p_{\bf k}$, of the mode with wave vector ${\bf k}$ and its coupling
$\lambda_{\bf k}({\bf r})$ to the chromophore which depends on the amplitude
of the fluctuation at the position ${\bf r}$ of the chromophore. We
explicitely coupled the environmental fluctuations only to the excited
state, $|e\rangle$, which expresses the fact that the electronic
ground state energy is defined by including all vibrational
equilibrium energies. We fixed $\hbar=k_B=1$ which we keep below.
We are not aiming at a complete description of exciton transfer
dynamics in complexes like FMO but are merely interested in the
question how spatially correlated environmental fluctuations influence
the transfer process between two excitonic sites.
For the sake of simplicity, we restrict the model under consideration
to two chromophore sites (acceptor and donor at ${\bf r}_{a/d}$) which
contain a single exciton. We do not consider different site energies
(i.e., $\epsilon_a=\epsilon_d$)
and are thus lead to the donor-acceptor Hamiltonian
\begin{equation}
\label{Hda} H_{\rm da} = \frac{1}{2}\Delta \left\{ |d\rangle\langle
a| \,+|a\rangle\langle d| \right\} \,+ \sum_{i=a/d}|i\rangle\langle i|
\sum_{\bf k}\lambda_{\bf k}({\bf r}_i) q_{\bf k} \,+\frac{1}{2}\sum_{\bf k}
\left(p_{\bf k}^2+\omega_{\bf k}^2q_{\bf k}^2 \right) \,
\end{equation}
of a single chromophore pair.
The state $|d\rangle$ ($|a\rangle$) denotes the exciton to be at the
donor (acceptor) and $\Delta$ is the respective dipole coupling
matrix element \cite{BioGi2005}.\\[3mm]
{\it Comparison of the donor-acceptor model with the spin-boson
model}\\[3mm]
The donor-acceptor Hamiltonian, Eq.~(\ref{Hda}), can easily be
transformed into a Hamiltonian which is closer to the widely
studied spin-boson model~\cite{SpiBoLe1987,SpiBoWeiss}
\begin{eqnarray}
\label{spibo}
H_{\rm da} &=& \Delta\, \frac{\sigma_x}{2} \,+
\frac{\sigma_z}{2} \sum_{\bf k} \left\{\lambda_{\bf k}({\bf r}_2) - \lambda_{\bf k}({\bf
r}_1) \right\} q_{\bf k} \,+ \frac{\bbbone}{2} \sum_{\bf k}
\left\{\lambda_{\bf k}({\bf r}_2) + \lambda_{\bf k}({\bf r}_1) \right\} q_{\bf k} \nonumber \\
& & +\frac{1}{2}\sum_{\bf k} \left(p_{\bf k}^2+\omega_{\bf k}^2q_{\bf k}^2 \right)
\end{eqnarray}
by introducing the Pauli matrices
$\{\bbbone,\sigma_x,\sigma_y,\sigma_z\}$ with
$\sigma_x=|d\rangle\langle a| +|a\rangle\langle d|$ and
$\sigma_z=|d\rangle\langle d| -|a\rangle\langle a|$.
One difference is given by the term proportional to $\lambda_{\bf k}({\bf
r}_1)+\lambda_{\bf k}({\bf r}_2)$ which couples to the identity operator
$\bbbone$ of the donor-acceptor system thus causing fluctuations of
the reference energy of the donor-acceptor system. Accordingly
it is irrelevant for its dynamics.
However, this term modifies the bath modes by shifting their
zero-point energies and thus changes their thermal equilibrium state.
In the spirit of dissipative quantum dynamics, the treatment of
system-bath problems typically rely on the assumption that the bath is
only weakly influenced by the coupling to the system itself and thus
the mentioned effects should not affect the dissipative dynamics
of the donor-acceptor system. However, at strong coupling or for a
slow bath with cut-off frequency $\omega_c\lesssim\Delta$ the validity
of these assumptions is questionable~\cite{Reinhold}. Thus the
dynamics generated by our donor-acceptor Hamiltonian in Eq.\
(\ref{Hda}) differs from the
standard spin-boson problem in three aspects. First, the thermal
equilibrium state to which the total system is driven is
different and, second, the factorized initial conditions for both
cases reflect two different initial conditions. Third, in the
spin-boson model, a single bath is coupled to the system whereas in
the donor-acceptor Hamiltonian, two baths are
coupled to the system states. Thus, even when both baths are
mutually uncorrelated, the resulting rates for the donor-acceptor
system are twice as large compared to the spin-boson model (assuming
that all baths are coupled with equal strength). This has to be taken
into account when comparing results from both approaches.
\subsection{The quasiadiabatic propagator
path integral for the multi-bath case}
The dynamics of the donor-acceptor is characterized by the time
evolution of the reduced density matrix $\rho(t)$, which is obtained
after tracing out the environmental (or bath) degrees of freedom,
i.e.,
\begin{equation}
\rho(t) \,=\, {\rm Tr} \left\{{U(t,0)W(0)U^{-1}(t,0)} \right\}_B \\
\end{equation}
and
\begin{equation}
U(t,0) \,=\, {\cal T} \exp\left\{ -\frac{i}{\hbar} \int_0^t ds H_{da}
\right\} \, .
\end{equation}
Here, $U(t,0)$ denotes the propagator of the full system plus
bath and ${\cal T}$ denotes the time-ordering operator. $W(0)$ is the
total density operator at initial time set at $t=0$. We assume
standard factorizing initial conditions~\cite{SpiBoWeiss}, i.e.
$W(0)\propto\rho(0)\exp(-H_B/T)$, where the bath with the Hamiltonian
$H_B=\frac{1}{2}\sum_k \left(p_{k}^2+\omega_k^2q_{k}^2 \right)$ is at
thermal equilibrium
at temperature $T$ and the system is prepared according
to $\rho(0)$. Throughout this work, we always start with the exciton
at the donor site, i.e., $\rho(0)=|d\rangle\langle d|$.
We calculate $\rho (t)$ using the numerically exact quasiadiabatic
propagator path-integral (QUAPI)~\cite{QUAPI1,QUAPI2,Bio11} scheme.
For details of the iterative technique, we refer to previous works
\cite{QUAPI1,QUAPI2,Bio11}. In brief, the algorithm is based on a
symmetric Trotter splitting
of the short-time propagator ${\cal K}(t_{k + 1}, t_k)$ for the full
Hamiltonian into a part depending on the system Hamiltonian and a part
involving the bath and the coupling term. The short-time propagator
describes time evolution over a Trotter time slice $\delta t$. This
splitting is by construction exact in the limit $\delta t \to 0$ but
introduces a finite Trotter error for a finite time increment,
which has to be eliminated by choosing $\delta t$ small enough such
that convergence is achieved. On the other side, the bath degrees of
freedom generate correlations being non-local in time. For any finite
temperature, these correlations decay exponentially fast at asymptotic
times, thereby setting the associated memory time scale. QUAPI now
defines an object called the reduced
density tensor, which lives on this memory time window and establishes
an iteration scheme in order to extract the time evolution of this
object. Within the memory time window, all correlations are included
exactly over the finite memory time $\tau_{\rm mem} = K \delta t$ and
can safely be neglected for times beyond $\tau_{\rm mem}$.
Then, the memory parameter $K$ has to be increased, until convergence
is found. The two strategies to achieve convergence are naturally
countercurrent, but nevertheless
convergent results can be obtained in a wide range of parameters.
For the purpose of this work, we have to extend the standard
formulation of QUAPI which only includes the coupling to
one bath. The entire influence of a single bath coupled via the
operator $\hat{s}$ to the donor-acceptor system is described in terms of the real-time
path-integral formulation by the influence functional
\begin{equation}\label{InfluenceFunc}
I(\{s_i^+,s_i^-\};\delta t) \,=\,
\exp\left\{ -\frac{1}{\hbar}\sum_{i=0}^N\sum_{i'=0}^i \,
[s_i^+-s^-_i]
\left[\eta_{ii'}s_{i'}^+-\eta_{ii'}^* s^-_{i'}\right] \right\}
\end{equation}
where the path segments $s_i^\pm$ associated to a Trotter time slice
$i$ given as interval
$[(i-\frac{1}{2})\delta t,(i+\frac{1}{2})\delta t]$ (with total time $t=N\delta t$)
are assumed to have constant values over a single time slice. The number
of path segments within a Trotter time slice is given by the dimension
of the Hilbert space in which the system-bath coupling operator lives.
The superscript $\pm$ denotes the propagation direction forward or backward in
time since we work with density operators. The total path integration over
all paths $s^\pm (t')$ has to be performed as the discrete sum over
all configurations $\{s_i^+,s_i^-\}$ of paths segments between initial and final
time. The time-discrete
bath correlators $\eta_{ii'}$ are defined in Ref.~\cite{QUAPI1} and the
superscript $^*$ denotes the complex conjugate.
Multiple independent baths, $H_{B\alpha}$, which couple to system operators
$\hat{s}_\alpha$ will simply cause a product of influence functionals
since each bath acts separately as described above. Thus, the
total influence functional assumes the form
\begin{equation}\label{InfluenceFunc2}
I(\{s_{i,\alpha}^+,s_{i,\alpha}^-\};\delta t) \,=\,
\exp\left\{ -\frac{1}{\hbar}\sum_\alpha\sum_{i=0}^N\sum_{i'=0}^i \,
[s_{i,\alpha}^+-s_{i,\alpha}^-]
\left[\eta_{ii'}^{(\alpha\alpha)}s_{i',\alpha}^+-\eta_{ii'}^{*
(\alpha\alpha)} s_{i',\alpha}^-\right] \right\}
\end{equation}
Here we denoted the bath correlators $\eta_{ii'}^{(\alpha\alpha)}$
with the additional superscripts since for differing baths the
correlators will differ.
The question whether the environmental fluctuations act {\it locally}
or in a correlated manner can be tackled by the following extension.
Local fluctuations couple to the donor and to the acceptor separately and
independently. This implies that
Eq.\ (\ref{InfluenceFunc2}) describes {\em all\/} effects due to
environmental fluctuations. If, however, the fluctuations are caused
by extended waves, like phonon modes or if the fluctuations rattling the
donor can at least partially still be felt at the acceptor site, then the
fluctuations at the various sites are no longer independent and spatial
correlations have to be taken into account.
Hence, Eq.\ (\ref{InfluenceFunc2}) has to be generalized to
\begin{equation}\label{InfluenceFunc3}
I(\{s_{i,\alpha}^+,s_{i,\beta}^-\};\delta t) \,=\,
\exp\left\{ -\frac{1}{\hbar}\sum_{\alpha,\beta}\sum_{i=0}^N\sum_{i'=0}^i
\, [s_{i,\alpha}^+-s_{i,\alpha}^-]
\left[\eta_{ii'}^{(\alpha\beta)}s_{i',\beta}^+-\eta_{ii'}^{*
(\alpha\beta)} s_{i',\beta}^-\right] \right\}
\end{equation}
where $\eta_{ii'}^{(\alpha\beta)}$ are the mixed bath correlators
expressing the correlations of the fluctuations acting at operator
$\hat{s}_\alpha$ and $\hat{s}_\beta$.
The detailed numerical evaluation of the influence functional in the
extended QUAPI scheme becomes more involved,
but the general procedure is not affected by this extension.
\subsection{Correlated environmental fluctuations at different sites}
In a crystal, environmental fluctuations acting, e.g., on electrons,
are generated by vibrations of the lattice atoms and are the well-known
phonons. Phonons are also present in disordered
media (condensed, soft or fluid). In the sense of propagating modes
of the host material which evolve with time through the medium,
they are commonly limited to the low energy sector or, more specifically, to
energies associated to wave lengths on which the material appears
homogeneous. Once the wave length becomes smaller than the disorder
length
scale,
the modes can generally be thought of as localized fluctuations. However,
their localization length (or radius) is still connected to the wave
length of the mode and accordingly even a {\it localized} mode extends
over some finite spatial range. The same picture holds for dipolar
fluctuations in solvents and vibrations of charged
macromolecular side chains forming
the bio-environment of light harvesting complexes.
\begin{figure}[t]
\begin{center}
\epsfig{file=fig1.eps,width=8cm}
\end{center}
\caption{\label{fig1} Donor and acceptor with two environmental field
with different wave length are illustrated. The donor holds an
electron in the excited state which forms with the hole left in the
ground state an exciton having a finite
electric dipole moment.}
\end{figure}
Excitons, as coupled electron hole pairs, have an electric dipole moment
$\vec\mu$ which couples to the electric field $\vec{E}({\bf r})$ at the
exciton position ${\bf r}$ generated by the environmental dipolar
fluctuations (as illustrated in Fig.~\ref{fig1}). This results in the
interaction energy $H_{SB}=\vec\mu\cdot \vec{E}({\bf r})$.
For simplicity, we ignore the angular dependence in the following which
only cause correction factors of the order of one~\cite{Peter}. Within
the donor-acceptor model the exciton dipole moment is described
by $\mu=|\vec\mu|=\mu_0|i\rangle\langle i|$ where we furthermore have assumed
that the dipole moments at each chromophore site are the same (again
neglecting factors of the order one due to angular dependencies).
The electric field is proportional to the amplitude of the propagating
normal modes of the medium, $E({\bf r})={\rm sgn}\{\vec{E}({\bf
r})\}\cdot |\vec{E}({\bf r})|=E_0(1/\sqrt{N})\sum_{\bf k} q_{\bf k} e^{i
{\bf kr}}$ finally leading to the interaction Hamiltonian in
Eq.~(\ref{Hda}) with $\lambda_{\bf k}({\bf r})=(\mu_0E_0/\sqrt{N})e^{i
{\bf kr}}$.
Similar ideas have recently been used to understand the phonon influence
on double quantum dot charge qubits~\cite{Bio10a,Bio10b} or tunneling
defects~\cite{Peter}.
For propagating modes in three spatial dimensions the spectral function
of intersite fluctuations between chromophores $i$ and $j$ becomes
\begin{eqnarray} J_{ij}(\omega) &=& \sum_{\bf k} \frac{\lambda_{\bf k} ({\bf r}_i)
\lambda_{-{{\bf k}}}({\bf r}_j) }{2\omega_{\bf
k}}\delta(\omega-\omega_{\bf k})
\,=\, 2\alpha\omega e^{-\omega / \omega_c} \frac{\sin(\omega
t_0)}{\omega t_0} \nonumber \\
& & \quad\mbox{with}\quad t_0 \,=\, \frac{r_{ij}}{v}
\end{eqnarray}
with the sound velocity $v$ (assuming linear dispersion $\omega_{\bf
k}=vk$ and $k=|{\bf k}|$), the distance $r_{ij}=|{\bf r}_i-{\bf r}_j|$
between site $i$ and $j$, coupling strength $\alpha$ and upper cut-off
$\omega_c$ using an exponential form for the cut-off function. For the
small cut-off frequencies typical for biomolecular environments
($\omega_c\simeq\Delta$) the cut-off function will modify quantitatively
but not qualitatively the results. However, no detailed information
about
the specific cut-off functions for biomolecular environments is
available in the literature. The (on-site) spectrum is Ohmic (linear in
$\omega$) for the FMO complex~\cite{BioCh2005,BioGi2007}.
Linear dispersion for the normal modes is a strong assumption and it is
not clear if the simple Debye picture holds in biological soft matter up
to energies $\Delta$.
For on-site fluctuations at site $i$ the spectral function simplifies
\begin{equation} J_{ii}(\omega) \,=\, \sum_{\bf k} \frac{|\lambda_{\bf k}({\bf
r}_i)|^2}{2\omega_{\bf k}}\delta(\omega-\omega_{\bf k})
\,=\, 2\alpha\omega e^{-\omega / \omega_c} \, . \label{onsite}
\end{equation}
Alternatively one might consider localized environmental fluctuations
with localization length $\xi$, which is taken to be
independent of the mode energy $\omega$, resulting in the spectral
density
\begin{equation} J_{ij}^{\rm loc}(\omega)
\,=\, 2\alpha\omega e^{-\omega / \omega_c} e^{-r_{ij}/\xi}
\end{equation}
for the intersite spectrum whereas the on-site spectrum,
Eq.\ (\ref{onsite}), is unaltered.
The intersite spectrum $J_{ij}^{\rm loc}(\omega)$ vanishes for sites far
apart $r_{ij}\gg\xi$ and the fluctuations at the donor and the acceptor
sites are
uncorrelated. The intersite spectrum becomes identical to the on-site
spectrum for close sites $r_{ij}\ll\xi$. In the later case, the
environmental fluctuations of both sites are fully correlated and thus
actually identical. As illustrated in Fig.~\ref{fig1} by the blue line,
both excitons then ``see'' the same electric field, which, in turn,
only modifies the total
energy but not the energy difference between donor and acceptor.
Accordingly, these fully correlated fluctuations cannot influence the
dynamics of the donor-acceptor system.
\begin{figure}[t]
\begin{center}
\epsfig{file= fig2.eps,width=8cm}
\end{center}
\caption{\label{fig2} Occupation difference of donor and acceptor versus
time and speed of sound for $\omega_c=\Delta=106$~cm$^{-1}$, $T=152$~K,
$\alpha=0.08$ and $r_{da}=3.8$~\AA~assuming propagating environmental
modes.}
\end{figure}
For the case of propagating modes, qualitatively the same holds.
When $\Delta t_0\gg 1$ all modes with $\omega\ge\Delta$ will not
contribute to the intersite spectral function. At least at weak coupling
($\alpha \ll 1$),
mainly the modes resonant with the tunneling splitting are relevant and
thus we expect the intersite spectrum to be irrelevant. The environment
acts as two independent baths at each chromophore site,
as usually assumed in the literature.
When the shortest wave length $\lambda_c$ in the spectrum
is larger than the distance between donor and acceptor,
we have that $\omega_ct_0=(r_{ij}/\lambda_c)\ll 1$.
Then, the environmental fluctuations are fully
correlated between the sites and thus
do not affect the donor-acceptor system.
For the donor-acceptor system we can transform the Hamiltonian as
discussed in Eq.~(\ref{spibo}) and then define a single effective
spectral function (as for a spin-boson
problem~\cite{SpiBoWeiss,SpiBoLe1987}) resulting in
\begin{equation} J_{\rm eff}(\omega) \,=\, 4\alpha\omega e^{-\omega / \omega_c}
\left( 1- \frac{\sin(\omega t_0)}{\omega t_0} \right) \, .
\end{equation}
In the limit $\omega_c t_0\ll 1$, we get
\begin{equation} J_{\rm eff}(\omega) \,\simeq\, 4\alpha t_0^2\omega^3 e^{-\omega /
\omega_c}
\end{equation}
which is of super-Ohmic form. Super-Ohmic environmental fluctuations can
neither cause overdamping (except at large temperatures) nor localization,
in clear qualitative contrast to pure Ohmic fluctuations.
This is a drastic qualitative effect which spatially
correlated environmental fluctuations cause
on coherent exciton transfer.
\subsection{Dynamics of a single transfer step}
Having determined the reduced density matrix by QUAPI, we can evaluate the
occupation difference $P(t)=\langle \sigma_z \rangle$ between donor and
acceptor.
In Fig.~\ref{fig2}, $P(t)$ is plotted over time versus sound velocity
$v$ with which the modes are assumed to propagate.
We have chosen all parameter to match rather closely the properties of
chromophores in the FMO complex~\cite{BioCh2005}. We have used
$\Delta=106$~cm$^{-1}$ as tunneling element
which corresponds to the largest coupling in
the FMO complex~\cite{BioCh2005} between chromophore $1$ and $2$ but
have neglected the energy difference between the two sites. The distance
between site $1$ and $2$ in the FMO complex of {\it Chlorobium
tepidum}~\cite{Bio4} is $r_{12}=3.8$~\AA~which are the closest two
chromophores. Site $2$ and $7$ are maximally apart, $r_{27}=11.3$~\AA.
The bath cut-off frequency varies in the
literature~\cite{BioCh2005,BioAd2006} between $\omega_c=32$~cm$^{-1}$
and $150$~cm$^{-1}$. To be specific, we choose $\omega_c=106$~cm$^{-1}$ and
temperature $T=152$~K$=\Delta/k_B$.
We find that quantum coherent oscillations occur which decay within about
1~ps for the smallest value of the sound velocity of several hundred m/s. The
environmental fluctuations are uncorrelated in this case.
For larger sound velocities,
meaning increasing correlations of the fluctuations, the decay slows down
considerably as expected since the wave length of the modes causing
decoherence becomes larger than the distance between the chromophores
and thus cannot harm coherence any longer. We are not aware of
experimental data regarding the precise values of sound velocities for the biological
embedding materials of the FMO complexes. As a guide we might use
the sound velocity of water, $v\simeq 1500$~m/s, which falls into the
range of our plot. When frozen to ice, as in the low temperature
experiments at $77$~K or $180$~K, one finds that $v\simeq
3150$~m/s and coherence lives considerably longer.
In order to elucidate the dependence of this effect on the system-bath coupling,
Fig.\ \ref{fig4} a) shows the result for $\alpha=0.2$, resulting in a
reorganization energy $\lambda\simeq 2\alpha\omega_c=42.5$~cm$^{-1}$.
This
compares to the case shown in Fig.\ \ref{fig2}, where
we have set $\alpha=0.08$, resulting in a
reorganization energy $\lambda\simeq 2\alpha\omega_c=17$~cm$^{-1}$.
The
stronger coupling results in faster decoherence and the crossover to
fully correlated environmental fluctuations causing long-time
coherence is pushed to larger values of the sound velocity.
To address the temperature dependence, we show in
Fig.\ \ref{fig4} b)
and c) the results for $P(t)$ for the same parameters as
in Fig.\ \ref{fig2}, except that
temperature is set to $T=76$~K in b) and $T=304$~K in c). At lower
temperatures coherence expectedly survives longer whereas
at higher temperatures coherence lives shorter.
Nevertheless, for all temperatures the
profound effect due to the finite sound velocity is present.
This turnover between uncorrelated (strong decoherence) and strongly
correlated (weak decoherence) environmental fluctuations thus is only
weakly dependent on temperature.
\begin{figure}[t]
\begin{center}
\epsfig{file=fig3.eps,width=16cm}
\end{center}
\caption{\label{fig4}
Occupation difference of donor and acceptor
versus time and speed of sound assuming propagating environmental
modes. Parameters are chosen to be $\omega_c=\Delta=106$~cm$^{-1}$
and $r_{da}=3.8$~\AA, and in a) $T=152$~K, $\alpha=0.2$, in b)
$T=76$~K, $\alpha=0.08$, and in c) $T=304$~K, $\alpha=0.08$.}
\end{figure}
Next, we discuss the dependence of the crossover on the localization
length $\xi$. Fig.~\ref{fig3} shows $P(t)$ for localized modes with localization
lengths between $0$ - $2$~nm. Again as expected, for small localization
lengths, the fluctuations at each site are uncorrelated and the
occupation difference decays in less than a picosecond. Assuming a
distance between donor and acceptor of $r_{12}=3.8$~\AA, the
localization lengths plotted in Fig.~\ref{fig3}, reach up to about
four times the donor-acceptor distance.
Then sizable correlations are expected and coherent
oscillations for more than 2 picoseconds occur.
\begin{figure}[t]
\begin{center}
\epsfig{file=fig4.eps,width=8cm}
\end{center}
\caption{\label{fig3} Occupation difference of donor and acceptor
versus time and localization length $\xi$ of localized environmental modes.
Parameters are chosen as $\omega_c=\Delta=106$~cm$^{-1}$,
$T=152$~K, $\alpha=0.08$ and $r_{da}=3.8$~\AA.}
\end{figure}
In conclusion, assuming localized or propagating modes results
qualitatively in the same behavior. The used parameters are all taken
for the case of the FMO complex and thus our results strongly suggest that
spatial correlations of the environmental fluctuations due
to finite propagation time of the modes strongly
influence the decay of coherence in exciton transfer processes. In
order to judge on the quantitative effect, a comprehensive experimental
investigation of the environmental modes is needed, in particular,
whether the modes are propagating or localized, and accordingly,
whether the sound velocity and/or the localization
length is relevant.
So far, we have discussed the influence of spatially correlated
environmental fluctuations on the coherence of a {\em single\/}
donor-acceptor pair.
These results can in principle be extended to a chain of
more chromophoric sites without changing the
physical picture qualitatively.
\section{Transfer in two chromophore pairs}
\subsection{Model}
The FMO complex consists of three identical subunits, each of which
consists of seven chromophoric sites and acts as a conductor for the
excitons. Most likely, this structure has been optimized with respect
to efficiency and seems to contain also some redundancy, which
might be a measure of reliability in
nature. In any case, the complex structure gives raise to the question
whether a crosstalk between the subunits exists, and this even in
a quantum coherent manner. In turn, the question whether this serves
any purpose for functionality of increased efficiency is reasonable.
In order to approach this question on a qualitative level, simple
low-dimensional effective models are necessary. We model a single
subunit by one
donor-acceptor pair and discuss in the following two such
donor-acceptor pairs located at a distance $r$. We assume that each
pair initially contains a single exciton at the respective donor site.
We explicitly suppress exciton transfer from one pair to the other and
start from the Hamiltonian
\begin{eqnarray}
\label{Hpda}
H_{\rm pda} &=& \frac{1}{2}\Delta \sum_{j=1}^2 \left\{
|d_j\rangle\langle a_j| \,+|a_j\rangle\langle d_j| \right\} \,+
\sum_{j=1}^2\sum_{i=a_j/d_j}|i\rangle\langle i| \sum_{\bf k}\lambda_{\bf k}({\bf
r}_i) q_{\bf k} \nonumber \\
& & +\frac{1}{2}\sum_{\bf k} \left(p_{\bf k}^2+\omega_{\bf k}^2
q_{\bf k}^2 \right) \, .
\end{eqnarray}
We assume in the following that
each chromophore couples separately to the environmental fluctuations
but in contrast to the previous section, here there are two distinct
distances involved. As before, each donor is separated by a distance
$r_{da}=|{\bf r}_{a_1}-{\bf r}_{d_1}|=|{\bf r}_{a_2}-{\bf r}_{d_2}|$
from its acceptor. We assume this distance to be identical
for both donor-acceptor pairs. The separation between both pairs is
$r=|{\bf r}_{a_1}-{\bf r}_{a_2}|=|{\bf r}_{d_1}-{\bf r}_{d_2}|$.
\subsection{Results}
\begin{figure}[t]
\begin{center}
\epsfig{file= fig5.eps,width=12cm}
\end{center}
\caption{\label{fig8}Probability $P_{d_1,d_2}(t)$ of both exciton
being at the
respective donor sites versus time and sound velocity assuming
propagating environmental modes. Parameters are chosen to be
$\omega_c=\Delta=106$~cm$^{-1}$, $T=15.2$~K, and $\alpha=0.04$.
The distances are for a) $r_{da}=3.8$~\AA~and $r=38$~\AA~, and for b) $r_{da}=38$~\AA~and $r=3.8$~\AA.}
\end{figure}
Since we have seen that propagating and localized (with a finite
localization length)
modes cause similar results for the decay of coherence, we restrict
the following investigation to propagating modes. As before we set the
tunneling element $\Delta=106$~cm$^{-1}$ and the
fluctuation cut-off frequency $\omega_c=106$~cm$^{-1}$.
Fig.~\ref{fig8} shows the probability $P_{d_1,d_2}(t)$ that both
excitons are located
at the donor sites versus time for various values of the sound
velocity for a weak coupling $\alpha=0.04$. Note that we start from
$P_{d_1,d_2}(t=0)=1$. The reorganization energy is then
$\lambda\simeq 2\alpha\omega_c=8.5$~cm$^{-1}$ and a rather low
temperature $T=15.2$~K is chosen. In Fig.~\ref{fig8} a), we fix the
distances $r_{da}=3.8$~\AA~and $r=38$~\AA~which reflects two rather
distant donor-acceptor pairs. Both donor-acceptor pairs should thus
be independent of each other since no direct coupling is assumed. We
find a qualitatively similar behavior as for the case of a single pair
as described in Section \ref{single}. Differences arise due to the
weaker coupling and lower temperature. In contrast to the former
case,
we show the results for two close-by donor-acceptor pairs,
$r=3.8$~\AA~but with large donor-acceptor distance $r_{da}=38$~\AA~
in Fig.~\ref{fig8} b). Here, we find that the effect of decoherence
increases with increasing sound velocity, in clear contrast
to the previous discussion in Section \ref{single}. Hence, increasing
spatial correlations between the two donor-acceptor pairs
destroy quantum coherence.
For a quantitative investigation of this observation, we determine
the associated decoherence rate $\Gamma$ by fitting an exponentially
damped
cosine to the data for $P_{d_1,d_2}(t)$ shown in Fig.~\ref{fig8}
and plot it in Fig.~\ref{fig6} versus the sound velocity for several
distance ratios.
As fundamental distance scale, we use $r_0=3.8$~\AA, the distance of
chromophore $1$ and $2$ in the FMO complex of {\it Chlorobium
tepidum}~\cite{Bio4}. At the sound velocity $v=7600$~m/s, a mode can
travel this distance within the time of $\Delta^{-1}=50$~fs.
When both donor-acceptor pairs are far apart, $r=10\,r_0$, and the
distance between donor and acceptor is $r_{da}=r_0$ (shown by the blue
up-triangle in Fig.~\ref{fig6}), we recover the result of the previous
Section \ref{single}. The decoherence rate decreases with increasing
sound velocity. When the donor-acceptor distance is also large,
$r_{da}=10\,r_0$, there is no dependence of $\Gamma$ on $v$
for the investigated range of sound velocities (yellow left triangles
in Fig.~\ref{fig6}). The spatial correlations of the fluctuations
simply do not extend from the donor to the acceptor site and thus the
fluctuations are uncorrelated. When the two pairs are close to each
other and the donor and acceptor sites are also close, the
decoherence rate $\Gamma$ decreases with increasing sound velocity
(black circles in Fig.~\ref{fig6}). A totally different case is
reached when donor and acceptor sites are well separated,
$r_{da}=10\,r_0$ and $r_{da}=100\,r_0$ (red squares and green diamonds
in Fig.~\ref{fig6}), but the two pairs are close, $r=r_0$. With
increasing sound velocity, the decoherence rate increases and
approximately doubles. The effect is slightly larger with larger
distance between donor and acceptor.
With increasing sound velocity, the wave lengths of the propagating
modes increase and thus spatial correlations of these modes reach
further. When the fluctuations at the sites of both donor-acceptor
pairs are fully correlated, which is approximately the case for sound
velocities of $v\approx 7600$~m/s for $r=r_0$, then the fluctuations
of the second donor (acceptor) equally influence the first one. Thus,
the influence of the
fluctuations effectively doubles which is reflected in doubling the
decoherence rate. When donor and acceptors are closer, they as well
become more correlated with increasing sound velocity, which, as
discussed in the Section \ref{single}, results in decreasing
decoherence rates. Thus, two effects are competing here. This also
explains why for $r_{da}=10\,r_0$ (red squares in Fig.~\ref{fig6}) the
increase of the rate $\Gamma$ is weaker than for $r_{da}=100\,r_0$
(green diamonds in Fig.~\ref{fig6}). When both distances are equal,
$r_{da}=r=r_0$ (black circles), the suppression of decoherence due to
spatial correlations between donor and acceptor sites is the dominant
effect, but it is weakened in comparison to the case of $r_{da}=r_0$
and $r_{da}=10\,r_0$ (blue up triangles). Hence, we find two regimes,
in which either one of the two effects dominates.
How sharp the crossover between the two regimes is, becomes visible when looking at the data for $r_{da}=2\,r_0$ and $r=r_0$. Here again, the decoherence rate $\Gamma$ is increased by spatial correlations. Similar behavior (not shown) is also found for a smaller fluctuation cut-off frequency $\omega_c=53$~cm$^{-1}$ and otherwise identical parameters. The inset of Fig.~\ref{fig6} shows the change of the decoherence rate $\Gamma$ when changing the donor-acceptor distance $r_{da}$ from $r_0$ to $2\,r_0$ for a fixed sound velocity
$v=4560$~m/s. We find that in this regime, the change is linear in $r_{da}$.
Qualitatively the same happens for very large fluctuation cut-off frequencies $\omega_c=1060$~cm$^{-1}$
(as shown in Fig.~\ref{fig7}a for otherwise identical parameters). Note that in this regime, non-Markovian effects are suppressed.
The increase (decrease) of $\Gamma$ for correlated donor-acceptor pairs (for correlated fluctuations at donor and acceptor site)
becomes sharper.
\begin{figure}[t]
\begin{center}
\epsfig{file= fig6.eps,width=9cm}
\end{center}
\caption{\label{fig6} Decoherence rate $\Gamma$ associated to the occupation probability $P_{d_1,d_2}(t)$ versus sound velocity $v$. Parameters are chosen as $\Delta=106$~cm$^{-1}$, $T=15.2$~K, $\alpha=0.04$ and $\omega_c=106$~cm$^{-1}$. Inset: $\Gamma$ versus
the ratio $r_{da}/r_0$ for $v=4560$~m/s in the range $r_0\le r_{da}\le 2r_0$.}
\end{figure}
The influence of temperature is studied in Fig.~\ref{fig7}b), which shows the decoherence rate $\Gamma$ for
a higher temperature, $T=152$~K, but otherwise for the same parameters as in Fig.~\ref{fig6}.
For small donor-acceptor distance, $r_{da}=r_0$, we find as before that the decoherence rate decreases with increasing sound velocity.
However, the picture changes for large donor-acceptor distance, $r_{da}=10r_0$. Then, the effect of decoherence is almost independent
of the sound velocity, irrespective of the distance $r$ between the donor-acceptor pairs.
\begin{figure}[t]
\begin{center}
\epsfig{file= fig7.eps,width=15cm}
\end{center}
\caption{\label{fig7} Decoherence rate $\Gamma$ associated to the occupation probability $P_{d_1,d_2}(t)$ versus sound velocity $v$. Parameters are $\Delta=106$~cm$^{-1}$, $\alpha=0.04$, and in a) $T=15.2$~K, $\omega_c=1060$~cm$^{-1}$, and in b) $T=152$~K, $\omega_c=106$~cm$^{-1}$.}
\end{figure}
\section{Discussion and conclusions}
To summarize, we have investigated the effect of spatially correlated
environmental fluctuations on the quantum coherent transfer dynamics of
excitons in donor-acceptor systems.
Technically, spatially correlated environmental fluctuations can be
included by adopting the numerical quasiadiabatic propagator path
integral scheme. The fluctuations at each chromophore
site generate a separate term in the Feynman-Vernon
influence phase. The spatial correlations
generate additional terms which then describe
bath modes at different positions. Nevertheless, only the
Feynman-Vernon influence phase is modified and the general QUAPI
approach still remains feasible.
When the spatial correlations of the environmental fluctuations extend
from the donor to the acceptor site, their decohering influence is strongly
reduced since then, the energies at both sites are identically modified.
The energy difference between donor and acceptor is not changed but
only the global reference energy fluctuates which does not influence the
dynamics. The spatial correlations of propagating modes are characterized
by their wave length which itself is determined by the sound velocity
assuming linear dispersion. In contrast, the spatial
correlations of localized modes are determined by
their localization length. Although these two cases are in
principle different, the qualitative effect on
quantum coherence is the same and depends only on the ratio of distance
between donor and acceptor and the correlation length given either by
the localization length or by the wave length of resonant modes.
Two donor-acceptor systems in close proximity in addition show an
increase in the decoherence rate in dependence on their spatial
distance. When the donor
and acceptor sites are far apart and their fluctuations are only weakly
correlated, correlations of close-by donor-acceptor pairs become relevant
when the distance between the two donor-acceptor systems becomes
small enough that their
environmental fluctuations are correlated. Then, each donor-acceptor
system ``sees'' the fluctuations at the site of the other one and
thus the decoherence rate is doubled. For
intermediate spatial distances, both effects are competing with each other.
The effect of an increased decoherence rate, however, is suppressed at
higher temperatures and is probably less relevant at room temperature.
Our results show that quantum coherence in the excitation transfer crucially
depends on spatial correlations in the environmental fluctuations as
soon as correlations lengths are of the order of the spatial distances of
the chromophores. We have shown that for realistic material parameters (in particular,
sound velocities), noticeable influence of the finite propagation times of
environmental modes occurs. In fact, correlated fluctuations can
reduce the decohering effect of the chromophore environment,
when the correlation range extends over typical
transfer distances within a chromophore chain. This effect even survives
(even though diminished) up to room temperature and
thus might be relevant for exciton transfer in biological systems. These
findings are in line with the experimental findings~\cite{Bio5,Fleming09a}.
At the same time, correlations between different chromophore chains
increase the decoherence rate and thus a close packing would be
disadvantageous. This increase is, however, suppressed at room temperature.
Thus, increased thermal fluctuations actually indirectly support quantum
coherence since they reduce spatial correlations which would increase
decoherence. This effect could be tested experimentally
by performing the reported electronic
coherence photon echo experiments at even lower temperature
at a full FMO complex with all three subunits.
At low temperature, spatial correlations between the
FMO subunits should reduce quantum coherence
more effectively than at higher temperatures.
\ack This work was supported by the Excellence Initiative of the
German Federal and State Governments.
\section*{References}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,884
|
\section{Introduction}
The unprecedented demands for high quality and ubiquitous wireless services impose enormous challenges to existing cellular networks. Applications like rate-centric enhanced mobile broadband (eMBB), ultra-reliable, low latency communications (URLLC), and massive machine-type communications (mMTC) services are the targets for designing the fifth-generation (5G) of communication systems. However, the goals of the sixth-generation (6G) of wireless communication systems are expected to be transformative and revolutionary encompassing applications like data-driven, instantaneous, ultra-massive, and ubiquitous wireless connectivity, as well as connected intelligence~\cite{letaief2019roadmap,saad2019vision}. Therefore, new transmission technologies are needed in order to support these new applications and services. Reconfigurable intelligent surfaces (RISs), also called intelligent reflecting surfaces (IRSs)~\cite{Qingqing2020Towards,cheng2020downlink} or large intelligent surfaces (LISs)~\cite{hou2019mimo,liang2019large}, comprise an array of reflecting elements for reconfiguring the incident signals. Owing to their capability of proactively modifying the wireless communication environment, RISs have become a focal point of research in wireless communications to mitigate a wide range of challenges encountered in diverse wireless networks~\cite{di2020smart,Yang_114}. The advantages of RISs are listed as follows:
\begin{itemize}
\item \textbf{Easy to deploy:} RISs are nearly-passive devices, made of electromagnetic (EM) material. As illustrated in Fig.~\ref{fourparts}, RISs can be deployed on several structures, including but not limited to building facades, indoor walls~\cite{perovic2019channel}, aerial platforms, roadside billboards, highway polls, vehicle windows, as well as pedestrians' clothes due to their low-cost.
\item \textbf{Spectral efficiency enhancement:} RISs are capable of reconfiguring the wireless propagation environment by compensating for the power loss over long distances. Virtual line-of-sight (LoS) links between base stations (BSs) and mobile users can be formed via passively reflecting the impinging radio signals. The throughput enhancement becomes significant when the LoS link between BSs and users is blocked by obstacles, e.g., high-rise buildings. Due to the intelligent deployment and design of RISs, a software-defined wireless environment may be constructed, which, in turn, provides potential enhancements of the received signal-to-interference-plus-noise ratio (SINR).
\item \textbf{Environment friendly:} In contrast to conventional relaying systems, e.g., amplify-and-forward (AF) and decode-and-forward (DF)~\cite{huang2019holographic}, RISs are capable of shaping the incoming signal by controlling the phase shift of each reflecting element instead of employing a power amplifier~\cite{ntontin2019reconfigurable,Bjrnson2019}. Thus, deploying RISs is more energy-efficient and environment friendly than conventional AF and DF systems.
\item \textbf{Compatibility:} RISs support full-duplex (FD) and full-band transmission due to the fact that they only reflect the EM waves. Additionally, RIS-enhanced wireless networks are compatible with the standards and hardware of existing wireless networks~\cite{zhou2020spectral}.
\end{itemize}
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=18cm]{fourparts.pdf}
\caption{RISs in wireless communication networks.}
\label{fourparts}
\end{center}
\end{figure*}
Due to the aforementioned attractive characteristics, RISs are recognized as an effective solution for mitigating a wide range of challenges in commercial and civilian applications. There have been many recent studies on RISs and their contributions focus on several application scenarios under different assumptions. As a result, the system models proposed by these research contributions tend to be different. Thus, there is an urgent need to categorize the existing research contributions, which is one of the main goals of this paper.
Fig.~\ref{fourparts} illustrates the applications of RISs in diverse wireless communication networks. In Fig.~\ref{fourparts}(a), RIS-enhanced cellular networks are illustrated, where RISs are deployed for bypassing the obstacles between BSs and users. Thus, the quality of service (QoS) in heterogeneous networks and the latency performance in mobile edge computing (MEC) networks are improved~\cite{cao2019intelligent,bai2019latency}. On the other hand, RISs can act as a signal reflection hub to support massive connectivity via interference mitigation in device-to-device (D2D) communication networks~\cite{cao2020sum}, or RISs can cancel undesired signals by smartly designing the passive beamforming in the context of physical layer security (PLS)~\cite{YangOn2020}. Additionally, RISs can be deployed to strengthen the received signal power of cell-edge users and mitigating the interference from neighbor cells~\cite{pan2019Intelligent}, and the power loss over long distances can be compensated in simultaneous wireless information and power transfer (SWIPT) networks~\cite{Wu_SWIPT_letter,Tang_SWIPT,Pan_SWIPT,Wu_SWIPT}. In Fig.~\ref{fourparts}(b), RIS-assisted indoor communications are illustrated, where RISs can be deployed on walls for enhancing the QoS in some rate-hungry indoor scenarios, such as virtual reality (VR) applications. Additionally, in order to guarantee no blind spots in the coverage area of some block-sensitive scenarios, such as visible light communications~\cite{wang2020performance} and wireless fidelity (WiFi) networks, a concatenated virtual RIS-aided LoS link between the access points (APs) and the users can be formed with the aid of RISs, which indicates that both the propagation links between the APs and the RISs, as well as between the RISs and the users can be in LoS. In Fig.~\ref{fourparts}(c), RIS-enhanced unmanned systems are illustrated. RISs can be leveraged for enhancing the performance of unmanned aerial vehicle (UAV) enabled wireless networks~\cite{li2019reconfigurable}, cellular-connected UAV networks~\cite{ma2019enhancing}, autonomous vehicular networks, autonomous underwater vehicle (AUV) networks, and intelligent robotic networks by fully reaping the aforementioned RIS benefits. For instance, in RIS-enhanced UAV-aided wireless networks, one can adjust the phase shifts of RISs instead of controlling the movement of the UAVs in order to form concatenated virtual LoS links between the UAVs and the users. Therefore, the UAVs can maintain the hovering status only when the concatenated virtual LoS links cannot be formed even with the aid of RISs, which reduces the movement manipulations and the energy consumption of UAVs. In Fig.~\ref{fourparts}(d), RIS-enhanced Internet of Things (IoT) networks are illustrated, where RISs are exploited for assisting intelligent wireless sensor networks~\cite{RIS_fading_to_gamma}, intelligent agriculture, and intelligent factory\cite{Mu_robot}.
There are some short magazine papers~\cite{Liaskos,Qingqing2020Towards,huang2019holographic,gacanin2020wireless}, surveys and tutorials~\cite{di2019smart,liang2019large,basar2019wireless,Gong2020Survey,di2020smart,Wu2020Tutorial} in the literature that introduced RISs and their variants, but the focus of these papers is different from our work. More specifically, Wu~\emph{et al.}~\cite{Qingqing2020Towards} provided an overview of the applications of RISs as reflectors in wireless communications, and identified some challenges and future research opportunities for implementing RIS-assisted wireless networks. Liang~\emph{et al.}~\cite{liang2019large} presented an overview of the reflective radio technology with a particular focus on the large intelligent surface/antennas. In~\cite{di2020smart}, Di Renzo~\emph{et al.} provided a comprehensive overview of employing RISs for realizing smart radio environments in wireless networks, where an electromagnetic-based communication-theoretic framework for analyzing and optimizing metamaterial-based RISs is presented and a survey of recent research contributions on RISs is given. Huang~\emph{et al.}~\cite{huang2019holographic} introduced the concept of holographic multiple-input and multiple-output (MIMO) surfaces (HMIMOS), and discussed both active and passive RISs, encompassing the hardware architectures, operation modes, and applications in communications. In~\cite{Liaskos}, Liaskos~\emph{et al.} presented one kind of RIS prototype, namely the HyperSurface tile, for realizing programmable wireless environments. Gacanin~\emph{et al.}~\cite{gacanin2020wireless} gave an overview of employing artificial intelligence (AI) tools in RIS-assisted radio environments. Di Renzo~\emph{et al.}~\cite{di2019smart} introduced the concept of smart radio environments empowered by RISs, and discussed recent research progresses and future potential challenges. Basar~\emph{et al.}~\cite{basar2019wireless} reviewed recent research efforts on RIS-empowered wireless networks, identified the differences between RISs and other technologies, and presented future research challenges and opportunities. Gong~\emph{et al.}~\cite{Gong2020Survey} surveyed recent research works on RIS-assisted wireless networks and discussed emerging applications and implementation challenges of RISs. From the perspective of enhancing the communication performance, Wu~\emph{et al.}~\cite{Wu2020Tutorial} gave a tutorial on design issues in RIS-assisted wireless networks, including passive beamforming optimization, channel estimation, and deployment design.
Although the aforementioned magazines/surveys/tutorials presented either general concepts or specific aspects of RISs (e.g., from a physics-based or a communication-based perspective), the fundamental performance limits of RISs and some potential applications in wireless networks are not covered. The comparison between widely employed mathematical tools for performance evaluations and optimizations in RIS-enhanced wireless networks is also not discussed. Moreover, a detailed framework based on machine learning (ML) tools for designing RIS-enhanced wireless networks is missing, except for a short magazine paper~\cite{gacanin2020wireless}. Motivated by all the aforementioned considerations, this paper provides a comprehensive discussion of RIS-enhanced wireless network principles, from physics to wireless communications, and discusses research opportunities for exploiting RISs in diverse applications, such as unmanned systems, non-orthogonal multiple access (NOMA), and ML. Table \ref{tab:comparision} illustrates the comparison of this treatise with the existing magazines/surveys/tutorials in the context of RISs.
Against the above observations, our main contributions are as follows.
\begin{table*}[ht!]\scriptsize
\begin{center}
\resizebox{\textwidth}{!}{\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}\hline
\multirow{3}{*}{---} & \multirow{3}{*}{Classifications} & \multirow{3}{*}{Key Contents} &Wu &Liang &Di Renzo &Huang &Liaskos &Gacanin &Di Renzo &Basar & Gong&Wu &This \\
& & &\emph{et al.} &\emph{et al.} &\emph{et al.} &\emph{et al.} &\emph{et al.} & \emph{et al.} & \emph{et al.} & \emph{et al.} & \emph{et al.} & \emph{et al.}&work\\
& & &\cite{Qingqing2020Towards} &\cite{liang2019large} &\cite{di2020smart} &\cite{huang2019holographic} &\cite{Liaskos} &\cite{gacanin2020wireless} &\cite{di2019smart} &\cite{basar2019wireless} &\cite{Gong2020Survey} \cite{Wu2020Tutorial} &\\
\hline
\multirow{10}{*}{Physics-based} & \multirow{6}{*}{Electromagnetics} & Distinguishing ray-optics \& wave-optics perspective & & &$\checkmark$ & & & & & & & &$\checkmark$ \\
\cline{3-14}
& & Macroscopic description of metasurface-based RIS & & &$\checkmark$ & & & &$\checkmark$ &$\checkmark$ & & & \\
\cline{3-14}
& &Surface equivalence theorems & & &$\checkmark$ & & & &$\checkmark$& & & &$\checkmark$ \\
\cline{3-14}
& & Distinguishing far-field \& near-field & & &$\checkmark$ & & & & & & & &$\checkmark$\\
\cline{3-14}
& & Formulating the reflection coefficient &$\checkmark$ &$\checkmark$ & & & & & &$\checkmark$ &$\checkmark$ & $\checkmark$ & $\checkmark$\\
\cline{3-14}
& & Power conservation principle & & &$\checkmark$ & & & & &&&&\\
\cline{2-14}
& \multirow{2}{*}{RIS Modeling} & RIS control mechanism &$\checkmark$ &$\checkmark$ &$\checkmark$ &$\checkmark$ &$\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$ &$\checkmark$ & $\checkmark$ & $\checkmark$\\
\cline{3-14}
& & Typical tunable functions & & &$\checkmark$ & & & & $\checkmark$ & $\checkmark$ &$\checkmark$ & & $\checkmark$\\
\cline{2-14}
& \multirow{2}{*}{Realizations} & RIS hardware and prototypes & &$\checkmark$ &$\checkmark$ &$\checkmark$ &$\checkmark$ & & $\checkmark$ & $\checkmark$ &$\checkmark$ &$\checkmark$ & $\checkmark$\\
\cline{3-14}
& & RIS synthesis methods & & &$\checkmark$ &$\checkmark$ &$\checkmark$ & &$\checkmark$ & && & $\checkmark$\\
\cline{1-14}
\multirow{22}{*}{Communication-based}
& & Path loss models & &$\checkmark$ &$\checkmark$ & & & & &$\checkmark$ &$\checkmark$ &$\checkmark$ & $\checkmark$\\
\cline{3-14}
& Performance & Stochastic geometry & & &$\checkmark$ & & & &$\checkmark$ &$\checkmark$ &$\checkmark$ & &$\checkmark$\\
\cline{3-14}
& evaluation & Discussion on stochastic analysis tools & & & & & & &$\checkmark$ & & & &$\checkmark$\\
\cline{3-14}
& & Discussion on small-scale fading channels & & & & & & & & & & &$\checkmark$\\
\cline{2-14}
& & Information-theoretic capacity limits & & & & & & & & & & & $\checkmark$\\
\cline{3-14}
& & Passive beamforming optimization &$\checkmark$ &$\checkmark$ &$\checkmark$ &$\checkmark$ & & & &$\checkmark$ & $\checkmark$&$\checkmark$ &$\checkmark$\\
\cline{3-14}
& RIS-aided & Resource management & & & &$\checkmark$ & & & & & & &$\checkmark$\\
\cline{3-14}
&communication & Comparison of employed mathematical approaches & & & & & & & & & & &$\checkmark$\\
\cline{3-14}
&design & Channel estimation &$\checkmark$ &$\checkmark$ &$\checkmark$ &$\checkmark$ & & & & $\checkmark$ & $\checkmark$ &$\checkmark$ &$\checkmark$\\
\cline{3-14}
& & Deployment design &$\checkmark$ &$\checkmark$ & & & & &$\checkmark$ & & &$\checkmark$ &$\checkmark$\\
\cline{3-14}
& & Modulation & & &$\checkmark$ & & & &$\checkmark$ &$\checkmark$ & & &\\
\cline{3-14}
& & Localization and sensing & & &$\checkmark$ & & & &$\checkmark$ & & & &\\
\cline{2-14}
& & Architecture& & & & & &$\checkmark$ & & & && $\checkmark$\\
\cline{3-14}
& ML-empowered & DL& & &$\checkmark$ &$\checkmark$ & & $\checkmark$ & &$\checkmark$ &$\checkmark$ & & $\checkmark$\\
\cline{3-14}
& RIS & RL & & & & & &$\checkmark$ & & & & & $\checkmark$\\
\cline{3-14}
& & Comparison of RL-based algorithms & & & & & & & & & & & $\checkmark$\\
\cline{3-14}
& & Supervised, unsupervised, and federated learning & & & & & &$\checkmark$ & & & & & $\checkmark$\\
\cline{2-14}
& & NOMA-RIS& & &$\checkmark$ & & & & & & & & $\checkmark$\\
\cline{3-14}
& & PLS-RIS& &$\checkmark$ &$\checkmark$ &$\checkmark$ & & & &$\checkmark$ &$\checkmark$ &$\checkmark$ & $\checkmark$\\
\cline{3-14}
& & SWIPT-RIS & &$\checkmark$ &$\checkmark$ &$\checkmark$ & & & & &$\checkmark$ &$\checkmark$ & $\checkmark$\\
\cline{3-14}
& Compatibility & UAV-RIS & & &$\checkmark$ & & & & & &$\checkmark$ &$\checkmark$ & $\checkmark$\\
\cline{3-14}
& & AV/CV-RIS & & & & & & & & & & & $\checkmark$\\
\cline{3-14}
& & MEC-RIS & & &$\checkmark$ & & & & & &$\checkmark$ &$\checkmark$ &$\checkmark$ \\
\cline{3-14}
& & mmWave-RIS & & &$\checkmark$ & & & & $\checkmark$ &$\checkmark$ &$\checkmark$ &$\checkmark$ &$\checkmark$ \\
\cline{2-14}
\hline
\end{tabular}}
\end{center}
\caption{Comparison of this work with available magazines/surveys/tutorials. Here, ``DL'' refers to ``Deep Learning'', ``RL'' refers to ``Reinforcement Learning'', ``AV'' refers to ``Autonomous Vehicle'', and ``CV'' refers to ``Connected Vehicle''.}
\label{tab:comparision}
\end{table*}
\begin{itemize}
\item We overview the fundamental principles that govern the operation of RISs and their interaction with the EM signals. We also survey typical RIS functions and their corresponding principles. Specifically, we focus on patch-array based implementation and compare the ray-optics perspective with the wave-optics perspective.
\item We develop performance evaluation techniques for multi-antenna assisted RIS systems. Research contributions are also summarized along with their advantages and limitations.
\item We investigate RISs from the information-theoretic perspective, based on which we review the protocols and approaches for jointly designing beamforming and resource allocation schemes with different optimization objectives. Additionally, the major open research problems are discussed.
\item We discuss the need of amalgamating ML and RISs. After reviewing the most recent research contributions, we propose a novel framework for optimizing RIS-enhanced intelligent wireless networks, where big data analytic and ML are leveraged for optimizing RIS-enhanced wireless networks.
\item We identify major research opportunities associated with the integration of RISs into other emerging technologies and discuss potential solutions.
\end{itemize}
As illustrated in Fig.~\ref{organization}, this paper is structured as follows. Section II elaborates on the fundamental operating principles of RIS-enhanced wireless networks. Section III focuses on the performance evaluation of multi-antenna RIS-assisted systems and the main advantages of using RISs in wireless networks. In Section IV, the latest research activities on the joint design of beamforming and resource allocation are discussed. The framework of ML-empowered RIS-enhanced intelligent wireless networks is presented in Section V. Finally, Section VI investigates the integration of RISs with other emerging technologies towards the design and optimization of 6G wireless networks.
\begin{figure} [t!]
\centering
\includegraphics[width=3.2in]{organization.pdf}
\caption{Organization of the present paper.}\label{organization}
\end{figure}
\begin{table*}[t!]\scriptsize
\caption{LIST OF ACRONYMS}
\begin{center}
\centering
\begin{tabular}{|l||l|}
\hline
AF & Amplify-and-Forward\\
AO & Alternating Optimization\\
AUV & Autonomous Underwater Vehicle\\
BC & Broadcast Channel\\
BS & Base Station\\
CSI & Channel State Information\\
D2D & Device-to-Device\\
DF & Decode-and-Forward\\
DL& Deep Learning\\
EE & Energy Efficiency\\
EM & Electromagnetic\\
FD & Full-Duplex\\
HD & Half-Duplex\\
IRS & Intelligent Reflecting Surface\\
IoT & Internet of Things\\
LIS & Large Intelligent Surface\\
LoS & Line-of-Sight\\
MEC & Mobile Edge Computing\\
MIMO & Multiple-Input and Multiple-Output\\
MISO & Multiple-Input and Single-Output\\
ML & Machine Learning\\
MOS & Mean Opinion Score\\
NP & Non-deterministic Polynomial-time \\
NOMA & Non-Orthogonal Multiple Access\\
OFDM & Orthogonal Frequency Division Multiplexing \\
OMA & Orthogonal Multiple Access\\
PDF& Probability Density Function \\
PLS & Physical Layer Security\\
QoS& Quality of Service\\
RIS& Reconfigurable Intelligent Surface\\
RL & Reinforcement Learning\\
SCA & Successive Convex Approximation\\
SE & Spectral Efficiency\\
SG & Stochastic Geometry\\
SIC& Successive Interference Cancelation\\
SIMO& Single-Input and Multiple-Output\\
SINR& Signal-to-Interference-plus-Noise Ratio\\
SISO& Single-Input and Single-Output\\
SNR& Signal-Noise Ratio\\
SWIPT& Simultaneous Wireless Information and Power Transfer\\
UAV & Unmanned Aerial Vehicle\\
VLC& Visible Light Communication\\
VR& Virtual Reality\\
WiFi &Wireless Fidelity\\
5G& Fifth-Generation\\
6G& Sixth-Generation\\
\hline
\end{tabular}
\end{center}
\label{table:abbre}
\end{table*}
\section{RIS: From Physics to Wireless Communications}
An RIS is a two-dimensional (2D) material structure with programmable macroscopic physical characteristics. The most important characteristic of an RIS is that its EM wave response can be reconfigured. In contrast to conventional wireless communication networks, the channels between the transmitters and the receivers can be controlled in RIS-aided networks. Thus, the strength of the desired received signal can be enhanced at the terminal devices. In this section, we introduce the fundamental principles which govern the operation of RISs and their interaction with the EM signals. We also survey typical RIS functions and their corresponding principles.
\subsection{Different Categories of RISs}\label{types}
\begin{figure}[t!]
\centering
\includegraphics[width =3.5in]{cat3.pdf}
\caption{Different types of metasurfaces.}
\label{cat}
\end{figure}
Considering their structures, RISs can be realized by using metamaterial or patch-array based technologies. Metamaterial-based RISs are referred to as metasurfaces. Deployed at different locations, RISs can be designed to work as reflecting/refracting surfaces between the BS and the user or waveguide surfaces operating at the BS. Considering the tuning mechanisms, RISs can be reconfigured electrically, mechanically, or thermally. Depending on their energy consumptions, RISs can be categorized as passive-lossy, passive-lossless, or active. The active or passive nature of RISs determines their ultimate performance capabilities. It is worth mentioning, however, that RISs cannot be completely passive because of their inherent property of being configurable. Here, we discuss three important RIS working operations: waveguide~\cite{smith2017analysis}, refraction~\cite{zhu2014dynamic}, and reflection~\cite{dai2018independent}. With the aid of Love's field equivalence principle~\cite{rengarajan2000field}, the reflected and refracted EM field can be studied by introducing equivalent surface electric and magnetic currents~\cite{zhu2014dynamic}. In the three working conditions, the RIS converts and radiates a wave (either induced by an incident wave or fed by a waveguide) into a desired propagating wave in free space. The surface equivalence principles (SEPs) including Love's field equivalence principle and Huygens' principle are introduced in Section \ref{mm}.
\subsubsection{Waveguide RIS}
R. Smith~\emph{et al.}~\cite{smith2017analysis} presented a theoretical study of waveguide-fed metasurfaces. The elements in the metasurface are modeled as uncoupled magnetic dipoles. The magnitude of each dipole element is proportional to the product of the reference wave and each element's polarizability. By tuning the polarizability, the metasurface antenna can perform beamforming. Each element on the metasurface serves as a micro-antenna. Compared to conventional antenna arrays, the compact waveguide metasurface occupies less space and can transmit towards wider angles.
\subsubsection{Refracting RIS}
Viktar S.~\emph{et al.}~\cite{asadchy2016perfect} proposed a theoretical design of a perfectly refracting and reflecting metasurfaces. The authors used an equivalent impedance matrix model so that the tangential field components at the two sides of the metasurface are appropriately optimized. Moreover, three possible device realizations are discussed: self-oscillating teleportation metasurfaces, non-local metasurfaces, and metasurfaces formed by only lossless components. The role of omega-type bianisotropy in the design of lossless-component realizations of perfectly refractive surfaces is discussed.
\subsubsection{Reflecting RIS}
Dai~\emph{et al.}~\cite{dai2018independent} designed a digital coding reflective metasurface. The elements in the metasurface contain varactor diodes with a tunable biasing voltage. By pre-designing several digitized biasing voltage levels, each element can apply discrete phase shifts and achieve beamforming for the reflected wave.
The rest of this section is focused on the operating principles for RISs that operate as reflectors.
\subsection{Understanding Metasurfaces From the View of Physics} \label{mm}
A wireless signal is essentially an EM wave propagating in a three-dimensional space. Attenuation or reduction of the signal strength occurs as the EM wave propagates through the space and interacts with the scattering objects. From basic principles of electromagnetism, the signal power per unit area is proportional to the square of the electric field of the corresponding wave in a given media. As far as reflective and refractive smart surfaces are concerned, this requires the understanding of how the EM waves interact with the surrounding objects. The equivalence principle, especially the SEP, is the building block for studying the EM wave transformations. Some authors also call it Love's field equivalence principle. The principle can be adopted for both external problems (source-free region) and internal problems. Love's field equivalence principle states that the EM field outside or inside a close surface can be uniquely determined by the electric and magnetic currents on the surface. As shown in Fig.~\ref{love}, the equivalent problem for the region \textrm{I} can be reformulated by placing equivalent currents on S that satisfy the boundary condition for each particular case and filling the region \textrm{II} with the same medium of constitutive parameters $\epsilon$ and $\mu$. Thus, the equivalent currents ($-J_s$,$-M_s$), together with the original source currents ($J_1$,$M_1$), radiate the correct fields in region \textrm{I}. The equivalent problem for region \textrm{II} can be formulated similarly.
Love's field equivalence principle is the theoretical foundation for analyzing the radiation pattern of RISs. However, the SEP does not specify how to calculate the EM field produced by the surface currents. To obtain the signal strength at an arbitrary field point, the Huygens-Fresnel principle can be employed. The Huygens-Fresnel principle is a method of analysis applied to problems of wave propagation, which states that every point on a wavefront is itself the source of spherical wavelets, and the secondary wavelets emanating from different points mutually interfere. The sum of these spherical wavelets forms the wavefront. Based on the Huygens-Fresnel principle, the EM field scattered by an RIS (in reflection and refraction) can be quantified analytically.
\begin{figure}[t!]
\centering
\includegraphics[width =3.5in]{love.pdf}
\caption {Love's equivalence internal problem (for region \textrm{I}). }
\label{love}
\end{figure}
As far as waveguide-based RISs are concerned, the operating principle can be summarized as follows. In \cite{fong2010scalar}, the EM wave manipulation of the waveguide metasurface performs the coupling between three-dimensional free space waves and two-dimensional surface waves. As a result, the metasurface can be regarded as a hologram, which carries additional information about its radiated signal propagating in the 3D space. After being excited by the source, this pre-designed information is coupled into the radiated field. Fig.~\ref{holo} conceptually represents a pre-designed holographic waveguide-based RIS.
\begin{figure}[t!]
\centering
\includegraphics[width =3.5in]{holo.pdf}
\caption {Conceptual illustration of the holographic impedance smart surface.}
\label{holo}
\end{figure}
\subsection{Analysis of RISs: Ray Optics Perspective v.s. Wave Optics Perspective}\label{design}
To characterize the interaction between an RIS and the impinging EM waves, one can adopt approximations and tools either from the perspective of ray-optics or wave-optics. These two perspectives have been used by physicists for a long time. Even though they are based on some approximations, these two methods of analysis are useful in order to obtain important insights into the interaction of light or radio waves with materials. In the RIS literature, both methods of analysis are often employed. However, the assumptions behind their use and their physical interpretations are intrinsically different. To shed light on their differences and similarities, we compare the two methods in this subsection. As shown in Fig~\ref{ray}(a), from the ray-optics perspective, an EM wave is modeled as a collection of geometrical rays with varying phases. The phase of each ray increases linearly with the optical path length as its traverses through the vacuum or other media. As a result, at each location of the $i$-th ray, a phase (denoted by $\phi_i$) can be defined. When the ray interacts with a material, the phenomenon is studied by determining the relationship between the change of the phase and the material refraction index. The desirable reflected wave is obtained if the ensemble of rays obey the proper co-phase condition. The wave-optics perspective is shown in Fig~\ref{ray}(b), where an EM wave is represented by the corresponding electric field and magnetic field. At each position, each of these two vector fields can be characterized by a time-varying complex-valued vector, with a direction, an amplitude, and a phase. From the wave-optics perspective, the interaction between the wave and the material can be studied using the equivalent principles discussed in the previous subsection. The points with equal phase values form a series of surfaces in space, which we refer to as wavefronts. As a result, the desired scattered waves (reflected or refracted) are obtained if proper wavefront transformations are performed by the RIS.
\begin{table*}[]\scriptsize
\begin{center}
\centering
\begin{tabular}{|l|l|l|}
\hline
& \textbf{Ray-optics} & \textbf{Wave-optics} \\ \hline
Wave representation & Geometrical rays & Vector fields \\ \hline
Theoretical foundation & Snell's law & Maxwell's equations \\ \hline
Surface profile & Phase discontinuity & Surface impedance \\ \hline
Requirement of the reflected wave & Co-phase condition & Proper wavefront \\ \hline
Power flow & Not accurate & Accurate \\ \hline
\end{tabular}
\caption{Comparing different wave representation perspectives}
\label{table:pers}
\end{center}
\end{table*}
\subsubsection{Comparing Ray-Optics and Wave-Optics Perspectives}
Table \ref{table:pers} highlights some of the differences between the two methods of analysis. Compared with wave-optics, the ray-optics perspective is a stronger simplification of the real system. As a result, it is easier to be adopted and can produce a quick prediction about the RIS design. However, ray-optics methods fail when considering the RIS power flow. Wave-optics methods, on the other hand, predict the power flow by using the Poynting vector, which enables us to study the local and overall RIS power consumption. This is an important issue to consider when designing and manufacturing RISs. For example, the authors of~\cite{asadchy2016perfect} and~\cite{estakhri2016wave} point out that it is impossible to realize lossless plane-wave beam steering with locally passive RISs. One has to adopt the wave-optics perspective to study the power flow of the system. Moreover, the authors of~\cite{estakhri2016wave} state that, according to their simulation results, one can expect increasingly improved performance if the RISs are designed based on wave-optics approximation in comparison to those designed based on the ray-optics approximation. In conclusion, both perspectives have their advantages and limitations. However, adopting the wave-optics perspective is the most appropriate choice for most cases. A study of the differences, in terms of power flow and reflection coefficient, between ray optics and wave optics methods of analysis is reported in \cite{di2020smart}.
\begin{figure}[t!]
\centering
\includegraphics[width =3.5in]{ray.pdf}
\caption {Comparison between ray-optics and wave-optics perspective.}
\label{ray}
\end{figure}
\subsection{Achieving Tunability: Patch Array RIS}\label{tune2}
The EM characteristics of an RIS, such as the phase discontinuity, can be reconfigured by tuning the surface impedance, through various mechanisms. Apart from electrical voltage, other mechanisms can be applied, including thermal excitation, optical pump, and physical stretching. Among them, electrical control is the most convenient choice, since the electrical voltage is easier to be quantized and controlled by field-programmable gate array (FPGA) chips. The choice of RIS materials include semiconductors~\cite{zhu2013active} and graphene~\cite{emani2015graphene}.
Regardless of the tuning mechanisms, we focus our attention on patch-array smart surfaces in the following text. The general geometry layout of this type of RIS can be modeled as a periodic (or quasi-periodic in the most general case) collection of unit cells integrated on a substrate. For ease of description, we limit our discussion to RISs that are based on a local design, in which the cells do not interact with each other. A local design usually results in the design of sub-optimal RISs. A comprehensive discussion about local and non-local designs can be found in \cite{di2020smart}. To characterize the tunability of the RIS, the method of equivalent lumped-element circuits can be adopted. As shown in Fig. \ref{vara}, the unit cell is equivalent to a lumped-element circuit with a load impedance $Z_l$. Particularly, the equivalent load impedance can be tuned by changing the bias voltage of the varactor diode. When modeling patch-array RISs in wireless communication systems, we can characterize, under a local design, each of its unit cells through an equivalent reflection coefficient. For example, the reflection coefficient of the $i$-th cell can be modeled as follows:
\begin{equation}\label{refl}
r_i=\beta_i \cdot e^{j\phi_i}
\end{equation}
where $\beta_i$ and $\phi_i$ correspond to the amplitude response and the phase response, respectively. As shown in~\cite{gradoni2020end}, the equivalent reflection coefficient depends on the tuning impedance of the lumped circuit that controls each unit cell, as well as the self and mutual impedances (if mutual coupling cannot be ignored) at the ports of the RIS. In particular, as shown in~\cite{abeywickrama2019intelligent} and~\cite{gradoni2020end}, $\beta_i$ and $\phi_i$ in \eqref{refl} are usually not completely independent with each other, i.e., $\beta_i=f(\phi_i)$. In the following sections, $\Phi(\vec{r_x})$ refers to the phase discontinuity introduced by the RIS as a function of the position on the RIS, and $\phi_{mn}$ refers to the phase discontinuity of the $(m,n)$-th element of a patch-array RIS.
Existing designs of patch-array RISs can apply discrete phase control and, in some cases, amplitude control. Arun~\emph{et al.}~\cite{arun2020rfocus} designed \textit{RFocus}, which is a two-dimensional surface with a rectangular array of passive antennas. The size of each passive unit is $\lambda/4 \times \lambda/10$ and the EM waves are either reflected or refracted. The authors show that the \textit{RFocus} surface can be manufactured at a low cost, and that it can improve the median signal strength by $9.5$ times. Welkie~\emph{et al.}~\cite{welkie2017programmable} developed a low-cost device embedded in the walls of a building to passively reflect or actively transmit radio signals. Dunna~\emph{et al.}~\cite{dunna2020scattermimo} realized \textit{ScatterMIMO}, which uses a smart surface to increase the scattering in the environment. In their hardware design, each reflector unit uses a patch antenna connected to four open-ended transmission lines. The transmission lines provide $0$, $\pi/2$, $\pi$ or $3\pi/2$ phase shifts. Based on measurements, it is shown that \textit{ScatterMIMO} increases the throughput by factor of two and the signal-noise ratio (SNR) by 4.5 dB as compared with baselines schemes. It is worth mentioning that tunability can be achieved with implementations other than patch-array based surfaces. For example, PIVOTAL COMMWARE proposed a new technique called \textit{Holographic Beam Forming (HBF)}. The proposed holographic beamformer has a low cost and power consumption, as compared with other transmission technologies, such as massive MIMO and phased arrays.
\begin{figure}[t!]
\centering
\includegraphics[width =3.5in]{varactor.pdf}
\caption{Schemetic diagram of the varactor RIS.}
\label{vara}
\end{figure}
\subsection{RIS Operating Principles}\label{prin}
Considering single-beam reflection, a patch-array based RIS can be configured to serve a terminal device in the far-field and near-field regions. Among the many operating functions and configurations of RISs, \textit{anomalous reflection} and \textit{beamforming} are widely used in the context of wireless communications. Adopting the wave-optics perspective, anomalous reflection is a wavefront transformation from a plane wave to another plane wave, while beamforming is a wavefront transformation from a plane wave to a desired wavefront. Adopting the ray-optics perspective, we present the operating principles of these two configurations in the following text. As far as anomalous reflection is concerned, the RIS is designed to reflect an incident beam to a far-field terminal, following the generalized laws of reflection~\cite{bell1969generalized}. As far as beamforming (also called focusing) is concerned, the incident wave is focused towards a targeted region, often referred to as the \textit{focal point}. The required RIS configuration follows the co-phase condition~\cite{huang2008reflectarray}. The relation between these two operating principles is discussed in detail in~\cite{Renzo_cylindrical_mirror_scatter}. Before presenting these two different principles, the physical distinction between the near-field region and the far-field region is clarified.
\subsubsection{Near Field v.s. Far Field}
In the spirit of dimensional analysis, the characteristics of a system can be represented by dimensionless numbers. In order to separate the near-field region from the far-field region, a proper dimensionless number is needed. Let $L$ and $R_F$ denote the antenna aperture size of the RIS and the focal distance, respectively. Assume that $z$ is the distance of a particular field point to the RIS. Theoretically, the far-field and near-field regimes can be differentiated as follows: The distance of $2L^2/\lambda$ is a commonly used criterion to decide the boundary between the near-field and far-field regions (see ~\cite{johnson1973determination}, equation (1)). The position corresponding to $z=2L^2/\lambda$ is the boundary between the near-field region and the far-field region. This result comes from the inspection of the power density variation with the distance between a field point and the RIS. Within the near field where $z<2L^2/\lambda$, the power density shows significant variations. The peak position of the power density in the near-field region, namely $R_F$, changes with different RIS configurations. Using proper co-phase conditions, beam focusing can be achieved within the near-field of the RIS. It is worth mentioning that, in general, the boundary between the near-field and far-field regimes depends on the specific configuration of the RIS, as it was recently remarked in~\cite{Renzo_cylindrical_mirror_scatter}.
In general, the essential difference between the near-field region and the far-field region is how the power density changes with distance. Consider, for example, that the RIS focuses the wave within an area $a$. The total energy incident on the RIS is proportional to the solid angle, $\Omega$ spanned by the surface area of the RIS with respect to the location of the transmitter. After the reflection, the transmitted energy is spread over the area $a$. Thus, the power density around the focal point is proportional to $\Omega /a$. Moreover, according to \cite{arun2020rfocus}, the area $a$ is proportional to $\lambda^2 (1+4z^2/L^2)$, as a result of the Abbe diffraction limit. In the far-field region, the second term inside the brackets dominates and $\Omega /a$ is proportional to $L^2\Omega /z^2$, which is the typical spherical dissipation of the signal power with the distance. In the near-field region, the first term dominates and the area $a$ becomes very small. As a result, a high focusing gain can be achieved.
With the aid of an RIS, in general, the signal can be enhanced in both the near-field and far-field regimes. However, the rationale of this enhancement is different. For near-field applications, the RIS is expected to enhance the signal strength for users located at targeted locations with respect to the RIS, while reducing the signal at other locations. For far-field applications, the RIS is typically expected to enhance the signal strength for users located at targeted angles with respect to the RIS.
\begin{figure}[t!]
\centering
\includegraphics[width =3.5in]{function.pdf}
\caption{Typical functions of reflecting surfaces.}
\label{func}
\end{figure}
In the following text, adopting the ray-optics perspective, we discuss the generalized laws of refraction and reflection, as well as the corresponding co-phase condition.
\subsubsection{The Generalized Laws of Refraction and Reflection}
From a geometrical optics perspective, anomalous reflection and refraction from an RIS can be described by using the generalized laws of refraction and reflection~\cite{bell1969generalized}, which is a natural derivation of both Fermat's principle and the boundary conditions governed by Maxwell's equations.
\begin{pbox}
\textit{Achieving anomalous reflection (Fig. \ref{func}(a))}\\
Suppose that the phase discontinuity at the boundary is a function of the position along the $x$ direction $\Phi(\vec{r_x})$, where $\vec{r_x}$ is the position vector on the boundary. Moreover, suppose that the derivative of the phase discontinuity exists. Then, the angle of reflection ($\theta_1$) and the angle of refraction ($\theta_2$) are~\cite{chen2016review}:
\begin{align}
\label{gen_law}
\theta_1 &= sin^{-1}\Big[sin \theta_i + \frac{\lambda}{2\pi n_1}\frac{d\Phi}{dx}\Big], \\
\theta_2 &= sin^{-1}\Big[\frac{n_1}{n_2}sin \theta_i + \frac{\lambda}{2\pi n_2}\frac{d\Phi}{dx}\Big],
\end{align}
where $\theta_i$ is the angle of incidence, $\lambda$ is the wavelength of the transmitted signal in vacuum, and $n_1$, $n_2$ are the refractive indexes, as shown in Fig.~\ref{ref}.
\end{pbox}
\begin{figure}[t!]
\centering
\includegraphics[width =3.5in]{ref.pdf}
\caption{Illustration of the generalized laws of refraction and reflection.}
\label{ref}
\end{figure}
There are other results related to the generalized laws of refraction and reflection, including the critical angles for total internal reflection or refraction. The main result just presented here states that, when a phase discontinuity is introduced at the boundary surface, the angles of reflection and refraction depend not only on the angle of incidence but also on the wavelength, refractive indexes, and the gradient of the phase discontinuity. This gives extra controllable parameters to manipulate the reflected and refracted EM waves. As a result, anomalous reflection can be achieved by tuning the phase gradient (${d\Phi}/{dx}$) based on \eqref{gen_law} or, in the discrete patch-array implementation, by tuning the length of the super-lattice. However, the assumption that the derivative of the phase discontinuity is constant ($d\Phi/dx = \text{const.}$) does not necessary hold if different wave transformations are needed.
\subsubsection{Co-phase condition}
Focusing is usually implemented when the RIS is within the near-field of the source or the terminal is close to the RIS. In these cases, the curvatures of the incident and reflected wavefront are non-negligible. The optimization of the surface aims to produce a pencil-beam pointing towards the direction of the terminal. When the link between the source and the RIS, as well as the link between the RIS and the terminal are in LoS, the following co-phase condition~\cite{huang2008reflectarray} can be applied.
\begin{figure}[t!]
\centering
\includegraphics[width =3.5in]{cophase.pdf}
\caption{Coordinate representation of co-phase condition.}
\label{coph}
\end{figure}
\begin{pbox}
\textit{Achieving beamforming (focusing) (Fig. \ref{func}(b))}\\
Let $r_{mn}$ denote the position of the $(m,n)$-th RIS element, $r_s$ denote the position of the source, and $\hat{u}$ denote the direction of the observer with respect to the surface. As shown in Fig.~\ref{coph}, $\phi_{mn}$ can be chosen as follows~\cite{huang1998ka}:
\begin{equation}\label{cophase}
-k_0(|\vec{r}_{mn}-\vec{r}_s|-\vec{r}_{mn}\cdot \hat{u})+\phi_{mn} = 2\pi \cdot t
\end{equation}
where $t=1,2,3...$ and $k_0 = 2\pi/\lambda_c$.
\end{pbox}
The two design principles just discussed provide guidance on how to configure the RIS phase shift patterns for typical applications. In more complex wireless communication systems, the RIS role is more intricate, thus cannot be categorized by the two working functions described in Fig.~\ref{func}. In these cases, to determine the RIS configuration, an optimization problem needs be formulated. These issues are elaborated in the following sections.
\subsection{Discussions and Outlook}
In the design and configuration of RISs, both theoretical limitations and hardware implementation limitations affect the overall performance of the resulting system. Theoretical limitations are, e.g., the result of considering simplified assumptions or adopting naive perspectives during the modeling of the RIS and its interaction with the EM wave. Hardware implementation limitations come, e.g., from the discretization of an ideally continuous RIS profile. In the following text, we elaborate on three major points.
\subsubsection{Hardware limitations}
In practical application scenarios, many hardware parameters significantly affect the achievable performance of the system. For example, the number of quantization levels of the RIS phase shifts, the maximum number of elements that is possible to integrate on the substrate, and the percentage of scattering environment that can be coated by an RIS. Existing research contributions studied the limitations and tradeoffs caused by these hardware constraints by analyzing their effect on the channel distribution~\cite{xu2020novel}, the power scaling law~\cite{qian2020beamforming}, and performance metrics such as the outage probability~\cite{zhang2019analysis} and ergodic capacity~\cite{boulogeorgos2020ergodic}. At the time of writing, the analysis of the impact of hardware limitations and how an RIS performs compared to other available technologies are open research issues.
\subsubsection{System design simplifications}
The adoption of oversimplified models for RIS hardware or channel models result in limitations on the system design. Because of the complex nature of RISs and their interaction with the environment, initial research contributions have adopted simple models. For example, using hardware models based on local designs, the ray-optics perspective for channel modeling, and decoupling reflection amplitude and phase shift of the RIS elements. Even though the ray-optics approximation can yield effective designs in some cases (as shown in Section \ref{prin}), it is preferable to adopt the wave-optics design in practical cases~\cite{Renzo_cylindrical_mirror_scatter}. Further research efforts are needed to bridge complex physical models of different RIS implementations with widely used communication models~\cite{gradoni2020end}.
\subsubsection{Optimization limitations}
To reap the benefits of deploying RISs in wireless networks, the RIS parameters (e.g., the reflection coefficient and deployment location) and network resource allocation (e.g., transmit beamforming and user scheduling) should be jointly optimized. However, the resulting joint optimization problems are normally non-convex and involve highly-coupled variables, which make it challenging to derive a globally optimal solution. Though some efficient algorithms have been recently proposed to compute high-quality suboptimal solutions~\cite{wu2019intelligent,huang2019reconfigurable,Mu_124}, the performance limits of RISs remain unknown. To fully understand the attainable performance limits, information-theoretic perspective investigations ~\cite{Karasik_Joint,Mu_Capacity} are important and sophisticated mathematical tools~\cite{liu2020ris} are expected to be employed. Further details are given in Sections VI and V.
\section{Performance Analysis of Multi-Antenna Assisted RIS Systems}\label{section:MIMO-RIS}
In Section II, we have discussed the fundamental physical properties of the RISs. However, how RISs affect the communication performance is still an open problem. To systematically survey existing designs for RIS-enhanced networks, we discuss the following topics: (1) channel models, (2) performance analysis, and (3) benchmark schemes.
\subsection{ \textbf{Channel Models}\label{large-scale-Hou}}
\textbf{1) Path Loss Models}
Some research contributions on the path loss for RISs are available in~\cite{di2020smart},~\cite{tang2019wireless} and~\cite{Renzo_cylindrical_mirror_scatter}, which showed that the power scattered by an RIS is usually formulated in terms of an integral that accounts, by leveraging the Huygens principle, for the impact of the entire surface in the free-space scenario, where scattering, shadowing, and reflection are ignored. Closed-form expressions of the integral are, on the other hand, difficult to obtain, except for some asymptotic regimes, which correspond to viewing the RIS as electrically small and electrically large (with respect to the wavelength and the transmission distances). It is worth mentioning, in addition, that the path-loss model depends on the particular phase gradient applied by the RIS. Notably, the scaling laws can be different if the RIS operates as an anomalous reflector and as a focusing lens. In the following, we briefly discuss two scaling laws that have recently been reported for RISs that operate as anomalous reflectors (described in the previous section). Further information and details can be found in~\cite{di2020smart} and~\cite{Renzo_cylindrical_mirror_scatter}.
\begin{itemize}
\item \textbf{Electrically Small RISs:} In this asymptotic regime, the RIS is assumed to be relatively small in size compared with the transmission distances. In this regime, the RIS can be approximated as a small-size scatterer. In general, the path-loss scales with the reciprocal of the product of the distance between the transmitter and the center of the RIS and the distance between the center of the RIS and the receiver. In addition, the received power usually increases with the size of the RIS. The received power is usually maximized in the direction of anomalous reflection, where the path loss through the RIS follows the ``product of distances'' models, which can be formulated as:
\begin{equation}\label{anomalous reflection}
L({d_{SR}},{d_{RU}})\approx {\lambda_S} ({d_{SR}}{d_{RU}})^{-1},
\end{equation}
where $\lambda_S$ denotes the coefficient of the electrically small scenario, $d_{SR}$ and $d_{RU}$ represent the distance of source-RIS and RIS-user links, respectively. A detailed discussion is given in~\cite[Secs. IV-A, IV-B, IV-C]{Renzo_cylindrical_mirror_scatter}.
\item \textbf{Electrically Large RISs:} In this asymptotic regime, the RIS is assumed to be large (ideally infinitely large) in size compared with the transmission distances and the wavelength. In this regime, the RIS can be approximated as a large flat mirror. Let us denote by $x_0$ the point of the RIS (if it exists) at which the first-order derivative of the total phase response of the combined incident signal, reflected signal, and the surface reflection coefficient of the RIS is equal to zero. In general, the path-loss asymptotically scales with the reciprocal of a weighted sum of the distance between the transmitter and $x_0$ and the distance between $x_0$ and the receiver. In addition, the received power is not dependent on the size of the RIS, which is viewed as asymptotically infinite. This result substantiates the fact that the power scaling law of the RIS is physically correct, since it does not grow to infinity as the size of the RIS goes to infinity. This is because the scaling law and the behavior of the RIS are different with respect to the electrically small regime. In this case, the path loss through the RIS follows the ``sum of distances'' models, which can be approximated as:
\begin{equation}\label{Elec_large}
L({d_{SR}},{d_{RU}})\approx {\lambda_L} (d_{SR}+d_{RU})^{-1},
\end{equation}
where $\lambda_L$ denotes the coefficient of the electrically large scenario. A detailed discussion is given in~\cite[Secs. IV-A, IV-B, IV-C]{Renzo_cylindrical_mirror_scatter}.
\end{itemize}
\textbf{2) Spatial Models}
Stochastic geometry (SG) tools are capable of capturing the location randomness of users thus enabling the derivation of computable or closed-form expressions of key performance metrics. Specifically, several spatial processes exist for modeling the locations of users in different wireless networks, i.e., the homogeneous Poisson point process (HPPP)~\cite{Liu_Coop_NOMA_SWIPT,Liu_physical_scurity_NOMA}, the Poisson cluster process (PCP)~\cite{PCP_Yi,PCP_Hou}, the Binomial point process (BPP), as well as the Hard core point process (HCPP)~\cite{SG_survey}. We list some promising approaches for analyzing the performance of RIS-enhanced networks by using SG in Table~\ref{table:RIS_SG}.
In~\cite{stochastic_twoLayer_PPP}, the RIS elements are employed on obstacles, and it is assumed that randomly distributed users are located in the serving area of the RISs. In~\cite{Renzo_random_RIS}, the objects are modeled by a modified random line process of fixed length and with random orientations and locations. Therein, the probability that a randomly distributed object that is coated with an RIS acts as a reflector for a given pair of transmitter and receiver was investigated. In~\cite{ChaoZ2021_PCP_RIS}, PCP was invoked in the RIS-enhanced large-scale networks, where the angle of reflection is constrained by the angle of incidence. Therefore, the randomly distributed users and BSs are located at the same side of the RIS.
\begin{table*}[htbp]\scriptsize
\caption{Summary of RIS-enhanced SG Networks}
\begin{center}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
\centering
\textbf{Approaches} & \textbf{Advantages} & \textbf{Disadvantages} & \textbf{Ref.} \\
\hline
\centering
RIS-enhanced HPPP & Fairness-oriented design & Restricted user distribution & \cite{hou2019mimo}\\
\hline
\centering
RIS-enhanced two layer HPPP & Coverage-hole enhancement & RISs are deployed at obstacles & \cite{stochastic_twoLayer_PPP}\\
\hline
\centering
HPPP conditioned on angle & Practical design & Not tractable & \cite{NOMA_poission}\\
\hline
\centering
RIS-enhanced HPPP conditioned on angle & Practical design for RIS-enhanced networks & Complicated & --\\
\hline
\end{tabular}
\end{center}
\label{table:RIS_SG}
\end{table*}
\textbf{3) Small-Scale Fading Models}
Currently, two main approaches have been used for analyzing the performance of RIS-aided systems in small-scale fading channels: (1) the central-limit-theorem-based (CLT-based) distribution and (2) the use of approximated distributions.
\begin{itemize}
\item \textbf{CLT-based Distribution:} Let us consider a single-antenna BS that communicates with a single-antenna user with the aid of an RIS of $N$ elements. If the two received signals, from the BS and from the RIS, can be coherently combined, the effective channel power gain is given by
\begin{equation}\label{Small_scale_CLT}
\begin{aligned}
& {\left| {{{\mathbf{r}}^H}{\mathbf{\Phi g}} + h} \right|^2}, \\
{\rm{s.t.}}\;\;& {\beta _{1}}, \cdots, {\beta _{N}} = 1\\
& {\phi _{1}}, \cdots, {\phi _{N}} \in \left[ {0,2\pi } \right),
\end{aligned}
\end{equation}
where $h\in {{\mathbb{C}}^{1 \times 1}}$, ${\mathbf{g}} \in {{\mathbb{C}}^{M \times 1}}$, ${\mathbf{r}} \in {{\mathbb{C}}^{M \times 1}}$ denote the channels of the BS-user, BS-RIS, and RIS-user links, respectively. ${\mathbf{\Phi }} = {\rm{diag}}\left( {{\beta _1}{e^{j{\phi _1}}},{\beta _2}{e^{j{\phi _2}}}, \ldots ,{\beta _N}{e^{j{\phi _N}}}} \right)$ denotes the reflection-coefficient matrix of the RIS, where $\left\{ {{\beta _1},{\beta _2} \ldots ,{\beta _N}} \right\}$ and $\left\{ {{\phi _1},{\phi _2} \ldots ,{\phi _N}} \right\}$ represent the amplitude coefficients and phase shifts of the RIS elements, respectively. In this setup, the CLT-based technique stands as an approximation tool for analyzing the performance in the low-medium-SNR regime. This is due to the fact that the distribution of the probability density function (PDF) in the range $0$ to $0+$ is not precise by using the CLT~\cite{DING_SISO_CLT}.
In Rayleigh fading channels, the distribution of an RIS-enhanced link follows a modified Bessel function~\cite{DING_SISO_CLT}. Since both transmitter and RISs are part of the infrastructure, and the RISs are typically positioned to exploit the LoS path with respect to the locations of the transmitters and the receivers for increasing the received signal power, Zhang~\emph{et al.}~\cite{zhang2019analysis} studied Rician fading channels, and the analysis showed that the signal power follows a non-central chi-squared distribution with two degrees of freedom. Ding and Poor~\cite{RIS_zhiguo_simple} proposed an RIS-enhanced network, where RISs are utilized for effectively aligning the directions of the users' channel gains. By utilizing the CLT-based technique, Cheng~\emph{et al.}~\cite{Yanyu2021_multi_RIS} studied the multi-RIS network, where the channel distributions were investigated with or without BS-user links.
\item \textbf{Approximated Distribution:} The exact distribution of the received SNR of the signal reflected from an RIS is non-trivial to be obtained, and hence the use of approximated distributions is often necessary.
Qian~\emph{et al.}~\cite{qian2020beamforming} proposed a simple approximated distribution of the received SNR, and proved that the received SNR can be approximated by two (or one) Gamma random variables and the sum of two scaled non-central chi-square random variables. A prioritized signal enhancement design was proposed by Hou~\emph{et al.}~\cite{Hou_SISO_NOMA_RIS}, where both the outage performance and ergodic rate of the user with the best channel gain were calculated. Lyu and Zhang~\cite{lyu2020spatial} proposed a single-input and single-output (SISO) network with multiple randomly deployed RISs, and showed that the exact distribution in terms of received signal power can be approximated by a Gamma distribution. Makarfi~\emph{et al.}~\cite{RIS_fading_to_gamma} proposed an RIS-enhanced network, whose equivalent channel is modeled by the Fisher-Snedecor $\mathcal{F}$ distribution.
\end{itemize}
Based on the above mentioned contributions~\cite{DING_SISO_CLT,zhang2019analysis,RIS_zhiguo_simple,Hou_SISO_NOMA_RIS,lyu2020spatial,RIS_fading_to_gamma}, where only approximated channel distributions are obtained, the exact channel distribution of RIS-enhanced networks is still an open problem. Based on recent research results, for example, a fundamental limitation lies in the calculation of the diversity order of RIS-enhanced networks under ideal operating conditions and in the presence of hardware limitations, e.g., quantized phase shifts. For example, the diversity order obtained by using the CLT-based distribution~\cite{DING_SISO_CLT} is $\frac{1}{2}$ in the high-SNR regime, whereas the diversity order is $\frac{N}{3}$ if an approximation based on the Gamma distribution is used, where $N$ denotes the number of RIS elements. However, the CLT-based and Gamma-based distributions are not exact, making the performance analysis of RIS-enhanced networks an interesting problem for future research.
Furthermore, since the exact distribution contains higher-order components, which approach zero in the high-SNR regime, most of the previous contributions~\cite{cheng2020downlink,Tang2020PLS} adopt the approximated distribution method for modeling small-scale fading channels, and the exact distribution of RIS-enhanced networks is still an open problem. For example, recent exact results on the impact of phase noise on the diversity order of RIS-enhanced transmission can be found in~\cite{discrete_RIS_order}.
\subsection{Performance Analysis}
In this subsection, we briefly discuss currently available papers on the performance analysis of RISs that are realized as large arrays of tiny and inexpensive antennas whose phase response is locally optimized. By offering extra diversity in the spatial domain, multi-antenna techniques are of significant importance. The application of multi-antenna enhanced RIS networks has attracted substantial interest from academia~\cite{lyu2020spatial,Hou_SISO_NOMA_RIS,hou2019mimo} and industry~\cite{RIS_industrial_linglong,Hou_MIMO_RIS_IC,Bjrnson2019}.
Given the increasing number of research contributions on RISs, its advantages are becoming more clear, especially in terms of spectral efficiency (SE) and energy efficiency (EE) enhancement. There are several key challenges for performance analysis in RIS-enhanced networks. One of the main challenges is to evaluate the exact distributions of the cascade channels between the BS and users through RISs. Another challenge is evaluating the effective channel gain after passive beamforming at the RIS. Table~\ref{table:RIS} summarizes the existing contributions on RISs with multiple antennas and illustrates their comparison. RIS-enhanced single user networks have been analyzed in \textbf{Section~III.A}. Hence, we turn out attention to RIS-enhanced multi-user networks. A prioritized signal-enhancement-based (SEB) was proposed by Hou~\emph{et al.}~\cite{Hou_SISO_NOMA_RIS}, where passive beamforming is designed for the user with the best channel gain, and all the other users rely on RIS-enhanced beamforming.
\begin{table*}[htbp]\scriptsize
\caption{Important contributions on RIS-enhanced networks. ``DL'' and ``UL'' represent downlink and uplink, respectively. The ``sum-rate gain'' implies that the gain brought by invoking RIS technique}
\begin{center}
\centering
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\centering
\textbf{Ref.} &\textbf{Scenarios} & \textbf{Direction}& \textbf{Users} & \textbf{Main Objectives} & \textbf{Techniques} \\
\hline
\centering
\cite{hou2019mimo} & MIMO & DL & Multiple users & OP and throughput & Fairness SEB \\
\hline
\centering
\cite{Bjrnson2019} & SISO & DL & Single user & sum-rate gain & Compare with relay\\
\hline
\centering
\cite{RIS_fading_to_gamma} & SISO & UL & Single user & OP and throughput & Effective channel gain \\
\hline
\centering
\cite{DING_SISO_CLT} & SISO & DL & Single user & Effective channel gain & Compare with random phase shifting\\
\hline
\centering
\cite{zhang2019analysis} & SISO & DL & Single user & Effective channel gain & SEB\\
\hline
\centering
\cite{RIS_zhiguo_simple} & SISO & DL & Single user & OP & SEB\\
\hline
\centering
\cite{Hou_SISO_NOMA_RIS} & SISO & DL & Multiple users & OP and throughput & Prioritized SEB \\
\hline
\centering
\cite{lyu2020spatial} & SISO & DL & Single user & Effective channel gain & SEB\\
\hline
\centering
\cite{qian2020beamforming} & MIMO & DL & Single users & Effective channel gain & Random matrix theory and CLT \\
\hline
\centering
\cite{Ntontin2019multi}& MIMO & DL & Multiple users & sum-rate gain & SEB \\
\hline
\centering
\cite{Hou_MIMO_RIS_IC} & MIMO & DL & Multiple users & Interference cancellation & SCB and less constraint at RAs\\
\hline
\centering
\cite{discrete_RIS_number_bit} & SISO & UL & Multiple users & Sum-rate & Minimum required finite resolution\\
\hline
\centering
\cite{CR_RIS_Rice} & MISO & DL & Multiple users & Sum-rate & Multi-RIS distribution\\
\hline
\centering
\cite{RIS_ruiZhang_discrete} & MISO & DL & Multiple users & Sum-rate & Discrete phase shifts \\
\hline
\end{tabular}
\end{center}
\label{table:RIS}
\end{table*}
\begin{itemize}
\item \textbf{RIS-enhanced Signal Enhancement Designs:} By assuming that multiple waves are co-phased at the users, the received signal can be significantly enhanced, which leads to the following optimization problem:
\begin{equation}\label{SEB}
\begin{aligned}
{\rm{max}}\;\;&{\left| {{{\mathbf{r}}^H}{\mathbf{\Phi g}} + h} \right|^2} \\
{\rm{s.t.}}\;\; & {\beta _{1}}, \cdots, {\beta _{N}} = 1\\
& {\phi _{1}}, \cdots, {\phi _{N}} \in \left[ {0,2\pi } \right).
\end{aligned}
\end{equation}
In order to further enhance the SE of RIS-enhanced networks, multiple antenna techniques can be employed at both the BS and users. Yuan~\emph{et al.}~\cite{CR_RIS_Rice} proposed a cognitive-radio-based RIS-enhanced multiple-input and single-output (MISO) network, where both perfect and imperfect channel state information (CSI) setups were considered. However, in many research works, continuous amplitude coefficients and phase shifts are assumed at the RISs, whilst in practice the phase shifts of RISs may not be continuous. Thus, You~\emph{et al.}~\cite{RIS_ruiZhang_discrete} proposed a discrete phase shifts model for a MISO enhanced RIS network. Zhang~\emph{et al.}~\cite{discrete_RIS_number_bit} then evaluated the required number of bits for finite-resolution RISs in an uplink SISO network. Hou~\emph{et al.}~\cite{hou2019mimo} investigated an RIS-enhanced MIMO network, where a fairness oriented design was considered by applying SG tools for modeling the impact of the users' locations.
\item \textbf{RIS-enhanced Signal Cancellation Designs:} Another application of deploying RISs in wireless networks is signal cancellation, where the reflected signals and the direct signals can be destructively combined. The corresponding optimization problem can be formulated as follows:
\begin{equation}\label{SCB}
\begin{aligned}
{\rm{min}}\;\;&{\left| {{{\mathbf{r}}^H}{\mathbf{\Phi g}} + h_I} \right|^2} \\
{\rm{s.t.}}\;\; & {\beta _{1}}, \cdots, {\beta _{N}} \le 1\\
& {\phi _{1}}, \cdots, {\phi _{N}} \in \left[ {0,2\pi } \right).
\end{aligned}
\end{equation}
where $h_I$ denotes the aggregate interference signals from other-cell BSs. By assuming that both the inter-cell and intra-cell interferences are perfectly known, the optimal solution to~\eqref{SCB} is to adjust both the signal phase and amplitude coefficients of the BS-RIS-user links to the opposite of the effective interference links with the same amplitude. By doing so, some promising applications can be realized, e.g. RIS-enhanced PLS and interference cancellation. On the one hand, by assuming that perfect CSI is available at the RIS controller, the inter-cell and intra-cell interferences can be eliminated. On the other hand, considering the PLS requirements, RISs also stand as a potential solution for cooperative jamming techniques, i.e., RISs act as artificial noise sources. By adopting this approach, several contributions have been made. Hou~\emph{et al.}~\cite{Hou_MIMO_RIS_IC} proposed an RIS-enhanced interference cancellation technique in a MIMO network, where the inter-cluster interference can be eliminated without active beamforming weights and detection vectors. Furthermore, this work can be adopted for application to coordinated multi-point (CoMP) networks for inter-cell interference cancellation in cellular networks. Shi~\emph{et al.}~\cite{PLS_RIS_SWIPT} investigated an RIS-enhanced secure beamforming technique, where the secrecy rate of the legitimate user was derived. Lyu~\emph{et al.}~\cite{PLS_RIS_Jammer} investigated an RIS jamming scenario, where RISs act as jammers for attacking legitimate communications without using any internal energy.
\end{itemize}
\subsection{Benchmark Schemes}
\begin{figure}[t!]
\centering
\includegraphics[width =3in]{rate_HD_FD.pdf}
\caption{Spectral efficiency of RIS-enhanced, FD-relay as well as HD-relay networks versus the number of RIS elements. Please refer to~\cite{Hou_SISO_NOMA_RIS} for simulation parameters.}
\label{Compare With AF_DF relay fading fig 6}
\end{figure}
In order to assess the advantages and limitations of RISs, two benchmark transmission technologies are usually considered: 1) surfaces with random phase shifts; and 2) relay networks.
\begin{itemize}
\item \textbf{Random Phase Shifts:} RISs are capable of shifting the phase of the incident signal, and hence multiple signals can be boosted or eliminated at the user side or at the BS side. Hence, a well-accepted benchmark scheme to quantify the performance enhancement by RIS elements is given by a surface that is not configurable and that can ideally be modeled by a surface with random phase shifts~\cite{DING_SISO_CLT}.
\item \textbf{Relay Networks:} Generally speaking, relay-aided networks can be classified into two pairs of classic relaying protocols, which are 1) FD and HD relay networks; and 2) AF and DF relay networks. By assuming that the optimal power split strategies of both the AF and DF relays are employed, the performance gain between RIS-enhanced and relay-aided networks can be compared. Specifically, Bjornson~\emph{et al.}~\cite{Bjrnson2019} compared the achievable data rate of both RIS-enhanced and DF-relay-aided SISO network, where the BS-user links are blocked. It was pointed out that when the number of tunable elements of the RISs is large enough, an RIS-enhanced network is capable of outperforming a DF-relay-aided network. In an effort to provide a comprehensive analysis for both RIS-enhanced and relay-aided networks, Ntontin~\emph{et al.}~\cite{Ntontin2019multi} compared the system performance of classic maximal ratio transmission (MRT) and maximal ratio combining (MRC) techniques. Fig.~\ref{Compare With AF_DF relay fading fig 6} illustrates the potential benefits of RIS-enhanced networks compared with both HD-relay and FD-relay networks in terms of network throughput~\cite{Hou_SISO_NOMA_RIS}. Here, the performance of HD-relaying is obtained for an equal time-split ratio. We can see that the network throughput gap between the RIS-enhanced network and the other pair of relay aided networks becomes smaller, when the number of RIS elements increases. For example, when the number of RIS elements $N=23$ and the transmit power $P=25$ dBm, the proposed RIS-enhanced network is capable of outperforming both FD and HD relay aided networks, which indicates that the RIS-enhanced network becomes more competitive, when the number of RIS elements is large enough.
Table~\ref{table:RIS_compare_benchmarks} provides a comparison between RIS-enhanced and relay aided networks in terms of advantages and limitations.
\end{itemize}
\begin{table*}[htbp]\scriptsize
\caption{Comparison of the RIS-enhanced and Benchmarks. The ``Power'' Implies Additional Power Supply at the RISs or at the Relay. The ``CT'' Denotes Concurrent Transmission}
\begin{center}
\centering
\begin{tabular}{|l|l|l|c|c|c|c|}
\hline
\centering
\textbf{Mode} & \textbf{Pros} & \textbf{Cons} &\textbf{Delay}& \textbf{Power} & \textbf{Interference} & \textbf{CT} \\
\hline
\centering
RIS-enhanced & High EE, simple device & CSI must be perfectly known & \ding{55} & \ding{55} & \ding{55} & \checkmark \\
\hline
\centering
HD relaying under AF protocol & No decoding at relay & Noise is amplified & \checkmark & \checkmark & \ding{55} & \ding{55} \\
\hline
\centering
HD relaying under DF protocol & No self-interference & Latency is high & \checkmark & \checkmark & \ding{55} & \ding{55} \\
\hline
\centering
FD relaying under AF protocol & No decoding at relay & Noise and interference are amplified & \ding{55} & \checkmark & Self-interference & \checkmark \\
\hline
\centering
FD relaying under DF protocol & No latency & Rate ceiling occurs & \ding{55} & \checkmark & Self-interference & \checkmark \\
\hline
\centering
MIMO relay & High SE & High cost, difficult to realize at mmWave & \ding{55} & \checkmark & \checkmark & \checkmark \\
\hline
\end{tabular}
\end{center}
\label{table:RIS_compare_benchmarks}
\end{table*}
\subsection{Discussions and Outlook}
Although previous research contributions have analyzed the approximated performance of RIS-enhanced networks, there are still three major open research problems.
1) Path loss experiments for outdoor scenarios: Since only~\cite{tang2019wireless} reported experimental measurements of the path loss in free-space environments, the development of experimentally-validated path-loss models in outdoor scenarios is an open research issue, especially in the presence of reflecting and scattering objects.
2) Exact distributions: Current research methods for performance analysis are based on the CLT-based distribution and approximated distribution~\cite{Yanyu2021_multi_RIS,RIS_fading_to_gamma}, which are, however, accurate only in the high-SNR regime. More advanced and accurate analytical models are needed for analyzing the diversity order and the network performance in the low-SNR regime.
3) Integrated application scenarios: To serve the desired users in different scenarios, signal enhancement and signal cancellation designs were well investigated in recent research works~\cite{Hou_SISO_NOMA_RIS,Ntontin2019multi,Hou_MIMO_RIS_IC}. However, the desired signals and interference signals can be simultaneously enhanced and mitigated, which constitutes an important future direction.
\section{RIS Beamforming and Resource Allocation}
As described in the previous section, deploying RISs enables high performance enhancements in wireless networks. Motivated by this benefit, how to jointly design the transmit and passive beamforming as well as the optimal allocation of the wireless resources has become an important task for RIS-enhanced wireless networks. In this section, we first discuss the information-theoretic performance limits of RISs. Then, we review recent research contributions with a particular focus on the joint beamforming optimization and the resource allocation design. Along the literature review, representative mathematical tools for facilitating the two types of design are also discussed along with their benefits and drawbacks.
\subsection{Information-Theoretic Perspective}
In order to unveil the fundamental performance limits of RISs, several research works~\cite{Karasik_Joint,Mu_Capacity,Zhang_MAC} have been devoted to investigating the RIS performance gains from an information-theoretic perspective.
\begin{figure}[t!]
\centering
\subfigure[Capacity regions with RIS-NOMA.]{\label{capacityNOMA}
\includegraphics[width= 3in]{CapacityNOMA.pdf}}
\subfigure[Rate regions with RIS-OMA.]{\label{rateOMA}
\includegraphics[width= 3in]{RateOMA.pdf}}
\setlength{\abovecaptionskip}{-0cm}
\caption{Illustration of the capacity and rate regions for a random channel realization with different RIS phase resolutions. $M_R$ denotes the number of RIS reflection elements. The full parameter settings can be found in \cite{Mu_Capacity}.}\label{RIS capacity}
\end{figure}
\begin{itemize}
\item \emph{Capacity-achieving design:} In~\cite{Karasik_Joint}, Karasik {\em et al.} derived the capacity for an RIS-aided single-input and multiple-output (SIMO) communication system. With finite input signal constellations, it was proved that a joint information encoding scheme at both the transmitted signals and the RIS configurations is necessary for achieving the channel capacity~\cite{Karasik_Joint}. Based on this insight, the authors further proposed a practical transmission strategy by utilizing layered encoding and successive cancellation decoding techniques. Numerical examples showed that the proposed joint encoding scheme outperforms the conventional max-SNR scheme.
\item \emph{Capacity region characterization:} The capacity region of the fading SISO broadcast channel (BC) was proved to be achieved by invoking the superposition coding (SC) at the transmitter and the successive interference cancellation (SIC) at multiple receivers~\cite{Ergodic}, i.e., employing the NOMA transmission. Inspired by these results, Mu {\em et al.}~\cite{Mu_Capacity} investigated the capacity and rate regions of RIS-enhanced multi-user wireless communication systems achieved by NOMA and orthogonal multiple access (OMA), respectively. The Pareto boundary of each region was characterized by solving a series of sum rate maximization problems via the rate-profile technique. As shown in Fig. \ref{capacityNOMA} and Fig. \ref{rateOMA}, by deploying an RIS, the NOMA capacity region and the OMA rate region can be improved. The capacity/rate regions are further enlarged by employing more precise phase resolutions and larger numbers of reflection elements. Furthermore, Zhang {\em et al.}~\cite{Zhang_MAC} investigated the capacity region of the multiple access channel (MAC) with two users in both centralized and distributed RIS deployment strategies. The results demonstrated that the centralized RIS deployment strategy can achieve a higher capacity gain than the distributed strategy.
\end{itemize}
\indent
\subsection{Joint Transmit and Passive Beamforming Design}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=2.8in]{jointBF.pdf}
\setlength{\abovecaptionskip}{-0cm}
\caption{Illustration of joint transmit and passive beamforming design.}
\label{BF}
\end{center}
\end{figure}
\subsubsection{Optimization objectives}As shown in Fig. \ref{BF}, an RIS is deployed to assist the transmission between the BS and the users by passively reflecting the signals. The RIS reflection coefficients can be adjusted by the BS through an RIS controller. Hence, the transmit beamforming at the BS and the passive beamforming at the RIS have to be jointly designed to improve the communication performance. In the following, we review the related research works in terms of their considered optimization objectives.
\begin{itemize}
\item \emph{Transmit power minimization or EE maximization:} In~\cite{wu2019intelligent}, Wu {\em et al.} minimized the transmit power for an RIS-enhanced MISO system in both single-user and multi-user scenarios. Alternating optimization (AO) based algorithms were developed to find locally-optimal solutions. The passive beamforming was designed by invoking the semidefinite relaxation (SDR) approach. It was revealed that an RIS can simultaneously enhance the desired signal strength and mitigate the interference for the multi-user scenario. The same problem was further investigated in~\cite{Wu_138} by taking discrete RIS phase shifts into consideration. The optimal solutions were derived by applying the branch-and-bound method and exhaustive search for single-user and multi-user scenarios, respectively. To reduce the computational complexity, efficient successive refinement algorithms were further designed. It was shown that the proposed low complexity algorithms are capable of achieving near-optimal performances. Han {\em et al.}~\cite{han2019intelligent} investigated physical-layer broadcasting in an RIS-aided network, where the total transmit power for satisfying QoS requirements of all users was minimized. Furthermore, Fu {\em et al.}~\cite{Fu_142} focused on an RIS-enhanced MISO downlink NOMA system, where the transmit power was minimized by jointly optimizing the transmit and passive beamforming vectors as well as user decoding orders. In order to overcome the drawbacks of the SDR approach, an alternating difference-of-convex (DC) method was proposed for handling the non-convex rank-one constraint. Zhu {\em et al.}~\cite{Zhu_NOMA} proposed an improved quasi-degradation condition for the RIS-enhanced MISO NOMA system to minimize the transmit power. Under this condition, NOMA is shown to be able to outperform the zero-forcing beamforming scheme. Zheng {\em et al.}~\cite{Zheng_NOMA} compared the minimum transmit power performance between OMA and NOMA in a discrete phase shift RIS-enhanced SISO system. A near-optimal solution was obtained by applying the linear approximation initialization and the AO method. The results showed that NOMA may perform worse than TDMA when the users have symmetric deployments and rate requirements, which revealed the importance of user pairing in the RIS-assisted NOMA system. Huang {\em et al.}~\cite{huang2019reconfigurable} solved the EE maximization problem in an RIS-enhanced multi-user MISO system, where a realistic RIS power consumption model was proposed in terms of the number of reflection elements and the phase resolutions at the RIS. An AO-based algorithm was designed for addressing the formulated problem, where the RIS phase shifts and the BS transmit power allocation were optimized by invoking the gradient descent method and the fractional programming method. The results demonstrated that the RIS achieves significantly better EE performance than the traditional active relay-assisted communication. In contrast to the aforementioned works based on the perfect CSI assumption, Zhou {\em et al.}~\cite{Zhou_141} investigated the robust beamforming design for an RIS-enhanced multi-user MISO system with imperfect CSI assumptions. The transmit power was minimized while satisfying QoS requirements of all users under all possible channel error realizations. The formulated non-convex problem was transformed into a sequence of semidefinite programming (SDP) subproblems, where the CSI uncertainties and the non-convex unit-modulus constraints were handled by applying approximation transformations and the convex-concave procedure~\cite{lipp2016variations}, respectively. In~\cite{Zhou_IP}, the robust beamforming design was further studied under two channel error models, namely the bounded CSI error model and the statistical CSI error model. The S-procedure and the Bernstein-Type inequality were applied in each model. The results unveiled that the RIS may degrade the system performance when the channel error is high. Zappone {\em et al.}~\cite{zappone2020overhead} modeled the overhead for carrying out the channel estimation and adjusting the RIS. Based on the proposed overhead model, the EE of an RIS-empowered MIMO communication network was maximized by jointly optimizing the RIS phase shifts as well as the transmitted and received filters.
\item \emph{SE or capacity maximization:} Yu {\em et al.}~\cite{Yu140} considered the SE maximization problem in an RIS-enhanced MISO system. Since the SDR approach only provides an approximate solution~\cite{wu2019intelligent}, two efficient algorithms were proposed by invoking the fixed-point iteration method and the manifold optimization method for the passive beamforming design. It was demonstrated that the proposed algorithms can achieve a higher performance and consume a lower complexity than the SDR approach. To solve the same problem, a branch-and-bound algorithm was further proposed by Yu {\em et al.}~\cite{Yu_optimal}, which is capable of obtaining a globally optimal solution. Though suffering from an extremely high computational complexity, the proposed branch-and-bound algorithm serves as a performance benchmark to verify the effectiveness of existing suboptimal algorithms. In~\cite{Ning137}, Ning {\em et al.} focused on an RIS-enhanced downlink MIMO system to maximize the SE. The passive beamforming was designed by using the sum of gains maximization principle and by utilizing the alternating direction method of multipliers. In~\cite{Ying_GMD}, Ying {\em et al.} considered an RIS-enhanced mmWave hybrid MIMO system, where the phase shifts at the RIS were designed by leveraging the angle information of the LoS BS-RIS channel. Moreover, Perovi{\'c} {\em et al.}~\cite{perovic2019channel} investigated RIS-assisted indoor mmWave communications, where two schemes were developed to maximize the channel capacity. Zhang {\em et al.}~\cite{Zhang_Capacity} characterized the fundamental capacity limit of RIS-aided point-to-point MIMO communication systems, by jointly optimizing the RIS reflection coefficients and the MIMO transmit covariance matrix. The communication capacity was maximized in both the narrowband transmission with frequency-flat fading channels and the broadband orthogonal frequency division multiplexing (OFDM) transmission over frequency-selective fading channels. Yang {\em et al.}~\cite{Yang_114} proposed a practical transmission protocol by considering the channel estimation for an RIS-enhanced OFDM system under frequency-selective channels. To reduce the required training overhead, the RIS reflection elements were divided into multiple groups and only the combined channel of each group has to be estimated. Based on the proposed grouping scheme, the achievable rate was maximized by jointly optimizing the power allocation at the transmitter and the phase shifts at the RIS with the AO method. In~\cite{You_113}, You {\em et al.} designed a transmission protocol by considering the channel estimation with discrete phase shifts at the RIS. To reduce the channel estimation errors, a low complexity discrete fourier transform (DFT)-Hadamard based reflection pattern scheme was developed. The achievable data rate was further maximized based on the estimated channel by designing the RIS phase shifts using the proposed successive refinement algorithm.
\item \emph{Sum rate maximization:} In~\cite{Huang_117}, Huang {\em et al.} maximized the sum rate in RIS-enhanced multi-user MISO downlink communications. By employing the zero-forcing precoding at the BS, the RIS reflection matrix and the power allocation were alternately optimized with the aid of the majorization-minimization approach. Moreover, the weighted sum rate maximization problem was investigated by Guo {\em et al.}~\cite{guo2019weighted}. Under the AO framework, the transmit beamforming was obtained using the fractional programming method, and three iterative algorithms were designed for optimizing the reflection coefficients in terms of different types of RIS reflection elements. In~\cite{Jung_96}, the asymptotic optimal discrete passive beamforming solution was derived and a modulation scheme was proposed to maximize the achievable sum rate for the RIS-enhanced multi-user MISO transmission. To further enhance the performance, Jung {\em et al.}~\cite{Jung_96} designed a joint user scheduling and transmit power control scheme, which can strike the tradeoff between the rate fairness and the maximum sum rate among the users. Mu {\em et al.}~\cite{Mu_124} focused their attention on the sum rate maximization problem in an RIS-enhanced MISO NOMA system with both ideal and non-ideal assumptions of RIS elements. The non-convex rank-one constraint of the passive beamforming design was handled by invoking the sequential rank-one constraint relaxation approach, which is guaranteed to obtain a locally optimal rank-one solution. Instead of optimizing the passive beamforming with the instantaneous CSI, Zhao {\em et al.} proposed a two-timescale transmission protocol for maximizing the achievable average sum rate in an RIS-enhanced multi-user system~\cite{Zhao_TwoTime}. To reduce the channel training overhead and complexity, the RIS phase shifts were firstly optimized with the statistical CSI. Then, the transmit beamforming was designed with the instantaneous CSI and the optimized RIS phase shifts.
\item \emph{User fairness:} Nadeem {\em et al.}~\cite{Nadeem_128} maximized the minimum SINR of an RIS-enhanced MISO system, where the BS-RIS-user link was assumed to be a LoS channel. A deterministic approximations was developed for the minimum SINR performance under the optimal linear precoder by employing the random matrix theory. As a result, the RIS phase shifts can be optimized using the channel's large-scale statistics, which can significantly reduce the overhead of the signal exchange~\cite{Nadeem_128}. Yang {\em et al.}~\cite{Yang_125} investigated the max-min rate problem in an RIS-enhanced NOMA system in both single-antenna and multi-antenna cases. A combined-channel-strength based user ordering scheme was proposed for achieving a near-optimal performance.
\end{itemize}
\begin{table*}[htbp]\scriptsize
\caption{Contributions on joint transmit and passive beamforming design}
\begin{center}
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\centering
\textbf{Ref.} &\textbf{Scenarios} & \textbf{Phase shifts} & \textbf{CSI} & \textbf{Main Objectives} & \textbf{Techniques/Characteristics} \\
\hline
\centering
\cite{wu2019intelligent} & SU/MU-DL-MISO & Continuous & Perfect & Transmit power & AO-SDR-based algorithm and two stage algorithm\\
\hline
\centering
\cite{Wu_138} & SU/MU-DL-MISO & Discrete & Perfect & Transmit power & Near-optimal ZF-based successive refinement method\\
\hline
\centering
\cite{han2019intelligent} & MU-DL-MISO & Continuous & Perfect & Transmit power & Physical-layer broadcasting\\
\hline
\centering
\cite{Fu_142} & MU-DL-MISO NOMA & Continuous & Perfect & Transmit power & Alternating DC method\\
\hline
\centering
\cite{Zhu_NOMA} & MU-DL-MISO NOMA & Continuous & Perfect & Transmit power & Improved quasi-degradation condition\\
\hline
\centering
\cite{Zheng_NOMA} & MU-DL-SISO NOMA & Discrete & Perfect & Transmit power & Asymmetric and symmetric user pairing\\
\hline
\centering
\cite{huang2019reconfigurable} & MU-DL-MISO & Continuous & Perfect & EE & RIS power consumption model\\
\hline
\centering
\cite{Zhou_141} & MU-DL-MISO & Continuous & Imperfect & Transmit power & The worst-case robust beamforming design\\
\hline
\centering
\cite{Zhou_IP} & MU-DL-MISO & Continuous & Imperfect & Transmit power & Imperfect cascaded channels at the transmitter\\
\hline
\centering
\cite{zappone2020overhead} & SU-DL-MIMO & Continuous & Estimated & EE & Overhead model for channel estimation and RIS configuration\\
\hline
\centering
\cite{Yu140} & SU-DL-MISO & Continuous & Perfect & SE & Fixed point iteration and manifold optimization methods\\
\hline
\centering
\cite{Yu_optimal} & SU-DL-MISO & Continuous & Perfect & SE & Branch-and-bound algorithm\\
\hline
\centering
\cite{Ning137} & SU-DL-MIMO & Continuous & Perfect & SE & Sum of gains principle\\
\hline
\centering
\cite{Ying_GMD} & SU-DL-MIMO mmWave & Continuous & Perfect & SE & Broadband hybrid beamforming\\
\hline
\centering
\cite{perovic2019channel} & SU-DL-MIMO mmWave & Continuous & Perfect & Channel capacity & RIS-assisted indoor mmWave environments\\
\hline
\centering
\cite{Zhang_Capacity} & SU-DL-MIMO & Continuous & Perfect & Channel capacity & Frequency-flat fading and frequency-flat selective channels\\
\hline
\centering
\cite{Yang_114} & SU-DL-SISO OFDMA & Continuous & Estimated & Achievable rate & RIS element grouping scheme\\
\hline
\centering
\cite{You_113} & SU-UL-SISO & Discrete & Estimated & Achievable rate & Near-orthogonal DFT-Hadamard based reflection patterns \\
\hline
\centering
\cite{Huang_117} & MU-DL-MISO & Continuous & Perfect & Sum rate & Majorization-minimization approach\\
\hline
\centering
\cite{guo2019weighted} & MU-DL-MISO & Continuous/Discrete & Perfect & Weighted sum rate & Iterative algorithms with closed-form expressions\\
\hline
\centering
\cite{Jung_96} & MU-DL-MISO & Discrete & Perfect & Sum rate & Interference-free modulation scheme\\
\hline
\centering
\cite{Mu_124} & MU-DL-MISO NOMA & Continuous/Discrete & Perfect & Sum rate & Sequential rank-one constraint relaxation approach\\
\hline
\centering
\cite{Zhao_TwoTime} & MU-DL-MISO & Discrete & Statistical CSI & Sum rate & Low channel training overhead\\
\hline
\centering
\cite{Nadeem_128} & MU-DL-MISO & Continuous & Statistical CSI & Max-min SINR & Signal exchange overhead reduction\\
\hline
\centering
\cite{Yang_125} & MU-DL-SISO/MISO NOMA & Continuous & Perfect & Max-min rate & Near-optimal NOMA user ordering scheme\\
\hline
\end{tabular}
}
\end{center}
\label{table:joint beamforming}
\end{table*}
All the aforementioned research contributions on the joint transmit and passive beamforming design are summarized in Table \ref{table:joint beamforming}. ``SU" and ``MU" represent single-user and multi-user, respectively. ``DL" and ``UL" represent downlink and
uplink, respectively.
\subsubsection{Approaches for passive beamforming design} An example of the joint transmit and passive beamforming design problem can be formulated as follows
\begin{subequations}\label{beamforming design}
\begin{align}
\mathop {\min /\max}\limits_{{\mathbf{w}},{\bm{\theta }}} &\;f\left( {{\mathbf{w}},{\bm{\theta }}|{\mathcal{H}}} \right) \\
\label{transmit}{\rm{s.t.}}\;\;&{\mathbf{w}} \in {\mathcal{T}},\\
\label{passive}&{\bm{\theta }} \in {\mathcal{P}},
\end{align}
\end{subequations}
where ${\mathcal{H}}$ denotes the set of given CSI, ${\mathbf{w}}$ and ${\bm{\theta }}$ denote the transmit beamforming vector and passive beamforming vector, receptively, ${\mathcal{T}}$ and ${\mathcal{P}}$ denote the corresponding feasible set for ${\mathbf{w}}$ and ${\bm{\theta }}$, respectively. Here, $f\left( {{\mathbf{w}},{\bm{\theta }}|{\mathcal{H}}} \right)$ denotes the objective function that depends on ${\mathbf{w}}$ and ${\bm{\theta }}$ given the CSI. Let ${\theta _i} = {\beta _i}{e^{j{\phi _i}}}$ be the $i$th element of the passive beamforming vector ${\bm{\theta }}$. Depending on the specific implementation of the RIS, three case studies can be considered~\cite{Mu_124}.
\begin{itemize}
\item \textbf{Continuous amplitude and phase shift}: In this case, it is assumed that the amplitude and phase shift of each RIS element can be adjusted continuously, which results in the following feasible set.
\begin{align}\label{phi 1}
{\mathcal{P}_1} = \left\{ {{\beta _i},{\phi _i}|{\beta _i} \in \left[ {0,1} \right],{\phi _i} \in \left[ {0,2\pi } \right)} \right\}.
\end{align}
\item \textbf{Constant amplitude and continuous phase shift}: In this case, it is assumed that the amplitude and phase shift of each RIS element are fixed, e.g., $\beta_i=1$, and can be adjusted continuously, respectively. The corresponding feasible set is given by
\begin{align}\label{phi 2}
{\mathcal{P}_2}= \left\{ {{\beta _i},{\phi _i}|{\beta _i} = 1,{\phi _i} \in \left[ {0,2\pi } \right)} \right\}.
\end{align}
\item \textbf{Constant amplitude and discrete phase shift}: In this case, it is assumed that the amplitude and phase shift of each RIS element are fixed, e.g., $\beta_i=1$, and can be adjusted based on a discrete set of values, respectively. The feasible set can be expressed as
\begin{align}\label{phi 3}
{\mathcal{P}_3} = \left\{ {{\beta _i},{\phi _i}|{\beta _i} = 1,{\phi _i} \in {{\mathcal{D}}}} \right\},
\end{align}
where ${\mathcal{D}} = \left\{ {0,\frac{{2\pi }}{N}, \cdots ,\left( {N - 1} \right)\frac{{2\pi }}{N}} \right\}$ and $N$ denotes the number of candidate phase shifts.
\end{itemize}
It is worth noting that the first two case studies are difficult to realize in practice. Due to the hardware constraints in fact, it is quite challenging to realize continuous amplitude and phase shift control. However, the first two case studies can be used to characterize the theoretical performance upper bounds of RISs.
By inspection of the three case studies just considered, we evince that the joint beamforming optimization problem is generally a non-convex problem since ${\mathbf{w}}$ and ${\bm{\theta }}$ are coupled together. The existing algorithms for solving the non-convex joint beamforming optimization in RIS-assisted wireless networks are mainly based on the AO method. The advantage of this approach is that, given the passive beamforming vector, the transmit beamforming design becomes a conventional problem, which has been extensively investigated. However, the passive beamforming design under given transmit beamforming vectors is still a non-trivial task to tackle. The main challenges to solve the problem include the unit modulus constraint and the discrete nature of the feasible set. In the following, we list the approaches employed in current research contributions for optimizing the passive beamforming. Table \ref{table:approach} summarizes the characteristics of those approaches.
\begin{table*}[htbp]\scriptsize
\caption{Summary of approaches to passive beamforming design}
\begin{center}
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
\centering
\textbf{Approaches} & \textbf{Phase shifts} & \textbf{Advantages} &\textbf{Disadvantages} & \textbf{Ref.} \\
\hline
\centering
SDR & Continuous & Relax to convex problem & Require rank-one solution construction & \cite{wu2019intelligent,Zhu_NOMA}, etc. \\
\hline
\centering
Quantization method & Discrete & Easy to implement & Substantial performance loss& \cite{Yang_125} \\
\hline
\centering
Branch-and-bound & Continuous/Discrete & Optimal solution & Relatively high complexity & \cite{Wu_138,Yu_optimal} \\
\hline
\centering
Iterative algorithms & Continuous/Discrete & Good complexity-performance tradeoff & Performance depends on initialization & \cite{Wu_138,Fu_142}, etc. \\
\hline
\end{tabular}
\end{center}
\label{table:approach}
\end{table*}
\begin{itemize}
\item \textbf{SDR:} A common method for handling the non-convex unit-modulus constraint is to transform the passive beamforming vector into a rank-one and positive semi-definite matrix. By applying the SDR approach, which ignores the non-convex rank-one constraint, the original non-convex problem becomes a convex SDP problem that can be solved by using many efficient convex optimization tools. If the obtained matrix solution is not rank-one, Gaussian randomization methods are usually used. However, the constructed rank-one solution is, in general, suboptimal and it may be even infeasible for the original passive beamforming design problem, which not only causes some performance degradation but cannot guarantee the convergence of the AO-based iterative algorithm either.
\item \textbf{Quantization method:} Under the assumption of finite resolution phase shifts, one method is to relax each discrete phase shift variable ${\phi _i} \in \left\{ {0,\frac{{2\pi }}{N}, \cdots ,\left( {N - 1} \right)\frac{{2\pi }}{N}} \right\}$ into a continuous variable ${\phi _i} \in \left[ {0,2\pi } \right)$. After solving the relaxed problem, the obtained continuous solutions are quantized to their nearest discrete values. However, the quantization method may lead to a substantial performance loss, especially for low-resolution phase shifts. Additionally, it is worth mentioning that the non-convex unit-modulus constraints still exist after applying the continuous relaxations.
\item \textbf{Branch-and-bound:} Due to the non-convex nature of the passive beamforming design problem, it is challenging to obtain the optimal solution with standard convex optimization techniques. The branch-and-bound approach has been applied for solving polynomial-time (NP)-hard discrete and combinatorial optimization problems and some specific continuous optimization problems. For example, the branch-and-bound approach was adopted for deriving the optimal solution of the discrete passive beamforming design~\cite{Wu_138} and the continuous passive beamforming design~\cite{Yu_optimal}.
\item \textbf{Iterative algorithms:} The main idea of iterative algorithms is to obtain a locally optimal or a high-quality suboptimal solution for the original problem at an acceptable computational complexity. Some iterative algorithms were developed for the passive beamforming design, such as the successive refinement algorithm~\cite{Wu_138,Zhang_Capacity,You_113,guo2019weighted}, the alternating DC algorithm~\cite{Fu_142}, the conjugate gradient search~\cite{huang2019reconfigurable}, the fixed point iteration method and the manifold optimization method~\cite{Yu140}, and the sequential rank-one constraint relaxation approach~\cite{Mu_124}. It was shown that these proposed iterative algorithms can achieve a good tradeoff between performance and computational complexity.
\end{itemize}
\subsection{Resource Management in RIS-enhanced Networks}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=3.5in]{resource_allocation.pdf}
\setlength{\abovecaptionskip}{-0cm}
\caption{Illustration of resource management in large-scale RIS-assisted networks.}
\label{resource-management}
\end{center}
\end{figure}
\subsubsection{Resource allocation problems} In Fig. \ref{resource-management}, a large-scale RIS-assisted transmission scenario is considered, where multiple BSs serve multiple users with the aid of multiple RISs. In this context, several key issues need to be discussed.
\begin{itemize}
\item \emph{Subchannel assignment:} The bandwidth efficiency can be improved by properly allocating users to different subchannels. If the RIS elements are not frequency-selective, one single common RIS reflection matrix needs to be shared among all subchannels, which makes the resulting optimization problems challenging to solve. To address this difficulty, Yang {\em et al.}~\cite{Yang_129} proposed a dynamic passive beamforming scheme, where the resource blocks are dynamically allocated to different user groups with varied RIS phase shifts over different time slots. In~\cite{Zuo_NOMA}, Zuo {\em et al.} investigated the joint subchannel assignment, power allocation, and passive beamforming design problem in a multi-channel downlink RIS-NOMA network.
\item \emph{User-RIS association:} In multi-RIS assisted multi-user communications, how to associate the users to different RISs is an interesting problem. The user-RIS association schemes in general determine the overall network performance. Considering a multi-RIS assisted massive MIMO system, Li {\em et al.}~\cite{Li_126} found that the automatic interference cancelation property holds for RISs with infinitely large sizes. Then, the considered max-min SINR problem can be transformed into a user-RIS association problem, which was efficiently solved by the proposed greedy search algorithm.
\item \emph{Multi-cell RIS association:} In multi-cell scenarios, the optimization problem becomes much sophisticated for jointly considering the user-BS association, the user-RIS association, and the subchannel assignment. In some initial works considering multi-cell scenarios~\cite{pan2019Intelligent,Xie_multi}, RISs were deployed to enhance the performance of cell-edge users for simultaneously improving the received desired signal power and mitigating the received interference from other cells.
\end{itemize}
\subsubsection{Approaches to resource allocation problems} Scheduling different users with different subchannels/RISs/BSs is an NP-hard problem. Though the optimal solution can be obtained by exhaustively searching over all possible association combinations, it requires a prohibitively high computational complexity, especially for the large-scale networks in Fig. \ref{resource-management}. Therefore, low complexity and efficient algorithms have to be developed for striking a performance-versus-complexity tradeoff. In the following, we present some promising approaches, and discuss their advantages and disadvantages. Table \ref{table:approach RA} summarizes the characteristics of those approaches.
\begin{table*}[htbp]\scriptsize
\caption{Summary of approaches to resource allocation problems}
\begin{center}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
\centering
\textbf{Approaches} & \textbf{Advantages} &\textbf{Disadvantages} & \textbf{Ref.} \\
\hline
\centering
Binary relaxation & Relax to convex feasible set & Existence of performance gap & - \\
\hline
\centering
Matching theory & Achieve near-optimal performance & Require predefined preference list& \cite{Zuo_NOMA} \\
\hline
\centering
Heuristic algorithms & Flexible complexity-performance tradeoff & Unstable performance & \cite{Li_126} \\
\hline
\end{tabular}
\end{center}
\label{table:approach RA}
\end{table*}
\begin{itemize}
\item \textbf{Binary relaxation:} One idea is to relax the binary variable $\alpha \in \left\{ {0,1} \right\}$ into the continuous variable $\alpha \in \left[ {0,1} \right]$, where $\alpha $ represents the user association state. By doing so, the non-convex integer constraint is relaxed to be convex, and conventional convex optimization techniques can be applied for solving the relaxed problem. It is worth pointing out that the relaxed problem might still be non-convex especially when the optimization variables are highly-coupled. Additional efforts, such as utilizing the successive convex approximation (SCA) method, are required to obtain an approximate solution. Moreover, this kind of relaxation may result in a considerable performance loss between the original integer problem and the relaxed one.
\item \textbf{Matching theory:} Matching theory is a powerful method developed for solving the combinatorial user association problems. The user association combinational optimization problem in RIS-enhanced networks can be modeled as a high dimensional Users-BSs-RISs-Subchannels matching problem. Though high dimensional matching is NP-hard, it can be decomposed into several 2D matching subproblems, which can be efficiently solved. For example, Zuo {\em et al.}~\cite{Zuo_NOMA} applied many-to-one matching theory for the subchannel assignment in an RIS-enhanced NOMA system, which can achieve a near-optimal performance. However, leveraging matching theory requires to establish a predefined preference list for both users and resources. As the channel conditions always fluctuate in RIS-enhanced networks, these preference lists may need to be dynamically updated, which needs further investigations.
\item \textbf{Heuristic algorithms:} For solving computationally complex problems, one commonly employed method is to develop heuristic algorithms, where approximate solutions for the original optimization problem can be obtained with an acceptable computational complexity. In the aforementioned works, the greedy search based heuristic algorithm was designed for solving the user-RIS association problem~\cite{Li_126}. However, the performance of heuristic algorithms is sensitive to the designed strategies, which is not always stable.
\end{itemize}
\subsection{Discussions and Outlook}
With the growing number of research contributions on RIS-enhanced communications, the advantages of RISs have been verified in terms of SE, EE, and user fairness. However, most of the existing treatises solved the non-convex joint beamforming optimization problem employing the AO method, which decouples the joint transmit and passive beamforming design into two subproblems. Though a high-quality suboptimal solution can be obtained, advanced optimization techniques are required for solving this problem to characterize the optimal performance gain introduced by RISs, which also provides an important benchmark for verifying the optimality of any other low complexity suboptimal algorithms. Furthermore, in RIS-enhanced communication systems, it is known that obtaining accurate CSI is rather challenging due to the nearly passive working mode of RISs. Investigating the robust joint beamforming design and resource allocation~\cite{Zhou_141,Zhou_IP} constitutes an important research direction for practical RIS implementations.
Besides passive beamforming, RISs also introduce the following DoFs, which can be further exploited to reap the benefits of RISs in the future work.
\indent
\begin{itemize}
\item \emph{RIS deployment design:} The reflection link via the RIS experiences a severer path loss than the direct link. Therefore, the deployment location of the RIS has to be carefully designed to achieve considerable performance enhancements. How to jointly optimize the passive beamforming and the deployment location at the RIS as well as the wireless resource allocation at the AP/BS is a non-trivial task, which deserves further research efforts. In particular, one of prominent challenges is that the deployment location of the RIS determines both the path loss and the LoS components of the reflection channels, which causes the optimization variables to be highly-coupled. Therefore, efficient algorithms need to be designed. An initial study~\cite{Mu_deployment} has investigated the optimal RIS deployment strategy for both NOMA and OMA transmissions, which showed that asymmetric and symmetric RIS deployment locations among users are preferable for NOMA and OMA, respectively. Additionally, the RIS is usually deployed to avoid signal blockage to achieve a LoS dominated channel, thus having a small path loss. However, such a LoS channel based RIS deployment strategy may be ineffective, especially for RIS-assisted multi-user communications. This is because the LoS dominated channels are low-rank, the resulting ill-condition channel matrices significantly limit the achievable capacity even with a relatively small path loss. How to deploy the RIS to strike a balance between the path losses and the Non-LoS (NLoS) components of channels is another interesting problem, which deserves further research interests.
\item \emph{Dynamical RIS configuration:} In \cite{Karasik_Joint,Mu_Capacity}, the authors revealed that, from an information-theoretic perspective, the capacity-achieving transmission schemes need to dynamically adjusting the RIS. However, most of the existing research contributions assumed that the RIS reflection coefficients can be only adjusted once for each channel coherence duration. Note that one of the most typical application scenarios of RISs is to assist transmission of the users, who are static or moving slowly in the vicinity of RISs. In this case, the duration of one channel coherence block is usually tens of milliseconds, which is much larger than the time duration for adjusting the RIS (e.g., 220 microseconds in~\cite{arun2020rfocus}). Therefore, adjusting RIS multiple times in one channel coherence duration, namely the dynamic RIS configuration, is practically valid. This unique characteristic opens up new research opportunities, such as dynamical passive beamforming and resource allocation schemes, which merit further investigations.
\end{itemize}
\section{Machine Learning for RIS-enhanced Communication Systems}
ML techniques have gained remarkable interests in wireless communications due to their learning capability and large search-space~\cite{gacanin2020wireless,Xiao2020Artificial,wang2020artificial}. We survey existing research contributions, which apply ML techniques for tackling challenges in RIS-enhanced wireless networks. Finally, potential research challenges and opportunities of ML-empowered RIS systems are presented.
\subsection{Motivations and Architecture for Integrating ML in RIS-enhanced Wireless Networks}
In this subsection, we first present the challenges of the conventional RIS-enhanced wireless networks and the motivations for integrating ML in these networks, followed by the system architecture of ML-empowered RIS-enhanced wireless networks.
To effectively exploit RISs for optimizing wireless networks, preliminary research contributions have studied a number of technical challenges that include channel estimation/modeling, joint transmit and passive beamforming design, as well as resource allocation from the BS to the users. Powerful optimization techniques, such as convex optimization~\cite{guo2019weighted}, iterative algorithm~\cite{wu2019intelligent}, gradient descent approach~\cite{huang2019reconfigurable}, and alternating optimization algorithm~\cite{shen2019secrecy} have been adopted for addressing the aforementioned fundamental challenges. Although important insights have been gained by these research contributions. the following limitations still exist in conventional RIS-enhanced wireless networks:
\begin{itemize}
\item The users are generally assumed to be static for simplicity, i.e., the dynamic mobility of users is typically ignored. Another limitation in the existing literature is that the communication environment is assumed to be perfectly known, the differentiation of users' demand is always ignored as well.
\item The RIS/BS are not capable of learning from the unknown environment or from the limited feedback of the users. In practical applications of RISs in wireless networks, the system parameters are treated as random variables, which naturally leads itself to the derivation of insightful joint probability distributions conditioned on the users' tele-traffic demand and mobility. However, this is a highly dynamic stochastic environment, which is difficult for employing conventional optimization approaches. Additionally, the feedback from the users is usually resource-hungry and limited, which aggravates the challenges for the conventional RIS-enhanced wireless networks.
\item Finally, instantaneous CSI of all the channels are assumed to be available at the BS. However, CSI acquisition in RIS-enhanced wireless networks becomes more challenging than that in the conventional relay systems due to the passive nature of RISs, which also aggravates the challenge imposed on the conventional RIS-enhanced wireless networks.
\end{itemize}
\subsection{Deep Learning for RIS-enhanced Communication Systems}
Deep learning (DL) has shown great potentials to revolutionize communication systems. It can be applied in diverse areas of RIS-enhanced wireless networks due to its powerful learning capabilities~\cite{zappone2019model,Qin2019Deep,zappone2019wireless}.
The acquisition of timely and accurate CSI plays a pivotal role in wireless systems, especially in MIMO networks. However, CSI acquisition becomes more challenging due to the large number of antennas in massive MIMO systems~\cite{wen2018deep}. In order to tackle this challenge, a number of research contributions have adopted DL for estimating the CSI, especially for exploiting CSI structures beyond linear correlations.
In contrast to the conventional AF relay-aided wireless networks, in RIS-enhanced wireless networks, the RIS is a passive device, which is not capable of performing active transmission/reception and signal processing~\cite{chen2019channel}. In an effort to estimate a large number of unknown parameters caused by RISs, Taha {\em et al.}~\cite{taha2019enabling} exploited the DL method for learning the RIS reflection matrices directly from the sampled channel knowledge without any knowledge of the RIS array geometry. Liu {\em et al.}~\cite{liu2020deep} proposed a deep denoising neural network assisted compressive channel estimation for RIS-assisted mmWave systems with a low training overhead. Elbir {\em et al.}~\cite{elbir2020deep} presented a DL framework for channel estimation in the RIS-enhanced MIMO system. It was shown that the proposed convolutional neural networks
(CNNs)-based approach achieves lower normalized mean-square-error (NMSE) and more robust performance than other benchmarks.
The data-driven DL approach has the advantage of model-free representation or function learning such that no explicit models of the complicated wireless channels are needed, at the expense of requiring large amounts of training data and corresponding computational power. Thus, the DL method can be adopted for estimating the CSI of RIS-enhanced wireless networks.
Apart from the aforementioned applications of DL in RIS-enhanced wireless networks, Huang {\em et al.}~\cite{huang2019indoor} leveraged a deep neural network (DNN)-based approach in the indoor communication environment for estimating the mapping between a user's position and the configuration of the RIS to maximize the received SNR. Additionally, DL can also be applied for learning the optimal RIS phase shift configuration. Gao {\em et al.}~\cite{gao2020unsupervised} proposed a DL-based algorithm for optimally designing the phase shift of the RIS by training the DL offline. It can be observed that the proposed unsupervised learning mechanism outperforms the conventional optimization approach in terms of computational complexity. Khan {\em et al.}~\cite{khan2019deep} investigated the signal estimation and detection in the RIS-enhanced wireless networks. A DL-based approach was proposed for estimating channels and phase angles from a reflected signal received by an RIS. With the aid of DL, the bit-error-rate (BER) performance of the system was improved.
\subsection{Reinforcement Learning for RIS-enhanced Communication Systems}
Reinforcement learning (RL) is a powerful AI paradigm that can be used to empower agents by interacting with the environment. More explicitly, by exploiting the learning capability (e.g., learning from the environment, learning from the feedback of users, and learning from its mistakes) of the RL model, the challenges encountered in the conventional RIS-enhanced wireless networks may be mitigated, thus leading to improved performance.
The core idea of employing RL techniques in the RIS-enhanced wireless networks is that they allow the BS/RIS to improve their service quality by learning from the environment, from their historical experience, and from the feedback of the users~\cite{liu2019trajectory}. More explicitly, RL models can be used for supporting the BS/RIS (agents) in their interactions with the environment (states), whilst finding the optimal behavior (actions) of the BS/RIS. Furthermore, the RL model can incorporate farsighted system evolution (long-term benefits) instead of only focusing on current states. Thus, it is applied for solving challenging problems in the RIS-enhanced wireless networks.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=16cm]{RL.pdf}
\caption{Key features, pros and cons of RL algorithms~\cite{liu2019trajectory,van2016deep,wang2015dueling,fortunato2017noisy,dabney2018distributional,mnih2016asynchronous,schulman2017proximal,hamalainen2018ppo,lillicrap2015continuous,haarnoja2018soft,luong2019applications}. (PG represents Policy Gradient, TRPO denotes Trust Region Policy Optimization, PPO represents Proximal Policy Optimization, AC denotes Actor-Critic, A3C represents Asynchronous Advantage Actor-Critic, DDPG denotes Deep Deterministic Policy Gradient, TD3 represents Twin Delayed DDPG, SAC denotes Soft Actor-Critic)}
\label{RL}
\end{center}
\end{figure*}
As illustrated in Fig.~\ref{RL}, the RL algorithms can be divided into three categories, namely, value-based algorithms, policy-based algorithms, and actor-critic algorithms. Both advantages and disadvantages exist in the RL algorithms. Since RISs have discrete phase shifts, the DQN algorithm is more suitable for tackling the corresponding phase shift design problem.
To fully reap the benefits of deploying RISs in wireless networks, the joint transmit and passive beamforming design of the RIS-enhanced system has been considered in MISO systems~\cite{huang2020reconfigurable,feng2020deep}, OFDM-based systems~\cite{taha2020deep}, wireless security systems~\cite{Helin2020Deep} and millimeter wave systems~\cite{zhang2020millimeter} with the aid of RL algorithms. In contrast to the AO method, which alternately optimizes the transmit beamforming at the BS and the passive beamforming at the RIS, the RL-based solution is capable of simultaneously designing them. More explicitly, Huang {\em et al.}~\cite{huang2020reconfigurable} applied a deep deterministic policy gradient (DDPG) based algorithm for maximizing the throughput by utilizing the sum rate as instant rewards for training the DDPG model. In the proposed model, the continuous transmit beamforming and RIS phase shift were jointly optimized with low complexity. Taha {\em et al.}~\cite{taha2020deep} proposed a deep reinforcement learning (DRL) based algorithm for maximizing the achievable communication rate by directly optimizing interaction matrices from the sampled channel knowledge. In the proposed DRL model, only one beam was utilized for each training episode. Thus, the training overhead was avoided, while the dataset collection phase was not required. Zhang {\em et al.}~\cite{zhang2020millimeter} presented a DRL based algorithm for maximizing the throughput with both perfect and imperfect CSI. A quantile regression method was applied for modeling a return distribution for each state-action pair, which modeled the intrinsic randomness in the MDP interaction between the RIS and communication environment. Helin {\em et al.}~\cite{Helin2020Deep} considered the application of RISs to PLS. The system secrecy rate was maximized with the aid of the DRL model by jointly optimizing the beamforming and phase shift matrices under different users' QoS requirements and time-varying channel conditions. Additionally, post-decision state and prioritized experience replay schemes were utilized to enhance the learning efficiency and secrecy performance.
\subsection{A Novel Architecture of ML-empowered RIS-enhanced Wireless Networks}
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=16cm]{architecture.pdf}
\caption{Architecture of ML-empowered RIS-enhanced wireless networks.}
\label{architecture}
\end{center}
\end{figure*}
As a benefit of the ML-based framework, many challenges in conventional wireless communication networks have been circumvented, leading to enhanced network performance, improved reliability, and agile adaptivity~\cite{Xiao2020Artificial}. Fig.~\ref{architecture} illustrates a novel ML-empowered architecture for RIS-enhanced wireless networks. As shown in this figure, RISs are installed on the facade of a building for enhancing the wireless performance~\cite{di2019smart,Qingqing2020Towards}. The RIS is linked with a controller, which controls the reflecting elements for hosting the functionality of phase-shifting and amplitude absorption. A two-step approach is applied in the proposed ML-empowered RIS-enhanced wireless networks.
\begin{itemize}
\item As illustrated in the data collection, data processing, and feature extraction parts in Fig.~\ref{architecture}. The associated user information (e.g., device type, position, data rate demand, mobility, caching demand, and computing ability) is collected, stored, and processed. Thus, the users' behaviors and requirements can be predicted for efficiently deploying and operating the RIS. Meanwhile, the predicted information can be modified online with the currently collected data as the input.
\item Given the extracted features, adaptive schemes are leveraged for controlling the RISs, designing the phase shifts, resource allocation, and interference cancellation.
\end{itemize}
In the proposed ML-empowered RIS-enhanced wireless networks, RISs are capable of rapidly adapting to the dynamic environment by learning both from the environment and from the feedback of the users.
Research on the RIS deployment is fundamental and essential. However, there is a paucity of research on the problem of RIS position determination. Additionally, current research contributions mainly consider the performance optimization for both single-user and multi-user scenarios by optimizing the phase shift and/or precoding solutions of the RIS-enhanced communication systems~\cite{Nadeem_128,ye2019joint,liang2019large,jung2019performance,pan2019Intelligent}.
Considering the RIS deployments based on the users' mobility information and particular data demand implicitly assumes that the long-term movement information and tele-traffic requirement of users are capable of being learned/predicted. With this proviso, the deployment and control method of RISs may be designed periodically for maximizing the long-term benefits and hence reducing the control overhead. By considering the long-term mobility and data demand of users, RIS-enhanced wireless networks become highly dynamic systems. Meanwhile, in an effort to maximize the service quality in an unknown environment, RISs are supposed to learn by interacting with the environment and adapting the control/deployment policy based on the limited feedback of the users to overcome the uncertainty of the environment.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=14cm]{DQN.pdf}
\caption{DRL model for RIS networks.}
\label{DQN}
\end{center}
\end{figure*}
In this subsection, an RL-based model is presented to jointly design the deployment policy and phase shift policy of RISs while considering the time-varying data demand of users. As illustrated in Fig.~\ref{DQN}, in the RL-based model, the BS acts as an agent. Since a controller is installed, the BS can control both resource allocation policy for users and the RIS's position and phase shifts. At each timeslot, the BS periodically observes the state of the RIS-enhanced system. The state space consists of the RIS phase shifts, the allocated power to each user, as well as coordinates of both the RIS and users. An action is carried out by the BS for selecting the optimal control policy. The actions contain changing positions and varying phase shifts of the RIS, as well as varying the allocated power. The key underlying principle of the decision policy is carrying out an action that makes the DQN model obtain the maximum Q-value at each time slot. Following each action, the BS receives a penalty/reward, $r_t$, determined by the formulated objective function.
\begin{itemize}
\item \textbf{State of the RL model.} The state space consists of four parts: 1) the current phase shift of each reflecting element at the RIS; 2) the current 3D position of the RIS; 3) the current 2D position of each user; 4) the current power allocated from the BS to each user.
\item \textbf{Action of the RL model.} The action space consists of three parts: 1) the variable quantity of the $n$-th reflecting element's phase shift; 2) the moving direction and distance of the RIS; 3) the variable quantity of the $k$-th user's transmit power.
\item \textbf{Reward of the RL model.} The reward function is decided by the EE of the system. When action taken by the BS improves EE, the BS obtains a reward. Otherwise, when a reduction occurs in EE, the BS receives a penalty.
\end{itemize}
\begin{figure} [t!]
\centering
\includegraphics[width=3.3in]{EEvsdeploy.pdf}
\caption{EE with and without RIS~\cite{liu2020ris}.}\label{EEvsnoIRS}
\end{figure}
Fig.~\ref{EEvsnoIRS} characterizes the EE of the system in networks both with and without the assistance of an RIS. The EE is defined as the ratio between the system achievable sum mean opinion score (MOS) and the sum energy dissipation in Joule. It was shown that the EE of the system is enhanced by employing an RIS. The RIS-barycenter line indicates that the RIS is placed at the barycenter of all users. The RIS-random line indicates that the RIS is randomly deployed, while the RIS-optimal line indicates that the RIS is deployed at the optimal position derived from the proposed decaying double deep Q-network (${{\text{D}}^{\text{3}}}{\text{QN}}$) algorithm. The results of Fig.~\ref{EEvsnoIRS} confirm that there exists an optimal position for the RIS as far as the EE of the RIS-enhanced system is concerned. The performance of the RIS-enhanced system is improved by deploying the RIS at the optimal position compared to the random deployment strategy and the strategy of placing it at the barycenter.
\subsection{Other ML Techniques for RIS-enhanced Communication Systems}
Besides DL and RL techniques, a range of supervised learning and unsupervised learning algorithms have been applied in the current generation wireless networks. Thus, these approaches can also be adopted for tackling challenges in the RIS-enhanced wireless systems.
\subsubsection{Supervised Learning techniques for RIS-enhanced Communication Systems}
As one of the key branches of ML, powerful supervised learning techniques, such as regression, decision tree and random forest, K-nearest neighbors (KNN), support vector machines (SVM), and Bayes classification, have been adopted in diverse scenarios for tackling challenges such as spectrum sensing~\cite{umebayashi2017efficient}, traffic/QoE prediction~\cite{feng2017proactive}, channel/antenna selection~\cite{thilina2015dccc}, and networking association~\cite{abouzar2011action}. In the RIS-enhanced wireless networks, supervised learning algorithms can also be applied for solving the related problems with sufficient training data due to their advantages of low complexity and fast convergence speed.
\subsubsection{Unsupervised Learning techniques for RIS-enhanced Communication Systems}
In contrast to the supervised learning techniques, unsupervised learning methods do not rely on prior knowledge, which is not data-hungry. Thus, the unsupervised learning algorithms~\cite{wang2019thirty} such as K-means clustering, expectation-maximization, principal component analysis (PCA), and independent component analysis (ICA) can be applied in the RIS-enhanced wireless networks for tackling challenges such as BS deployment, user clustering/association~\cite{liu2019reinforcement}, channel/network state detection~\cite{assra2015approach}, data aggregation~\cite{morell2016data}, and interference cancellation~\cite{li2017digital}.
\subsubsection{Federated Learning techniques for RIS-enhanced Communication Systems}
Federated learning, which explores training statistical models directly on remote devices, has become a focal point in the area of large-scale ML and distributed optimization~\cite{niknam2019federated}. Since the federated learning algorithm is trained at the edge in distributed networks, the inaccessibility of private data is no longer a problem. Due to its privacy-preserving nature, federated learning algorithms can be applied for the deployment and design of multiple RISs, where each RIS can act as a distributed learner, trains its generated data and transfers its local model parameters instead of the raw training dataset to an aggregating unit. Thus, the deployment and design policy can be learned in a decentralized manner.
\subsection{Discussions and Outlook}
By exploiting ML learning capabilities, the aforementioned challenges encountered in RIS-enhanced wireless networks may be mitigated. This is due to the reason that RISs can learn by interacting with the environment and adapt the control/deployment policy based on the feedback of users to overcome the dynamic/uncertainty of the environment. It can be learned that the RL model can incorporate farsighted system evolution instead of only focusing on current states, which can reap long-term benefits for RIS-enhanced wireless networks. However, ML models will also pose some new challenges, including the layer design for the DL model, the state-action construction, and the reward function design for the RL model. In addition, simultaneously employing multiple RISs becomes more challenging due to the cooperation amongst RISs. Hence, intelligent deployment and design for multi-RIS enhanced wireless networks is highly desired. Finally, in the current ML models, either discrete or continuous state space is modeled for optimizing parameters in RIS-enhanced wireless networks, while joint discrete and continuous parameters exist in the networks. Hence, the joint discrete and continuous state space design in ML-enabled RIS-enhanced wireless networks is still challenging and constitutes an interesting topic.
\section{Integrating RISs with Other Technologies Towards 6G}
Current research contributions have proved that RIS-enhanced wireless networks are capable of obtaining tuned channel gains, improved QoS, enhanced coverage range, and reduced energy dissipation. These significant performance enhancements can be applied to diverse wireless communication networks. In this section, we identify the major issues and research opportunities on the path to 6G associated with the integration of RISs and other emerging technologies, such as NOMA, PLS, SWIPT, UAV-enabled wireless networks, and autonomous driving networks.
\subsection{NOMA and RIS}
In an effort to improve the SE and user connectivity of RIS-enhanced wireless networks, power-domain NOMA technology is adopted, whose key idea is to superimpose the signals of two users at different powers for exploiting the spectrum more efficiently by opportunistically exploring the users' different channel conditions~\cite{Liu_physical_scurity_NOMA,liu2017non}. Li {\em et al.}~\cite{li2019joint} considered a MISO-NOMA downlink communication network for minimizing the total transmit power by jointly designing the transmit precoding vectors and the reflecting coefficient vector. In~\cite{Yang_114}, Yang {\em et al.} jointly optimized the phase shifts matrix of the RIS, as well as the power allocation from the BS to the users. Thus, the minimum decoding SINR of all users was maximized for optimizing the throughput of the system by considering user fairness. Ding {\em et al.}~\cite{RIS_zhiguo_simple} proposed a novel design of RIS assisted NOMA networks. It can be observed in~\cite{RIS_zhiguo_simple} that, the directions of users' channel vectors can be aligned with the aid of the RIS, which emphasizes the importance of implementing NOMA technology. For an RIS-NOMA system, the core challenge is that the decoding order is dynamically changed due to the configuration of phase shifts of the RIS. Mu {\em et al.}~\cite{Mu_124} proposed an RIS-enhanced multiple-antenna NOMA transmission framework to maximize the throughput of the system by considering the NOMA SIC decoding order condition. The SCA technique and sequential rank-one constraint relaxation based algorithm were applied to obtain a locally optimal solution. Ni {\em et al.}~\cite{ni2020resource} proposed a resource allocation framework in multi-cell RIS-NOMA networks, where the achievable sum rate was maximized by solving the joint optimization problem of user association, sub-channel assignment, power allocation, phase shifts design, and decoding order determination.
In contrast to the conventional MIMO-NOMA systems, RIS-NOMA technology can overcome the challenges of the dynamic environment such as random fluctuation of wireless channels, blocking, and user mobility in an energy-efficient manner. The NOMA system can obtain tuned channel gains, improved fair resource allocation, enhanced coverage range, and high EE with the aid of RISs~\cite{sousa2020role}. However, NOMA also gives rise to new challenges when integrated with RISs. For multi-antenna NOMA transmission, the decoding order is not determined by the users' channel gains order, since additional decoding rate conditions need to be satisfied to guarantee successful SIC~\cite{Mu_124}. Additionally, both the active beamforming and passive phase shift design affect the decoding order among users and user clustering, which makes the decoding order design, user clustering, and joint beamforming design highly-coupled in RIS-NOMA networks.
\subsection{PLS and RIS}
It has been shown that RISs are capable of simultaneously enhancing the desired signal power at the intended user and mitigating the interference power at other unintended users~\cite{wu2019intelligent}. Inspired by this result, several researchers explored the potential performance gain in the context of PLS by applying the RIS~\cite{Yu_Secure,Cui_149,shen2019secrecy,Chu_Secure,Dong_Secure,Chen_150,Guan_AN,Yu_151}. Yu {\em et al.}~\cite{Yu_Secure} considered an RIS-enhanced multiple-input single-output single eavesdropper (MISOSE) channel, where the eavesdropper is equipped with a single antenna. The secrecy rate was maximized by jointly optimizing the transmit beamforming and the RIS phase shift matrix by using an AO-based algorithm. It was demonstrated that the secrecy performance can be significantly improved by deploying the RIS. Cui {\em et al.}~\cite{Cui_149} focused on the scenario where the eavesdropper has a better direct channel condition than that of the legitimate receiver and they are also highly correlated in space, where the achievable secrecy rate is rather limited in conventional communications. However, it was shown that the direct signals and the reflected signals can be destructively combined at the eavesdropper with the aid of RISs, thus significantly improving the secrecy rate. The same problem was further investigated in~\cite{shen2019secrecy,Dong_Secure} by considering a multi-antenna eavesdropper or legitimate receiver. Chu {\em et al.}~\cite{Chu_Secure} minimized the transmit power while satisfying the secrecy rate requirement in the RIS-enhanced MISOSE system. Chen {\em et al.}~\cite{Chen_150} studied the minimum secrecy rate maximization problem in the RIS-enhanced multi-user multiple-input single-output multiple eavesdropper (MISOME) system by considering both the continuous and discrete RIS phase shifts. Injecting artificial noise (AN) is an effective technique to enhance the secrecy rate~\cite{AN}. Motivated by this result, Guan {\em et al.}~\cite{Guan_AN} examined the effectiveness of employing AN in an RIS-enhanced MISOME system. The achievable secrecy rate was maximized by jointly optimizing the transmit beamforming, the passive beamforming, and AN. The results verified the necessity of using AN, especially for systems with a large number of eavesdroppers. Yu {\em et al.}~\cite{Yu_151} considered an RIS-enhanced multi-user MISOME system under imperfect CSI with the aim of maximizing the sum rate, subject to the maximum information leakage constraint. An efficient AO-based algorithm was developed to optimize the transmit beamforming, the AN covariance matrix, and the RIS phase shifts. Numerical results showed that significant secrecy performance gains can be achieved by the RIS.
One critical issue of the RIS-enhanced PLS is that the joint design of transmit and passive beamforming requires the CSI of both AP-eavesdropper and RIS-eavesdropper links, which is quite challenging to obtain. This is because besides the nearly passive working mode of RISs, in practice, eavesdroppers usually stay almost silent to hide their positions and only detect signals in the air. Therefore, robust joint beamforming designs under the imperfect CSI of the eavesdropper are essential to guarantee secure transmission. Moreover, given the uncertainty of eavesdroppers, deploying the RIS may increase the probability of information leakage since the eavesdropper can receive not only the direct signals from the AP but also the reflected signals from the RIS. The situation may become even worse when there are multiple cooperative eavesdroppers. In this case, setting a protected zone to establish an eavesdropper-exclusion area with carefully deployed RISs would help to enhance the secrecy performance, which deserve further investigations.
\subsection{SWIPT and RIS}
SWIPT is an attractive technique for future IoT networks. However, the low EE at the energy receivers is the main bottleneck in practical SWIPT systems. To overcome this limitation, deploying the RIS is a promising solution and the RIS-assisted SWIPT has been investigated in~\cite{Wu_SWIPT_letter,Tang_SWIPT,Pan_SWIPT,Wu_SWIPT}. In~\cite{Wu_SWIPT_letter}, Wu {\em et al.} investigated an RIS-assisted SWIPT system, subject to individual SINR requirements of information receivers. The weighted sum power received by energy receivers was maximized by jointly optimizing the transmit and passive beamforming with the proposed AO-based algorithm. Moreover, Tang {\em et al.}~\cite{Tang_SWIPT} maximized the minimum received power among energy receivers. The results in~\cite{Wu_SWIPT_letter} and~\cite{Tang_SWIPT} showed that deploying an RIS can improve the energy harvesting efficiency. Pan {\em et al.}~\cite{Pan_SWIPT} studied the weighted sum rate maximization problem in the RIS-assisted SWIPT MIMO system, subject to the energy harvesting requirement of each energy receiver. A block coordinate descent (BCD)-based algorithm was designed to find a Karush-Kuhn-Tucker (KKT) stationary point of the original optimization problem. Wu {\em et al.}~\cite{Wu_SWIPT} extended the RIS-assisted SWIPT system into a multi-RIS case, where the transmit power was minimized while satisfying the different QoS constraints at information users and energy users. It was shown that the RIS enlarges the wireless power transfer range and reduces the number of required energy beams.
Note that the above research contributions studied performance gain of deploying RISs for SWIPT mainly from the communication perspective and ignored the EM characteristic of RISs. As discussed in previous sections, there are substantial differences between the near-field region and the far-field region of RISs. Therefore, sophisticated EM-based wireless power transfer models are required for fully reaping the benefits of RISs, which need to be investigated in future work.
\subsection{UAV and RIS}
RISs can be applied in UAV-enabled wireless networks, where UAVs are employed to complement and/or support the existing terrestrial cellular networks~\cite{wang2018joint,osseiran2014COM}. An RIS enhances the UAV coverage and service quality by compensating for the power loss over long distances, as well as forming virtual LoS links between UAVs and mobile users via passively reflecting their received signals. Li {\em et al.}~\cite{li2019reconfigurable} jointly optimized the UAV trajectory and the RIS phase shifts in an iterative manner. It was shown in~\cite{li2019reconfigurable} that, the average achievable rates of the users were significantly improved with the aid of RISs. On the other hand, RISs can also be applied in UAV-aided wireless relay networks for enhancing performance. Zhang {\em et al.}~\cite{zhang2019reflections} considered the effective placement of a single UAV, which was equipped with an RIS to assist the mmWave downlink transmission while considering user mobility. By jointly designing the UAV trajectory and the RIS reflection parameters, a virtual LoS connection between the BS and users was guaranteed. Thus, both the average data rate and the achievable downlink LoS probability were improved. Yang {\em et al.}~\cite{YangOn2020_UAV} derived the analytical expressions of outage probability, BER, and average capacity by approximating the PDF of the instantaneous SNR in RIS-assisted UAV relaying systems. Mu {\em et al.}~\cite{Mu_UAV} proposed a novel RIS-aided multi-UAV NOMA transmission framework, where an RIS was deployed to enhance the desired signal strength between UAVs with their served ground users while mitigating the inter-UAV interference. Liu {\em et al.}~\cite{Xiao2020RISUAV} integrated UAVs in RIS-enhanced wireless networks for enhancing the service quality of the UAV. With the aid of RISs, the energy consumption of the UAV was significantly reduced.
Due to the fact that UAVs are battery-powered, how to reduce their energy consumption is one of the key challenges. The limited flight-time of UAVs (usually under 30 minutes) hampers the wide commercial roll-out of UAV-aided networking. By deploying RISs, one can adjust the RIS phase shift instead of controlling the UAV movement for forming virtual LoS links between the UAV and the users. Therefore, the UAV can maintain hovering status only when the virtual LoS links can not be formed even with the aid of the RIS. By invoking the aforementioned protocol, the total energy consumption of the UAV is minimized, which in turn, maximizes the UAV endurance. Additionally, by mounting a compact distributed laser charging (DLC) receiver or wireless power transmission (WPT) receiver antenna inside the UAVs, while a DLC/WPT transmitter is deployed on the ground or the building roof, the UAVs can be charged as long as they are flying within the coverage range of the DLC/WPT transmitter~\cite{Liu2016Charging,liu2019trajectory}. However, the LoS connection between UAVs and the charging stations/vehicles have to be guaranteed, which is challenging in the urban scenario when the LoS link between UAVs and charging stations/vehicles are blocked by high-rise buildings with a high probability. RISs are capable of smartly reconfiguring the wireless propagation environment by forming virtual LoS links between UAVs and the charging stations/vehicles via passively reflecting their received signals. Thus, the quality of charging service is enhanced with the aid of the RIS.
The RIS-enhanced UAV communication scenario is naturally a highly dynamic one, which falls into the field of ML. When considering both the trajectory design of UAVs and the phase shift design of RISs, the former one can be formed as a continuous state space while the latter one is usually formed as a discrete one. Hence, how to simultaneously deal with both continuous and discrete state space is challenging in ML-empowered RIS-enhanced UAV networks.
\subsection{Autonomous Driving/Connected Vehicles and RIS}
RISs can also be deployed in vehicle-to-infrastructure (V2I) assisted autonomous driving systems, where V2I components are employed to complement the costly onboard units (OBUs). V2I networks enable autonomous vehicles (AVs) or connected vehicles (CVs) to receive reliable real-time traffic information from BSs, the information is collected by roadside base stations (RBSs) and transmitted from RBSs to BSs, which facilitates the interaction among AVs/CVs and road users, hence enhancing their safety and traffic efficiency~\cite{yao2018v2x,vivacqua2018self,liu2020enhancing,williams2018information}. Since AVs/CVs quality and reliability are non-negotiable, the AVs/CVs system must be real-time, while the transmission is supposed to be 100\% reliable. However, the service quality of current V2I communication systems cannot be guaranteed due to the complex channel terrain in the urban environment and the complexity of road conditions, such as bad weather. Makarfi {\em et al.}~\cite{Makarf2020Reconfigurable} and Wang~\cite{Wang2020Outage} proved that the performance of vehicular networks can be significantly improved with the aid of RISs. Since the RISs are made of EM material, which can be installed on key surfaces, such as building facades, highway polls, advertising panels, vehicle windows, and even pedestrians' clothes. With the massive deployment of RISs, a virtual LoS connection between the BSs and AVs, as well as between the RSUs and BSs will be guaranteed, which enhances the reliability of V2I communications.
In RIS-enhanced autonomous driving systems, the driving safety of AVs is the primary consideration. In terms of safety, collisions have to be avoided, while the traffic rules also need to obey. Additionally, in RIS-enhanced V2I-assisted autonomous driving systems, the wireless service quality for AVs has to be guaranteed at each timeslot. Hence, how to improve the reliability of RIS-enhanced autonomous driving systems is an open and challenging problem.
\subsection{Discussions and Outlook}
The studies of RISs have unveiled promising research opportunities, such as NOMA, PLS, SWIPT, UAV-enabled wireless networks, and autonomous driving networks. Recent research contributions have proved that RIS-enhanced wireless networks can achieve tuned channel gains, improved QoS, enhanced coverage range, and reduced energy dissipation. However, the network and beamforming designs are highly coupled due to dynamic control of the RIS phase shifts, which brings challenges to these new research directions.
\section{Conclusions, Challenges, and Potential Solutions}
\subsection{Concluding remarks}
In this paper, recent research works on RIS-enhanced wireless networks proposed for applications to next-generation networks have been surveyed with an emphasis on the following aspects: operating principles of RISs, performance evaluation of multi-antenna assisted RIS systems, joint beamforming design and resource allocation for RISs, ML in RIS-enhanced wireless networks, and their integration with other key 6G technologies. We have highlighted the advantages and limitations of employing RISs for communication applications. Further research efforts are needed to bridge the complex physical models of the different RISs implementations with widely used communication models. We have considered the performance evaluation of multi-antenna assisted RIS systems by systematically surveying existing designs for RIS-enhanced wireless networks from the views of performance analysis, information theory, and optimization. In addition, we have discussed existing research contributions that applying ML tools for tackling the dynamic essence of the wireless environment such as random fluctuations of wireless channels and user mobility. Design guidelines for ML-empowered RIS-enhanced wireless networks have also been discussed. The benefits of integrating RISs with NOMA, UAV-terrestrial networks, PLS, SWIPT, and AVs/CVs have been discussed. However, the research of RIS-enhanced wireless networks is still at a very early stage and there are ample opportunities for important contributions and advances in this field. Some of them are listed as follows.
\subsection{Challenges and Potential Solutions}
\subsubsection{CSI Acquisition}
The acquisition of timely and accurate CSI plays a pivotal role in RIS-enhanced wireless systems, especially in MIMO-RIS and MISO-RIS networks. The majority of current research contributions assume perfect CSI available at the BS, RISs controllers, as well as the users. However, obtaining CSI in RIS-enhanced wireless networks is a non-trivial task, which requires a non-negligible training overhead. Additionally, in RIS-assisted NOMA networks, users in each cluster have to share the CSI with each other for implementing SIC. However, due to the passive characteristic of RISs, the CSI acquisition and exchanging are non-trivial. Potential solutions can be developed by employing DL methods for exploiting CSI structures beyond linear correlations.
\subsubsection{Pareto-Optimization for Satisfying Multiple Objectives}
In contrast to the conventional wireless networks, RIS-enhanced wireless networks are characterized by more rapidly fluctuating network topologies and more vulnerable communication links. Furthermore, RISs are more likely to be deployed in an environment with heterogeneous mobility profiles. Hence, the networks operate in a complex time-variant hybrid environment, where the classic mathematical models have limited accuracy. Additionally, the challenging optimization problems encountered in RIS-enhanced wireless networks usually have to satisfy multiple objectives (e.g., delay, throughput, BER, and power) in order to arrive at an attractive solution. To elaborate, by definition it is only possible to improve any of the metrics considered at the cost of degrading at least one of the others. The collection of Pareto-optimal points is referred to as the Pareto front. However, determining the entire Pareto-front of optimal solutions is still challenging. Potential solutions may be investigating near-real-time ML-aided Pareto-optimization for tackling the high-dynamic adaptation of RIS-enhanced wireless networks.
\begin{spacing}{1.75}
\bibliographystyle{IEEEtran}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Coronavirus: Scientists say a more contagious mutant strain has been sweeping globe
"The story is worrying, as we see a mutated form of the virus very rapidly emerging," the study's authors say.
Alexander Martin
Technology reporter @AlexMartin
Wednesday 6 May 2020 14:24, UK
Image: An illustration of coronavirus, showing the spike proteins in red. Pic: US CDC
Scientists say they have identified a mutation in coronavirus which they believe means a more contagious strain has been sweeping Europe and the US - and could even reinfect those who already have antibodies.
Researchers at Los Alamos National Laboratory in the US detected 14 mutations in the COVID-19 virus spike proteins, one of which - known as Spike D614G - they said was of "urgent concern".
Their research paper suggests the mutated strain of coronavirus that has become dominant across the world was first indentified in Europe and is different to those which spread early on in the pandemic.
Image: A graphic shows the mutation, in blue, becoming the most dominant strain
So urgent is the issue that the research paper describing their findings has been made available before being peer-reviewed, although this has caused concern among some observers.
By analysing more than 6,000 genetic sequences of coronavirus samples taken from patients globally, the researchers found the mutated strain was persistently becoming the most dominant version of the virus in every region it was detected in.
While first discovered in Europe in early February, the researchers believe the coronavirus mutation has now become the most prevalent strain across the whole of the world.
The study indicates it has been consistently out-competing the original strain detected in Wuhan, which spread through that region of China and some other Asian countries before March.
Dr Bette Korber, the study's lead author, said: "The story is worrying, as we see a mutated form of the virus very rapidly emerging, and over the month of March becoming the dominant pandemic form.
"When viruses with this mutation enter a population, they rapidly begin to take over the local epidemic, thus they are more transmissible."
The mutation to the spike protein has caused concern because this is one of the most effective parts of the virus, and the aspect which most treatments and vaccines are targeting.
For instance, the discovery of a "groundbreaking" antibody - that prevents the virus from infecting human cells - works by binding to this protein instead of allowing it to bind to cells and replicate.
:: Listen to the Daily podcast on Apple Podcasts, Google Podcasts, Spotify, Spreaker
The spike proteins are molecules on the outside of the virus which it uses to grab hold of and then penetrate the outer walls of human and animal cells.
There are two key features in the protein which have been attributed with its enormous infectious ability.
The first is called the receptor-binding domain (RBD), which they describe as "a kind of grappling hook that grips on to host cells", while the second is known as the cleavage site, "a molecular can opener that allows the virus to crack open and enter host cells".
The researchers acknowledge they do not know how the mutations have changed these key features.
Hate crimes against Chinese people soar in UK during COVID-19 crisis
However, the fact the team's findings have not yet been peer-reviewed has concerned some observers, who fear the potentially alarming report should be rigorously analysed before being made public.
On the website hosting the study, one user suggested the "title of the manuscript seems a bit disingenuous" and warned that the prevalence of the mutated spike protein could be a matter of correlation rather then causation.
Although the mutated form of the virus is quickly out-competing the one initially detected in Wuhan, the researchers say they are not certain of their hypothesis that this is due to the spike protein mutation rather than another mutation.
Given the spike protein's "vital importance both in terms of viral infectivity and as an antibody target, we felt an urgent need for an 'early warning' pipeline to evaluate spike pandemic evolution", the study's authors said.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Il Canon de 75 mm M (montagne) Modèle 1928 era un cannone da montagna francese, derivato dal famoso 75 mm Mle. 1897, impiegato durante la seconda guerra mondiale. Il pezzo equipaggiò le batterie della 4ª Divisione da montagna marocchina. Venne inoltre acquistato dalla Polonia. Dopo la resa della Francia, i pezzi di preda bellica furono riutilizzati dalla Wehrmacht come 7,5 cm GebK 283(f).
Altri progetti
Artiglierie di calibro da 65 mm a 85 mm
Armi d'artiglieria della seconda guerra mondiale
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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We are proud to be on the recommended reading lists of many university and colleges around the world, and are regularly used by TV and film companies conducting research. We are delighted to be acknowledged in many leading crime books. We are the original murder site since 1993.
Murder UK is a site dedicated to documenting and investigating murder in the UK. We aim to be precise with facts and avoid speculation. If however you find discrepancies please contact us
Web www.murderuk.com
Child Killers
Female Killers
Mass Murderers
One Off Killers
ANTHONY COSTA
Steven Bigby
Kill Total: 1 Kill place: Oxford St. London
Kill date: 12th May 2008 Victim(s): Steven Bibgy
Date of Birth: 1990 Marital Status: Single
AKA: Occupation:
Monday 12th May 2008,a 22 year old man was stabbed to death after an argument at McDonalds just by Oxford Circus tube station. He was stabbed at 4:45pm, in broad daylight in front of horrified shoppers, by the time paramedics arrived the victims heart had stopped, he was pronounced dead on arrival at hospital.
Wednesday 14th May 2008, Police named the dead man as 22 year old Steven Bigby. Steven was known to police, and was one of ten men charged with the rape of a girl in North London in January 2008. Bigby was also due to go on trial in June for wounding with intent and violent disorder, over a stabbing in October 2006 in north London.
Police have confirmed that 2 gangs clashed in a spontaneously incident outside McDonald in Oxford Street. The post mortem revealed that Bigby died from a single stab wound to the heart, police have recovered the weapon.
Thursday 15th May 2008, Anthony Costa, has been charged with Steven Bigby's murder and appeared at Waltham Forest Magistrates Court.
23rd December 2008, Costa told the Old Bailey Bigby was stabbed by accident after he took out his knife during the incident.
Anthony Costa, was given an indeterminate sentence for public protection and told he would serve at least five years for the attack.
Judge Hawkins said he accepted there had been a lack of intent to kill Bigby, but said Costa had little insight in to what he had done
Court - Old Bailey
Judge - Richard Hawkins
Have your say about this case in our USER FORUM..
Copyright Murderuk.com
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,832
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Q: openssl get certificate extensions I'm using openssl to parse X509 certificate.
I manage to get extensions, but I don't know how to extract the extension value.
The code I am using is:
X509_EXTENSION *extension = sk_X509_EXTENSION_pop(exts);
int critical = X509_EXTENSION_get_critical(extension);
ASN1_OBJECT *obj = extension-> object;
ln = OBJ_nid2ln(OBJ_obj2nid(obj));
if( !ln ) ln = "";
OBJ_obj2txt(objbuf,sizeof(objbuf),obj,1);
int nid = OBJ_txt2nid(ln);
This code tells me whether the extension is critical or not and gives me nid of the extension.
I suppose value can be obtained by:
ASN1_OCTET_STRING *data= X509_EXTENSION_get_data(extension);
but I am not sure how to handle the retrieved data object. The data object is supposed to be der-encoded.
Any idea on how to get the extension data?
EDIT:
As suggested here, I was trying to do:
ASN1_OCTET_STRING* octet_str = X509_EXTENSION_get_data(extension);
const unsigned char* octet_str_data = octet_str->data;
long xlen;
int tag, xclass;
int ret = ASN1_get_object(&octet_str_data, &xlen, &tag, &xclass, octet_str->length);
printf(@"value: %s\n", octet_str_data);
but the string I get after decoding is the same as prior to it - something like:
4Á˃◊∫Ns∑äP∂W≠%£A
A: This is very interesting. I've been watching the conversation transpire across the link you provided and here, and I concur with Roger Dahl on his comment in his answer's errata:
It appears that you are seeking a general solution on how to extract
the information in an x509v3 extension. As the title of this question
states, it is only about the special case of decoding a single OCTET
STRING. Beyond that, I can only point you to the original suggestion
of @Francois, which is to use an external ASN.1 parser, such as SNACC
or libtasn1 and to look at the OpenSSL source code (around
crypto/asn1/asn1_par.c:135).
(Link additions mine.)
The trick is you're trying to work with a modal extension model that isn't handled well generically. This answer provides a good overview on how that works. As for the rest of the answer, I concur with his observation that where the docs fail, reading the source is honestly the best solution.
A: If I was in your position I would seriously consider using one of the dedicated ASN.1 libraries to decode the certificates. Let OpenSSL do what it's good at, which is to validate your certificates against a trust chain. Once you know that you have a good certificate, pass it to an ASN.1 library and let it handle the rest. (SNACC looks good.) Note that the entire certificate is itself encoded with ASN.1, so the same library that you would use for handling the v3 extensions can be used for handling the entire certificate.
A: If you want to have a textual representation, you can use a memory BIO. I've answered how to this for a Key Usage extension in this thread, but the way to do it is the same for every extension.
Regards.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,452
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Onion Creek is a neighborhood in Austin.
Onion Creek is also the name of a creek that starts in Hays County, Texas and empties into the Colorado River (Texas).
Get to know Onion Creek better with our featured images and videos.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,104
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Hello Dies R Us friends. Lena here today to share a card I made for my daughter and her hubby as they celebrate their 9th wedding anniversary.
Now for a quick pictorial ~ I began by cutting out my background die in red and the base die in white.
Using Memento London Fog ink, I sponged all the embossed areas. It's always better to do this while the card stock is still in the die.
Next I cut my leaves and my flowers. I used Memento Bamboo Leaves ink to highlight the embossed areas. For these tiny areas, I always use a little make up applicator as I feel it works better in transferring the ink to cardstock.
All that was left was to build the tulip bouquet. I should have been smart and applied double sided adhesive to the card stock before die cutting the little tulips, but since I used my FineLine Applicator Bottle, it didn't take too long before they were all adhered and assembled.
On the inside, I used the largest circle from Anja's Circle by Marianne Designs to cut the red mat.
I used the circle that comes with Petra's Amazing Circle to cut the white mat where I will be writing my message. I cut out another bunch of tulips for a little something extra. Oh, yes, I almost forgot ... I stamped the greeting and cut out two little birds to hold it in place on the front of the card.
Here is a quick recap of the materials I used.
Stunning and creative card, Lena! So inspirational!
Absolutely GORGEOUS Lena! I'm sure your daughter LOVED it and will treasure it always!
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{
"redpajama_set_name": "RedPajamaC4"
}
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Tag Archives: Deal
3 Net Monitoring Metrics to Deal with Performance Degradation | IT Infrastructure Advice, Discussion, Community
Network Performance Monitoring using flow data (NetFlow) is an approach to isolate the root cause of performance issues related to network traffic by measuring a set of characteristics across L2-L7 layers.
There are three basic causes of performance issues: round trip time, server response time, and jitter. Each can contribute to low performance and downtimes. Let's examine each one.
1. Round trip time
Also called network delay, round trip time represents a data transfer time of a packet being transmitted from client to server and back. It is a single value that models the performance of the network itself, calculated by observing the time needed to establish a TCP session. A typical value in enterprise networks in one location is less than 1 ms (even tens of microseconds) as on the local network. An application has no impact on the TCP handshake as this is part of the TCP/IP stack implemented in the operating system itself. It would require an operating system malfunction to influence this metric which won't happen in practice. Here are some typical root causes of network delays.
Overload of network devices: High packet rates impact buffers in network devices where packets need to wait to be dispatched. QoS can help to prioritise critical services to a certain extent but experiencing a DDoS attack may lead to network congestion and increased values of RTT.
Clients working from remote locations: Complaining about slow application responses might not always be the case. Having an RTT of 500ms when connecting from home through a VPN to a company data centre means that just to transmit the packet takes half a second and any application will look slow from a user's perspective.
Cloud applications: To lower the delay, SaaS providers use CDNs and proxy servers to host the application as close to customers as possible. For the same reason large companies purchase dedicated lines to connect their infrastructure directly to cloud providers.
Ethernet vs. Wi-Fi: In my practical experience, the usual performance difference between wired Ethernet connection and WiFi is around 10ms. So 10ms is the average penalty you get when going through WiFi instead of wired Ethernet connection. And we are still talking about ideal conditions.
Performance bottleneck caused by heterogeneous port speeds: Imagine a 10G backbone while servers are connected through 1G, especially when multiple servers share such a 1G uplink. Numerous clients can easily generate traffic that will spike above 1G port capacity, saturating switch buffers, which leads to packet drops. Such packets need to be retransmitted and consecutively users experience a network delay.
2. Server response time
This metric represents the request processing time on the server side and so represents the delay caused by the application itself. The measured server response time expresses the time difference between the predicted observation time of the server's ACK packet (prediction based on observation time of the client request and previously measured RTT value) and the actual observation time of the server's response. The measurement can't rely on observing an ACK packet from the server since the ACK packet might be merged with the server's response.
SRT enables a performance measurement of the whole application, per application server, per client network range or even individual clients. This enables finding correlations between application performance and a number of clients or a specific time of the day. Using this metric together with RTT answers the ultimate question. Is it a network issue or application issue?
3. Jitter – variance of delay between packets
Jitter can show irregularities in packet flow by calculating the variance of individual delays between the packets. In an ideal case, delay between the individual packets is a constant value, which means that jitter is 0. In reality, having a jitter value of 0 doesn't occur as a variety of parameters might influence the data stream. Why should we measure jitter anyway? Jitter is critical and has the main value for assessing the quality of real-time applications, such as conference calls and video streaming. But also when downloading, e.g. a Linux distribution ISO file of Linux distribution from a mirror, jitter may indicate an unstable network connection.
Continuous monitoring and baselining of network performance monitoring metrics by using flow data helps network administrators to identify an issue in the network itself, specific connections or applications. It's valuable to reveal problems before users do and prevent complaints on performance degradation. Long term monitoring of network performance metrics (RTT, SRT, Jitter) can help to predict future needs (capacity planning) and incidents.
Network performance monitoring metrics can considerably improve the performance of the network as well as contributing to the improvement of the application side.
Let's Make a Deal: Negotiating with IT Vendors | IT Infrastructure Advice, Discussion, Community
If you work in an enterprise IT organization that's been around for a more than a decade or two, it's a pretty good bet that you do business with one of the so-called megavendors — IBM, Microsoft, Oracle, and/or SAP. These established vendors have had deep roots in enterprise businesses for many years with ERP systems, databases, and more.
And if you think that your business is under pressure from market and industry disruptors, you should also realize that the same market forces are in play for these megavendors. The cloud has changed their licensing and business models. They are also under strain as these models have shifted, and their sales tactics have shifted, too, as they look to drive upsell revenue. These companies want to increase your spending year-over-year in the cloud with subscription licensing. They have a strategic product set that promotes the sale of other products, too.
Before you head into your contract negotiations with these megavendors, you need to prepare your own tactics and strategies. That's according to Melanie Alexander, a director analyst at Gartner specializing in vendor contract negotiations. She provided some perspective on the best ways to prepare for your negotiations with these vendors during a session at the recent Gartner Data and Analytics Summit in Orlando, Florida.
"Their main purpose in life is for you to spend more money with them," Alexander said. They will want to upsell you to use the full platform — to get you on the hardware and middleware and application stack, she said. They have a strategic product set and those products help them promote the sales of their other products. They want to get you into their cloud and lock you into subscription pricing. They want you to increase your spending with them year over year.
Read the rest of this article on InformationWeek.
HPE Inks Deal For SimpliVity
29/01/2017 Stiri IT Externe No comments
Hewlett-Packard Enterprise on Tuesday announced an agreement to buy hyperconverged startup SimpliVity for $650 million in cash to bolster its hybrid IT strategy.
Founded in 2009, SimpliVity was an early player in the fast-growing hyperconverged infrastructure market. The startup came out of stealth in 2012 with its OmniStack platform that combines compute, storage services, and network switching. The platform, which is composed of SimpliVity's Data Virtualization Platform software and purpose-built Accelerator Card, includes data compression, deduplication, and built-in backup.
Gartner labeled SimpliVity a leader in hyperconvergence, along with Cisco, EMC, Nutanix, and NetApp, in its Magic Quadrant for Integrated Systems last fall. In addition to offering an OmniCube appliance, SimpliVity teams with Cisco, Dell, Huawei, and Lenovo to integrate OmniStack into their servers.
"This transaction expands HPE's software-defined capability and fits squarely within our strategy to make hybrid IT simple for customers," Meg Whitman, HPE president and CEO, said in a statement.
HPE said it will continue to offer its own hyperconverged products, the HC 380 and HC 250, for existing customers and partners. The company jumped into the hyperconvergence nearly a year ago with the HC 380. SimpliVity customers and partners shouldn't expect any immediate changes in product roadmap, according to HPE, which said it will continue to support them.
Within 60 days of the deal closing — which HPE expects in the second quarter of its fiscal year 2017 — the company plans to offer SimpliVity's software qualified for its ProLiant DL380 servers. By the second half, it expects to offer a range of integrated HPE SimpliVity systems on ProLiant servers.
Dan Conde, an analyst at Enterprise Strategy Group and Interop ITX Review Board member, told me in an email that SimpliVity provides HPE with better differentiation in the hyperconverged infrastructure market. HPE's own products aren't built from the ground-up for hyperconvergence to the same extent as SimpliVity's, he said.
"I think they [HPE] wanted some 'secret sauce'," Conde said.
Technology Business Research recently estimated that the market for hyperconverged platforms will reach $7.2 billion by 2020.
SimpliVity's OmniCube made its way to Hollywood last year, when it was disguised as the Pied Piper box in HBO's "Silicon Valley" television show.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Now That I Found You may refer to:
"Now That I Found You" (Terri Clark song), a 1998 song by Terri Clark
"Now That I Found You", a 1999 song by Mytown
"Now That I Found You", a 2013 song by Britney Spears from her album Britney Jean
"Now That I Found You" (Carly Rae Jepsen song), a 2019 song by Carly Rae Jepsen
See also
"Baby Now That I've Found You", a 1967 song by The Foundations
"Now That I've Found You", a 2016 song by Martin Garrix
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,851
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\section{Introduction}
The problem of estimating the underlying probability distribution
from an observed data sequence arises in a variety of fields
such as compression, adaptive control, and linguistics.
The most familiar technique is to use
the empirical distribution of the data, also known
as the type. This approach has a number of virtues.
It is the maximum likelihood (ML) distribution,
and if each symbol appears frequently in the string,
then the law of large numbers guarantees that the
estimate will be close to the true distribution.
In some situations, however, not all symbols will appear
frequently in the observed data.
One example is a digital image with the pixels themselves, rather than bits,
viewed as the symbols~\cite{Orlitsky:Unknown}. Here the size of the
alphabet can meet or exceed the total number of observed
symbols, i.e., the number of pixels in the image. Another example is
English text. Even in large corpora, many words will appear once
or twice or not at all~\cite{Efron:Thisted}. This makes
estimating the distribution of English words using the type
ineffective. This problem is particularly pronounced when
one attempts to estimate the distribution of bigrams, or pairs of words,
since the number of bigrams is evidently the square of the
number of words.
To see that the empirical distribution is lacking as an estimator
for the probabilities of uncommon symbols,
consider the extreme situation in which the
alphabet is infinite and we observe a length-$n$ sequence
containing $n$ distinct symbols~\cite{Orlitsky:Science}.
The ML estimator will assign probability $1/n$
to the $n$ symbols that appear in the string and zero
probability to the rest. But common sense suggests that
the $(n+1)$st symbol in the sequence is very likely to be one that
has not yet appeared. It seems that the ML estimator is
overfitting the data. Modifications to the ML estimator such
as the Laplace ``add one'' and the Krichevsky-Trofimov
``add half''~\cite{Krichevsky:Trofimov} have
been proposed as remedies, but these only alleviate the
problem~\cite{Orlitsky:Science}.
In collaboration with Turing, Good~\cite{Good:Turing} proposed an estimator
for the probabilities of rare symbols that differs
considerably from the ML estimator. The Good-Turing
estimator has been shown to work well in practice~\cite{Gale:One},
and it is now used in several application areas~\cite{Orlitsky:Science}.
Early theoretical work on the estimator focused on its
bias~\cite{Good:Turing,Robbins:Unseen,Juang:Bias}.
Recent work has been directed toward developing confidence
intervals for the estimates using central limit
theorems~\cite{Esty:Efficiency,Mao:Poisson} or
concentration inequalities \cite{McAllester:COLT,Drukh:Concentration}.
Orlitsky, Santhanam, and Zhang~\cite{Orlitsky:Science}
showed that the estimator has a pattern redundancy that is
small but not optimal. None of these works,
however, have shown that the estimator is strongly consistent.
We show that the Good-Turing estimator is strongly consistent
under a natural formulation of the problem. We consider the
problem of estimating the total probability of all symbols
that appear $k$ times in the observed string
for each nonnegative integer $k$. For $k = 0$, this is
the total probability of the unseen symbols, a quantity that
has received particular attention~\cite{Robbins:Unseen,McAllester:Missing}.
Estimating the total
probability of all symbols with the same empirical
frequency is a natural approach when the symbols appear
infrequently so that there is insufficient data to accurately
estimate the probabilities of the individual symbols.
Although the total probabilities
are themselves random, we show that under our model
they converge to a deterministic limit, which we
characterize.
Note that if the alphabet is small and the block length
is large, then the problem effectively reduces to the
usual probability estimation problem since it is unlikely
that multiple symbols will have the same empirical frequency.
It is known that the Good-Turing estimator performs poorly
for high-probability symbols~\cite{Orlitsky:Science},
but this is not a problem
since the ML estimator can be employed to estimate the probabilities
of symbols that appear frequently in the observed string.
We therefore focus on the situation in which the
symbols are unlikely, meaning that they have probability
$O(1/n)$. We allow the underlying distributions to vary
with the block length
$n$ in order to maintain this condition, and we assume that,
properly scaled, these distributions converge. This model
is discussed
in detail in the next section, where we also describe the
Good-Turing estimator. In Section~\ref{sumsection}, we establish
the convergence of the total probabilities.
Section~\ref{GTsection} uses this convergence result to show strong
consistency of the Good-Turing estimator. Some comments regarding
how to estimate other quantities of interest are made
in the final section.
\section{Preliminaries}
Let $(\Omega_n,\mathcal{F}_n,P_n)$ be a sequence of probability spaces.
We do not assume that $\Omega_n$ is finite or even countable.
Our observed string is a sequence of $n$ symbols
drawn i.i.d.\ from $\Omega_n$ according to $P_n$.
Note that the alphabet and the underlying distribution are
permitted to vary with $n$. This allows us to model the
situation in which the block length is large while the
number of occurrences of some symbols is small.
\subsection{Total Probabilities}
For each nonnegative integer $k$, let $A_k^n$ denote the
set of symbols in $\Omega_n$ that appear exactly $k$ times
in the string of length $n$. We call
\begin{equation*}
\xi_k^n := P_n(A_k^n)
\end{equation*}
the \emph{total probability} of symbols that appear $k$ times.
Of course, for $k \ge 1$, $\xi_k^n$ is simply the sum of
the probabilities of the symbols with frequency $k$.
On the other hand, $A_0^n$ will be uncountable if $\Omega_n$ is.
We view $\xi_k^n$ as a random probability distribution
on the nonnegative integers. Our goal is to estimate this
distribution.
\subsection{The Good-Turing Estimator}
The Good-Turing estimator is normally viewed as an estimator
for the probabilities of the individual symbols. Let
$\varphi_k^n = |A_k^n|$ denote the number of
symbols that appear exactly $k$ times in the
observed sequence. The basic Good-Turing estimator
assigns probability
\begin{equation*}
\frac{(k+1)\varphi_{k+1}^n}{n \varphi_k^n}
\end{equation*}
to each symbol that appears $k \le n-1$ times~\cite{Good:Turing}.
The case $k = n$ must be handled separately, but this case
is unimportant to us since under our model it is unlikely
that only one symbol will appear in the string.
This formula
can be naturally viewed as a total
probability estimator since the $\varphi_k^n$
in the denominator is merely dividing the total
probability equally among the $\varphi_k^n$ symbols
that appear $k$ times. Thus the Good-Turing total probability
estimator assigns probability
\begin{equation*}
\zeta_k^n := \frac{(k+1)\varphi_{k+1}^n}{n}
\end{equation*}
to the aggregate of symbols that have appeared $k$ times for
each $k$ in $\{0,\ldots,n-1\}$.
As a convention, we shall always assign
zero probability to the set of symbols that appear $n$ times
\begin{equation*}
\zeta_n^n := 0.
\end{equation*}
Like $\xi_k^n$, $\zeta_k^n$ is a random probability distribution
on the nonnegative integers.
As a total probability estimator, $\zeta_k^n$ is not ideal.
For one thing, $\zeta_k^n$ can be positive even when $A_k^n$
is empty, in which case $\xi_k^n$ is clearly zero.
A similar problem
arises when estimating the probabilities of individual symbols,
and modifications to the basic Good-Turing estimator
have been proposed to avoid it~\cite{Good:Turing}. But we shall
show that even the basic form of the Good-Turing
estimator is strongly consistent for total probability
estimation.
\subsection{Shadows}
The distributions of the total probability,
$\xi_k^n$, and the Good-Turing estimator, $\zeta_k^n$,
are unaffected if one relabels the symbols in
$\Omega_n$. This fact makes it convenient
in what follows to consider the probabilities
assigned by $P_n$ without reference to the labeling
of the symbols.
\begin{defn}
Let $X_n$ be a random variable on $\Omega_n$ with
distribution $P_n$. The \emph{shadow} of $P_n$ is
defined to be the distribution of the random
variable $P_n(\{X_n\})$.
\end{defn}
As an example, if $\Omega_n = \{a,b,c\}$ and
\begin{equation*}
P_n(\{a\}) = P_n(\{b\}) = \frac{1}{2} P_n(\{c\}) = \frac{1}{4}, \\
\end{equation*}
then the shadow of $P_n$ would be uniform over
$\{1/4,1/2\}$. If $P_n$ is itself uniform, then
its shadow is deterministic. Note that the
discrete entropy of a distribution only depends
on the distribution through its shadow. We will
write $P_n(X_n)$ as a shorthand for $P_n(\{X_n\})$
in what follows.
For finite alphabets, specifying the
shadow is equivalent to specifying the unordered
components of $P_n$, viewed as a probability vector.
This is clearly seen in the above example, since the
shadow is uniformly distributed over $\{1/4,1/2\}$
if and only if the underlying distribution has two
symbols with probability $1/4$ and one with
probability $1/2$.
If $P_n$ has a continuous component, then the shadow
will have a point mass at zero equal to the probability
of this component. The shadow reveals nothing
more about the continuous component than its
total probability, but
we shall have no need for such information. Indeed,
the distributions of both $\xi_k^n$ and $\zeta_k^n$
depend on $P_n$ only through its shadow.
\subsection{Unlikely Symbols}
To prove strong consistency, we assume
that the scaled profiles, $n \cdot P_n(X_n)$,
converge to a nonnegative random variable $Y$
with distribution $Q$. This implies, in particular,
that asymptotically almost every symbol has
probability $O(1/n)$ and therefore appears
$O(1)$ times in the sequence on average. As an
example, if $P_n$ is a uniform distribution
over an alphabet of size $n$, then
the scaled shadow, $n \cdot P_n(X_n)$, equals
one a.s.\ for each $n$
(and hence it converges in distribution).
More complicated examples can be constructed by
quantizing a fixed density more and more finely
to generate the sequence of distributions.
\section{Total Probability Convergence}
\label{sumsection}
Before considering the performance of the Good-Turing
estimator, we study the asymptotics of the total
probabilities themselves. Under our assumption that
the scaled shadows converge, we show that the
total probabilities converge almost surely to
a deterministic Poisson mixture.
\begin{prop}
\label{sumprop}
The random distribution $\xi^n$ converges to
\begin{equation*}
\lambda_k := \int_0^\infty \frac{{y}^k \exp(-y)}{k!} \; dQ(y)
\quad k = 0,1,2,\ldots
\end{equation*}
in $L^1$ almost surely as $n \rightarrow \infty$.
\end{prop}
We prove this result by first showing that the mean
of $\xi^n$ converges to $\lambda$ and then proving
concentration
around the mean. To show convergence of the mean, it
is convenient to make several definitions. Let
\begin{equation*}
g_k^n(y) = {n \choose k} \left(\frac{y}{n}\right)^k
\left(1 - \frac{y}{n}\right)^{n - k}
\end{equation*}
and
\begin{equation*}
g_k(y) = \frac{y^k \exp(-y)}{k!}.
\end{equation*}
Since
\begin{equation*}
{n \choose k} \frac{1}{n^k} \rightarrow \frac{1}{k!} \quad \text{as
$n \rightarrow \infty$}
\end{equation*}
and
\begin{equation*}
\left(1 + \frac{y_n}{n}\right)^n
\rightarrow \exp(y) \quad \text{if $y_n \rightarrow y$},
\end{equation*}
it follows that for all sequences
$y_n \rightarrow y$, $g_k^n(y_n) \rightarrow g_k(y)$. Note also
that $g_k^n(y) \le 1$ if $0 \le y \le n$ by the binomial theorem.
Let
\begin{equation*}
C^n = \{\omega \in \Omega_n: P_n(\omega) > 0\}
\end{equation*}
and note that $C^n$ is countable for each $n$.
\begin{lemma}
\label{summean}
For all nonnegative integers $k$,
\begin{equation*}
\lim_{n \rightarrow \infty} E[\xi_k^n] = \lambda_k.
\end{equation*}
\end{lemma}
\begin{proof}
We shall show that
\begin{equation}
\label{sumexpformula}
E[\xi_k^n] = E[g_k^n(nP_n(X_n))].
\end{equation}
First consider the case $k \ge 1$. Here
\begin{align*}
\xi_k^n & = P_n(A_k^n \cap C^n) \\
& = \sum_{\omega \in C^n} 1(\omega \in A_k^n) P_n(\omega)
\end{align*}
so by monotone convergence
\begin{align*}
E[\xi_k^n] & = \sum_{\omega \in C^n} {n \choose k} P_n(\omega)^k
(1 - P_n(\omega))^{n - k} P_n(\omega) \\
& = \sum_{\omega \in C^n} g_k^n(n P_n(\omega)) P_n(\omega) \\
& = E[g_k^n(nP_n(X_n)) 1(X_n \in C^n)] \\
& = E[g_k^n(nP_n(X_n))].
\end{align*}
Next consider the case $k = 0$. Here
\begin{align*}
\xi_0^n & = P_n(A_0^n) \\
& = P_n(A_0^n \cap C^n) + P_n(A_0^n - C^n) \\
& = \sum_{\omega \in C^n} 1(\omega \in A_0^n) P_n(\omega) +
P_n(\Omega_n - C^n).
\end{align*}
So again by monotone convergence,
\begin{align*}
E[\xi_0^n] & = \sum_{\omega \in C^n} (1- P_n(\omega))^n P_n(\omega) +
P_n(\Omega_n - C^n) \\
& = \sum_{\omega \in C^n} g_0^n(nP_n(\omega)) P_n(\omega) +
P_n(\Omega_n - C^n) \\
& = E[g_0^n(nP_n(X_n))1(X^n \in C^n)] \\
& \phantom{= E[g_0^n(} +
E[g_0^n(nP_n(X_n))1(X_n \notin C^n)] \\
& = E[g_0^n(nP_n(X_n))].
\end{align*}
This establishes~(\ref{sumexpformula}).
Since
$n P_n(X_n)$ converges in distribution to $Y$, we can create a
sequence of random variables $\{Y_n\}_{n = 1}^\infty$ such that
$Y_n$ has the same distribution as $n P_n(X_n)$ and $Y_n$ converges
to $Y$ almost surely~\cite[Theorem~4.30]{Kallenberg:Foundations:2}. Then
\begin{equation*}
g_k^n(Y_n) \rightarrow g_k(Y) \quad \text{a.s.}
\end{equation*}
Since $g_k^n(Y_n) \le 1$ a.s.,
the bounded convergence theorem implies
\begin{align*}
\lim_{n \rightarrow \infty}
E[g_k^n(Y_n)] & = E[g_k(Y)] \\
& = \int_0^\infty g_k(y) \; dQ(y)
= \lambda_k.
\end{align*}
\end{proof}
\begin{lemma}
\label{sumconc}
For all nonnegative integers $k$,
\begin{equation*}
\lim_{n \rightarrow \infty} |\xi^n_k - E[\xi_k^n]| = 0 \quad \text{a.s.}
\end{equation*}
\end{lemma}
\begin{proof}
Let
\begin{equation*}
B^n = \left\{ \omega \in \Omega_n : P_n(\omega) \ge \frac{1}{n^{3/4}}
\right\}
\end{equation*}
and note that $|B^n| \le n^{3/4}$. Then let
\begin{equation*}
\tilde{\xi}_k^n = P_n(A_k^n \cap B^n),
\end{equation*}
and note that
\begin{equation*}
|\xi^n_k - E[\xi_k^n]| \le
\left|(\xi^n_k - \tilde{\xi}^n_k) - E[\xi_k^n - \tilde{\xi}^n_k]\right|
+ \tilde{\xi}^n_k + E[\tilde{\xi}_k^n].
\end{equation*}
Now if we change one symbol in the underlying sequence, then
$\xi^n_k - \tilde{\xi}^n_k$ can change by at most $2/n^{3/4}$.
By the Azuma-Hoeffding-Bennett concentration
inequality~\cite[Corollary~2.4.14]{Dembo:LD}, it follows
that for all $\epsilon > 0$
\begin{equation*}
\Pr\left(\left|(\xi^n_k - \tilde{\xi}^n_k) - E[\xi_k^n - \tilde{\xi}^n_k]
\right| \ge \epsilon\right)
\le 2 \exp\left[- \frac{\epsilon^2 \sqrt{n}}{8}\right].
\end{equation*}
Since the right-hand side is summable over $n$, this implies that
\begin{equation*}
\left|(\xi^n_k - \tilde{\xi}^n_k) -
E[\xi_k^n - \tilde{\xi}^n_k]\right| \rightarrow 0
\quad \text{a.s.}
\end{equation*}
Now
\begin{equation*}
\tilde{\xi}_k^n = \sum_{\omega \in B^n} P_n(\omega) 1(\omega \in A_k^n)
\end{equation*}
so
\begin{align*}
E[\tilde{\xi}_k^n] & = \sum_{\omega \in B^n} P_n(\omega) {n \choose k}
(P_n(\omega))^k (1 - P_n(\omega))^{n - k} \\
& \le \sum_{\omega \in B^n} {n \choose k}
(P_n(\omega))^k (1 - P_n(\omega))^{n - k}.
\end{align*}
But
\begin{multline*}
{n \choose k} (P_n(\omega))^k (1 - P_n(\omega))^{n - k} \\
= \exp\left[-n\left(H\left(\frac{k}{n}\right) +
D\left(\frac{k}{n}\Big|\Big|P_n(\omega)\right)\right)\right],
\end{multline*}
where $H(\cdot)$ denotes the binary entropy function and
$D(\cdot||\cdot)$
denotes binary Kullback-Leibler divergence, both with natural
logarithms~\cite[Theorem~12.1.2]{Cover:IT}.
For all sufficiently large $n$, $k/n < 1/n^{3/4}$, which implies
that for all $\omega \in B^n$,
\begin{equation*}
D\left(\frac{k}{n}\Big|\Big|P_n(\omega)\right) \ge
D\left(\frac{k}{n}\Big|\Big|\frac{1}{n^{3/4}}\right).
\end{equation*}
This gives
\begin{multline*}
{n \choose k} (P_n(\omega))^k (1 - P_n(\omega))^{n - k} \\
\le {n \choose k} \left(\frac{1}{n^{3/4}}\right)^k
\left(1 - \frac{1}{n^{3/4}}\right)^{n - k},
\end{multline*}
so
\begin{equation*}
E[\tilde{\xi}_k^n]
\le n^{3/4} {n \choose k} \left(\frac{1}{n^{3/4}}\right)^k
\left(1 - \frac{1}{n^{3/4}}\right)^{n - k}.
\end{equation*}
Since
\begin{equation*}
{n \choose k} \le \frac{n^k}{k!},
\end{equation*}
this implies
\begin{equation}
\label{tildeexp}
E[\tilde{\xi}_k^n]
\le \frac{n^{(k+3)/4}}{k!} \left(1 - \frac{1}{n^{3/4}}\right)^{n - k}.
\end{equation}
Now the right-hand side tends to zero as $n \rightarrow \infty$, so
\begin{equation*}
\lim_{n \rightarrow 0} E[\tilde{\xi}_k^n] = 0.
\end{equation*}
In fact, the right-hand side of~(\ref{tildeexp}) is summable over
$n$. By Markov's inequality,
\begin{equation*}
\Pr(\tilde{\xi}_k^n > \epsilon) \le \frac{E[\tilde{\xi}_k^n]}{\epsilon},
\end{equation*}
this implies that $\tilde{\xi}_k^n \rightarrow 0$ a.s.
The conclusion follows.
\end{proof}
\emph{Proof of Proposition~\ref{sumprop}:}
It follows from Lemmas~\ref{summean} and~\ref{sumconc} that for each $k$,
\begin{equation*}
\lim_{n \rightarrow \infty} \xi_k^n = \lambda_k \quad \text{a.s.}
\end{equation*}
That is, $\xi^n$ converges pointwise to $\lambda$ with probability
one. The strengthening to $L^1$ convergence follows from Scheff\'{e}'s
theorem~\cite[Theorem 16.12]{Billingsley:PM}, but we shall give a
self-contained proof since it is brief.
Observe that with probability one,
\begin{align*}
0 & = \sum_{k = 0}^\infty \left[\lambda_k - \xi_k^n\right] \\
& = \sum_{k = 0}^\infty \left[\lambda_k - \xi_k^n\right]^+ -
\sum_{k = 0}^\infty \left[\lambda_k - \xi_k^n\right]^-,
\end{align*}
where $[\cdot]^+$ and $[\cdot]^-$ represent the positive and
negative parts, respectively. Thus
\begin{equation*}
\sum_{k = 0}^\infty \left| \lambda_k - \xi_k^n\right| =
2 \sum_{k = 0}^\infty \left[ \lambda_k - \xi_k^n\right]^+ \quad \text{a.s.}
\end{equation*}
But $[\lambda_k - \xi_k^n]^+$ converges pointwise to 0
a.s.\ and is less than or equal to $\lambda_k$. The dominated
convergence theorem then implies that
\begin{equation*}
\lim_{n \rightarrow \infty} \sum_{k = 0}^\infty \left[\lambda_k
- \xi_k^n\right]^+ = 0 \quad \text{a.s.}
\end{equation*}
\mbox{ } \hfill \QED
\section{Strong Consistency}
\label{GTsection}
The key to showing strong consistency is to establish
a convergence result for the Good-Turing estimator
that is analogous to Proposition~\ref{sumprop} for
the total probabilities.
\begin{prop}
\label{GTprop}
The random distribution $\zeta^n$ converges to $\lambda$ in $L^1$ almost
surely as $n \rightarrow \infty$.
\end{prop}
The desired strong consistency follows from
this result and Proposition~\ref{sumprop}.
\begin{theorem}
The Good-Turing total probability estimator is strongly consistent, i.e.,
\begin{equation*}
\lim_{n \rightarrow \infty} \sum_{k = 0}^n
|\xi_k^n - \zeta_k^n| = 0 \quad \text{a.s.}
\end{equation*}
\begin{proof}
We have
\begin{equation*}
\sum_{k = 0}^n |\xi_k^n - \zeta_k^n| \le
\sum_{k = 0}^\infty |\xi_k^n - \lambda_k| +
\sum_{k = 0}^\infty |\lambda_k - \zeta_k^n|.
\end{equation*}
We now let $n \rightarrow \infty$ and invoke Propositions~\ref{sumprop}
and~\ref{GTprop}.
\end{proof}
\end{theorem}
The proof of Proposition~\ref{GTprop} parallels that of
Proposition~\ref{sumprop} in the previous section. In
particular, we first show that the mean of $\zeta^n$
converges to $\lambda$ and
then establish concentration around the mean.
\begin{lemma}
\label{GTmean}
For all nonnegative integers $k$,
\begin{equation*}
\lim_{n \rightarrow \infty} E[\zeta_k^n] = \lambda_k.
\end{equation*}
\end{lemma}
\begin{proof}
We shall show that
\begin{equation}
\label{GTexpformula}
E[\zeta_k^n] = E[g_k^{n-1}((n-1)P_n(X_n))].
\end{equation}
First consider the case $k \ge 1$. Here
\begin{equation*}
\zeta_k^n = \sum_{\omega \in C^n} \frac{k+1}{n}
1(\omega \in A_{k+1}^n).
\end{equation*}
So by monotone convergence,
\begin{align*}
E[\zeta_k^n] & = \sum_{\omega \in C^n} \frac{k+1}{n} {n \choose k + 1}
(P_n(\omega))^{k+1} (1 - P_n(\omega))^{n - k - 1} \\
& = \sum_{\omega \in C^n} {n - 1 \choose k}
(P_n(\omega))^{k} (1 - P_n(\omega))^{n - k - 1} P_n(\omega) \\
& = \sum_{\omega \in C^n} g_k^{n-1}((n-1)P_n(\omega)) P_n(\omega) \\
& = E[g_k^{n-1}((n-1)P_n(X_n))1(X_n \in C^n)] \\
& = E[g_k^{n-1}((n-1)P_n(X_n))].
\end{align*}
Next consider the case $k = 0$. Here
\begin{align*}
\zeta_0^n & = \frac{1}{n} |A_1^n| \\
& = \frac{1}{n} |A_1^n \cap C^n| + \frac{1}{n} |A_1^n - C^n| \\
& = \frac{1}{n} \sum_{\omega \in C^n} 1(\omega \in A_1^n)
+ \frac{1}{n} |A_1^n - C^n|.
\end{align*}
Again invoking monotone convergence,
\begin{align*}
E[\zeta_0^n] & = \frac{1}{n} \sum_{\omega \in C^n} {n \choose 1}
P_n(\omega) (1- P_n(\omega))^{n-1} \\
& \phantom{= \frac{1}{n} \sum_{\omega \in C^n}}
+ P_n(\Omega_n - C^n) \\
& = \sum_{\omega \in C^n}
g_0^{n-1}((n-1)P_n(\omega)) P_n(\omega) \\
& \phantom{= \sum_{\omega \in C^n}}
+ P_n(\Omega_n - C^n) \\
& = E[g_0^{n-1}((n-1)P_n(X_n))1(X_n \in C^n)] \\
& \phantom{= E[g_0} + E[g_0^{n-1}((n-1)P_n(X_n))1(X_n \notin C^n)] \\
& = E[g_0^{n-1}((n-1)P_n(X_n))].
\end{align*}
This establishes~(\ref{GTexpformula}). Following the reasoning
in the proof of Lemma~\ref{summean}, this implies
\begin{equation*}
\lim_{n \rightarrow \infty} E[\zeta_k^n] = E[g_k(Y)] = \lambda_k
\end{equation*}
for all $k$.
\end{proof}
\begin{lemma}
\label{GTconc}
For all nonnegative integers $k$,
\begin{equation*}
\lim_{n \rightarrow \infty} |\zeta_k^n - E[\zeta_k^n]| = 0 \quad \text{a.s.}
\end{equation*}
\end{lemma}
\begin{proof}
Observe that if we alter one symbol in the underlying
i.i.d.\ sequence, then $\zeta_k^n$ will change by at
most $2(k+1)/n$. As in the proof of Lemma~\ref{sumconc}, the
Azuma-Hoeffding-Bennett concentration
inequality~\cite[Corollary~2.4.14]{Dembo:LD} then implies
that
\begin{equation*}
\Pr(|\zeta_k^n - E[\zeta_k^n]| > \epsilon) \le
2\exp\left[- \frac{\epsilon^2 n}{8 (k+1)^2}\right].
\end{equation*}
Since the right-hand side is summable over $n$, the conclusion follows.
\end{proof}
\emph{Proof of Proposition~\ref{GTprop}:}
The result follows from
Lemma~\ref{GTmean}, Lemma~\ref{GTconc}, and Scheff\'{e}'s
theorem~\cite[Theorem 16.12]{Billingsley:PM}
as in the proof of Proposition~\ref{sumprop}.
\hfill \QED
\section{Shadow Estimation}
Proposition~\ref{sumprop} shows that the total
probabilities converge to a deterministic limit, which
is a function of the limit of the scaled shadows, $Q$. In fact,
the total probabilities converge to a Poisson mixture,
with $Q$ being the mixing distribution. The functional
form of the Poisson distribution enables us to create a
simple function of the observed string, the Good-Turing
estimator, that has the same limit as the total
{probabilities.~In}
particular, we can consistently estimate
the total probabilities
\newpage
\noindent
without having to explicitly estimate
$Q$.
In general, such a shortcut might not be
available. It is of interest therefore to study how to
estimate $Q$ itself from the observed string. With
an estimator for $Q$, one could create a ``plug-in''
estimator for other quantities of interest.
\section*{Acknowledgment}
This research was supported in part by
the Army Research Office under grant DAAD19-00-1-0466
and the National Science Foundation under grants
CAREER-0237549, ITR-0325924, and CCR-0312413.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,963
|
Here are some of the most recent companies we have worked with. We have a transparent pricing policy so you can be sure there will be no nasty surprises along the way.
Have a browse, take a look around and then contact usto see our complete portfolio and discuss your requirements.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,407
|
{"url":"https:\/\/cran-r.c3sl.ufpr.br\/web\/packages\/TransPhylo\/vignettes\/multitree.html","text":"# Simultaneous Inference of Multiple Transmission Trees\n\n#### 2021-04-20\n\nTransPhylo can draw inference on transmission events from a dated phylogenetic tree. In this tutorial, we demonstrate an extension, infer_multittree_shar_param, that allows simultaneous inference of multiple transmission trees from corresponding phylogenetic trees, with the possibility of sharing any subset of TransPhylo parameters. This may be useful when one is faced with multiple transmission clusters, which may be define genetically through SNP cutoff coupled with epidemiological data, and it may be the case that jointly analyzing these clusters is desirable assuming they share the same underlying epidemiological parameters. It is also computational efficient since fewer number of parameters will be estimated. Another use case might be that sometimes it is difficult to summarise a single representative tree from a collection of phylogenetic trees, such as BEAST trees. In this case, one could run multiple tree inference on a subsample of BEAST trees. This has additional benefit of incorporating some degree of uncertainty from the BEAST posterior.\n\n## Simulated Data\n\nlibrary(ape)\nlibrary(TransPhylo)\n\nWe simulate two outbreaks with the same set of parameters defined below, using TransPhylo\u2019s outbreak simulator.\n\nneg <- 100\/365\noff.r <- 1.5\nw.shape <- 10\nw.scale <- 0.1\nws.shape <- w.shape\nws.scale <- w.scale\npi <- 0.8\n\nset.seed(1234)\nsimu1 <- simulateOutbreak(neg=neg, off.r=off.r, pi=pi, w.shape=w.shape,\nw.scale=w.scale, dateStartOutbreak=2000,dateT=2005)\nsimu2 <- simulateOutbreak(neg=neg, off.r=off.r, pi=pi, w.shape=w.shape,\nw.scale=w.scale, dateStartOutbreak=2000,dateT=2005)\n\nWe plot the combined phylogenetic and transmission trees, each colour represents an infected host, colour changes when there is a transmission event.\n\nplot(simu1)\n\nplot(simu2)\n\n## Run single and multiple tree inference algorithm\n\nThe corresponding phylogenetic trees can be extracted. We first use single tree routine inferTTree separately for each phylogenetic tree, then use the multiple tree routine infer_multittree_shar_param to jointly infer transmission trees while sharing all parameters. The second parameter of offspring distribution, off.p, has no effect because it is not updated by default, but we still include it in the set of shared parameters.\n\nWe compare the posterior estimates of the parameters with those obtained from running inferTTree separately. In the multiple tree routine, we specify the two parameters of beta prior to be both 1, in order to get a uniform prior for the sampling rate that is consistent with inferTTree, where uniform prior is assumed.\n\nptree1 <- extractPTree(simu1)\nptree2 <- extractPTree(simu2)\n\niters <- 2e3; thin <- 10\nrecord_tp1 <- inferTTree(ptree1, w.shape, w.scale, ws.shape, ws.scale,\nmcmcIterations = iters, thinning = thin, dateT = 2005)\nrecord_tp2 <- inferTTree(ptree2, w.shape, w.scale, ws.shape, ws.scale,\nmcmcIterations = iters, thinning = thin, dateT = 2005)\nrecord_tpj <- infer_multittree_share_param(list(ptree1,ptree2), w.shape, w.scale, ws.shape, ws.scale,\nmcmcIterations = iters, thinning = thin, dateT = 2005,\nshare = c(\"neg\", \"off.r\", \"off.p\", \"pi\"))\n\nTake the last 50% trees. Note that the multiple tree inference returns list of length two where each element contains the result corresponding to the input phylogenetic tree.\n\nbegin <- (iters * 0.5) \/ thin + 1\nend <- iters \/ thin\nrecord_tp1 <- record_tp1[begin:end]\nrecord_tp2 <- record_tp2[begin:end]\nrecord_tpj[[1]] <- record_tpj[[1]][begin:end]\nrecord_tpj[[2]] <- record_tpj[[2]][begin:end]\n\n## Compare parameter estimates\n\nWe visualize the estimates from the three runs. Note that because we have shared the parameters, we can pick either element from the list result record_tpj.\n\nget_param_estimates <- function(record, p){\nsapply(record, function(x) x[[p]])\n}\n\ndf <- data.frame(run = rep(c(\"tp1\",\"tp2\",\"tp_multitree\"), each = length(record_tp1)),\npi = c(get_param_estimates(record_tp1, \"pi\"),\nget_param_estimates(record_tp2, \"pi\"),\nget_param_estimates(record_tpj[[1]], \"pi\")),\noff.r = c(get_param_estimates(record_tp1, \"off.r\"),\nget_param_estimates(record_tp2, \"off.r\"),\nget_param_estimates(record_tpj[[1]], \"off.r\")),\nneg = c(get_param_estimates(record_tp1, \"neg\"),\nget_param_estimates(record_tp2, \"neg\"),\nget_param_estimates(record_tpj[[1]], \"neg\")))\n\nIn this example where we explicitly set the parameters when simulating the outbreaks, we see that the posterior estimates from the multiple tree inference generally have lower variance than those resulting from single tree inference. The true parameter values are shown in gray reference line.\n\nboxplot(df$pi~df$run,ylab='pi')\nlines(c(0,4),rep(0.8,2),col='grey')\n\nboxplot(df$off.r~df$run,ylab='off.r')\nlines(c(0,4),rep(1.5,2),col='grey')\n\nboxplot(df$neg~df$run,ylab='Ne*g')\nlines(c(0,4),rep(100\/365,2),col='grey')","date":"2023-03-31 10:08:40","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\": 0.4614296555519104, \"perplexity\": 5286.3088850774075}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296949598.87\/warc\/CC-MAIN-20230331082653-20230331112653-00624.warc.gz\"}"}
| null | null |
Dipartimento di Capital – dipartimento argentino nella provincia di Corrientes
Dipartimento di Capital – dipartimento argentino nella provincia di Córdoba
Dipartimento di Capital – dipartimento argentino nella provincia di La Rioja
Dipartimento di Capital – dipartimento argentino nella provincia di Mendoza
Dipartimento di Capital – dipartimento argentino nella provincia di Salta
Dipartimento di Capital – dipartimento argentino nella provincia di Catamarca
Dipartimento di Capital – dipartimento argentino nella provincia di San Juan
Dipartimento di Capital – dipartimento argentino nella provincia di Tucumán
Dipartimento di Capital – dipartimento argentino nella provincia di La Pampa
Dipartimento di Capital – dipartimento argentino nella provincia di Misiones
Dipartimento di Capital – dipartimento argentino nella provincia di Santiago del Estero
Dipartimento di La Capital – vecchio nome del Dipartimento di Juan Martín de Pueyrredón, dipartimento argentino nella provincia di San Luis
Dipartimento di Capital – dipartimento argentino nella provincia di Santa Fe
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,532
|
Prorophora curvibasella is a species of snout moth. It is found in Uzbekistan.
References
Phycitinae
Moths described in 1887
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,871
|
Covered Bridges of Vermont, Illustrated map & Guide to 107 authentic Vermont Covered Bridges, Hartnett House Map Publishers, P.O. Box 41, Freeport ME 04032, (207) 865- 4037.
Covered Bridges of Vermont, by Ed Barna, published by The Countryman Press, P.O. Box 748, Woodstock VT 05091-0748. Also available through the author. write: Ed Barna, 7 Forbes Circle, Middlebury, VT 05753-1128 ($17.00).
Spanning Time: Vermont's Covered Bridges, by Joseph C. Nelson, published by The New England Press, P.O. Box 575, Shelburne, VT 05482-0575. Also available through the author. Write Joe Nelson, P.O. Box 267, Jericho, VT 05465-0267, ($40.00) or through web page www.vermontbridges.com/thebook. The book features 102 color photographs of Vermont's covered bridges in fifteen guided tours. An appendix provides: A Summary of Vermont's Covered Bridges; A Covered Bridge Glossary; A Bridge Truss section explaining how trusses work; The Bridge Builders, providing thumbnail biographies of the people who designed and built the bridges; A Covered Bridge Reading List, for those who want to read more.
The Vermont Covered Bridge Society will continue to answer questions about covered bridges informally by surface mail: VCBS, Inc. P.O. Box 97, Jeffersonville, VT 05464, or by email: jcnelson@together.net, and through this website.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,638
|
Q: why does function work on element but not array? I'm getting confused about when I can apply a function to an array element but not to the entire array. For exapmle:
I have a dataframe of time stamps where I'd just like to get the unique dates and day of weeks into a dataframe. Pulling the dates out is pretty easy:
dates_only = df.dtime.dt.date.unique()
at that point:
print type(dates_only) -> <type 'numpy.ndarray'>
print type(dates_only[0]) -> <type 'datetime.date'>
print dates_only[0].weekday() -> 4
but I can't create a new series by applying the weekday function:
dow = dates_only.weekday()
---------------------------------------------------------------------------
AttributeError Traceback (most recent call last)
<ipython-input-10-76986b9b07a2> in <module>()
----> 4 dow = dates_only.weekday()
AttributeError: 'numpy.ndarray' object has no attribute 'weekday'
Do I need to cast the ndarray somehow or should I be processing these in a loop like:
for i in range(len(dates_only)):
dow[i] = dates_only[i].weekday()
I think it will work but it just seems so wrong...
Thanks for looking!
A: There is problem unique return numpy array, but you can convert it to DatetimeIndex and then use DatetimeIndex.weekday, last use Series constructor:
start = pd.to_datetime('2015-02-24')
rng = pd.date_range(start, periods=10)
df = pd.DataFrame({'dtime': rng, 'a': range(10)})
print (df)
a dtime
0 0 2015-02-24
1 1 2015-02-25
2 2 2015-02-26
3 3 2015-02-27
4 4 2015-02-28
5 5 2015-03-01
6 6 2015-03-02
7 7 2015-03-03
8 8 2015-03-04
9 9 2015-03-05
dates_only = pd.DatetimeIndex(df.dtime.dt.date.unique())
dow = dates_only.weekday
print (dow)
[1 2 3 4 5 6 0 1 2 3]
dow = pd.Series(dates_only.weekday, index=df.index)
print (dow)
0 1
1 2
2 3
3 4
4 5
5 6
6 0
7 1
8 2
9 3
dtype: int32
Another solution is convert numpy array to_datetime and then use Series constructor:
dates_only = pd.Series(pd.to_datetime(df.dtime.dt.date.unique()), index=df.index)
print (dates_only)
dow = dates_only.dt.weekday
print (dow)
0 1
1 2
2 3
3 4
4 5
5 6
6 0
7 1
8 2
9 3
dtype: int64
A: You have an array of time objects.
And time objects have that method week_day():
print dates_only[0].weekday() -> 4
But that doesn't turn that ARRAY of time objects into a time object!
In other words: when you have a box of apples, does that mean you can eat the box itself?
See here for potential solutions.
A: You can not handle an array like this. You have to loop over the array as you wrote (does not look wrong to me) or use map:
dow = map(lambda date: date.weekday(), dates_only)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,476
|
{"url":"https:\/\/brilliant.org\/problems\/modular-arithmetic-problem\/","text":"# Why is Six Afraid of Forty Nine?\n\nFind the remainder when $$6^{273}+8^{273}$$ is divided by $$49$$:\n\n\u00d7","date":"2018-12-16 07:12:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.3601468801498413, \"perplexity\": 2074.03846626372}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-51\/segments\/1544376827281.64\/warc\/CC-MAIN-20181216051636-20181216073636-00519.warc.gz\"}"}
| null | null |
{"url":"https:\/\/scikit-learn.org\/stable\/modules\/generated\/sklearn.decomposition.fastica.html","text":"# sklearn.decomposition.fastica\u00b6\n\nsklearn.decomposition.fastica(X, n_components=None, algorithm=\u2019parallel\u2019, whiten=True, fun=\u2019logcosh\u2019, fun_args=None, max_iter=200, tol=0.0001, w_init=None, random_state=None, return_X_mean=False, compute_sources=True, return_n_iter=False)[source]\n\nPerform Fast Independent Component Analysis.\n\nRead more in the User Guide.\n\nParameters: X : array-like, shape (n_samples, n_features) Training vector, where n_samples is the number of samples and n_features is the number of features. n_components : int, optional Number of components to extract. If None no dimension reduction is performed. algorithm : {\u2018parallel\u2019, \u2018deflation\u2019}, optional Apply a parallel or deflational FASTICA algorithm. whiten : boolean, optional If True perform an initial whitening of the data. If False, the data is assumed to have already been preprocessed: it should be centered, normed and white. Otherwise you will get incorrect results. In this case the parameter n_components will be ignored. fun : string or function, optional. Default: \u2018logcosh\u2019 The functional form of the G function used in the approximation to neg-entropy. Could be either \u2018logcosh\u2019, \u2018exp\u2019, or \u2018cube\u2019. You can also provide your own function. It should return a tuple containing the value of the function, and of its derivative, in the point. The derivative should be averaged along its last dimension. Example: def my_g(x): return x ** 3, np.mean(3 * x ** 2, axis=-1) fun_args : dictionary, optional Arguments to send to the functional form. If empty or None and if fun=\u2019logcosh\u2019, fun_args will take value {\u2018alpha\u2019 : 1.0} max_iter : int, optional Maximum number of iterations to perform. tol : float, optional A positive scalar giving the tolerance at which the un-mixing matrix is considered to have converged. w_init : (n_components, n_components) array, optional Initial un-mixing array of dimension (n.comp,n.comp). If None (default) then an array of normal r.v.\u2019s is used. random_state : int, RandomState instance or None, optional (default=None) If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by np.random. return_X_mean : bool, optional If True, X_mean is returned too. compute_sources : bool, optional If False, sources are not computed, but only the rotation matrix. This can save memory when working with big data. Defaults to True. return_n_iter : bool, optional Whether or not to return the number of iterations. K : array, shape (n_components, n_features) | None. If whiten is \u2018True\u2019, K is the pre-whitening matrix that projects data onto the first n_components principal components. If whiten is \u2018False\u2019, K is \u2018None\u2019. W : array, shape (n_components, n_components) Estimated un-mixing matrix. The mixing matrix can be obtained by: w = np.dot(W, K.T) A = w.T * (w * w.T).I S : array, shape (n_samples, n_components) | None Estimated source matrix X_mean : array, shape (n_features, ) The mean over features. Returned only if return_X_mean is True. n_iter : int If the algorithm is \u201cdeflation\u201d, n_iter is the maximum number of iterations run across all components. Else they are just the number of iterations taken to converge. This is returned only when return_n_iter is set to True.\n\nNotes\n\nThe data matrix X is considered to be a linear combination of non-Gaussian (independent) components i.e. X = AS where columns of S contain the independent components and A is a linear mixing matrix. In short ICA attempts to un-mix' the data by estimating an un-mixing matrix W where S = W K X.`\n\nThis implementation was originally made for data of shape [n_features, n_samples]. Now the input is transposed before the algorithm is applied. This makes it slightly faster for Fortran-ordered input.\n\nImplemented using FastICA: A. Hyvarinen and E. Oja, Independent Component Analysis: Algorithms and Applications, Neural Networks, 13(4-5), 2000, pp. 411-430","date":"2019-05-24 08:56:10","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\": 0.19935955107212067, \"perplexity\": 5566.449204665636}, \"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-2019-22\/segments\/1558232257601.8\/warc\/CC-MAIN-20190524084432-20190524110432-00156.warc.gz\"}"}
| null | null |
\section{Introduction and primary methods}
It is well known that harmonic measure on domains in $\mathbb{C}$ can be interpreted in terms of exit distributions of planar Brownian motion. The conformal invariance of harmonic measure therefore implies a conformal invariance principle for exit distributions as well, and this principle can also be deduced directly from L\'evy's theorem on the conformal invariance of Brownian motion. This invariance allows in many instances for exit distributions to be calculated on simply connected domains for which a conformal equivalence with the disk is known. However, L\'evy's theorem in fact does not require maps to be injective, permitting general nonconstant analytic functions as well. We will show how the conformal invariance of the exit distribution of Brownian motion can be extended to nonconstant analytic functions, as well as to more general stopping times. This allows us to derive new expressions for many exit times and to calculate the distributions of Brownian motion at the exit time of certain non-simply connected domain, as well as at certain stopping times which are not exit times. We will give a number of illustrative examples, and show how certain identities can result from the appropriate choice of stopping times. Perhaps most notably, we will see a number of different ways in which the values for $\sum_{n=1}^{\infty} \frac{1}{n^{2m}}$ and $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2m+1}}$ can be deduced.
\vspace{12pt}
L\'evy's theorem is as follows (see \cite{durBM} or \cite{revyor} for a proof).
\begin{theorem} \label{holinv}
Let $f$ be analytic and nonconstant on a domain $U$, and let $a \in U$. Let $B_t$ be a Brownian motion starting at $a$, and $\tau$ a stopping time such that the set of Brownian paths $\{B_t: 0 \leq t \leq \tau\}$ lie within $U$ a.s. Then the process $f(B_{t})$ stopped at $\tau$ is a time-changed Brownian motion.
\end{theorem}
It should be noted that the time change referenced in the theorem can be expressed explicitly, but is not important for our purposes. Let $\gamma$ be a smooth curve parameterized by arclength, $B_t$ a Brownian motion starting at $a$, and $\tau$ a stopping time such that $B_\tau \in \gamma$ a.s. $\rho^a_\tau(w) ds$ will denote the density of $B_\tau$ on $\gamma$, when it exists, with $ds$ denoting the arclength element. We then have the following identity, valid for any measurable subset $A$ of $\gamma$:
\begin{equation} \label{}
P_a(B_\tau \in A) = \int_{A} \rho^a_\tau(\gamma_s) ds.
\end{equation}
Now, Levy's Theorem provides us with a Brownian motion $\hat B_t$ which is a time-change of $f(B_t)$ and a stopping time $\hat \tau$, which is the image under the time change of $\tau$, so that $\hat B_{\hat \tau} \in f(\gamma)$ a.s. We will also use the notation $\rho^a_{\hat \tau}(w) ds$ to denote the density of $\hat B_{\hat \tau} = f(B_\tau)$. Our method of projection is contained in the following theorem.
\begin{theorem} \label{massage}
Let $U$ be a domain, and suppose $f$ is a function analytic on $U$. Let $B_t$ be a Brownian motion starting at $a$, and $\tau$ a stopping time such that the set of Brownian paths $\{B_t: 0 \leq t \leq \tau\}$ lie within $U$ a.s. Suppose that $\gamma$ is a smooth curve in $U$ such that $B_\tau \in \gamma$ a.s. Then for any $a \in V$ and $w \in f(\gamma)$ we have
\begin{equation} \label{cherry}
\rho^{f(a)}_{\hat \tau} (w)ds = \sum_{z \in f^{-1}(w)\cap \gamma} \frac{\rho^a_\tau (z)}{|f'(z)|} ds.
\end{equation}
\end{theorem}
The proof is essentially immediate, since any Brownian path which finishes at $f^{-1}(w) \cap \gamma$ at time $\tau$ will be mapped under $f$ to a path finishing at $w$. The $|f'(z)|$ in the denominator on the right side of \rrr{cherry} is the scaling factor required for the change in the arclength element mapped under the analytic function $f$. We also remark that in the case that $f$ is conformal, it is often easier to use $g = f^{-1}$, and \rrr{cherry} becomes
\begin{equation} \label{cherry2}
\rho^{a}_{\hat \tau} (w)ds = \rho^{g(a)}_{\tau} (g(w))|g'(w)| ds.
\end{equation}
In the next section we will proceed through a series of examples which illustrate the use of the theorem.
\section{Examples}
The first two examples, concerning the disk and half-plane, are certainly known but are included for completeness and for their use in the later examples. In what follows, we will use the notation $T_U$ to denote the exit time of any domain $U$; that is, $T_U = \inf\{t \geq 0 | B_t \in U^c\}$.
\vspace{12pt}
\ccases{disk}(disk) There is only one exit distribution in $\mathbb{C}$ which is obvious: that of a disk in which the Brownian motion starts at the center. With ${\mathbb D} = \{|z|<1\}$, rotational invariance shows immediately that $\rho^0_{T_{\mathbb D}}(e^{i \theta})ds = \frac{ds}{2\pi}$. For $a \in {\mathbb D}$ consider the M\"obius transformation
\begin{equation} \label{diskmap}
\psi_a(z) = \frac{z-a}{1-\bar a z}.
\end{equation}
It is well-known that $\psi_a$ is a conformal self-map of ${\mathbb D}$ sending $a$ to 0. Using $\psi'_a = \frac{1-|a|^2}{(1-\bar a z)^2}$ and \rrr{cherry2} we obtain the following identity:
\begin{equation} \label{mae}
\rho^a_{T_{\mathbb D}}(e^{i \theta})ds = \frac{1}{2\pi} \frac{1-|a|^2}{|1-\bar a e^{i\theta}|^2} ds.
\end{equation}
We can also calculate the exit density from disks with radius other than one, as well as the hitting density of the circle when the Brownian motion begins at a point outside the circle. Let $m{\mathbb D} = \{|z|<m\}$ and $m{\mathbb D}^c = \{|z|>m\}$; Note that $T_{m{\mathbb D}}$ and $T_{m{\mathbb D}^c}$ both signify the first hitting time of $\{|z|=m\}$, but the former is of interest when the initial point of the Brownian motion has modulus less than $m$, and the latter when it is greater. If $|a|<m$, then we can project \rrr{mae} using the map $z \longrightarrow mz$ to obtain
\begin{equation} \label{}
\rho^a_{T_{m{\mathbb D}}}(me^{i \theta})ds = \frac{1}{m}\rho^{a/m}_{T_{{\mathbb D}}}(e^{i \theta})ds = \frac{1}{2\pi m} \frac{1-|\frac{a}{m}|^2}{|1-\frac{\bar a e^{i \theta}}{m}|^2}ds = \frac{1}{2\pi m} \frac{m^2-|a|^2}{|m-\bar a e^{i \theta}|^2}ds,
\end{equation}
while if $|a|>m$ then we use the map $z \longrightarrow \frac{m}{z}$ to get
\begin{equation} \label{}
\rho^a_{T_{m{\mathbb D}}}(me^{i \theta})ds = \frac{1}{m}\rho^{m/a}_{T_{{\mathbb D}}}(e^{-i \theta})ds = \frac{1}{2\pi m} \frac{1-|\frac{m}{a}|^2}{|1-\frac{m}{\bar a} e^{-i \theta}|^2}ds = \frac{1}{2\pi m} \frac{|a|^2-m^2}{|a-m e^{i \theta}|^2}ds.
\end{equation}
As a side note, the Poisson Integral Formula for harmonic functions in the disk ${\mathbb D}$ can be derived from \rrr{mae}, as we have by Dynkin's formula for harmonic $h$ (see \cite{fima})
\begin{equation} \label{}
h(a) = E_a[h(B_{T_{\mathbb D}})] = \frac{1}{2\pi} \int_{0}^{2\pi} h(e^{i\theta})\frac{1-|a|^2}{|1-\bar a e^{i\theta}|^2} d\theta.
\end{equation}
The reader may check that setting $a = re^{it}$ and performing a few simple manipulations yield a more standard form of the formula.
\vspace{12pt}
\ccases{halfplane} (half-plane) We can also easily calculate the exit distribution of a half-plane, as follows. Let $T_{\mathbb H}$ be this exit time. The conformal map taking ${\mathbb H} = \{y>0\}$ to ${\mathbb D}$ is given by $f(z) = \frac{z-i}{z+i}$, with $f'(z) = \frac{2i}{(z+i)^2}$. We obtain
\begin{equation} \label{}
\rho^i_{T_{\mathbb H}}(x)ds = \frac{1}{\pi} \frac{1}{1+x^2}ds.
\end{equation}
To find the distribution from a more general point $a = u+vi$ use the map $f(z) = u+vz$, which fixes ${\mathbb H}$ and maps $i$ to $a$, to obtain
\begin{equation} \label{}
\rho^a_{T_{\mathbb H}}(x)ds = \frac{1}{\pi} \frac{v}{v^2+(x-u)^2}ds.
\end{equation}
As in the case of the unit disk, this distribution leads via Dynkin's formula to the Poisson Integral Formula for the upper half-plane.
\vspace{12pt}
\ccases{punctdisk} (punctured disk) Now let $V = \{0<|z|<1\}$ be the punctured disk. We can calculate the exit distribution of $V$ by projecting from ${\mathbb H}$ via the covering map $f(z) = e^{iz}$ (note that $|f'(z)| = 1$ on $\delta {\mathbb H}$). For any $a \in V$ we obtain
\begin{equation} \label{page}
\rho^a_{T_V}(e^{i \theta})ds= \sum_{k=-\infty}^{\infty} \frac{-\ln |a|}{\pi ((\ln |a|)^2 + (arg(a)-(\theta+2\pi k))^2)}ds.
\end{equation}
Planar Brownian motion does not see points, i.e. $P_a(B_t = 0 \mbox{ for some }t \geq 0) = 0$. Thus, the exit distribution for the disk and the punctured disk agree. The difference then between the expressions \rrr{mae} and \rrr{page} is that each term in the sum in \rrr{page} corresponds to a different homotopy class of paths in the punctured disk terminating at $e^{i\theta}$, while \rrr{mae} does not differentiate between the homotopy classes. Furthermore, equating \rrr{mae} and \rrr{page}, assuming $a \in (0,1)$ for simplicity, gives the identity
\begin{equation} \label{effyou}
\sum_{k=-\infty}^{\infty} \frac{-\ln a}{\pi ((\ln a)^2 + (\theta+2\pi k)^2)} = \frac{1}{2\pi} \frac{1-a^2}{|1-a e^{i\theta}|^2} = \frac{1-a^2}{2\pi(1+a^2-2a\cos \theta)}.
\end{equation}
This identity can be manipulated into a more easily recognized identity, as follows. Divide both sides by $-\ln a$ and simplify. This gives
\begin{equation} \label{cthg}
\sum_{k=-\infty}^{\infty} \frac{1}{\pi((\ln a)^2 + (\theta + 2\pi k)^2) } = \frac{1-a^2}{2(-\ln a)(1+a^2-2a\cos \theta)}.
\end{equation}
Assuming $\theta \neq 0$, we can now let $a \nearrow 1$ using $\lim_{a \longrightarrow 1} \frac{1-a^2}{\ln a} = -2$, and obtain
\begin{equation} \label{cthg2}
\sum_{k=-\infty}^{\infty} \frac{1}{(\theta + 2\pi k)^2 } = \frac{1}{2(1-\cos \theta)}.
\end{equation}
Subtract $\frac{1}{\theta^2}$ from both sides (the term $k=0$), take the limit as $\theta \longrightarrow 0$ using $\lim_{\theta \longrightarrow 0} \frac{1}{2(1-\cos \theta)} - \frac{1}{\theta^2} = \frac{1}{12}$, and simplify. We obtain
\begin{equation} \label{jill}
\sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}.
\end{equation}
This is Euler's celebrated Basel sum, which has many other existing proofs, including a different probabilistic proof making use of planar Brownian motion (see \cite{meecp}). Note also that if we differentiate \rrr{cthg2} $2m-2$ times and let $\theta \longrightarrow 0$ we will be able to obtain the well-known values of $\sum_{k=1}^{\infty} \frac{1}{k^{2m}}$. Furthermore, if we take $a = e^{-1}$, then $\theta = 0, \pi$ successively, we obtain two identities, which may be added to obtain a third as follows.
\begin{equation} \label{mapleton}
\begin{gathered}
\sum_{k=-\infty}^{\infty} \frac{1}{\pi (1 + (2\pi k)^2)} = \frac{1}{2}\coth(\frac{1}{2}), \\
\sum_{k=-\infty}^{\infty} \frac{1}{\pi (1 + (\pi+2\pi k)^2)} = \frac{1}{2}\tanh(\frac{1}{2}), \\
\sum_{k=-\infty}^{\infty} \frac{1}{\pi (1 + (\pi k)^2)} = \coth(1).
\end{gathered}
\end{equation}
The final identity in \rrr{mapleton} is a standard identity which arises as an example of several different techniques for summing series, for instance using the residue theorem \cite[Ch. 7]{schaum}.
\vspace{12pt}
\ccases{strip} (infinite strip) We now calculate the exit distribution on an infinite strip. We let $W = \{-1 < Re(z) < 1\}$ and take the starting point of the Brownian motion $a$ to lie on the real interval $(-1,1)$; it is clear that the distribution with any other starting point can be obtained from this merely by translation. Apply our theorem to the function $\tan (\frac{\pi}{4} z)$, which maps $W$ conformally to ${\mathbb D}$, to get
\begin{equation} \label{is1}
\rho^a_{T_W}(\pm 1+yi)ds = |\frac{\pi}{4}\sec^2(\frac{\pi}{4}(\pm 1+yi)))|\rho^{\tan (\frac{\pi}{4} a)}_{T_{\mathbb D}}(\tan (\pm \frac{\pi}{4} + \frac{\pi}{4}yi))ds.
\end{equation}
We can simplify
\begin{equation} \label{}
\begin{split}
|\frac{\pi}{4}\sec^2(\frac{\pi}{4}(\pm 1+yi)))| & = \pi |e^{-\frac{\pi}{4}y+ \frac{\pi}{4}i} + e^{\frac{\pi}{4}y- \frac{\pi}{4}i}|^{-2} \\
& = \pi |\frac{1}{\sqrt{2}}(e^{-\frac{\pi}{4}y}+ e^{\frac{\pi}{4}y}) + \frac{i}{\sqrt{2}}(e^{-\frac{\pi}{4}y}- e^{\frac{\pi}{4}y})|^{-2} \\
& = \frac{\pi}{2}(\cosh^2 \frac{\pi}{4}y+\sinh^2 \frac{\pi}{4}y)^{-1} \\
& = \frac{\pi}{2}(\cosh \frac{\pi}{2}y)^{-1}.
\end{split}
\end{equation}
Furthermore, using the distribution calculated in Example \ref{disk}, we have
\begin{equation} \label{}
\rho^{\tan (\frac{\pi}{4} a)}_{T_{\mathbb D}}(\tan (\pm \frac{\pi}{4} + \frac{\pi}{4}yi))ds = \frac{1-\tan^2(\frac{\pi}{4} a)}{2\pi|1-\tan (\frac{\pi}{4} a)\tan (\pm \frac{\pi}{4} + \frac{\pi}{4}yi)|^2}ds.
\end{equation}
Using $\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1-\tan \alpha \tan \beta}$ and the identity $\tan(\frac{\pi}{4}yi)= i \tanh(\frac{\pi}{4}y)$ gives
\begin{equation} \label{}
\begin{split}
\tan (\pm \frac{\pi}{4} + \frac{\pi}{4}yi) &= \frac{\pm 1 + i \tanh(\frac{\pi}{4}y)}{1 \mp i \tanh(\frac{\pi}{4}y)} = \frac{\pm(1 \pm i \tanh(\frac{\pi}{4}y))^2}{1 + \tanh^2(\frac{\pi}{4}y)} \\
& = \frac{\pm(1 - \tanh^2(\frac{\pi}{4}y)) + 2i\tanh(\frac{\pi}{4}y) }{1 + \tanh^2(\frac{\pi}{4}y)},
\end{split}
\end{equation}
which yields
\begin{equation} \label{is2}
\rho^{\tan (\frac{\pi}{4} a)}_{T_{\mathbb D}}(\tan (\pm \frac{\pi}{4} + \frac{\pi}{4}yi))ds = \frac{1-\tan^2(\frac{\pi}{4} a)}{2\pi((1\mp\tan (\frac{\pi}{4} a)\frac{1 - \tanh^2(\frac{\pi}{4}y)}{1 + \tanh^2(\frac{\pi}{4}y)})^2+(\tan (\frac{\pi}{4} a)\frac{2\tanh(\frac{\pi}{4}y) }{1 + \tanh^2(\frac{\pi}{4}y)})^2)}ds.
\end{equation}
Combining \rrr{is1}-\rrr{is2} gives
\begin{equation} \label{is3}
\rho^a_{T_W}(\pm 1+yi)ds = \frac{\mbox{sech} (\frac{\pi}{2}y)(1-\tan^2(\frac{\pi}{4} a))}{4((1\mp\tan (\frac{\pi}{4} a)\frac{1 - \tanh^2(\frac{\pi}{4}y)}{1 + \tanh^2(\frac{\pi}{4}y)})^2+(\tan (\frac{\pi}{4} a)\frac{2\tanh(\frac{\pi}{4}y) }{1 + \tanh^2(\frac{\pi}{4}y)})^2)}ds.
\end{equation}
There is another method for calculating $\rho^a_{T_W}(\pm 1+yi)ds$ which uses a form of the reflection principle. This method will be applied in other examples below, namely Examples \ref{halfstrip} and \ref{rectangle}, and we prove its validity carefully for this example, while in the later ones merely referencing this one. Let us define $\tau(b) = \inf\{t \geq 0: Re(B_t) = b\}$. It is clear with comparison with the half-plane example above that $\rho^a_{\tau(b)}(b+yi)ds = \frac{1}{\pi}\frac{|a-b|}{|a-b|^2+y^2}ds$. It will turn out that
\begin{equation} \label{tats}
\begin{gathered}
\rho^a_{T_W}(1+yi)ds = \rho^a_{\tau(1)}(1+yi)ds - \rho^a_{\tau(-3)}(-3+yi)ds + \rho^a_{\tau(5)}(5+yi)ds - \rho^a_{\tau(-7)}(-7+yi)ds + \ldots, \\
\rho^a_{T_W}(-1+yi)ds = \rho^a_{\tau(-1)}(1+yi)ds - \rho^a_{\tau(3)}(3+yi)ds + \rho^a_{\tau(-5)}(-5+yi)ds - \rho^a_{\tau(7)}(7+yi)ds + \ldots.
\end{gathered}
\end{equation}
We will prove the first equation in \rrr{tats}, and for the proof of this it will help to isolate several lemmas. Let us extend the definition of $\tau$ by recursively defining $\tau(b_1, b_2, \ldots, b_{n+1}) = \inf\{t \geq \tau(b_1, \ldots, b_{n}): Re(B_t) = b_{n+1}\}$; that is, $\tau(b_1, b_2, \ldots, b_{n})$ is the first time at which $Re(B_t)$ has visited the sequence $b_1, b_2, \ldots , b_n$ in order. The first lemma should be clear upon consideration, and we therefore omit the proof.
\begin{lemma} \label{still}
On the event $\{\tau(b_1) < \tau(b_2)\}$, we have $\tau(b_1,b_2, \dots, b_n) = \tau(b_2, \dots, b_n)$.
\end{lemma}
That is, on the set of all Brownian paths which hit $b_1$ before $b_2$, we can drop $b_1$ from the start of the sequence without changing the value of the stopping time. The following is an immediate consequence.
\begin{lemma} \label{mang}
For $A \subseteq {\mathbb R}$, we have
\begin{equation} \label{alive}
\begin{split}
&\Big( 1_{\{Im(B_{\tau(1)}) \in A\}} - 1_{\{Im(B_{\tau(-1,1)}) \in A\}}\Big) + \Big(1_{\{Im(B_{\tau(1,-1,1)}) \in A\}} - 1_{\{Im(B_{\tau(-1,1,-1,1)}) \in A\}}\Big) \\
&+ \ldots + \Big(1_{\{Im(B_{\tau(1,-1, \ldots, -1, 1)}) \in A\}} - 1_{\{Im(B_{\tau(-1,1,-1,\ldots, 1, 1)}) \in A\}}\Big) \\
& \qquad \leq 1_{\{Re(B_{T_W})=1,Im(B_{T_W}) \in A\}} \\
& \qquad \leq 1_{\{Im(B_{\tau(1)}) \in A\}} -\Big( 1_{\{Im(B_{\tau(-1,1)}) \in A\}} - 1_{\{Im(B_{\tau(1,-1,1)}) \in A\}}\Big) \\
& \qquad \qquad - \ldots - \Big( 1_{\{Im(B_{\tau(-1,1, \ldots ,-1,1)}) \in A\}} - 1_{\{Im(B_{\tau(1,-1,1, \ldots ,-1,1)}) \in A\}}\Big).
\end{split}
\end{equation}
\end{lemma}
{\bf Remark:} In the sums in \rrr{alive}, it should be understood that the sequences defining the $\tau$'s alternate and increase in length by one with each successive term.
\vspace{12pt}
{\bf Proof:} We begin by noting that $\min(1_{\{Im(B_{\tau(1)}) \in A\}}, 1_{\{\tau(1) < \tau(-1)\}}) = 1_{\{Re(B_{T_W})=1,Im(B_{T_W}) \in A\}}$. By Lemma \ref{still}, on the set $\{\tau(-1) < \tau(1)\}$ we have $\tau(-1,1,-1,\ldots, -1, 1) = \tau(1,-1, \ldots, -1, 1)$, and thus each positive term in the leftmost sum in \rrr{alive} is canceled by the subsequent negative term and therefore the sum is 0. On the other hand, on the set $\{\tau(1) < \tau(-1)\}$ each negative term in the leftmost sum except the last is canceled by the subsequent positive one, again by Lemma \ref{still}. Thus, on $\{\tau(1) < \tau(-1)\}$, the leftmost side of \rrr{alive} is equal to $1_{\{Im(B_{\tau(1)}) \in A\}} - 1_{\{Im(B_{\tau(-1,1,-1,\ldots, 1, 1)}) \in A\}} \leq 1_{\{Im(B_{\tau(1)}) \in A\}} = 1_{\{Re(B_{T_W})=1,Im(B_{\tau(1)}) \in A\}}$. It follows that the leftmost side is less than or equal to $\min(1_{\{Im(B_{\tau(1)}) \in A\}}, 1_{\{\tau(1) < \tau(-1)\}}) = 1_{\{Re(B_{T_W})=1,Im(B_{T_W}) \in A\}}$. The second inequality follows similarly from Lemma \ref{still}, for on the set $\{\tau(1) < \tau(-1)\}$ the difference inside each set of parentheses on the rightmost side is zero yielding a value of $1_{\{Im(B_{\tau(1)}) \in A\}} \geq 1_{\{Re(B_{T_W})=1,Im(B_{T_W}) \in A\}}$, while on $\{\tau(-1) < \tau(1)\}$ we have $1_{\{Re(B_{T_W})=1,Im(B_{T_W}) \in A\}}=0$, while the right side is equal to $1_{\{Im(B_{\tau(1,-1,1, \ldots ,-1,1)}) \in A\}} \geq 0$. {\hfill $\Box$ \bigskip}
If we let $A$ be a small interval on the line $\{Re(z)=1\}$ centered at $1+yi$, divide by the length of the interval, and then let this length go to 0 we obtain
\begin{equation} \label{denim}
\rho^a_{T_W}(1+yi)ds = \rho^a_{\tau(1)}(1+yi)ds - \rho^a_{\tau(-1,1)}(1+yi)ds + \rho^a_{\tau(1,-1,1)}(1+yi)ds - \rho^a_{\tau(-1,1,-1,1)}(1+yi)ds + \ldots.
\end{equation}
We should mention that, intuitively, \rrr{denim} is very simple: in order to calculate $\rho^a_{T_W}(1+yi)ds$ we want to count paths which leave $\{Re(z)<1\}$ at $1+yi$, however we want to remove the contribution from paths which strike $\{Re(z)=-1\}$ first, so we consider $\rho^a_{\tau(1)}(1+yi)ds - \rho^a_{\tau(-1,1)}(1+yi)ds$; however, by subtracting $\rho^a_{\tau(-1,1)}(1+yi)ds$ we have subtracted too much, as we have incorrectly subtracted the contribution from paths which hit $\{Re(z)=1\}$ before $\{Re(z)=-1\}$, so we must add $\rho^a_{\tau(1,-1,1)}(1+yi)ds$ to compensate; however, by an analogous argument we have overcompensated, and must therefore subtract $\rho^a_{\tau(-1,1,-1,1)}(1+yi)ds$, and so forth. It remains only to understand how to calculate the density $\rho^a_{\tau(b_1, \ldots, b_{n})}(b_n+yi)ds$.
\begin{lemma} \label{}
For any sequence of real numbers $a=b_0, b_1, \ldots, b_n$ we have
\begin{equation} \label{}
\begin{split}
\rho^a_{\tau(b_1, \ldots, b_n)}(b_n + yi)ds &= \rho^a_{\tau(a + \sum_{j=1}^{n}|b_j-b_{j-1}|)}(a + \sum_{j=1}^{n}|b_j-b_{j-1}| + yi)ds \\
&= \rho^a_{\tau(a - \sum_{j=1}^{n}|b_j-b_{j-1}|)}(a - \sum_{j=1}^{n}|b_j-b_{j-1}| + yi)ds.
\end{split}
\end{equation}
\end{lemma}
{\bf Proof:} By induction on $n$. The case $n=1$ follows from the symmetry of Brownian motion over the line $\{Re(z)=a\}$. Suppose that the result holds for $n$, and consider a sequence $b_1, \ldots, b_n, b_{n+1}$. If $b_n$ lies between $b_{n-1}$ and $b_{n+1}$, then $\tau(b_1, \ldots, b_{n-1}, b_n, b_{n+1}) = \tau(b_1, \ldots, b_{n-1}, b_{n+1})$, since the real part of the Brownian motion must hit $b_n$ in passing from $b_{n-1}$ to $b_{n+1}$, so the result follows from the induction hypothesis (since then also $|b_{n+1}-b_n| + |b_n-b_{n-1}| = |b_{n+1}-b_{n-1}|$). If, on the other hand, $b_n$ does not lie between $b_{n-1}$ and $b_{n+1}$, then it must lie between $b_{n-1}$ and $b_n -(b_{n+1}-b_n)$. However, we must have $\rho^a_{\tau(b_1, \ldots, b_n, b_{n+1})}(b_{n+1} + yi)ds = \rho^a_{\tau(b_1, \ldots, b_n, b_n-(b_{n+1}-b_n))}(b_n-(b_{n+1}-b_n) + yi)ds$, since the reflection principle implies that the process
\begin{equation} \label{}
\tilde B_t = \left \{ \begin{array}{ll}
B_t & \qquad \mbox{if } t \leq \tau(b_1, \ldots, b_n) \\
b_n - (Re(B_t) - b_n) + i Im(B_t) & \qquad \mbox{if } t > \tau(b_1, \ldots, b_n)\;,
\end{array} \right.
\end{equation}
is also a Brownian motion; this is the reflection of $B_t$ over the line $\{Re(z) = b_n\}$ for $t > \tau(b_1, \ldots, b_n)$. By the same argument as before, $\tau(b_1, \ldots,b_{n-1}, b_n, b_n-(b_{n+1}-b_n)) = \tau(b_1, \ldots,b_{n-1}, b_n-(b_{n+1}-b_n))$, and furthermore $|b_n-(b_{n+1}-b_n) - b_n| = |b_{n+1}-b_n|$, so the result again follows from the induction hypothesis. {\hfill $\Box$ \bigskip}
Using this lemma, we see that \rrr{denim} reduces to the first equation in \rrr{tats}, with the other equation in \rrr{tats} following by the symmetric argument. Using the expression immediately preceding \rrr{tats}, we see
\begin{equation} \label{tats2}
\begin{gathered}
\rho^a_{T_W}(1+yi)ds = \frac{ds}{\pi} \Big( \frac{1-a}{(1-a)^2+y^2} - \frac{3+a}{(3+a)^2+y^2} + \frac{5-a}{(5-a)^2+y^2} - \frac{7+a}{(7+a)^2+y^2} + \ldots \Big), \\
\rho^a_{T_W}(-1+yi)ds = \frac{ds}{\pi} \Big( \frac{1+a}{(1+a)^2+y^2} - \frac{3-a}{(3-a)^2+y^2} + \frac{5+a}{(5+a)^2+y^2} - \frac{7-a}{(7-a)^2+y^2} + \ldots \Big).
\end{gathered}
\end{equation}
Equating the expression for $\rho^a_{T_W}(1+yi)ds$ in \rrr{tats2} with that in \rrr{is3} gives the identity
\begin{equation} \label{nosering}
\begin{split}
\frac{1}{\pi}\sum_{j=1}^{\infty} \frac{(-1)^{j+1}((2j-1) + (-1)^j a)}{((2j-1) + (-1)^j a)^2 +y^2} = \frac{\mbox{sech} (\frac{\pi}{2}y)(1-\tan^2(\frac{\pi}{4} a))}{4((1-\tan (\frac{\pi}{4} a)\frac{1 - \tanh^2(\frac{\pi}{4}y)}{1 + \tanh^2(\frac{\pi}{4}y)})^2+(\tan (\frac{\pi}{4} a)\frac{2\tanh(\frac{\pi}{4}y) }{1 + \tanh^2(\frac{\pi}{4}y)})^2)}.
\end{split}
\end{equation}
Needless to say, our lives are considerably simplified by setting $y=0$ or $a=0$. For $a=0$, we obtain
\begin{equation} \label{}
\sum_{j=1}^{\infty} \frac{(-1)^{j+1}(2j-1)}{(2j-1)^2 +y^2} = \frac{\pi}{4}\mbox{sech}(\frac{\pi}{2}y),
\end{equation}
which can be obtained by other methods, for instance the residue theorem (\cite{schaum}). Taking $y=0$ gives us Leibniz's representation for $\pi$:
\begin{equation} \label{propasi}
\frac{\pi}{4} = 1-\frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots
\end{equation}
Returning to \rrr{tats2}, set now $y=0$ to obtain
\begin{equation} \label{mei}
\sum_{j=1}^{\infty} \frac{(-1)^{j+1}}{(2j-1) + (-1)^j a } = \frac{1}{1-a} - \frac{1}{3+a} + \frac{1}{5-a} - \frac{1}{7-a} + \ldots= \frac{\pi}{4} \Big(\frac{1+\tan (\frac{\pi}{4} a)}{1-\tan (\frac{\pi}{4} a)}\Big).
\end{equation}
This identity is somewhat unusual, and may be new. It can be manipulated to obtain a number of other identities, as follows. Let $g(a) = \frac{\pi}{4} \Big(\frac{1+\tan (\frac{\pi}{4} a)}{1-\tan (\frac{\pi}{4} a)}\Big)$. Then, if $r$ is a positive integer, differentiating \rrr{mei} $r-1$ times yields the identity
\begin{equation} \label{mei2}
\sum_{j=1}^{\infty} \frac{(r-1)!(-1)^{r(j+1)}}{((2j-1) + (-1)^j a)^r} = g^{(r-1)}(a).
\end{equation}
Note that if $r$ is even then all terms in the sum will be positive, while if $r$ is odd then the sum will be alternating. Setting $a = 0$ in this identity gives the values of all sums of the form $\sum_{j=1}^{\infty} \frac{1}{(2j-1)^{2m}}$ or $\sum_{j=1}^{\infty} \frac{(-1)^j}{(2j-1)^{2m+1}}$, which are well-known with the first few equal to $\sum_{j=1}^{\infty} \frac{(-1)^{j+1}}{(2j-1)} = \frac{\pi}{4}, \sum_{j=1}^{\infty} \frac{1}{(2j-1)^{2}} = \frac{\pi^2}{8}, \sum_{j=1}^{\infty} \frac{(-1)^{j+1}}{(2j-1)^{3}} = \frac{\pi^3}{32}, \sum_{j=1}^{\infty} \frac{1}{(2j-1)^{4}} = \frac{\pi^4}{96}, \ldots$. These values have been known since the time of Euler, and it should be noted that the values of the even-powered identities easily give the values of $\zeta(2k)$, the Riemann $\zeta$ function evaluated at the even integers.
\vspace{12pt}
\rrr{mei2} can also be manipulated into a different identity, as follows. Let $r$ be odd and let us refer to the identity \rrr{mei2} as $I(a)$. Let $q$ be a positive integer, and consider the sum $I(\frac{q-1}{q}) + I(\frac{q-3}{q}) + \ldots + I(\frac{-q+3}{q}) + I(\frac{-q+1}{q})$. The left side of this new identity is a sum, and we will group the terms in this new sum according to their place in the original sums; that is, the first term will be the sum of the first terms in $I(\frac{q-1}{q}), \ldots, I(\frac{-q+1}{q})$, the second term will be the sum of the corresponding second terms (each of which is negative), and so forth. Using $\frac{1}{((2j-1)+ (-1)^j (\frac{p}{q}))^r} = \frac{q^r}{((2j-1)q+ (-1)^j p)^r}$, it may be verified that the first term in the new sum is $q^r$ times the sum of the reciprocals of the $r$-th power of the odd integers from $1$ to $2q-1$, the second term (which is negative) is $q^r$ times the sum of the reciprocals of the $r$-th power of the odd integers from $2q+1$ to $4q-1$, and so forth. We obtain the identity
\begin{equation} \label{armdark}
\begin{split}
\Big( \frac{1}{1} + \frac{1}{3^r} &+ \ldots + \frac{1}{(2q-1)^r}\Big) - \Big( \frac{1}{(2q+1)^r} + \ldots + \frac{1}{(4q-1)^r}\Big) + \Big( \frac{1}{(4q+1)^r} + \ldots + \frac{1}{(6q-1)^r}\Big) \\
&- \Big( \frac{1}{(6q+1)^r} + \ldots + \frac{1}{(8q-1)^r}\Big) + \ldots \\
& \qquad \qquad = \frac{1}{(r-1)!q^r} \sum_{k=0}^{q-1}g^{(r-1)}(\frac{q-1-2k}{q}).
\end{split}
\end{equation}
As a representative sample of the sums that are obtained for various choices of $q$ and $r$, we have:
\begin{equation} \label{}
\begin{gathered}
\frac{1}{1} + \frac{1}{3} - \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - \frac{1}{13} - \frac{1}{15} + \ldots = \frac{\pi \sqrt{2}}{4},\\
\frac{1}{1} + \frac{1}{3} + \frac{1}{5} - \frac{1}{7} - \frac{1}{9} - \frac{1}{11} + \frac{1}{13} + \frac{1}{15} + \frac{1}{17} - \ldots = \frac{5 \pi}{12},\\
\frac{1}{1} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} - \frac{1}{9} - \frac{1}{11} - \frac{1}{13} - \frac{1}{15} + \ldots = \pi \sqrt{2+ \sqrt{2}},\\
\frac{1}{1^3} + \frac{1}{3^3} - \frac{1}{5^3} - \frac{1}{7^3} + \frac{1}{9^3} + \frac{1}{11^3} - \frac{1}{13^3} - \frac{1}{15^3} + \ldots = \frac{3 \pi^3 \sqrt{2}}{128},\\
\frac{1}{1^3} + \frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} - \frac{1}{9^3} - \frac{1}{11^3} + \frac{1}{13^3} + \frac{1}{15^3} + \frac{1}{17^3} - \ldots = \frac{29 \pi^3}{864},\\
\frac{1}{1^5} + \frac{1}{3^5} - \frac{1}{5^5} - \frac{1}{7^5} + \frac{1}{9^5} + \frac{1}{11^5} - \frac{1}{13^5} - \frac{1}{15^5} + \ldots = \frac{\pi^5 57 \sqrt{2}}{24576},\\
\frac{1}{1^5} + \frac{1}{3^5} + \frac{1}{5^5} - \frac{1}{7^5} - \frac{1}{9^5} - \frac{1}{11^5} + \frac{1}{13^5} + \frac{1}{15^5} + \frac{1}{17^5} - \ldots = \frac{1225 \pi^5}{373248}.
\end{gathered}
\end{equation}
If we desire the analogous sums which include the even integers, we may argue as follows. Let $q \geq 2$ be a positive integer, and consider $I(\frac{q-2}{q}) + I(\frac{q-4}{q}) + \ldots + I(\frac{-q+4}{q}) + I(\frac{-q+2}{q})$. Ordering the terms as before, it may be checked that the left side will be $q^r$ times the sum of the $r$-th powers of the reciprocals of the even integers other than the multiples of $2q$. We therefore obtain
\begin{equation} \label{armdark2}
\begin{split}
\Big( \frac{1}{2^r} + \frac{1}{4^r} &+ \ldots + \frac{1}{(2q-2)^r}\Big) - \Big( \frac{1}{(2q+2)^r} + \ldots + \frac{1}{(4q-2)^r}\Big) + \Big( \frac{1}{(4q+2)^r} + \ldots + \frac{1}{(6q-2)^r}\Big) \\
&- \Big( \frac{1}{(6q+2)^r} + \ldots + \frac{1}{(8q-2)^r}\Big) + \ldots \\
& \qquad \qquad = \frac{1}{(r-1)!q^r} \sum_{k=1}^{q-1} g^{(r-1)}(\frac{q-2k}{q}).
\end{split}
\end{equation}
In order to include the multiples of $2q$, add $\frac{1}{(2q)^r}$ times the series $\Delta_r :=\frac{1}{1^r}-\frac{1}{2^r} + \frac{1}{3^r} - \frac{1}{4^r} + \ldots$. Multiplying both sides by $2^r$, we obtain
\begin{equation} \label{embeth}
\begin{split}
\Big( \frac{1}{1^r} + \frac{1}{2^r} &+ \ldots + \frac{1}{q^r}\Big) - \Big( \frac{1}{(q+1)^r} + \ldots + \frac{1}{(2q)^r}\Big) + \Big( \frac{1}{(2q+1)^r} + \ldots + \frac{1}{(3q)^r}\Big) \\
&- \Big( \frac{1}{(3q+1)^r} + \ldots + \frac{1}{(4q)^r}\Big) + \ldots \\
& \qquad \qquad = \frac{\Delta_r}{q^r} + \frac{2^r}{(r-1)!q^r} \sum_{k=1}^{q-1} g^{(r-1)}(\frac{q-2k}{q}).
\end{split}
\end{equation}
The formula holds for $q=1$ as well, provided that the final sum on the right side is taken to be empty and therefore 0. Note that $\Delta_1 = \ln 2$, but that no other closed-form values of $\Delta_r$ for $r$ odd are known (nor are any likely to be known soon, as they can be expressed in terms of the Riemann zeta function evaluated at $r$). For $r=1, q=2,3,4$ we obtain
\begin{equation} \label{}
\begin{gathered}
\frac{1}{1} + \frac{1}{2} - \frac{1}{3} - \frac{1}{4} + \frac{1}{5} + \frac{1}{6} - \frac{1}{7} - \frac{1}{8} + \ldots = \frac{\pi}{4} + \frac{\ln 2}{2},\\
\frac{1}{1} + \frac{1}{2} + \frac{1}{3} - \frac{1}{4} - \frac{1}{5} - \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} - \ldots = \frac{2\pi}{3\sqrt{3}} + \frac{\ln 2}{3},\\
\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{5} - \frac{1}{6} - \frac{1}{7} - \frac{1}{8} + \ldots = \frac{\pi(1+2\sqrt{2})}{8} + \frac{\ln 2}{4}.
\end{gathered}
\end{equation}
\vspace{12pt}
\ccases{halfstrip} (half strip) Let $W$ now be the semi-infinite strip $\{-1<Re(z)<1, Im(z) > 0\}$. If $x \in (-1,1)$ and $a = \alpha + \beta i \in W$, we can calculate $\rho_{T_W}^{\alpha+\beta i}(x)ds$ by using the map $f(z) = \sin (\frac{\pi}{2}z)$, which maps $W$ conformally onto ${\mathbb H} = \{Im(z) > 0\}$, fixing $-1$ and $1$. A simple calculation shows $\sin (\frac{\pi}{2}(\alpha + \beta i)= \cosh(\frac{\pi \beta}{2})\sin(\frac{\pi \alpha}{2}) + i \sinh(\frac{\pi \beta}{2})\cos(\frac{\pi \alpha}{2}),$ and Theorem \ref{massage} and the calculation in Example \ref{halfplane} combine to give
\begin{equation} \label{michelle}
\begin{split}
\rho_{T_W}^{\alpha+\beta i}(x)ds & = \frac{1}{2} \Big(\frac{\sinh(\frac{\pi \beta}{2})\cos(\frac{\pi \alpha}{2})\cos(\frac{\pi x}{2})}{\sinh^2(\frac{\pi \beta}{2})\cos^2(\frac{\pi \alpha}{2}) + (\cosh(\frac{\pi \beta}{2})\sin(\frac{\pi \alpha}{2}) - \sin (\frac{\pi x}{2}))^2} \Big)ds \\
& = \frac{1}{2} \Big(\frac{\sinh(\frac{\pi \beta}{2})\cos(\frac{\pi \alpha}{2})\cos(\frac{\pi x}{2})}{\sinh^2(\frac{\pi \beta}{2}) + \sin^2(\frac{\pi \alpha}{2}) + \sin^2(\frac{\pi x}{2}) -2\cosh(\frac{\pi \beta}{2}) \sin(\frac{\pi \alpha}{2}) \sin (\frac{\pi x}{2})} \Big)ds,
\end{split}
\end{equation}
where the identity $\sinh^2(\frac{\pi \beta}{2})\cos^2(\frac{\pi \alpha}{2}) + \cosh^2(\frac{\pi \beta}{2})\sin^2(\frac{\pi \alpha}{2}) = \sinh^2(\frac{\pi \beta}{2}) + \sin^2(\frac{\pi \alpha}{2})$ was used to simplify the denominator. On the other hand, we can calculate this density using reflection as well. We claim that
\begin{equation} \label{}
\begin{split}
\rho_{T_W}^{\alpha+\beta i}(x)ds &= \rho_{T_{\mathbb H}}^{\alpha+\beta i}(x)ds - \Big(\rho_{T_{\mathbb H}}^{\alpha+\beta i}(2-x)ds +\rho_{T_{\mathbb H}}^{\alpha+\beta i}(-2-x)ds\Big) \\
& \qquad \qquad + \Big(\rho_{T_{\mathbb H}}^{\alpha+\beta i}(4+x)ds +\rho_{T_{\mathbb H}}^{\alpha+\beta i}(-4+x)ds\Big) - \ldots \\
& = \frac{ds}{\pi} \Big(\frac{\beta}{\beta^2 + (\alpha - x)^2} - \Big(\frac{\beta}{\beta^2 + (\alpha - (2-x))^2} + \frac{\beta}{\beta^2 + (\alpha - (-2-x))^2}\Big) \\
& \qquad \qquad+ \Big(\frac{\beta}{\beta^2 + (\alpha - (4+x))^2} + \frac{\beta}{\beta^2 + (\alpha - (-4+x))^2}\Big) - \ldots \Big)
\end{split}
\end{equation}
In order to justify this, we note first that $T_W = \min (T_{\mathbb H}, \tau(-1), \tau(1))$, where $\tau$ was defined in the previous example. If a Brownian path exits $W$ at $x$, then it also exits ${\mathbb H}$ at $x$, but must also not have hit $\{Re(z) = \pm 1\}$ before leaving $W$. The likelihood of exiting ${\mathbb H}$ at $x$ after first striking $\{Re(z) = 1\}$ is the same, by reflection, as the likelihood of exiting ${\mathbb H}$ at the reflection of $x$ over $\{Re(z) = 1\}$, which is $2-x$. Thus we must subtract $\rho_{T_{\mathbb H}}^{\alpha+\beta i}(2-x)ds$, and by the symmetric argument must also subtract the quantity obtained by reflection over $\{Re(z) = -1\}$, which is $\rho_{T_{\mathbb H}}^{\alpha+\beta i}(-2-x)ds$. However, we have now twice subtracted the contribution from paths which hit both of $\{Re(z) = \pm 1\}$ before hitting $\{Im(z)=0\}$, and we must therefore add $\rho_{T_{\mathbb H}}^{\alpha+\beta i}(4+x)ds +\rho_{T_{\mathbb H}}^{\alpha+\beta i}(-4+x)ds$; however, we have again overcompensated, and therefore must reflect again and subtract, and so forth (this argument can be made rigorous by adapting the methods given in Example \ref{strip}). Equating the two values obtained for the density, we obtain the identity
\begin{equation} \label{}
\begin{split}
\frac{1}{\pi} &\Big(\frac{\beta}{\beta^2 + (\alpha - x)^2} - \Big(\frac{\beta}{\beta^2 + ((\alpha+x) - 2)^2} + \frac{\beta}{\beta^2 + ((\alpha+x) + 2)^2}\Big) \\
& \qquad + \Big(\frac{\beta}{\beta^2 + ((\alpha - x) - 4)^2} + \frac{\beta}{\beta^2 + ((\alpha-x) + 4)^2}\Big) - \ldots \Big) \\
& \qquad \qquad = \frac{1}{2} \Big(\frac{\sinh(\frac{\pi \beta}{2})\cos(\frac{\pi \alpha}{2})\cos(\frac{\pi x}{2})}{\sinh^2(\frac{\pi \beta}{2}) + \sin^2(\frac{\pi \alpha}{2}) + \sin^2(\frac{\pi x}{2}) -2\cosh(\frac{\pi \beta}{2}) \sin(\frac{\pi \alpha}{2}) \sin (\frac{\pi x}{2})} \Big),
\end{split}
\end{equation}
where $x, \alpha \in (-1,1)$, and $\beta > 0$. This identity is symmetric in $\alpha$ and $x$, and if we set for instance $x=0$ we get
\begin{equation} \label{rach}
\begin{split}
\frac{1}{\pi} &\Big(\frac{\beta}{\beta^2 + \alpha^2} - \Big(\frac{\beta}{\beta^2 + (\alpha - 2)^2} + \frac{\beta}{\beta^2 + (\alpha+ 2)^2}\Big) + \Big(\frac{\beta}{\beta^2 + (\alpha - 4)^2} + \frac{\beta}{\beta^2 + (\alpha + 4)^2}\Big) - \ldots \Big) \\
& \qquad \qquad = \frac{1}{2} \Big(\frac{\sinh(\frac{\pi \beta}{2})\cos(\frac{\pi \alpha}{2})}{\sinh^2(\frac{\pi \beta}{2}) + \sin^2(\frac{\pi \alpha}{2})} \Big).
\end{split}
\end{equation}
Setting $\alpha=0$ as well gives
\begin{equation} \label{}
\frac{1}{\pi} \Big(\frac{1}{\beta} - \frac{2\beta}{\beta^2 + 2^2} + \frac{2\beta}{\beta^2 + 4^2} - \frac{2\beta}{\beta^2 + 6^2} + \frac{2\beta}{\beta^2 + 8^2} - \ldots \Big) = \frac{1}{2\sinh(\frac{\pi \beta}{2})}.
\end{equation}
If we subtract $\frac{1}{\pi \beta}$ from both sides, divide both sides by $\beta$, take the limit as $\beta \longrightarrow 0$, and simplify, we obtain
\begin{equation} \label{}
-1 + \frac{1}{2^2} - \frac{1}{3^2} + \frac{1}{4^2} - \ldots = \frac{-\pi^2}{12},
\end{equation}
which is easily seen to be equivalent to Euler's Basel sum \rrr{jill}. Returning to \rrr{rach}, if we assume $\alpha \neq 0$, divide both sides by $\beta$, and take the limit as $\beta \longrightarrow 0$, we get
\begin{equation} \label{clea}
\frac{1}{\alpha^2} - \Big(\frac{1}{(2-\alpha)^2} + \frac{1}{(2+\alpha)^2} \Big) + \Big(\frac{1}{(4-\alpha)^2} + \frac{1}{(4+\alpha)^2} \Big) - \ldots = \frac{\pi^2\cos(\frac{\pi \alpha}{2})}{4 \sin^2(\frac{\pi \alpha}{2})}.
\end{equation}
As before, the identity \rrr{clea} may be differentiated (and integrated) to obtain new identities if desired, and furthermore similar sums to \rrr{armdark} can be deduced. For example, taking an even number of derivatives and setting $x=0$ gives a sum equivalent to $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^{2m-1}}$ for integer $m$, while taking an odd number of derivatives and letting $x \longrightarrow 1$ reduces to a sum equivalent to $\sum_{n=1}^\infty \frac{1}{n^{2m}}$ for integer $m$.
\vspace{12pt}
Let us now calculate the exit distribution from $W$ on the rays $\{Re(z) = \pm 1, Im(z) > 0\}$. The identity \rrr{michelle} holds upon replacing $x$ with $\pm 1 + yi$, provided that we recall that the $\cos(\frac{\pi x}{2})$ in the numerator came from the $|f'(z)|$ term in Theorem \ref{massage}, and therefore must be replaced by $|\cos(\frac{\pi}{2}(\pm 1 + yi))|$. We may also use $|\cos(\frac{\pi}{2}(\pm 1 + yi))| = \sinh (\frac{\pi}{2}y)$ for $y>0$ and $\sin(\frac{\pi}{2}(\pm 1 + yi)) = \pm \cosh (\frac{\pi}{2}y)$, and we obtain
\begin{equation} \label{mich2}
\rho_{T_W}^{\alpha+\beta i}(\pm 1 + yi)ds = \frac{1}{2} \Big(\frac{\sinh(\frac{\pi \beta}{2})\cos(\frac{\pi \alpha}{2})\sinh(\frac{\pi y}{2})}{\sinh^2(\frac{\pi \beta}{2}) + \sin^2(\frac{\pi \alpha}{2}) + \cosh^2(\frac{\pi y}{2}) \mp2\cosh(\frac{\pi \beta}{2}) \sin(\frac{\pi \alpha}{2}) \cosh (\frac{\pi y}{2})} \Big)ds.
\end{equation}
This density can also be obtained by reflection, in several different ways. For example, if $W' = \{-1 < Re(z) < 1\}$ is the infinite strip of Example \ref{strip}, then $\rho_{T_W}^{\alpha+\beta i}(\pm 1 + yi)ds = \rho_{T_{W'}}^{\alpha+\beta i}(\pm 1 + yi)ds - \rho_{T_{W'}}^{\alpha+\beta i}(\pm 1 - yi)ds$. This is because a Brownian path that leaves $W'$ at $\pm 1 + yi$ will also leave $W$ at that point, provided it does not first strike the real axis; however, the paths that strike the real axis and then proceed to $\pm 1 + yi$ will contribute the same probability, by reflection, as those that leave $W'$ at $\pm 1 - yi$. On the other hand, if $\tilde \tau(b) = \inf\{t \geq 0: Re(B_t) = b \mbox{ or } Im(B_t)=0\}$, then arguing similarly as in Example \ref{strip} we have
\begin{equation} \label{tats22}
\begin{gathered}
\rho^{\alpha+\beta i}_{T_W}(1+yi)ds = \rho^{\alpha+\beta i}_{\tilde \tau(1)}(1+yi)ds - \rho^{\alpha+\beta i}_{\tilde \tau(-3)}(-3+yi)ds + \rho^{\alpha+\beta i}_{\tilde \tau(5)}(5+yi)ds - \rho^{\alpha+\beta i}_{\tilde \tau(-7)}(-7+yi)ds + \ldots, \\
\rho^{\alpha+\beta i}_{T_W}(-1+yi)ds = \rho^{\alpha+\beta i}_{\tilde \tau(-1)}(1+yi)ds - \rho^{\alpha+\beta i}_{\tilde \tau(3)}(3+yi)ds + \rho^{\alpha+\beta i}_{\tilde \tau(-5)}(-5+yi)ds - \rho^{\alpha+\beta i}_{\tilde \tau(7)}(7+yi)ds + \ldots.
\end{gathered}
\end{equation}
It might seem as though here we have fertile ground for other identities, however it is easy to see, again by reflection, that $\rho^{\alpha+\beta i}_{\tilde \tau(b)}(b+yi)ds = \rho^{\alpha+\beta i}_{\tau(b)}(b+yi)ds - \rho^{\alpha+\beta i}_{\tau(b)}(b-yi)ds$, where $\tau$ is the stopping time defined in Example \ref{strip}. Thus, any identities obtained here could just be obtained directly from \rrr{nosering}, evaluated at the proper values for $y$.
\vspace{12pt}
\ccases{Cminusslit} ($\mathbb{C} \backslash [-1,1]$) Consider again the map $f(z) = \sin (\frac{\pi}{2}z)$ from the previous example, but this time let us examine the projection of the stopping time $T_{\mathbb H}$ under this map. As stated before, this function maps the domain $\{-1<Re(z)<1, Im(z) > 0\}$ conformally onto ${\mathbb H}$, and it does so by taking $\{Re(z)=-1,Im(z)>0\}$ onto $(-\infty,-1)$, $[-1,1]$ onto itself, and $\{Re(z)=1,Im(z)>0\}$ onto $(1,\infty)$; this can be verified by noting that $\sin(\frac{\pi}{2}(\pm 1+yi)) = \pm \cosh (\frac{\pi}{2} y)$. Schwarz reflection tells us then that $f$ maps $\{-3<Re(z)<-1, Im(z) > 0\}$ and $\{1<Re(z)<3, Im(z) > 0\}$ conformally onto $\{Im(z)<0\}$, and then $\{-5<Re(z)<-3, Im(z) > 0\}$ and $\{3<Re(z)<5, Im(z) > 0\}$ again conformally onto $\{Im(z)>0\}$, with every point on ${\mathbb R}$ of course being mapped to a point in $[-1,1]$. It follows that the projection of $T_{\mathbb H}$ under this map will be the first hitting time of the set $[-1,1]$, and we will abuse notation somewhat to refer to this stopping time as $T_{[-1,1]}$. If $x \in [-1,1]$, then the point $\sin(\frac{\pi}{2}x)$ will have preimages at $x, \pm 2 - x, \pm 4 + x, \pm 6 - x, \ldots$. Applying Theorem \ref{massage}, it follows that, for $\alpha+\beta i \in {\mathbb H}$, we have
\begin{equation} \label{yi}
\begin{split}
(\frac{\pi}{2} \cos(\frac{\pi}{2}x)) \rho^{\sin(\frac{\pi}{2}(\alpha +\beta i))}_{T_{[-1,1]}} (\sin(\frac{\pi}{2}x)) ds & = \rho^{\alpha + \beta i}_{T_{\mathbb H}}(x)ds + \rho^{\alpha + \beta i}_{T_{\mathbb H}}(2-x)ds + \rho^{\alpha + \beta i}_{T_{\mathbb H}}(-2-x)ds\\
& \qquad + \rho^{\alpha + \beta i}_{T_{\mathbb H}}(4+x)ds + \rho^{\alpha + \beta i}_{T_{\mathbb H}}(-4+x)ds + \ldots \\
& = \frac{ds}{\pi}\Big(\frac{\beta}{\beta^2 + (\alpha-x)^2} + \frac{\beta}{\beta^2 + (\alpha-(2-x))^2} + \frac{\beta}{\beta^2 + (\alpha-(-2-x))^2}\\
& \qquad + \frac{\beta}{\beta^2 + (\alpha-(4+x))^2} + \frac{\beta}{\beta^2 + (\alpha-(-4+x))^2} + \ldots \Big)
\end{split}
\end{equation}
Note that the term $(\frac{\pi}{2} \cos(\frac{\pi}{2}x))$ comes from the $|f'|$ in Theorem \ref{massage}, which by periodicity is equal at all preimages of $\sin(\frac{\pi}{2}x)$. The identity \rrr{yi} is similar in spirit to the identity obtained in Example \ref{punctdisk}, as every term on the right side of \rrr{yi} corresponds to a different homotopy class of Brownian curves hitting $[-1,1]$, some from above and some from below. In order to obtain the value for the right side we need to calculate $\rho_{T_{[-1,1]}}$ in a different way, and we can argue as follows. The M\"obius transformation $\phi(z) = \frac{1+z}{1-z}$ maps $[-1,1]$ to $[0,+\infty]$, and therefore maps $\hat \mathbb{C} \backslash [-1,1]$ conformally onto $\hat \mathbb{C} \backslash [0,+\infty]$, where $\hat \mathbb{C}$ denotes the Riemann sphere. We can therefore use Theorem \ref{massage} to project the density for the first hitting time of $[-1,1]$ to the hitting time of $[0,+\infty)$. We obtain, for $\bar x = \sin(\frac{\pi}{2}x) \in [-1,1]$,
\begin{equation} \label{}
\rho^{\omega}_{T_{[-1,1]}}(\bar x) ds = \frac{2}{(1-\bar x)^2}\rho^{\phi(\omega)}_{T_{[0,+\infty)}}(\phi(\bar x)) ds.
\end{equation}
We can calculate $\rho_{T_{[0,+\infty)}}$ by projecting $\rho_{T_{\mathbb H}}$ via the map $z \longrightarrow z^2$, which maps ${\mathbb H}$ conformally onto $\mathbb{C} \backslash [0,+\infty)$. The point $\phi(\bar x)$ will have two preimages, at $\pm \sqrt{\phi(\bar x)}$, and we obtain
\begin{equation} \label{yima}
\begin{split}
\rho^{\phi(\omega)}_{T_{[0,+\infty)}}(\phi(\bar x)) ds & = \frac{ds}{2\pi \sqrt{\phi(\bar x)}}\Big( \frac{Im \sqrt{\phi(\omega)}}{(Im \sqrt{\phi(\omega)})^2 + (Re \sqrt{\phi(\omega)}-\sqrt{\phi(\bar x)})^2} \\
& \qquad + \frac{Im \sqrt{\phi(\omega)}}{(Im \sqrt{\phi(\omega)})^2 + (Re \sqrt{\phi(\omega)}+\sqrt{\phi(\bar x)})^2} \Big),
\end{split}
\end{equation}
where the branch of the square root is chosen that takes values in ${\mathbb H} \cup [0,+\infty)$. Combining \rrr{yi}-\rrr{yima} yields an identity, but this identity is fairly complex for arbitrary choice of $\alpha + \beta i$. We can simplify considerably by taking $\alpha$ to be an integer, and by periodicity we need only really consider $\alpha=-1$ and $\alpha = 0$. Let us begin with $\alpha=-1$. As before we note that $\sin(\frac{\pi}{2}(-1+\beta i)) = -\cosh(\frac{\pi}{2} \beta)$, and thus $\phi(\sin(\frac{\pi}{2}(-1+\beta i))) = - \frac{\cosh(\frac{\pi}{2} \beta) - 1}{\cosh(\frac{\pi}{2} \beta) + 1}$, so that $\sqrt{\phi(\sin(\frac{\pi}{2}(-1+\beta i)))} = i \sqrt{\frac{\cosh(\frac{\pi}{2} \beta) - 1}{\cosh(\frac{\pi}{2} \beta) + 1}}$, a purely imaginary number. \rrr{yi} therefore simplifies to become
\begin{equation} \label{ima1}
\begin{split}
&\frac{\cos(\frac{\pi}{2}x)}{(1-\sin(\frac{\pi}{2}x))^2\sqrt{\frac{1+\sin(\frac{\pi}{2}x)}{1-\sin(\frac{\pi}{2}x)}}} \Big(\frac{\sqrt{\frac{\cosh(\frac{\pi}{2} \beta) - 1}{\cosh(\frac{\pi}{2} \beta) + 1}}}{\frac{\cosh(\frac{\pi}{2} \beta) - 1}{\cosh(\frac{\pi}{2} \beta) + 1} + \frac{1+\sin(\frac{\pi}{2}x)}{1-\sin(\frac{\pi}{2}x)}}\Big)ds\\
& \qquad \qquad \qquad = \frac{2}{\pi} \Big(\frac{\beta}{\beta^2 + (1+x)^2}ds + \frac{\beta}{\beta^2 + (3-x)^2}ds + \frac{\beta}{\beta^2 + (5+x)^2}ds + \frac{\beta}{\beta^2 + (7-x)^2}\Big)ds;
\end{split}
\end{equation}
\vspace{12pt}
note that the set of preimages in the sum in \rrr{yi} are symmetric around $-1$, thus the multiplicative factor of $2$ and the one-sided sum on the right side of \rrr{ima1}. Dividing both sides by $\beta$ and letting $\beta \searrow 0$, using $\lim_{\beta \searrow 0} \frac{\sqrt{\cosh(\frac{\pi}{2} \beta) - 1}}{\beta} = \frac{\pi}{2\sqrt{2}}$, gives
\begin{equation} \label{ima2}
\begin{split}
\frac{1}{(1+x)^2} + \frac{1}{(3-x)^2} + \frac{1}{(5+x)^2} + \frac{1}{(7-x)^2} + \ldots & = \frac{\pi^2 \cos(\frac{\pi}{2}x)}{8(1-\sin(\frac{\pi}{2}x))^2\Big(\frac{1+\sin(\frac{\pi}{2}x)}{1-\sin(\frac{\pi}{2}x)}\Big)^{3/2}} \\
& = \frac{\pi^2}{8(1+\sin(\frac{\pi}{2}x))}.
\end{split}
\end{equation}
If we take for instance $x=0$, we obtain
\begin{equation} \label{}
\frac{1}{1^2}+\frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + \ldots = \frac{\pi^2}{8},
\end{equation}
which is easily seen to be equivalent to Euler's Basel sum \rrr{jill}. In fact, it is not hard to see that \rrr{ima2} and \rrr{mei} are equivalent, with \rrr{ima2} simply being the derivative of \rrr{mei} (with $x = -a$).
\vspace{12pt}
Returning to \rrr{yi}, let us now see what happens when we take $\alpha=0$. We need to calculate $\rho^{\sin(\frac{\pi}{2}(\beta i))}_{T_{[-1,1]}} (\sin(\frac{\pi}{2}x)) ds$, and we will use the same maps as before. We have $\sin(\frac{\pi}{2}\beta i) = i \sinh(\frac{\pi}{2}\beta i)$, and it may be checked that
\begin{equation} \label{}
\sqrt{\phi(i\sinh(\frac{\pi}{2}\beta))} = \sqrt{\frac{1+i\sinh(\frac{\pi}{2}\beta)}{1-i\sinh(\frac{\pi}{2}\beta)}} = \frac{1+i\sinh(\frac{\pi}{2}\beta)}{\sqrt{1+\sinh^2(\frac{\pi}{2}\beta)}} = \mbox{sech}(\frac{\pi}{2}\beta) + i \tanh(\frac{\pi}{2}\beta).
\end{equation}
It may also be checked by simple manipulations that $\sqrt{\phi(\sin(\frac{\pi}{2}x))} = \sec(\frac{\pi}{2}x) + \tan(\frac{\pi}{2}x)$. These calculations, together with \rrr{yi}-\rrr{yima}, give the identity
\begin{equation} \label{}
\begin{split}
&\frac{\cos(\frac{\pi}{2}x)}{2(\sec(\frac{\pi}{2}x) + \tan(\frac{\pi}{2}x))(1-\sin(\frac{\pi}{2}x))^2} \Big( \frac{\tanh(\frac{\pi}{2}\beta)}{\tanh^2(\frac{\pi}{2}\beta) + (\mbox{sech}(\frac{\pi}{2}\beta)-(\sec(\frac{\pi}{2}x) + \tan(\frac{\pi}{2}x)))^2} \\
& \qquad + \frac{\tanh(\frac{\pi}{2}\beta)}{\tanh^2(\frac{\pi}{2}\beta) + (\mbox{sech}(\frac{\pi}{2}\beta)+(\sec(\frac{\pi}{2}x) + \tan(\frac{\pi}{2}x)))^2} \Big) \\
& = \frac{1}{\pi}\Big(\frac{\beta}{\beta^2 + x^2} + \frac{\beta}{\beta^2 + (2-x)^2} + \frac{\beta}{\beta^2 + (2+x)^2} + \frac{\beta}{\beta^2 + (4-x)^2} + \frac{\beta}{\beta^2 + (4+x)^2} + \ldots \Big).
\end{split}
\end{equation}
Divide both sides by $\beta$ and let $\beta \longrightarrow 0$ to obtain
\begin{equation} \label{}
\begin{split}
&\frac{1}{x^2} + \frac{1}{(2-x)^2} + \frac{1}{(2+x)^2} + \frac{1}{(4-x)^2} + \frac{1}{(4+x)^2} + \ldots \\
& \qquad =\frac{\pi^2 \cos(\frac{\pi}{2}x)}{4(\sec(\frac{\pi}{2}x) + \tan(\frac{\pi}{2}x))(1-\sin(\frac{\pi}{2}x))^2} \Big( \frac{1}{(1-(\sec(\frac{\pi}{2}x) + \tan(\frac{\pi}{2}x)))^2} + \frac{1}{(1+(\sec(\frac{\pi}{2}x) + \tan(\frac{\pi}{2}x)))^2} \Big) \\
& \qquad = \frac{\pi^2 \cos(\frac{\pi}{2}x)(1+(\sec(\frac{\pi}{2}x) + \tan(\frac{\pi}{2}x))^2)}{8\tan^2(\frac{\pi}{2}x)(\sec(\frac{\pi}{2}x) + \tan(\frac{\pi}{2}x))^3(1-\sin(\frac{\pi}{2}x))^2} \\
& \qquad = \frac{\pi^2 \cos(\frac{\pi}{2}x)(1+(\sec(\frac{\pi}{2}x) + \tan(\frac{\pi}{2}x))^2)}{8\sin^2(\frac{\pi}{2}x)(\sec(\frac{\pi}{2}x) + \tan(\frac{\pi}{2}x))}
\end{split}
\end{equation}
As with the earlier examples, a number of other identities can be deduced by evaluating the sum at particular values of $x$ and by differentiation. For example, taking the limit as $x \longrightarrow 1$ reduces to $\sum_{n=1}^{\infty} \frac{1}{(2n-1)^2} = \frac{\pi^2}{8}$, and subtracting $\frac{1}{x^2}$ from both sides and letting $x \longrightarrow 0$ leads to $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$. Similarly, taking the suitable number of derivatives and letting $x$ tend to either $1$ or $0$ allow one to calculate the values of $\sum_{n=1}^\infty \frac{1}{n^{2m}}$ for integer $m$.
\vspace{12pt}
\ccases{rectangle} (rectangle) Let $W = \{-1<Re(z)<1,-k<Im(z)<k\}$ for some $k>0$. Let us calculate $\rho^{\alpha+\beta i}_{T_W}(1+yi)ds$ in two different ways by reflection. Let us first note that, arguing as in Example \ref{halfstrip}, if $W' = \{-1 < Re(z) < 1\}$ then
\begin{equation} \label{aunt13}
\begin{split}
\rho_{T_W}^{\alpha+\beta i}(1+yi)ds &= \rho_{T_{W'}}^{\alpha+\beta i}(1+yi)ds - \Big(\rho_{T_{W'}}^{\alpha+\beta i}(1+(2k-y)i)ds +\rho_{T_{W'}}^{\alpha+\beta i}(1+ (-2k-y)i)ds\Big) \\
& \qquad \qquad + \Big(\rho_{T_{W'}}^{\alpha+\beta i}(1+(4k+y)i)ds +\rho_{T_{W'}}^{\alpha+\beta i}(1+ (-4k+y)i)ds\Big) - \ldots
\end{split}
\end{equation}
On the other hand, if $\hat \tau(b) = \inf\{t \geq 0: Re(B_t) = b \mbox{ or } Im(B_t)= \pm k\}$, then arguing similarly as in Example \ref{strip} we have
\begin{equation} \label{tats3}
\rho^{\alpha+\beta i}_{T_W}(1+yi)ds = \rho^{\alpha+\beta i}_{\hat \tau(1)}(1+yi)ds - \rho^{\alpha+\beta i}_{\hat \tau(-3)}(-3+yi)ds + \rho^{\alpha+\beta i}_{\hat \tau(5)}(5+yi)ds - \rho^{\alpha+\beta i}_{\hat \tau(-7)}(-7+yi)ds + \ldots
\end{equation}
The two series given in \rrr{aunt13} and \rrr{tats3} must therefore be equal, with the terms in \rrr{aunt13} given by $\rrr{is3}$ and the terms in \rrr{tats3} given by an appropriate scaling, rotation, and translation of the corresponding density \rrr{michelle} for the domain $W''=\{-1<Re(z)<1, Im(z)>0\}$ considered in Example \ref{halfstrip}, since $\hat \tau(b)$ is the exit time of the domain bounded by $\{Re(z) = b\}$ and $\{Im(z) = \pm k\}$ containing $\alpha+\beta i$, and this domain is clearly conformally equivalent to $W''$. Suffice it to say that the resulting identity is quite complex and not particularly illuminating. Let us therefore simplify things by setting $\alpha=\beta=0$; the identity which results is then
\begin{equation} \label{}
\begin{split}
\mbox{sech}& (\frac{\pi}{2} y) -\Big(\mbox{sech} (\frac{\pi}{2} (2k-y)) + \mbox{sech} (\frac{\pi}{2} (-2k-y))\Big) + \Big(\mbox{sech} (\frac{\pi}{2} (4k+y)) + \mbox{sech} (\frac{\pi}{2} (-4k+y))\Big) - \ldots \\
& = \frac{2\cos(\frac{\pi y}{2k})}{k}\Big(\frac{\sinh(\frac{\pi}{2k})}{\sinh^2(\frac{\pi}{2k}) + \sin^2(\frac{\pi y}{2k})} - \frac{\sinh(\frac{3\pi}{2k})}{\sinh^2(\frac{3\pi}{2k}) + \sin^2(\frac{\pi y}{2k})} + \frac{\sinh(\frac{5\pi}{2k})}{\sinh^2(\frac{5\pi}{2k}) + \sin^2(\frac{\pi y}{2k})} - \ldots \Big)
\end{split}
\end{equation}
Taking $y=0$, using the fact that $\mbox{sech}$ is an even function, gives the identity
$$
\frac{1}{2} - \mbox{sech} (\pi k) + \mbox{sech} (2\pi k) - \mbox{sech} (3\pi k) + \mbox{sech} (4\pi k) - \ldots = \frac{1}{k} \Big( \mbox{csch} (\frac{\pi}{2k}) - \mbox{csch} (\frac{3\pi}{2k}) + \mbox{csch} (\frac{5\pi}{2k}) - \mbox{csch} (\frac{7\pi}{2k})+ \ldots\Big)
$$
\ccases{annulus} (annulus) Now let us consider the annulus $A_r=\{e^{-r}<|z|<e^{r}\}$, where $r>0$ is real. We map the strip $W=\{-1<Re(z)<1\}$ in Example \ref{strip} to $A_r$ via the function $f(z)=e^{rz}$, noting that $|\frac{d}{dz} e^z| = re^{r}$ on $\{Re(z)=1\}$ and $|\frac{d}{dz} e^z| = r e^{-r}$ on $\{Re(z)=-1\}$. We'll assume $a \in (e^{-r},e^r)$ is real. Theorem \ref{massage}, using \rrr{is3}, then gives
\begin{equation} \label{skyfall}
\begin{split}
\rho^a_{T_{A_r}}(e^{\pm r}e^{i \theta})ds &= \frac{1}{re^{\pm r}}\sum_{k=-\infty}^{\infty} \rho^{\frac{\ln a}{r}}_{T_W}(\pm 1+i(\frac{\theta + 2\pi k}{r}))ds \\
& = \frac{1}{re^{\pm r}}\sum_{k=-\infty}^{\infty} \frac{\mbox{sech} (\frac{\pi}{2}(\frac{\theta + 2\pi k}{r}))(1-\tan^2(\frac{\pi\ln a}{4r}))}{4((1\mp\tan (\frac{\pi\ln a}{4r})\frac{1 - \tanh^2(\frac{\pi}{4}(\frac{\theta + 2\pi k}{r}))}{1 + \tanh^2(\frac{\pi}{4}(\frac{\theta + 2\pi k}{r}))})^2+(\tan (\frac{\pi\ln a}{4r})\frac{2\tanh(\frac{\pi}{4}(\frac{\theta + 2\pi k}{r})) }{1 + \tanh^2(\frac{\pi}{4}(\frac{\theta + 2\pi k}{r}))})^2)}ds.
\end{split}
\end{equation}
As with the punctured disk example, each term in the sum in \rrr{skyfall} corresponds to a homotopy class of curves leaving the annulus at that point. The formula is far simpler if we take $a=1$, in which case we have
\begin{equation} \label{}
\rho^a_{T_{A_r}}(e^{\pm r}e^{i \theta})ds = \frac{ds}{4 re^{\pm r}} \sum_{k=-\infty}^{\infty} \mbox{sech} (\frac{\pi}{2}(\frac{\theta + 2\pi k}{r}))
\end{equation}
We now show how applying Dynkin's formula with this expression yields an identity. If $h$ is harmonic on $A_r$, continuous on $\bar A_r$, then we obtain
\begin{equation} \label{chess}
\begin{split}
h(1) & = E_1[h(B_{T_{A_r}})] \\
&= \int_{0}^{2\pi} h(e^{-r + i\theta}) \Big( \frac{1}{4 re^{-r}}\sum_{n=-\infty}^{\infty}\mbox{sech} (\frac{\pi}{2}(\frac{\theta + 2\pi k}{r}))\Big) (e^{-r} d\theta) \\
& \qquad + \int_{0}^{2\pi} h(e^{r + i\theta}) \Big( \frac{1}{4 r e^r}\sum_{n=-\infty}^{\infty}\mbox{sech} (\frac{\pi}{2}(\frac{\theta + 2\pi k}{r})) \Big) (e^{r}d\theta) \\
& = \frac{1}{4r} \int_{-\infty}^{\infty} (h(e^{-r + i\theta}) + h(e^{r + i \theta}))\mbox{sech} (\frac{\pi \theta}{2r})d\theta\\
& = \frac{1}{4} \int_{-\infty}^{\infty} (h(e^{-r + ir \theta}) + h(e^{r + i r \theta}))\mbox{sech} (\frac{\pi \theta}{2})d\theta;
\end{split}
\end{equation}
note that the identity $ds=(e^{\pm r}d\theta)$ was used on the curves $\{|z|=e^{\pm r}\}$. If we put in $h(z) = z$ and rearrange we obtain the identity
\begin{equation} \label{sweet}
\int_{-\infty}^{\infty} e^{i r \theta} \mbox{sech}(\frac{\pi \theta}{2}) d\theta = 2 \mbox{sech} (r).
\end{equation}
It should be noted that the same identity with $r < 0$ is obtained by setting $h(z) = \bar z$. We have therefore derived the Fourier transform of the $\mbox{sech}$ function.
\vspace{12pt}
\noindent {\bf Remark:} It may be tempting to look for other harmonic functions on the annulus in order to derive new identities from \rrr{chess}, in particular we might hope that the argument given here provides a general method for evaluating Fourier transforms involving the function $\mbox{sech}$. However, the reader should be aware that a search for further identities using this exit distribution will lead to nothing else substantial. This is because for any harmonic function $h$ on $A_r$ an analytic function $g$ and real constant $C$ can be found such that $h = Re(g(z)) + C\log|z|$ (see for example \cite[Ex. III.3.4]{gamcomp}). $\log |z|$ in \rrr{chess} yields a triviality, and analytic functions on annuli are Laurent series in $z$, so that if we write $h-C\log|z| = \frac{g + \bar g}{2}$ we see that $h-C\log|z|$ can be expressed as a Laurent series in $z$ and $\bar z$. Applying \rrr{chess} to $h$ would therefore not yield anything more than would a term by term application with $h(z) = z^q$ or $h(z) = \bar z^q$, each of which give simply \rrr{sweet}.
\vspace{12pt}
It is also interesting to see what happens if we try to adapt the reflection technique from Example \ref{strip} to this case. If the analogous argument to that in Example \ref{strip} applied, we would have
\begin{equation} \label{thai}
\rho^a_{T_{A_r}}(e^{r}e^{i \theta})ds = \rho^a_{\hat \tau(e^r)}(e^{r}e^{i \theta})ds -\rho^a_{\hat \tau(e^{-r},e^r)}(e^{r}e^{i \theta})ds + \rho^a_{\hat \tau(e^r,e^{-r},e^r)}(e^{r}e^{i \theta})ds - \rho^a_{\hat \tau(e^{-r},e^r,e^{-r},e^r)}(e^{r}e^{i \theta})ds + \ldots
\end{equation}
where $\hat \tau(m_1,m_2,\ldots, m_k)$ would be the first time the Brownian motion has hit all of the curves $\{|z|=m_1\}, \{|z|=m_2\}, \ldots, \{|z|=m_k\}$ in order. The quantities on the right side of \rrr{thai} are easy to find using reflection (note that reflection over $\{|z|=m\}$ is given by the function $z \longrightarrow \frac{m^2}{\bar z}$) and the calculations in Example \ref{disk}, and if we consider the simplest case, when $\theta = 0$, $a=1$, we would obtain the identity
\begin{equation} \label{coco} \nonumber
\frac{1}{4re^{r}} \sum_{k=-\infty}^{\infty} \mbox{sech} (\frac{\pi^2 k}{r}) = \frac{1}{2\pi e^r} \Big( \frac{e^r + 1}{e^r-1} - \frac{1+e^{-3r}}{1-e^{-3r}} + \frac{e^{5r} + 1}{e^{5r}-1} - \frac{1+e^{-7r}}{1-e^{-7r}} + \ldots \Big)
\end{equation}
However, this sum cannot be valid, since the sum on the right does not converge (the terms inside the parentheses approach 1). The reason that the argument fails in this case is that the densities $\rho^a_{\hat \tau(e^{-r},e^r, \ldots,e^r)}$ and $\rho^a_{\hat \tau(e^{r},e^{-r}, \ldots,e^r)}$ do not approach $0$ as the length of the sequence goes to infinity, but rather approach the uniform density on the circle. On the other hand, the analogous statement to Lemma \ref{mang} will still hold, so it might be interesting to see whether any sense can be made of \rrr{coco} or the reflection argument.
\vspace{12pt}
\ccases{wind1} We now consider a stopping time which is not the exit time of a domain. Start a Brownian motion at $1$, and let $\tau_r = \inf\{t: arg(B_t) = \pm r\pi\}$, with the branch of the argument chosen so that $arg(B_0)=0$ a.s. We can calculate the distribution of $B_{\tau_r}$, and arrive at
\begin{equation} \label{mae2}
\rho^1_{\tau_r}(ye^{\pm r\pi i})ds = \frac{ds}{2r\pi y(y^{\frac{1}{2r}}+y^{\frac{-1}{2r}})},
\end{equation}
when $r$ is not an integer. When $r$ is an integer, then $e^{r\pi i} = e^{-r\pi i}$, so doubling \rrr{mae2} we have
\begin{equation} \label{}
\rho^1_{\tau_r}(ye^{r\pi i})ds = \frac{ds}{r\pi y(y^{\frac{1}{2r}}+y^{\frac{-1}{2r}})}.
\end{equation}
To see this, we project the density found earlier of the half-plane via the map $f(z)=z^{2r}$ with $|f'(yi)| = 2ry^{2r-1}$ to get
\begin{equation} \label{}
\rho^1_{\tau_r}(y^{2r}e^{\pm \pi r i}) ds = \frac{ds}{\pi(1+y^2)2ry^{2r-1}}.
\end{equation}
Replace $y$ with $y^{\frac{1}{2r}}$ to arrive as claimed at
\begin{equation} \label{}
\rho^1_{\tau_r}(y e^{\pm \pi r i}) ds = \frac{ds}{2\pi r(1+y^{\frac{2}{2r}})y^{\frac{2r-1}{2r}}} = \frac{ds}{2\pi ry(y^{\frac{1}{2r}}+y^{\frac{-1}{2r}})}.
\end{equation}
This density admits the antiderivative $\frac{1}{\pi} \tan^{-1}(y^{\frac{1}{2r}})$, and this allows us in particular to note that
\begin{equation} \label{}
P_1(|B_{\tau_r}| \in (0,\varepsilon)) = \frac{2}{\pi} \tan^{-1}(\varepsilon^{\frac{1}{2r}}) = P_1(|B_{\tau_r}| \in (\frac{1}{\varepsilon}, \infty)),
\end{equation}
and this probability approaches $\frac{1}{2}$ as $r$ approaches $\infty$. Thus, as $r \longrightarrow \infty$, the distribution of $|B_{\tau_r}|$ approaches $\frac{1}{2}(\delta_0 + \delta_\infty)$ in distribution. This may seem surprising at first, especially the mass accumulating at $0$; but in fact it is to be expected since any mass at infinity must correspond to a mass at 0 due to the fact that $\frac{1}{B_t}$ is a (time-changed) Brownian motion. Evidently when $B_t$ comes close to $0$ it winds many times around the origin, analogously to how a one-dimensional Brownian motion hits $0$ infinitely often in any neighborhood of a visit to $0$, and this results in a large change in argument and hence a high likelihood of being near 0 at the first time attaining a prescribed argument.
\vspace{12pt}
We also remark briefly that the symmetry imposed upon $\tau_r$ can easily be dispensed of. In other words, we may let $\tau_{r_1,r_2} = \inf\{t: arg(B_t) = r_1\pi \mbox{ or } -r_2\pi\}$ for $r_1, r_2 >0$, and calculate
\begin{equation} \label{lamchop}
\begin{gathered}
\rho^1_{\tau_{r_1,r_2}}(ye^{r_1 \pi i})ds = \frac{\cos \theta \quad \! \! \! ds}{\pi (r_1+r_2)y^{1-\frac{1}{r_1+r_2}}(\cos ^2 \theta+(y^{\frac{1}{r_1+r_2}} - \sin \theta)^2)},\\
\rho^1_{\tau_{r_1,r_2}}(ye^{-r_2 \pi i})ds = \frac{\cos \theta \quad \! \! \! ds}{\pi (r_1+r_2)y^{1-\frac{1}{r_1+r_2}}(\cos ^2 \theta+(-y^{\frac{1}{r_1+r_2}} - \sin \theta)^2)},
\end{gathered}
\end{equation}
where $\theta = \frac{\pi}{2} \Big(\frac{r_2-r_1}{r_2+r_1}\Big)$. This is obtained by first noting by rotational invariance that we may obtain the distribution by starting a Brownian motion at $e^{\frac{\pi}{2}(r_2-r_1)i}$ and stopping it at the first time the argument reaches $\pm \frac{\pi}{2}(r_2+r_1)$, and then projecting the density from the right half-plane as before by the function $f(z)=z^{r_1+r_2}$. Note that if $e^{r_1 i} = e^{-r_2 i}$ then these two quantities must be added in order to find the density of $B_{\tau_{r_1,r_2}}$ on the ray $\{arg(z) = r_1\} = \{arg(z) = -r_2\}$.
\vspace{12pt}
We may also stop the Brownian motion at a prescribed argument. Let $\hat \tau_r = \inf\{t: arg(B_t) = r\}$, where we will assume $r>0$. If we let $W = \{Im(z) < r\}$, then the stopping time $T_W$ projects to $\hat \tau_r$ under the exponential map $z \longrightarrow e^z$. Theorem \ref{massage} gives
\begin{equation} \label{}
\rho_{\hat \tau_r}^1(ye^{ir})ds = \frac{1}{y}\rho^0_{T_W}(\ln y + ri) ds = \frac{1}{y\pi} \frac{r}{r^2 + (\ln y)^2} ds.
\end{equation}
It is straightforward to find the corresponding distribution function if desired, and to note that the accumulation of mass at $0$ and $\infty$ holds for this example as well; that is, $P_1(|B_{\hat \tau_r}| \in (0,\varepsilon)) = P_1(|B_{\hat \tau_r}| \in (\frac{1}{\varepsilon}, \infty)) \longrightarrow \frac{1}{2}$ as $r \longrightarrow \infty$. It may also be noted that reflection may be used in order to express $\rho_{\tau_r}$ or $\rho^1_{\tau_{r_1,r_2}}$ as an alternating infinite sum of terms of the form $\rho_{\hat \tau_r}^1$, exactly in the same manner as in Exercise \ref{strip}. The identities obtained in this manner are precisely the same as in Example \ref{strip}, since the exponential function will map the properly chosen strip or half-plane into the winding stopping times.
\vspace{12pt}
\ccases{what} Now let $\hat \tau = \inf_{t \geq 0} (B_t \in (-1,1), \{B_s\}_{0 \leq s \leq t} \cup [0,B_t] \mbox{ is not homotopic to a point} )$; that is, $t$ is the first time that $B_t$ lies on $(-1,1)$ simultaneously with the curve traced by $B_s$ up to time $t$ being wound around at least one of $-1$ and $1$. We will calculate the distribution of $B_{\hat \tau}$. First we set $\tau = \inf\{t: B_t \in (-\infty,-1] \cup [1,+\infty) \}$. We will show that $\rho^{0}_\tau(w) ds = \frac{1}{\pi |w| \sqrt{w^2-1}}$, and then project this density to $\hat \tau$. Let $\tau_1 = \inf\{t: B_t \in (-\infty,+\infty) \}$ and $\tau_0 = \inf\{t: B_t \in [0,+\infty) \}$. Using the map $f(z)=z^2$ and Theorem \ref{massage}, we project the density for $\tau_1$ of Example \ref{halfplane} via the conformal map $f(z)=z^2$ to get $\rho^{(-1)}_{\tau_2}(v) ds = \frac{1}{\pi (1+v)\sqrt{v}}$ (the $-1$ is placed in parentheses to prevent any confusion with an inverse map). We can now project this density via the transformation $w=\phi(v)=\frac{1+v}{1-v}$ to obtain
\begin{equation} \label{}
\begin{split}
\rho^{(0)}_\tau(v) ds & = \rho^{(-1)}_{\tau_1}(\frac{w-1}{w+1}) (\phi^{-1})'(w) ds \\
& = \frac{1}{\pi \sqrt{\frac{w-1}{w+1}}(1+\frac{w-1}{w+1})}\times \frac{2}{(w+1)^2} ds \\
& = \frac{1}{\pi |w|\sqrt{w^2-1}} ds,
\end{split}
\end{equation}
as claimed. We now will project the density of $\tau$ to $\hat \tau$ using the entire function $f(z) = \sin(\frac{\pi}{2}z)$. To see that this does the job, note that $\sin(\frac{\pi}{2}(-1+yi))=-\cosh(\frac{\pi}{2}y)$ and $\sin(\frac{\pi}{2}(1+yi))=\cosh(\frac{\pi}{2}y)$. $f$ therefore maps the boundary of the half-infinite strip $\{Im(z)>0, -1 < Re(z) <1\}$ injectively onto the boundary of $\{Im(z)> 0\}$. The argument principle allows us to conclude that $f$ maps $\{Im(z)>0, -1 < Re(z) <1\}$ conformally onto $\{Im(z)> 0\}$. The Schwarz reflection principle now informs us that $f$ maps $\{Im(z)>0, -3 < Re(z) <-1\}$ conformally onto $\{Im(z)< 0\}$, with $(-3,-1)$ being mapped to $(-1,1)$. Reflecting in this manner to the left and right, as well as below to $\{Im(z)<0, -1 < Re(z) <1\}$ and thence left and right to fill out the plane, the structure of $f$ can be understood; in particular we see that a closed curve ${\cal C}$ beginning and ending at $0$ (but otherwise not touching $(-1,1)$) is not homotopic to a point if and only if there is a curve $\gamma$ traveling from $0$ to some non-zero even integer such that $f(\gamma) = {\cal C}$. This tells us that $\tau$ is mapped to $\hat \tau$ under $f$. If $w \in (-1,1)$, then the preimages under $f$ of $w$ on $(-\infty,-1] \cup [1,+\infty)$ are all points of the form $(4n-2) - \frac{2}{\pi}\sin^{-1} w$ for integer $n$ or $\frac{2}{\pi}\sin^{-1} w+4n$ for integer $n \neq 0$. Note that the value of $|f'|$ at these points is $\frac{\pi}{2}\cos(\sin^{-1} w) = \frac{\pi}{2}\sqrt{1-w^2}$. Applying our theorem therefore gives
\begin{equation} \label{}
\begin{split}
\rho^0_{\hat \tau}(w)ds = &\frac{2ds}{\pi^2 \sqrt{1-w^2}} \Big( \sum_{\substack{n=-\infty \\ n\neq 0}}^\infty \frac{1}{|\frac{2}{\pi}\sin^{-1} w+4n|\sqrt{(\frac{2}{\pi}\sin^{-1} w+4n)^2-1}} \\
&+ \sum_{n=-\infty}^\infty \frac{1}{|\frac{2}{\pi}\sin^{-1} w-(4n+2)|\sqrt{(\frac{2}{\pi}\sin^{-1} w-(4n+2))^2-1}}\Big)\\
= & \frac{2ds}{\pi^2 \sqrt{1-w^2}} \sum_{\substack{n=-\infty \\ n\neq 0}}^\infty \frac{1}{|\frac{2}{\pi}\sin^{-1} w+2n|\sqrt{(\frac{2}{\pi}\sin^{-1} w+2n)^2-1}}.
\end{split}
\end{equation}
Somewhat similarly to earlier examples, each term in the sum here corresponds to homotopy classes of Brownian curves, with each term corresponding to two different classes: one in which $B_t$ approaches $B_\tau \in (-\infty,-1] \cup [1,+\infty)$ from above, and one in which it approaches from below. Note that
\begin{equation} \label{}
\frac{d}{dw} \Big(\frac{-1}{\pi} \cot^{-1} \sqrt{(\frac{2}{\pi}\sin^{-1} w+2n)^2 - 1}\Big) = \frac{2ds}{\pi^2 \sqrt{1-w^2}(\frac{2}{\pi}\sin^{-1} w+2n)\sqrt{(\frac{2}{\pi}\sin^{-1} w+2n)^2-1}}.
\end{equation}
The distribution function is therefore explicitly calculable, if desired.
\section{Acknowledgements} The author is grateful for support from Australian Research Council Grants DP0988483 and DE140101201.
\bibliographystyle{alpha}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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TV Columnist Michael Ausiello Knew Recording His Heartbreaking Memoir 'Spoiler Alert: The Hero Dies' Meant Not Sugarcoating A Single Thing
The TV news impresario opens up about celebrating the life and love of his husband Kit after his devastating death.
-Minute Listen
Abby West
Michael Ausiello is no stranger to hard work. He first built a name for himself as a television columnist at TV Guide and Entertainment Weekly, before building powerhouse TV news website, TVLine.com. None of that prepared him for the biggest battle of his life—seeing his husband Kit through his fatal neuroendocrine cancer diagnosis, mourning his death, and then writing and recording a memoir chronicling their years together and their love. Michael spoke with Audible's Abby West, a friend and former colleague, and shared how the book (which has now been optioned for a film) came together and what the recording wrung out of him. Listen in on their sometimes funny, sometimes emotional conversation.
Note: Text has been edited and may not match audio exactly.
Abby West: Hi, I'm Abby West, Audible editor, and a big fan of my guest here today, Michael Ausiello, author of Spoiler Alert: The Hero Dies.
Michael Ausiello: Hi, Abby.
AW: Hi. How are you?
Michael: Good to see you.
AW: Good to see you too. Michael and I worked together about 10 years ago. How crazy is that?
MA: Oh my God. We're old.
AW: When we were both at Entertainment Weekly. And Michael since went on to found and become editor-in-chief of TVLine, which is a premier TV news and entertainment site. I keep doing that. I say entertainment because I feel like it covers a lot, but he does a lot of TV news and scoops, and it's really for anyone obsessed with television. That is one of the go-to spots. Michael has really bared his soul in his memoir, talking about his husband, Kit. We're going to get into that, but he's going to tell us a little bit about his history leading up to writing this book and then recording this book, which is profound, and poignant, and will move anyone who listens to it and gets to know both Michael and Kit. Welcome, Michael.
MA: Thank you for having me.
AW: Happy to! So could you sort of tee up the book and tell us about Kit for people who have not read the book yet and don't necessarily know the full premise?
MA: The story is my husband, Kit, was diagnosed with an incredibly serious form of neuroendocrine cancer about 13 years into our relationship, and a couple doctors told him he basically had a year to live. Much of the book documents that year—that last year of his life—and the ups and downs, and what it did for our relationship, and what it did for him personally, what it did for me personally. But then the book also flashes back throughout to the first year of our relationship. There are some parallels to what happened in year one to what happened in year 13, and we go back and forth a little bit. Throughout we also touch on some of the years in between, but mostly the book is structured about year one and year 13.
AW: It was a really smart interplay. How did it come to you to do that parallel? Did it feel natural as you started writing?
MA: It evolved. You know, I knew going in that it was going to go back and forth. Initially, I thought it was going to be every other chapter. It was going to be year one and then it was going to be year 13, and then we're going to go back to year one. I just found that there was just too much. Too much happened in that last year. Too much important stuff happened that needed to get more of the focus, so that's where I arrived at. It's every couple chapters we journey back in time and get a little glimpse of Kit and Mike in the beginning.
"It's hard to psych yourself up in the morning to write about some of the worst moments of your life, but I kept my mind focused on the big picture."
AW: When did you start thinking about the book? We'll get more into the content, but when did you start thinking about writing the book?
MA: When Simon & Schuster approached me and said, "Will you write a book?"
AW: That's a good time.
MA: Yeah. That's when I was like, "Hmm, I guess I'll think about this now." A couple months after Kit passed away, an editor at Simon & Schuster approached me because he had been reading my dispatches on Facebook. I'd been sort of chronicling Kit's battle with cancer on Facebook to our friends and family. Some of the posts were lengthy. There's humor in there, and some of them were pretty candid about the struggles that he was facing. Anyway, this editor, Rakesh, was reading the dispatches and was moved by them and thought that there might be a book in there somewhere, and approached me a couple months after Kit passed and asked if I would be interested.
AW: Did you balk at first, or did it feel right?
MA: I balked a little bit, only because I didn't know if I had it in me. I didn't know if I had the strength to dive back into this horrible year that I just went through, and also I didn't know if practically speaking I had the time to do it. I have a full-time job, a demanding full-time job, so it would require me finding time—weekends mostly—to work on the book.
After I thought about all of that, the other piece of it was I knew it was an incredible opportunity. I knew people don't get approached to write a book all the time. It's not usually how it works, and it felt like an incredible honor and an incredible opportunity to tell this story. I felt deep down that there was a special story here to be told about Kit, about our relationship, not just the year he was sick but the 13 years that came before. That feeling of this is an opportunity I can't pass up ultimately sort of won out any of the concerns I had about time and strength, and I said yes.
AW: How long did it take to write?
MA: 18 months. Initially, I was like, "Oh, I'll have this done for you in eight months," because part of me was thinking the sooner I write it, the sooner I get through it and can sort of get on with my life. I was very optimistic with the initial timeframe, then it kept getting pushed back when I just realized what a huge undertaking it was. Also, I was only writing on weekends, which drags the process out a little bit.
AW: Yeah. When you thought about doing the book, did you also think about the fact that you're going to have to not only write the book and then promote the book and sort of live in this space for a couple of years, or did that realization happen afterwards, and how did that feel? Because now you have to talk to me about it, talk to a lot of people about it, and you're really in the space for a while.
MA: I was going to be in the space regardless. You know, he was such a huge part of my life and the loss was so profound that it's not like had this opportunity not come up I would just sort of be able to move on. I would be thinking about him constantly anyway. That said, this was a very immersive experience. I literally had to put myself back into our relationship, back into a lot of the painful moments from the past, from the year that he was sick, his death. Like some of the worst moments of my life, I had to immerse myself in them in order to write this story. I knew that that was going to be difficult, and it was. You know, it's hard to psych yourself up in the morning to write about some of the worst moments of your life, but I kept my mind focused on the big picture and the good that the book could potentially do, and also the good it could potentially do me to have this experience.
AW: Now, you are a public personality. You are.
MA: That's nice of you to say.
AW: Beloved by your fans and friends.
MA: All true. All true.
"I'm going to be honest and tell the whole story. I'm not going to sugarcoat it. I'm not going to sugarcoat me. I'm not going to sugarcoat Kit."
AW: Yeah, all totally true. You're public, but you're not always so personally public as this book has led you to have to be. How was that? Did you have a conversation with yourself to say, "I have to reveal it all in order to make this authentic"?
MA: I had a conversation with myself that if I'm going to do this, I'm going to be honest and tell the whole story. I'm not going to sugarcoat it. I'm not going to sugarcoat me. I'm not going to sugarcoat Kit. I'm not going to sugarcoat our relationship. I found the process of just being completely honest about my own issues very freeing and liberating in writing the book. It was really nice to not give a shit what people thought, because I'm one of those people who worry what people think about me—probably too much and worry people will judge me, and I didn't care when I was writing this book. I was just, "Let them judge me. I'm just going to put it all out there and be honest about who I am and let the chips fall where they may." That was a really nice place to be.
AW: Still in that place when it comes to the world?
MA: Yeah. It's slightly different now. I am still in that place. I absolutely still have no regrets about how candid I was, but it's a different experience now as people are reading the book. They know a lot about me, way more about me than I probably know about them, and that's a little awkward I think, especially like my brothers reading about some really intimate aspects of my sex life and friends of mine reading about body issues that I've had my whole life. It's weird. It's a weird spot to be in that now people know so much about me, but it's also actually I think brought me closer to a lot of my friends and my family, and in some cases I think made them more comfortable to be more open with me.
AW: I think you handled the book the same way I think I've always seen you handle things, with a touch of humor at all times even in the roughest moments without detracting from the realness and the rawness of the emotion. That takes a deft touch.
MA: Thank you, yeah.
AW: Both on paper and in audio. So, tell us about the audiobook process for you.
MA: Scary.
AW: Why was it scary?
MA: I'd never done anything like that before, and I'm not a performer. I mean I perform for my friends and I like to think of myself as entertaining at times, but I'm not … there are professionals who do this for a living and do it well. I'm not one of those people, and I wasn't sure if I would be able to do it justice. Also backing up a little bit, I wasn't sure that there was even going to be an audiobook for the book. I knew I was writing a book, but it wasn't until toward the end of the writing process that I found out that there was going to be an audiobook and that they wanted me to be the one to do it.
Of course, I agreed without even really giving it too much thought, because of course if there's going to be an audiobook and they want me to do it, I'm going to do it. I'm not going to say no. It really wasn't until I got into the studio and started that I realized what a huge undertaking it is emotionally and physically.
AW: Had you reread the book since you'd written it? Before going into the studio to record it?
MA: No. I had never read the book full from beginning to end. Most of my reading was a chapter at a time, because I wrote chapter by chapter, and then reread sections, and edited sections, but I never sat down and read the entire story from beginning to end. That was fraught for me, the idea of doing that at this stage, because what if I read it through and I was like, "Oh, this is not good. I want to change this. I want to change this. I cannot believe I … I need to take this out."
AW: See, that's the editor in you.
MA: Right, but it would have been too late in the process to make those kind of changes, so I was afraid that … I had been avoiding reading it from beginning to end for that reason, because I just felt like, "Well, what if it's terrible?" Even though my editors and my book coach were giving me all really positive feedback—like everything everybody was telling me was great—but still part of me was not believing that and was thinking, "I'll see for myself." So I was concerned about reading it for the first time. I had a lot of anxiety about that, but I will say one of the most surprising things about it was how little I wanted to change as I was reading it and how pleased I was with it, which really surprised me because I am my worst critic about everything. Everything. The fact that I came out of that experience feeling good about the book I wrote was really one of the best gifts that that experience gave me.
AW: That's amazingly gratifying. Yeah, that's fantastic.
MA: But the physical part of it was hard. Like I had never exerted my vocal cords to that extent before. A couple hours into the first session, my throat was hurting, and that surprised me. I was like, "Oh, my God. I can't just do this constantly without there being any physical challenges." It's like I have to give myself a break. I have to take care of my voice. I'm a big Diet Coke drinker. And I found that Diet Coke is probably not the best thing to be drinking while you're recording a book because you burp sometimes.
AW: There might be some gas.
MA: It might be a little gas, and there might be … you know, there's something about the way it affected my throat, which just didn't feel good. So I had to give up Diet Coke for all the days that I recorded the book, and I went to drinking iced tea. That was a big change.
AW: The sacrifice.
MA: I know, the sacrifices I made. By the second or third session, I felt like my throat adjusted to the demands of the job and it was fine, but it still took a little bit of a toll on my voice.
AW: Have you listened to the audiobook? [Editor note: This interview took place as the book launched.]
MA: No, I have not. I went in to do some pickups. There were a couple areas that we needed to tweak, so I went in for an hour and recorded some pickups. I briefly heard a snippet, and it startled me. First of all the quality was so good. I wasn't used to that. You know, I'm used to hearing my voice on a tape recorder from when I'm interviewing an actor when I'm transcribing it back. The quality was so crisp and so clean, but it was jarring just to hear my voice. I was like, "Please shut that off." That's not to say I'm not going to listen to it ever. Just, I'm not ready quite yet.
AW: That's understandable for sure. What was the first difficult passage for you in reading, difficult for whatever reason?
MA: The day we got married. When I talk about this in the book we'd been together for 13 years, but marriage was something we were just never interested in for a number of reasons. But when Kit was diagnosed with cancer, suddenly marriage became the most important thing to me and ultimately to him too. It became something like, "We need to do this. I want to be married to him. I want to go through this process—whatever it's going to be—as his husband." It was this crazy day, which I chronicle in the book, where we went from Sloan Kettering in the morning to City Hall in the afternoon, and recounting that day was the most emotionally difficult for me.
AW: You went someplace afterward right?
MA: We went back to our apartment. But what we did was we walked through the city. We did sort of a little bit of a tour of downtown Manhattan. We walked from City Hall to our apartment in the West Village, because it was an unseasonably comfortable winter day. Then we had a little get together at our apartment later that night. Anyway, that day and recounting the ceremony and our vows. Our vows—that was a little wrenching.
AW: Did you tear up there? Are there periods in the book where you're full-on emotional?
MA: Yes. There are a couple periods where I'm not like a blubbering mess, but there are a couple periods where you can tell that I'm crying when I'm speaking. They didn't stop me. They didn't yell, "cut!" I trusted them that if it didn't interfere with the book and that it worked that we would just keep going, and we did. There was one spot where we needed to stop just because my voice was cracking and people wouldn't have been able to understand what I was saying, and that came later when—I believe—the night he died, when I was going through some of the passages there. They were like, "Let's cut and give you a couple minutes to catch your breath and collect yourself and get your shit together." Yeah, that made sense.
AW: Totally. So we work in entertainment. Calling them friends is a stretch, but you have a lot of celebrity friends.
MA: I have some famous friends. Yes, Abby.
AW: You may have some famous friends.
MA: Are we going to drop names now?
AW: Sure. I could drop a certain curly-haired wonder that is a fave of yours.
MA: Keri Russell.
AW: Keri Russell. You know, in the book you do talk about moments in which you're dealing with your work life while your private life is so tumultuous, and stressful, and sad, and taking so much of your attention but you have to focus and do an interview and be the normal, upbeat, droll self that you are. Talk a little bit about what that was like and how it was putting this book out into the world, and then having to interact with some of these same people who now know more about you.
MA: My celebrity friends who are mentioned in the book, I made sure to get them all a copy in advance and I sent them a note and just said, "Hey, you're in this. FYI, this was going on in my life when I interviewed you on that day." That was extremely surreal and challenging for me on the days where I had to function as if everything was fine when in reality my world was falling apart. It was so early in the process that it wasn't something I could really talk to people about because we didn't know the full picture of what Kit's illness was. We didn't even know that it was in fact cancer in some of the instances I recount.
I talk about in the book how I had a set visit at The Americans on the day that Kit was going to one of his first appointments to find out why he was experiencing this discomfort. I was literally waiting to interview Keri Russell and Matthew Rhys on the set while I was staring at my phone also waiting for Kit to text me from the doctors. That was just a jarring sort of experience juggling those two things. Then interviewing Keri and Matthew and then not knowing what was going on, I don't know if they picked up on anything, if they could tell that my mind was elsewhere, because I wasn't fully present at all when I spoke to them. I did the best I could.
There's another experience in the book where I had to interview Kristen Bell for the Veronica Mars movie, and this was after we found out that Kit's condition was serious but still before we were comfortable telling anyone. No one at my work knew. Some of my family members knew, but that was about it. Kristen Bell comes into our offices, and I'm interviewing her on camera too. This wasn't a situation where I was off camera like with Keri and Matthew where I just put a tape recorder down. There's only so much that's expected of me in that kind of interview, but when you're on camera you have to be super present and on. Throughout the interview with Kristen, it was just like my mind was, again, just elsewhere. The interview was scheduled. It was for the Veronica Mars movie. It was a big deal. I debated saying no, but ultimately I went through with it.
"..words that I thought I knew, words that I had every confidence I knew how to pronounce. Then I'd come out with it and he'd have to stop. He's like, "That's actually not how you pronounce that." A couple of them were really embarrassing."
AW: Have you gotten any feedback or reactions from sending out the books?
MA: Yeah, all really positive.
AW: Surprised that you were going through that?
MA: Yeah, surprised and some said that, "I wish I had known because I would have given you a hug," or something like that. The response has been really, really positive and people have just been really supportive, which has been nice.
AW: Yes. You're kind of on the other side and being the talent now.
MA: Mm-hmm, It's uncomfortable. It's so weird. I love asking questions.
AW: I know you do.
MA: I love asking questions. It's a completely different experience being on the other side. How am I doing?
AW: You're doing great.
MA: You have to say that.
AW: No, I don't. I could tell you.
MA: All right, yeah. You would. You would tell me.
AW: What is your biggest takeaway from recording the audiobook? I know you haven't listened to it yet, but thinking about having people who thought they knew you already, have followed your career, or are fans of the site and your work. And there are a lot of people. I've seen the letters Ask Ausiello gets. They think they know you already, and now some of these people are going to have you in their ear. What does that feel like?
MA: I'm excited for them to get to know me better. A little nervous, but mostly I'm proud of the book. I'm proud of getting through the audiobook and doing what I think was a good job. I'm mostly excited for people to see different sides of me and to learn a little bit more about me and maybe even more than they want to know about me. I'm excited to hear from them and see what they thought.
AW: Did anything about this process make you want to write another book about anything?
MA: Mostly no. The thought of taking on a project like this again just seems like no, it's too much work. Especially again having a full time job, it's a lot. That said though, in the last couple months, I have started to think about other ideas I have for books, ideas that came up in the writing of this book, ideas I had before but I put aside because I didn't think I could write a book. Now I know I can, so I'm thinking about it. I'm excited at the possibility of writing something that isn't as serious. Even though there's so much humor in this book, it's obviously a very weighty subject. Some of the ideas I'm tossing around are a little lighter.
AW: Is there anything else we should know about this process from you?
MA: Yes. One thing I wanted to mention that I wanted to give a shout out to the engineer director who worked with me on this audiobook. Hold on. I want to make sure his name. It's Tyler. I want to make sure I get the pronunciation of his last name right. Tyler Whitlatch. He was awesome by the way. It's such an interesting experience. He's a stranger when I walk in, and we have this really intimate experience over four sessions. This book, it's a very intimate book. Here it's just the two of us in a studio for four days, and I'm crying, and I'm talking about just some really private things.
He was just so wonderful, so compassionate just in terms of giving me time and space and letting me … he had tissues ready for me in the studio in case I needed them, and I needed them. After each session, he'd give me a hug. It was just really … that part of it, of the experience, was really kind of beautiful just forming this relationship with this total stranger, and he was wonderful. One of the great things about Tyler was he really did his research. I mean he read the book, but he also predicted potential landmines in the form of words that I might … can I say the F-word?
"The thought of someone else doing the audiobook would have been so wrong. Unless it was like a star, like a huge star like Brad Pitt was doing it. That probably might have felt right."
AW: Go ahead.
MA: In the form of words that I could f—- up in the pronunciation of, words that I thought I knew, words that I had every confidence I knew how to pronounce. Then I'd come out with it and he'd have to stop. He's like, "That's actually not how you pronounce that." A couple of them were really embarrassing, because they're words that I should actually know how to pronounce. One of them was Downton Abbey's "Dowager ('dau̇-i-gər') Countess." Apparently it's dowager ('dau̇-i-jər'). I was not aware of this until I recorded my audiobook and was corrected, and there were a bunch of those. There were a bunch of those words, impressive 50 cent words, that I had in the book that I was like … thought I knew how to pronounce and I didn't. That was humbling, and it was an education, and it was also really funny. It was really funny, and it made me really grateful to have someone like Tyler on the other side of the glass wall just looking out for me and having my back. Yeah, so that was something.
AW: Yeah. It is quite the process, and especially for people who think we know words.
MA: Yeah.
AW: We do words all the time. I know that word—no.
MA: Yeah. Here's the thing. Writing it and saying it, two different things.
AW: Totally different.
MA: Different experience, so a little bit of a learning curve there.
AW: I found it interesting reading your book again, because I knew you I read it in your voice to begin with, so it felt like you were writing it very conversationally to begin with. Did you find that when you read it that felt natural?
MA: Yes. Yeah, absolutely. It all felt natural, and I think that made the process of it a lot easier too. It's like it was my voice on the paper, it was my voice literally, and it just felt completely organic. It felt right. Like the thought of someone else doing the audiobook would have been so wrong. Unless it was like a star, like a huge star like Brad Pitt was doing it. That probably might have felt right.
AW: It's cool since you'd get your pick, right?
MA: Brad Pitt?
AW: Yeah.
MA: No, no, no, no. Not Brad Pitt. I just threw out that name. I don't know who it would be.
AW: In the movie adaptation of this book, the TV adaptation of the book, who would play Michael Ausiello?
MA: God, that's a whole other can of worms. I was so glad that I did it and I can't imagine anyone else doing the audiobook. It was perfect.
AW: Fantastic. Well Michael, thank you so much for coming out here and talking to us about the process and the book.
MA: Thank you for having me. It's great to see you.
AW: You too. Thank you so much.
Authorrators
David Sedaris Doesn't Give a Damn if He Ever Reads a Book With His Eyes Again
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Fire and Fury and the 25th Amendment
This week an explosive new book called Fire and Fury: Inside the White House was published. The book was written by journalist Michael Wolff who claimed to have 'fly on the wall' access to the White House and its staff.
Wolff's book has caused a storm across Washington
Among the key claims in the book are:
Members of Trump's campaign team were certain he would not win. It is claimed that Kellyanne Conway, his campaign manager, was already job-hunting before Election Day.
Trump eats so much fast-food because he is worried about being poisoned.
That the First Lady, Melania Trump, was disappointed by Trump's election victory.
Staff in Trump's White House describe him as 'Child-Like'.
The book featured 'on the record' interviews with former Trump campaign strategist Steve Bannon. Among the claims about the Trump administration by Bannon were:
Trump's former strategist has now been labelled 'Sloppy Steve' by the President
That a meeting between Trump's son, Donald Trump Jr, and Lawyers with links to the Russian Government were "treasonous" and "unpatriotic".
That Ivanka Trump was "dumb" and that she had ambitions to be the first female President of the United States.
Trump's response was ferocious. As expected, the first salvos were fired on Twitter:
There have been a number of factual errors in the book. Some commentators have been extremely critical of the author. Trump himself has already questioned the access that Wolff had to him in making the book. Wolff has said that he has spent around 3 hours with Trump, the President has disputed this, saying he hardly met him. The book is now No.1 on the Amazon Best-Sellers List. Arguably, this is largely because of the reaction of the Trump Administration.
Among the things being talked about this week is the 25th Amendment. This was passed in February 1967 and changed a number if important things:
Section 1 – Confirmed that the Vice-President would assume the presidency if the President resigned or died.
Therefore, when Richard Nixon resigned on the 9th August 1974, Gerald Ford became President.
Section 2 – If there was a vacancy for Vice-President the President would nominate a Vice-President who would be confirmed by the House of Representatives.
This happened in 1974 when Spiro Agnew was forced to resign as Vice-President after pleading guilty to tax evasion. Gerald Ford was nominated by Nixon as the next Vice-President and the House of Representatives confirmed his appointment by 387-5.
When Gerald Ford became President in August 1974 he was the first to assume the position having not won a popular election. He was an unelected President.
Section 3 – This allowed for the President to temporarily hand his power to the Vice-President by signing a letter that is sent to the Speaker of the House of Representatives and Senate Pro Tempore. The President can return his powers by submitting a second letter.
George W. Bush twice used Section 3 to hand power to Dick Cheney whilst he was under general anaesthetic.
Dick Cheney (left) twice became Acting President while George W. Bush was indisposed
Section 4 – Of these, one that has been talked about this week is Section 4. Section 4 of the 25th Amendment allows a President to be removed from office if they are deemed by the Vice-President and the majority of the cabinet to be unfit to 'discharge their duties'. If this were agreed, the President would be removed from Office, either permanently or temporarily, and the Vice-President would assume the presidency.
Section 4 acts essentially as a Check on the Executive by the Executive. Section 4 has never been invoked. However, in 1981 it probably should have been. In March 1981 President Ronald Reagan was shot in an attempted assassination. As Vice-President George H.W Bush was not in Washington the Cabinet could not meet in time to invoke Section 4. As a result, whilst Reagan was in surgery America was technically without an 'Acting President'.
The idea of removing a President due to incapability was also rumoured to have been considered under Ronald Reagan. It is suggested that members of his staff, including Chief of Staff, Howard Baker, had doubts about Reagan's lazy and inattentive attitudes towards his duties.
In his book Wolff suggests that the notion of invoking Section 4 is often discussed in the Trump White House, however, it is unlikely a 'coup' would come from inside the White House. A Democratic Congressman, Jamie Raskin has proposed a bill called the Oversight Commission of the Presidential Capacity Act, this would take the power to check the President's capacity away from the President and give it to a congressional Commission of 11 members. The bill now has 56 co-sponsors.
There is no doubt that Wollf's book is damaging to Trump. However, claims it will result in his removal from Office are far-fetched. What is most damaging to Trump is that the claims seem to reinforce what people already think. This could be damaging to the Republicans in the November 2018 Mid-Term Elections and the 2020 Presidential Election.
This entry was posted in US Executive, US Government and Politics on January 8, 2018 by politicsteaching.
← What is Gerrymandering? What are the 'Great Offices of State' →
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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YEREVAN, SEPTEMBER 8, ARMENPRESS. Former Prime Minister of Armenia Hovik Abrahamyan is charged with part 1 of Article 309 of the criminal code of Armenia (abuse of power) and Article 310 ) Illegal participation in entrepreneurial activity), ARMENPRESS was informed from the press service of the Special Investigation Service. Written obligation not to leave the place of residence has been chosen as a preventive measure for him.
Earlier, Hovik Abrahamyan informed about accusation against him. He wrote on his Facebook page that on September 4 the Special investigation Service sent him a notification for interrogation as a witness, afterwards charges were pressed.
The investigation into the criminal case revealed that "Avazahatik" company, established in 2000 in Ararat Province, was engaged in mining. But in 2008 high ranking officials created artificial obstacles and demanded 60% shares of the company.
The founder of "Avazahatik" company and other witnesses have indicated the names of Hovik and Jonik Abrahamyans and Alik Sargsyan in their testimonies among the individuals linked with power abuse.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 3,795
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using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using Tridion.Extensions.DynamicDelivery.ContentModel.Factories;
using Tridion.Extensions.DynamicDelivery.ContentModel;
using System.ComponentModel.Composition;
namespace Tridion.Extensions.DynamicDelivery.Factories.LocalDatabase
{
[Export(typeof(IBinaryFactory))]
public class BinaryFactory : IBinaryFactory
{
#region IBinaryFactory Members
public bool TryFindBinary(string url, out IBinary binary)
{
throw new NotImplementedException();
}
public IBinary FindBinary(string url)
{
throw new NotImplementedException();
}
public bool TryGetBinary(string tcmUri, out IBinary binary)
{
throw new NotImplementedException();
}
public IBinary GetBinary(string tcmUri)
{
throw new NotImplementedException();
}
public bool TryFindBinaryContent(string url, out byte[] bytes)
{
throw new NotImplementedException();
}
public byte[] FindBinaryContent(string url)
{
throw new NotImplementedException();
}
public bool TryGetBinaryContent(string tcmUri, out byte[] bytes)
{
throw new NotImplementedException();
}
public byte[] GetBinaryContent(string tcmUri)
{
throw new NotImplementedException();
}
public bool HasBinaryChanged(string url)
{
throw new NotImplementedException();
}
#endregion
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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\section{Introduction}
In humans and other vertebrates the immune system is of crucial importance for
protecting an individual from dangers such as pathogens, toxins and cancer.
(For background information on immunology we refer to \cite{murphy12}.)
The central players in the immune system are the white blood cells (leukocytes)
and it is important that these cells be able to distinguish between dangerous
substances and host tissues. This is often referred to as the distinction
between non-self and self. A failure to combat dangerous substances may lead
to infectious diseases becoming life-threatening. On the other hand, if the
immune system attacks host tissues this can lead to autoimmune disease. The
task of discrimination is complicated. An important element of the process of
distinction between self and non-self is the activity of the class of leukocytes
called T cells. An individual T cell is supposed to recognize a particular
substance (antigen) and take suitable action if that substance is dangerous.
Recognition is based on the binding of the antigen to a molecule on the T cell
surface, the T cell receptor (TCR). It is believed that the most important
aspect of this process is the time the antigen remains bound before being
released (the dissociation time), an idea which has been called the 'lifetime
dogma' \cite{feinerman08a}. When it recognizes its antigen the T cell changes
its behaviour and is said to be activated. In what follows we study a
mathematical model for what happens in the first few minutes after a T cell
recognizes its antigen.
In \cite{altanbonnet05} Altan-Bonnet and Germain introduced a model for the
initial stage of T cell activation. Simulations using this model gave results
which fitted a number of experimental findings. On the other hand it was too
elaborate to be readily accessible to a mathematical analysis of its dynamics.
In \cite{francois13} the authors introduced a radically simplified version of
the model of \cite{altanbonnet05}. The new model includes the essential
explanatory power of the old one while being much more transparent and
tractable for analytical investigation. It also made some new predictions
which were confirmed experimentally. In \cite{francois13} a number of
interesting analytical calculations were performed but the
mathematical conclusions which can be drawn from these were not worked out
in detail.
The aim of the present paper is to obtain results about the qualitative
behaviour of solutions of the model of \cite{francois13} which are as
general as possible. In Section \ref{def} the model is defined and some
of its basic properties are derived. The model describes a situation where
both an agonist (the antigen which should be recognized) and an
antagonist (a competing antigen) are present. Section \ref{steady} is concerned
with the number of steady states and their stability. After some general
results have been derived, the discussion turns to more detailed properties
of the solutions in the case that the antagonist is absent and treats cases
where the number $N$ of phosphorylation sites included in the model is small.
In particular it is shown that when $N=3$ there are parameters for which
three positive steady states exist (Theorem 1). A numerical calculation
reveals that for a specific choice of these parameters two of the steady
states are stable while the third is a saddle. For $N\le 2$ there is a unique
steady state and in the case $N=1$ it is proved to be globally asymptotically
stable. There are parameter values for which the approach to this steady state
is oscillatory.
The qualitative behaviour of the steady state concentration of the maximally
phosphorylated state, which expresses the degree of activation of the T cell,
as a function of the antigen concentration and the dissociation time, is
investigated in the case where only the agonist is present in Section
\ref{response}. Let us consider the function $f(L_1,\nu_1)$, which expresses
the degree of activation in terms of the parameters $L_1$ (concentration
of agonist ligand) and $\nu_1$ (reaction rate for the dissociation of the
ligand from the receptor, i.e. the reciprocal of the dissociation time).
It is shown that the dependence exhibits certain types of non-monotone
behaviour in some cases. The results obtained include both rigorous results
on general features of the function $f$ (Theorem 2) and simulations which
reveal more detailed features. In particular it is found that are values of the
parameters in the model for which the function $f$ has a maximum as a function
of $\nu_1$ for fixed $L_1$. In other words, there is a value of the
dissociation time which is optimal for T cell activation. Thus the model
studied here is able to reproduce this fact which has been experimentally
observed \cite{lever16}.
The analysis of the response function is extended to cover the effects of the
antagonist in Section \ref{antagonist}. The last section is devoted to
conclusions and an outlook.
\section{Definition of the model}\label{def}
In the introduction it was stated that a T cell recognizes an antigen. In
more detail the molecule concerned is a peptide (a small protein) which is
bound to a host molecule called an MHC molecule. Thus we talk about a pMHC
complex as the object to be recognized. In the model of \cite{francois13} two
types of pMHC complexes are considered. The first, called an agonist,
represents the case where the antigen comes from a pathogen and should
activate the T cell. The second, called an antagonist, represents the case of
a self-antigen, which should not activate the T cell. Detection takes place
through the binding of a pMHC complex to the T cell receptor.
When this happens certain proteins associated to the T cell receptor are
phosphorylated, i.e. phosphate groups become attached to them. For simplicity we
will describe this by saying that the receptor-pMHC complex is phosphorylated.
The reaction network for the model of \cite{francois13} is shown in
Figure~\ref{fig:reactionnetwork}.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.5]{tcell_rendall_fig.pdf}
\end{center}
\caption{The model considered in this paper. The species $R$ represents the T
cell
receptor, and $L_1$ and $L_2$ are the two ligands, i.e. the agonist and
antagonist. The species $C_0$ represents unphosphorylated complexes of the T
cell receptor with the agonist, and the $C_j$'s are the $j$-phosphorylated
complexes. The $D_j$'s are the analogous complexes for the antagonist.
The phosphatase SHP-1 provides a negative feedback, and is represented by $S$.
The different reactions represent receptor complex phosphorylation with
rate constant $\phi$ and dephosphorylation with rate constant $b$, as well
as receptor complex dephosphorylation by $S$ with rate constant $\gamma$
and dissociation rate constants $\nu_1$ and $\nu_2$.
Antigens bind to $R$ with rate constant $\kappa$, and $S$ is activated by
the singly phosphorylated complexes with rate constant $\alpha$ and
deactivated with rate constant $\beta$.
}
\label{fig:reactionnetwork}
\end{figure}
The state variables will now be listed. The
concentration of
unphosphorylated complexes of the T cell receptor with the agonist will be
denoted by $C_0$ and the concentration of unphosphorylated complexes of the T
cell receptor with the antagonist will be denoted by $D_0$. $C_j$ and $D_j$
are the corresponding quantities for the case of $j$ phosphorylations, up to a
maximum value $N$. The specific value of $N$ has little influence in what
follows but it may be worth to note that a biologically reasonable value of
$N$ could be as large as 20 while in the model of \cite{altanbonnet05} we
have $N=9$. $R$, $L_1$ and $L_2$ are the total concentrations of receptors and
the two ligands, i.e. the agonist and antagonist. Another important element of
the system is SHP-1. This substance is a phosphatase which means that when
active it can remove phosphate groups from the receptor-pMHC complex. It
contributes a negative feedback loop to the system. $S$ is the concentration
of active SHP-1. The receptor complexes are subject to phosphorylation with
rate constant $\phi$ and dephosphorylation with rate constant $b$. They are
also dephosphorylated by SHP-1 with rate constant $\gamma $ and dissociate
with rate constants $\nu_1$ and $\nu_2$. Antigens bind to the receptor with
rate constant $\kappa$. SHP-1 is activated by the singly phosphorylated
complexes with rate constant $\alpha$ and deactivated with rate constant
$\beta$. All the rate constants are assumed positive. $S_T$ is the total
concentration of SHP-1. It is assumed that all reactions exhibit mass action
kinetics and this leads to the following system of equations
\begin{eqnarray}
&&\dot S=\alpha (C_1+D_1)(S_T-S)-\beta S,\label{phen1}\\
&&\dot C_0=\kappa (L_1-\sum_{j=0}^NC_j)(R-\sum_{j=0}^N (C_j+D_j))
+(b+\gamma S)C_1-(\phi+\nu_1)C_0,\label{phen2}\\
&&\dot C_j=\phi C_{j-1}+(b+\gamma S)C_{j+1}-(\phi+b+\gamma S+\nu_1)C_j,
\ \ 1\le j\le N-1,\label{phen3}\\
&&\dot C_N=\phi C_{N-1}-(b+\gamma S+\nu_1)C_N,\label{phen4}\\
&&\dot D_0=\kappa (L_2-\sum_{j=0}^ND_j)(R-\sum_{j=0}^N (C_j+D_j))
+(b+\gamma S)D_1-(\phi+\nu_2)D_0,\label{phen5}\\
&&\dot D_j=\phi D_{j-1}+(b+\gamma S)D_{j+1}-(\phi+b+\gamma S+\nu_2)D_j,
\ \ 1\le j\le N-1,\label{phen6}\\
&&\dot D_N=\phi D_{N-1}-(b+\gamma S+\nu_2)D_N.\label{phen7}
\end{eqnarray}
In a direct formulation of the system as arising from the reaction network
it is necessary to include the concentrations of free ligands, free
receptors and inactive phosphatase. This extended system has four conservation
laws corresponding to the total amounts of ligands, receptors and phosphatase.
Using these to eliminate the additional variables leads to the system
(\ref{phen1})-(\ref{phen7}).
The right hand sides of the equations are Lipschitz and so there is a unique
solution corresponding to each choice of initial data. To have a biologically
relevant solution the quantities in the extended system should be non-negative.
It is a well-known fact for reaction networks of this type that data for which
all concentrations are positive give rise to solutions with the same property
and that data for which all concentrations are non-negative give rise to
non-negative solutions. In terms of (\ref{phen1})-(\ref{phen7}) this implies
statements about the positivity of the quantities $S$, $C_j$ and $D_j$ and of
the differences $S_T-S$, $R-\sum_{j=0}^N(C_j+D_j)$, $L_1-\sum_{j=0}^NC_j$ and
$L_2-\sum_{j=0}^ND_j$. Let us call the region where all these quantities are
strictly positive the biologically feasible region. Note that due to the
conservation laws this region is bounded. Let $\Sigma_1=\sum_{j=0}^NC_j$ and
$\Sigma_2=\sum_{j=0}^NC_j$. Then it follows from (\ref{phen1})-(\ref{phen7})
that
\begin{eqnarray}
&&\dot\Sigma_1=\kappa(L_1-\Sigma_1)(R-\Sigma_1-\Sigma_2)-\nu_1\Sigma_1,
\label{total1}\\
&&\dot\Sigma_2=\kappa(L_2-\Sigma_2)(R-\Sigma_1-\Sigma_2)-\nu_2\Sigma_2.
\label{total2}
\end{eqnarray}
\noindent
{\bf Lemma 1} Consider a solution
$(S(t),C_0(t),\ldots,C_N(t), D_0(t),\ldots,D_N(t))$ in the closure of
the biologically feasible region. Then if
$(S^*,C_0^*,\ldots,C_N^*,D_0^*,\ldots,D_N^*)$ is an $\omega$-limit point
of this solution it
is also in the biologically feasible region. In particular, any
steady state is in the biologically feasible region.
\noindent
{\bf Proof} If $S^*=S_T$ we can consider the solution starting at that point
at some time $t_0$. Since the $\omega$ limit set of a given solution is
invariant the solution under consideration lies entirely in the $\omega$-limit
set of the original solution. In particular, it is contained in the closure of
the biologically feasible region. The solution starting at the point with
$S^*=S_T$ satisfies $\dot S(t_0)<0$ and therefore the inequality $S(t)>S_T$ for
$t$ slightly less than $t_0$, a contradiction. In a similar way equation
(\ref{total1}) implies that $\sum_{j=0}^NC^*_j$ cannot attain the value $L_1$
and equation (\ref{total2}) implies that $\sum_{j=0}^ND^*_j$ cannot attain the
value $L_2$. Summing (\ref{total1}) and (\ref{total2}) shows that
$\sum_{j=0}^NC^*_j+\sum_{j=0}^ND^*_j$ cannot attain the value $R$.
Note next that $C_0$ cannot be zero at an $\omega$-limit point. For if it
were zero at such a point we could consider the solution passing through that
point at a time $t_0$. The equation (\ref{phen2}) would imply that
$\dot C_0(t_0)>0$ and that $C_0(t)<0$ for $t$ slightly less than $t_0$, a
contradiction. Once the positivity of $C_0$ has been proved we can use
equation (\ref{phen3}) with $j=1$ to show that $C_1$ cannot be zero at an
$\omega$-limit point. This in turn allows us to prove using (\ref{phen1}) that
$S$ can never be zero at an $\omega$-limit point.
In a similar way it can be concluded successively that $C_2,\ldots,C_N$ and
$D_0,\ldots,D_N$ are positive at any $\omega$-limit point of a non-negative
solution. This concludes the proof of the lemma.
The fact that all $\omega$-limit points of solutions in the closure of the
biologically feasible region are in the biologically feasible region together
with the fact that the closure of that region is compact implies that the
infimum of the distance of a given solution to the boundary in the limit
$t\to\infty$ is strictly positive. When this last property holds the system
is said to be persistent \cite{butler86}. Note in addition that the
closure of the biologically feasible region is convex and hence homeomorphic
to a closed ball in a Euclidean space. It follows from the Brouwer fixed
point theorem that a steady state exists (cf. \cite{hale09}, Chapter I,
Theorem 8.2). Since steady states on
the boundary have already been excluded we can conclude that there is at least
one steady state in the biologically feasible region for any fixed choice of
parameters. That this is the case was assumed implicitly in \cite{francois13}.
\section{Multiplicity of steady states}\label{steady}
A question not addressed in \cite{francois13} is whether there might exist
more than one positive steady state for a fixed choice of parameters. In this
section it will be shown that for some values of $N$ and the reaction
constants this is the case. The aim is to find any parameter values with this
property while not worrying for the moment how biologically relevant this
choice of parameters is. Let $f_1$ and $f_2$ denote the right hand sides of
equations (\ref{total1}) and (\ref{total2}). Then
$\frac{\partial f_1}{\partial\Sigma_2}$ and
$\frac{\partial f_2}{\partial\Sigma_1}$ are negative and hence the system
(\ref{total1})-(\ref{total2}) is competitive. It follows that every solution
of this system converges to a steady state as $t\to\infty$ \cite{hirsch05}.
A steady state $(\Sigma_1^*,\Sigma_2^*)$ of (\ref{total1})-(\ref{total2})
satisfies the equations
\begin{eqnarray}
&&\kappa(L_1-\Sigma_1^*)(R-\Sigma_1^*-\Sigma_2^*)-\nu_1\Sigma_1^*=0
\label{totalstat1},\\
&&\kappa(L_2-\Sigma_2^*)(R-\Sigma_1^*-\Sigma_2^*)-\nu_2\Sigma_2^*=0
\label{totalstat2}.
\end{eqnarray}
We can solve for $\Sigma_1^*$ and $\Sigma_2^*$ as functions of
$\Sigma_1^*+\Sigma_2^*$.
\begin{eqnarray}
&&\Sigma_1^*=\frac{\kappa L_1(R-\Sigma_1^*-\Sigma_2^*)}
{\kappa (R-\Sigma_1^*-\Sigma_2^*)+\nu_1},\\
&&\Sigma_2^*=\frac{\kappa L_2(R-\Sigma_1^*-\Sigma_2^*)}
{\kappa (R-\Sigma_1^*-\Sigma_2^*)+\nu_2}.
\end{eqnarray}
Hence
\begin{equation}
\kappa (L_1+L_2-\Sigma_1^*-\Sigma_2^*)
=\frac{\kappa L_1\nu_1}{\kappa (R-\Sigma_1^*-\Sigma_2^*)+\nu_1}
+\frac{\kappa L_2\nu_2}{\kappa (R-\Sigma_1^*-\Sigma_2^*)+\nu_2}.
\end{equation}
The function of $\Sigma_1^*+\Sigma_2^*$ on the left hand side of this equation
is decreasing on the interval $[0,L_1+L_2]$. The function on the right hand
side is increasing on the interval $[0,R]$. Their graphs can intersect in at
most one point. We already know that they must intersect since a positive
steady state of the full system exists. That they intersect can also be seen
directly. For in all cases the left hand side is greater than the right hand
side for $\Sigma_1^*+\Sigma_2^*=0$ and the opposite inequality holds for
$\Sigma_1^*+\Sigma_2^*=\min\{L_1+L_2,R\}$. Thus the equation has a unique
solution for $\Sigma_1^*+\Sigma_2^*$ in the interval $[0,\min\{L_1+L_2,R\}]$
From this it is possible to compute values of $\Sigma_1^*$ and
$\Sigma_2^*$ which solve (\ref{totalstat1}) and (\ref{totalstat2})
and lie in the intervals $[0,\min\{L_1,R\}]$ and $[0,\min\{L_2,R\}]$,
respectively. The quantities $\Sigma_1^*$ and $\Sigma_2^*$ are functions of the
parameters $R$, $L_1$, $L_2$, $\kappa$, $\nu_1$ and $\nu_2$.
It can be concluded that the solution passing through an $\omega$-limit point
of a solution of the original system satisfies a simplified system containing
$\Sigma_1^*$ and $\Sigma_2^*$ as parameters. $C_0$ and $D_0$ can be eliminated
from this system in favour of the other $C_j$ and $D_j$. The result is
\begin{eqnarray}
&&\dot S=\alpha (C_1+D_1)(S_T-S)-\beta S,\label{phenlim1}\\
&&\dot C_1=\phi\Sigma_1^*+(b+\gamma S-\phi)C_2
-(2\phi+b+\gamma S+\nu_1)C_1-\phi\sum_{j=3}^N C_j\label{phenlim2}\\
&&\dot C_j=\phi C_{j-1}+(b+\gamma S)C_{j+1}-(\phi+b+\gamma S+\nu_1)C_j,
\ \ 2\le j\le N-1,\label{phenlim3}\\
&&\dot C_N=\phi C_{N-1}-(b+\gamma S+\nu_1)C_N,\label{phenlim4}\\
&&\dot D_1=\phi\Sigma_2^*+(b+\gamma S-\phi)D_2
-(2\phi+b+\gamma S+\nu_2)D_1-\phi\sum_{j=3}^N D_j,\label{phenlim5}\\
&&\dot D_j=\phi D_{j-1}+(b+\gamma S)D_{j+1}-(\phi+b+\gamma S+\nu_2)D_j,
\ \ 2\le j\le N-1,\label{phenlim6}\\
&&\dot D_N=\phi D_{N-1}-(b+\gamma S+\nu_2)D_N.\label{phenlim7}
\end{eqnarray}
This form of the equations is valid for $N\ge 3$. In the case $N=2$ it is still
correct if it is taken into account that the condition $2\le j\le N-1$ is never
satisfied so that the equations containing that condition are absent. The sum
from $j=3$ to $N$ is zero in that case. The case $N=1$ is exceptional from the
point of the notation.
In order to get more information we will restrict in the remainder of this
section to what we call the agonist-only case. This is obtained from the system
(\ref{phen1})-(\ref{phen7}) by setting $L_2$ and the $D_i$ to zero. There
is a corresponding limiting system, which is obtained from
(\ref{phenlim1})-(\ref{phenlim7}) by setting $\Sigma_2^*$ and the $D_i$ to
zero. In this case we write $\Sigma^*$ instead of $\Sigma_1^*$ for brevity.
Consider the limiting system in the agonist-only case with $N=1$. This is
\begin{eqnarray}
&&\dot S=\alpha C_1 (S_T-S)-\beta S\label{red1s},\\
&&\dot C_1=\phi\Sigma^*-(\phi+b+\gamma S+\nu_1)C_1\label{red1c}.
\end{eqnarray}
Solving the equation $\dot S=0$ for $C_1$ and substituting the result
into the equation $\dot C_1=0$ gives the quadratic equation
\begin{equation}\label{S1}
\beta\gamma S^2+[\beta (\phi+b+\nu_1)+\alpha\phi\Sigma^*]S
-\alpha\phi\Sigma^* S_T=0.
\end{equation}
Since the quadratic polynomial has positive leading term and is negative for
$S=0$ it is clear that it has a unique positive root. It follows from
(\ref{S1}) that this root is less than $S_T$. Equation (\ref{red1c}) implies
that $C_1<\Sigma^*$ at a steady state and so these quantities can be completed
to a steady state of the original system by defining $C_0=\Sigma^*-C_1$. The
steady state is unique in this case.
In the case $N=2$ the equations are
\begin{eqnarray}
&&\dot S=\alpha C_1(S_T-S)-\beta S,\\
&&\dot C_1=\phi\Sigma^*-(2\phi+b+\gamma S+\nu_1)C_1+(-\phi+b+\gamma S)C_2,\\
&&\dot C_2=\phi C_1-(b+\gamma S+\nu_1)C_2.
\end{eqnarray}
Proceeding in a manner analogous to what we did in the case $N=1$ it is possible
to get a cubic equation for $S$ in the case $N=2$, which we can write
schematically in the form $p(S)=\sum_{k=0}^Na_kS^k$. We have
\begin{eqnarray}
&&a_0=-\alpha S_T(b+\nu_1)\phi\Sigma^*,\nonumber\\
&&a_1=\beta [b(\phi+b+\nu_1)+\nu_1(2\phi+b+\nu_1)+\phi^2]
+\alpha(b+\nu_1)\phi\Sigma^*-\alpha\gamma S_T\phi\Sigma^*,\nonumber\\
&&a_2=\beta\gamma (\phi+2b+2\nu_1)+\alpha\gamma\phi\Sigma^*,\nonumber\\
&&a_3=\beta\gamma^2.\nonumber
\end{eqnarray}
The sequence of signs of the coefficients $a_i$ is either $(-,-,+,+)$ or
$(-,+,+,+)$. There is precisely one change of sign and thus by Descartes'
rule of signs the polynomial has precisely one positive root. Once a value
of $S$ is given the values of $C_1$ and $C_2$ at the steady state can be
determined successively. Following the arguments in the case $N=1$ we see
that $S<S_T$, $C_1+C_2<\Sigma^*$ and that the steady state is unique.
In the case $N=3$ the system is
\begin{eqnarray}
&&\dot S=\alpha C_1(S_T-S)-\beta S,\label{N31}\\
&&\dot C_1=\phi\Sigma^*-(2\phi+b+\gamma S+\nu_1)C_1
+(-\phi+b+\gamma S)C_2-\phi C_3,
\label{N32}\\
&&\dot C_2=\phi C_1-(\phi+b+\gamma S+\nu_1)C_2+(b+\gamma S)C_3,\label{N33}\\
&&\dot C_3=\phi C_2-(b+\gamma S+\nu_1)C_3.\label{N34}
\end{eqnarray}
A calculation for $N=3$ analogous to those already done gives a quartic
polynomial. Its coefficients are
\begin{eqnarray}
&&a_0=-[(b+\nu_1)^2+\phi\nu_1]\alpha\phi\Sigma^* S_T,\nonumber\\
&&a_1=\beta\gamma\{(\phi+b+\nu_1)[(b(b+\nu_1)+\nu_1(\phi+b+\nu_1)]
+\nu_1(\phi+b+\nu_1)\nonumber\\
&&+\phi^2(b+\nu_1)+\phi^3\}
+[(b+\nu_1)^2+\nu_1\phi]\alpha\phi\Sigma^*
-2(b+\nu_1)\alpha\gamma\phi\Sigma^* S_T,\nonumber\\
&&a_2=\beta\gamma\{b(b+\nu_1)+\nu_1(\phi+b+\nu_1)
+2(\phi+b+\nu_1)(b+\nu_1)+\phi\nu_1+\phi^2\}\nonumber\\
&&+2(b+\nu_1)\alpha\gamma\phi\Sigma^*
-\gamma^2\alpha\phi\Sigma^* S_T,\nonumber\\
&&a_3=\beta\{2\gamma (b+\nu_1)+\gamma^2(\phi+b+\nu_1)\}
+\gamma^2\alpha\phi\Sigma^*,\nonumber\\
&&a_4=\beta\gamma^3.\nonumber
\end{eqnarray}
The coefficient $a_0$ is negative while $a_3$ and $a_4$ are positive. Unless
$a_1>0$ and $a_2<0$ Descartes' rule of signs implies that the polynomial
only has one positive root. Otherwise the rule implies that it has one or
three positive roots (counting multiplicity) but does not decide between
these two cases.
It will now be shown that in the case $N=3$ there are values of the
coefficients for which the polynomial $p(S)$ has three positive roots. To do
this we vary the coefficients $S_T$ and $\nu_1$ in the system
(\ref{N31})-(\ref{N34}) and keep all others fixed. Note that these coefficients
come from the parameters in the agonist-only case of
(\ref{phen1})-(\ref{phen4}). To obtain the desired variation of the coefficients
we fix all parameters in (\ref{phen1})-(\ref{phen4}) except $S_T$, $\nu_1$ and
$\kappa$ and vary $\kappa$ in such a way that $\frac{\nu_1}{\kappa}$ does not
change. This ensures that $\Sigma^*$ does not change. In fact we may simplify
the calculations by setting $b=0$ since if three positive roots can be obtained
in that case the same thing can be obtained for $b$ small and positive by
continuity. Suppose that $S_T$ and $\nu_1$ depend on a parameter $\epsilon$
with both of them being positive for $\epsilon>0$. Suppose in addition that
in the limit $\epsilon\to 0$ we have the asymptotic relations
$S_T=\bar S_T\epsilon^{-1}+o(\epsilon^{-1})$ and
$\nu_1=\bar\nu_1\epsilon^4+o(\epsilon^4)$ for constants $\bar S_T$ and
$\bar\nu_1$. Then we obtain asymptotic expansions
$a_4=A_4$, $a_3=A_3+o(1)$, $a_1=A_1+o(1)$ for positive constants $A_4$,
$A_3$ and $A_1$, $a_0=A_0\epsilon^3+o(\epsilon^3)$ for a constant $A_0<0$ and
$a_2=A_2\epsilon^{-1}+o(\epsilon^{-1})$ for a constant $A_2<0$. Let
$q(S)=\epsilon p(S)$. Then $q(1)$ converges to $A_2$ for $\epsilon\to 0$
and is thus negative for $\epsilon$ small enough. The same is true for
$p(1)$. On the other hand
\begin{equation}
p(\epsilon^2)=A_0\epsilon^3+A_1\epsilon^2+A_2\epsilon^3
+A_3\epsilon^6+A_4\epsilon^8+o(\epsilon^2)=A_1\epsilon^2+o(\epsilon^2).
\end{equation}
Hence for $\epsilon$ sufficiently small $p(\epsilon^2)>0$. Putting these facts
together shows that $p$ has three positive roots when $\epsilon$ is small. For
each of these roots the values of $C_1$, $C_2$ and $C_3$ at the steady state
can be determined successively. $S<S_T$, $C_1+C_2+C_3<\Sigma^*$ and defining
$C_0=\Sigma^*-(C_1+C_2+C_3)$ gives a steady state of the original system.
It has already been noted that $p$ cannot have more than three positive roots.
There are parameter values for which the positive steady
state is unique. To see this it is enough to assume that $S_T$ is small
while keeping the other parameters fixed. Then $a_i>0$ for all $i>0$ and the
polynomial can have no more that one positive root since its derivative has
no positive root. These results can be summed up as follows:
\noindent
{\bf Theorem 1} The agonist-only case of the system (\ref{phen1})-(\ref{phen7})
has exactly one positive steady state for $N=1$ and $N=2$. In the case $N=3$
there are parameters for which it has three positive steady states and it can
never have more than three.
A concrete example of parameters for which there are three positive steady
states is obtained by setting $\alpha$, $\beta$, $\gamma$, $\phi$, $L_1$ and
$R$ equal to one and choosing $S_T=10$, $\kappa=2\times 10^{-4}$,
$\nu_1=10^{-4}$. A computer calculation shows that the coordinates
$(S^*,C_0^*,C_1^*,C_2^*,C_3^*)$ of the steady states are approximately
\begin{eqnarray}
&&(1.1769,0.1570,0.1334,0.1133,0.0963),\\
&&(0.0005,0.0001,0.0001,0.0003,0.4996),\\
&&(0.2860,0.0085,0.0294,0.1028,0.3593).
\end{eqnarray}
It shows in addition that while the first and second of these steady states
are asymptotically stable the third is a saddle with a one-dimensional
unstable manifold. A plot of the steady states as a function of the parameter
$L_1$, see Figure~\ref{fig:multistability}, suggests that there is a fold bifurcation.
\begin{figure}[ht]
\begin{center}
\mbox{%
\includegraphics[scale=0.5]{rendall_07april2017_c3L1_color_R1_crop.png}%
\ \ \
\includegraphics[scale=0.5]{rendall_07april2017_c3L1_log_color_R1_crop.png}
\end{center}
\caption{%
Multistability of steady states as a function of $L_1$. Shown is the coordinate
$C_3$, but other coordinates behave similarly.
Stable branches are shown in green colour and unstable in red.
Left: linear scale, right: log-log scale.
Parameters are
$\alpha = 1$,
$S_T = 10$,
$\beta = 1$,
$\kappa = 2\times10^{-4}$,
$R = 1$,
$b = 0$,
$\gamma = 1$,
$\phi = 1$,
$\nu_1 = 1$.
}
\label{fig:multistability}
\end{figure}
For higher values of $N$ it is possible to derive a polynomial equation of
degree $N+1$ for $S$. There is no obvious reason why this polynomial should
not have an arbitrarily large number of positive roots for $N$ arbitrarily
large. A simple upper bound is that the polynomial can have no more than
$N$ positive roots for $N$ odd and no more than $N+1$ for $N$ even.
In general it is difficult to obtain information about the stability of the
steady states by analytic methods. In the case $N=1$ the vector field
defining the dynamical system has negative divergence and so by Dulac's
criterion und Poincar\'e-Bendixson theory all solutions converge to the
steady state as $t\to\infty$. The system can exhibit damped
oscillations as will now be shown. To do this we choose
parameters so that
\begin{equation}\label{balance}
\alpha C_1+\beta=\phi+b+\gamma S+\nu_1.
\end{equation}
For fixed values of the quantities $R$ and $S_T$ the quantities $C_1$ and $S$
are bounded uniformly in the quantities appearing in (\ref{balance}). Thus
if we make $\alpha$ and $\beta$ small while fixing the other parameters we can
arrange that the left hand side is smaller than the right hand side. If
starting from there we make $\beta$ large while fixing the other parameters
we can arrange that the left hand side of (\ref{balance}) is
greater than the right hand side. It follows that parameter values exist for
which (\ref{balance}) holds. The reason why this is interesting is that the
discriminant of the characteristic equation of the linearization is the
sum of a term which vanishes when (\ref{balance}) holds and the expression
$-4\alpha\gamma (S_T-S)C_1$. Thus when (\ref{balance}) holds the linearization
has eigenvalues with negative real part and non-zero imaginary part and there
are damped oscillations.
An interesting limiting case of the agonist-only system is obtained by assuming
that $\alpha=0$ and $S=0$. We refer to this as the kinetic proofreading
system since it is closely related to McKeithan's kinetic proofreading model
\cite{mckeithan95}. In fact McKeithan only considered the case $b=0$ but
this makes no essential difference for the analysis which follows. It was
observed by Sontag \cite{sontag01} that the deficiency zero theorem of chemical
reaction network theory can be applied to McKeithan's system to conclude that
there is a unique steady state in each stoichiometric compatibility
class and that this solution is asymptotically stable in its class. Strictly
speaking chemical reaction network theory is applied to the extended system
which includes free receptors and free ligand as variables. To show that the
steady state is globally asymptotically stable it suffices to show that
there are no $\omega$-limit points on the boundary. That this is the case can
be proved just as we did for the full system above. The steady state is
hyperbolic as follows from Appendix C of \cite{feinberg95a}.
Consider now the full agonist-only system. Setting $\alpha=0$ gives a system
where the kinetic proofreading system is coupled to a system describing the
decay of $S$. The steady state of the kinetic proofreading system gives rise to
a steady state of the agonist-only system with $\alpha=0$ which is on
the boundary of the biologically feasible region and is a hyperbolic sink.
Denote its coordinates by $(0,C_j^*)$. For $\alpha$ small and positive there
exists a hyperbolic sink which is a small perturbation of that for $\alpha=0$.
It must be in the biologically feasible region since $C_1>0$ there and
equation (\ref{phen1}) would imply that $\dot S>0$ there if $S$ were negative.
Thus for sufficiently small values of $\alpha$ there exists a
positive steady state which is a hyperbolic sink
$(S^*(\alpha),C_j^*(\alpha))$ close to $(0,C_j^*)$. There exists a
positive number $r$ such that for $\alpha$ sufficiently small, say
$\alpha\le\alpha_0$, $(S^*(\alpha),C_j^*(\alpha))$ is the only $\omega$-limit
point of any solution in the open ball of radius $r$ about that steady
state.
Let $h(C_j)$ be the Lyapunov function in the proof of the deficiency
zero theorem. It is known from general arguments that $\dot h\le 0$ with
equality only for $C_j=C_j^*$. It follows that on the complement of the ball
of radius $r$ about the steady state the function $\dot h$ has a
strictly negative maximum. We can consider the behaviour of the function $h$
for solutions of the system for positive $\alpha$. For
small $\alpha$ it is still a Lyapunov function on the complement of a small
ball about the steady state while there are no $\omega$-limit points
except the steady state itself within that ball. Hence for $\alpha$
sufficiently small a solution can have no $\omega$-limit points other than
the steady state. It follows that for $\alpha$ small the steady state
is globally asymptotically stable. Of course this means that the
limiting system obtained from the agonist-only system by passing to a solution
through an $\omega$-limit point also has a unique steady state which is
globally asymptotically stable for $\alpha$ sufficiently small. A similar
argument applies in the case of the full system (\ref{phen1})-(\ref{phen7})
since in that case the system obtained by setting $\alpha$ and $S$ to zero
is just the product of two copies of the corresponding system in the
agonist-only case.
\section{The response function}\label{response}
This section is concerned with the agonist-only system. From a biological
point of view the essential input parameters to the system are the ligand
concentration $L_1$ and the binding time of the ligand to the receptor,
which in the model corresponds to $\nu_1^{-1}$. The latter is a measure of
the signal strength. The essential output is the value of $C_N$ which is
a measure of the activation of the T cell. Given values of $L_1$, $\nu_1$
and the other parameters we can consider the value of $C_N$ in a steady
state. In fact it is more convenient to use the quantities $\log C_N$ and
$\log L_1$. This leads to a response function $\log C^*_N=F(\log L_1,\nu_1)$.
If there is more than one steady state for a given choice of the
parameters this has to be thought of as a multi-valued function. It might
naively be thought that $F$ should be an increasing function of $L_1$ and a
decreasing function of $\nu_1$: more antigen leads to more activation of the T
cell and a longer binding time leads to more activation. This turns out not to
be the case and the function $F$ is not a monotone function of its arguments.
This was observed in the case of the dependence on $L_1$ in the simulations of
\cite{francois13}. It is possible to understand intuitively how this
situation can arise. An increase in the stimulation of the T cell leads to
activation of SHP-1 and that in turn has a negative effect on the activation
of the T cell. Many of the calculations in this section are guided by those in
\cite{francois13}.
The behaviour of the response function will be estimated in various parameter
ranges. In order to do this it is useful to parametrize the solutions in a
certain manner which will now be described. In the case of a steady state the
equation (\ref{phen3}) is a linear difference equation for the $C_j$ with
constant coefficients. This suggests looking for power-law solutions, an
idea which motivates the following result.
\noindent
{\bf Lemma 2} Steady state solutions of equations (\ref{phen2})-(\ref{phen4})
in the agonist-only case can be parametrized in the form
\begin{equation}\label{powerlaw}
C_j=a_+r_+^j+a_-r_-^j
\end{equation}
where the coefficients
$r_{\pm}$ and $a_{\pm}$ are positive and depend on $S$. The quantities $r_+$ and
$r_-$ are given by
\begin{equation}
r_{\pm}=\frac{\phi+b+\gamma S+\nu_1\pm\sqrt{(\phi+b+\gamma S+\nu_1)^2
-4\phi(b+\gamma S)}}{2(b+\gamma S)}\label{rpm}
\end{equation}
and satisfy $r_-<1<r_+$.
\noindent
{\bf Proof} Note first that the quantities $r_{\pm}$ in (\ref{rpm}) are
the roots of the characteristic equation
\begin{equation}\label{char}
\phi+(b+\gamma S)r^2-(\phi+b+\gamma S+\nu_1)r=0
\end{equation}
associated to the difference equation already mentioned and it is obvious
that they are positive. The fact that they satisfy the characteristic equation
is equivalent to the condition that the $C_j$ defined by (\ref{powerlaw})
satisfy the steady state form of equation (\ref{phen3}). That $r_-<1<r_+$ can
be seen by noting that the function on the left hand side of (\ref{char}) is
negative at $r=1$. The condition that the quantities $C_j$ satisfy the equations
(\ref{phen2})-(\ref{phen4}) with $\dot C_j=0$ is equivalent to the conditions
that they satisfy (\ref{powerlaw}) with $r_{\pm}$ as in (\ref{rpm}) and certain
coefficients $a_-$ and $a_+$ together with the equations obtained by
substituting (\ref{powerlaw}) into the equations $\dot C_0=0$ and
$\dot C_N=0$. The explicit form of these last equations is
\begin{eqnarray}
&&[(b+\gamma S)r_--(\phi+\nu_1)]a_-+[(b+\gamma S)r_+-(\phi+\nu_1)]a_+
=-\nu_1\sum_{j=0}^NC_j\label{c0eq}\\
&&r_-^{N-1}[\phi-(b+\gamma S+\nu_1)r_-]a_-+r_+^{N-1}[\phi-(b+\gamma S+\nu_1)
r_+]a_+=0.\label{c1eq}
\end{eqnarray}
It follows from the discussion in Section \ref{steady} that $\sum_{j=0}^NC_j$,
which was denoted there by $\Sigma_1^*$, is uniquely determined for fixed values
of the parameters in $(\ref{phen2})-(\ref{phen4})$ and fixed $S$. Thus for
fixed values of these parameters and $S$ all quantities in (\ref{c0eq}) and
(\ref{c1eq}) except $a_-$ and $a_+$ are known. It will now be shown that these
equations have a unique solution for $a_-$ and $a_+$. Note that
\begin{equation}\label{product}
[\phi-(b+\gamma S+\nu_1)r_-][\phi-(b+\gamma S+\nu_1)r_+]=
-\frac{\phi^2\nu_1}{b+\gamma S},
\end{equation}
as can most easily be seen by multiplying out the left hand side of this
equation and substituting for $r_+r_-$ and $r_++r_-$, which are the sum and
product of the roots of the characteristic equation (\ref{char}). Thus equation
(\ref{c1eq}) gives a positive expression for $a_+/a_-$. Note also that
(\ref{product}) implies that the factors in the product on the left hand side
of that equation have opposite signs. Since $r_-<r_+$ the first factor is
positive and the second negative. Substituting the expression for $a_+/a_-$
into (\ref{c0eq}) gives an equation of the form
\begin{equation}\label{ABeq}
a_-[A-B(r_-/r_+)^{N-1}]=-\nu_1\Sigma_1^*[\phi-(b+\gamma S+\nu_1)r_+]
\end{equation}
whose right hand side is positive. Here
\begin{eqnarray}
&&A=[(b+\gamma S)r_--(\phi+\nu_1)][\phi-(b+\gamma S+\nu_1)r_+]\\
&&B=[(b+\gamma S)r_+-(\phi+\nu_1)][\phi-(b+\gamma S+\nu_1)r_-]
\end{eqnarray}
It follows from the fact that the first factor on the left hand side of
(\ref{product}) is positive that the first factor in the expression for $A$ is
negative and hence that $A$ itself is positive. In addition, a straightforward
computation shows that $A>B$. If $B$ were
not positive then the quantity in square brackets on the left hand side of
(\ref{ABeq}) would be positive. If $B$ is positive then the fact that
$r_-<r_+$ implies that the quantity in square brackets is again positive.
Hence in any case (\ref{ABeq}) can be solved to give a unique positive value
of $a_-$. Then $a_+$ can be determined in such a way that (\ref{c0eq}) and
(\ref{c1eq}) hold. This completes the proof of Lemma 2.
Lemma 2 shows that for fixed parameters in (\ref{phen2})-(\ref{phen4}) and a
fixed value of $S$ the steady state values of all the $C_j$ are determined but
this does not yet give expressions for the $C_j$ which can be directly applied
to study the properties of the response function. For the purposes of what
follows it is convenient to rewrite (\ref{total1}) in the form
\begin{equation}\label{ctotaleq}
\kappa (L_1-\sum_{j=0}^NC_j)(R-\sum_{j=0}^N C_j)-\nu_1\sum_{j=0}^N C_j=0.
\end{equation}
The equation for $S$ can be solved to give the relation
$S=S_T\frac{C_1}{C_1+C_*}$ with $C_*=\frac{\beta}{\alpha}$. Summing the
expression for $C_j$ given in Lemma 2 over $j$ gives
\begin{equation}
\sum_{j=0}^N C_j=a_+\frac{r_+^{N+1}-1}{r_+-1}+a_-\frac{r_-^{N+1}-1}{r_--1}.
\end{equation}
The following equation relating $a_-$ and $a_+$ is equation (21) of
\cite{francois13}.
\begin{equation}\label{aplusminus}
a_+=-a_-\left(\frac{r_-}{r_+}\right)^{N+1}\frac{r_+-1}{r_--1}.
\end{equation}
Combining the last two equations gives
\begin{equation}\label{sumcj}
\sum_{j=0}^NC_j=\frac{a_-}{1-r_-}\left[1-\left(\frac{r_-}{r_+}\right)^{N+1}
\right].
\end{equation}
Having completed the necessary preliminaries we now proceed to study the
qualitative behaviour of the reponse function in different regimes. When $L_1$
is small it is to be expected that the concentration of the
phosphatase is small and that the response function resembles that of the
kinetic proofreading model. It will now be shown that when $L_1$ is small
the leading term in the function $F$ depends linearly on $\log L_1$ with slope
one and the additive constant in this linear function will be determined. The
equation (\ref{ctotaleq}) can be written in the form
\begin{equation}
\sum_{j=0}^NC_j=\frac{\kappa RL_1}{\kappa R+\nu_1}\left[1+
\frac{L_1}{R}\left((\sum_{j=0}^NC_j/L_1)^2
-(\sum_{j=0}^N C_j/L_1)\right)\right].
\end{equation}
Note that $\sum_{j=0}^N C_j\le L_1$ so that this equation implies that
\begin{equation}\label{sumleading}
\sum_{j=0}^NC_j=\frac{\kappa RL_1}{\kappa R+\nu_1}(1+qL_1/R).
\end{equation}
where $-\frac14<q<0$. Using (\ref{aplusminus}) it is possible to write down an
explicit expression for $C_N$, namely
\begin{equation}
C_N=\frac{a_-r_-^N(r_+-r_-)}{r_+(1-r_-)}.
\end{equation}
It follows from (\ref{sumcj}) that
\begin{equation}
C_N=r_-^N\frac{1-\frac{r_-}{r_+}}{1-(\frac{r_-}{r_+})^{N+1}}\sum_{j=0}^N C_j.
\end{equation}
Combining these equations gives
\begin{equation}
C_N=\left\{
r_-^N\frac{1-\frac{r_-}{r_+}}{1-(\frac{r_-}{r_+})^{N+1}}
\frac{\kappa R}{\kappa R+\nu_1}\right\}L_1(1+qL_1/R).
\end{equation}
The function of $r_-$ and $r_+$ in this equation defines a function of $S$.
This function of $S$ tends to a positive limiting value as $S\to 0$. Now
$C_1\le\sum_{j=0}^N C_i=O(L_1)$ and $S=O(C_1)$. Hence for $R$ fixed we can
replace the function of $r_+$ and $r_-$ in the above expression by its
limiting value for $S\to 0$. If the resulting relation is plotted
logarithmically it gives a straight line of slope one as the leading order
approximation in the limit $\log L_1\to -\infty$.
Next we look at an intermediate regime where the amount of activated SHP-1
is well away from both zero and $S_T$. As a first step, we obtain an estimate
for $r_-$ which is sharper than that in Lemma 2. To do this we compute the
left hand side of the characteristic equation (\ref{char}) for
$r=\frac{\phi}{\phi+\nu_1}$. The result is
$-\frac{\phi\nu_1(b+\gamma S)}{(\phi+\nu_1)^2}<0$. It follows that
$r_-<\frac{\phi}{\phi+\nu_1}$. Hence $1-r_->\frac{\nu_1}{\phi+\nu_1}$.
Substituting this into (\ref{sumcj}) gives
$a_->\frac{\nu_1}{\phi+\nu_1}\left(\sum_{j=0}^NC_j\right)$. Note that
$\frac{S}{S_T}\ge\min\left\{\frac{C_1}{2C_*},\frac12\right\}$. Hence a positive
lower bound for $C_1$ implies a positive lower bound for $\frac{S}{S_T}$.
Next we will derive a lower bound for $\gamma S$ in the case that $S_T$ is
large. This will be proved by contradiction. Suppose that $\gamma S\le\rho$
for some $\rho>0$. Then it follows from the characteristic equation that
$r_-\ge\frac{\phi}{\phi+\rho+\nu_1}$. Using this in the equation for $C_1$ gives
$C_1\ge\frac{\phi\nu_1}{(\phi+\nu_1)(\phi+\rho+\nu_1)}
\left(\sum_{j=0}^NC_j\right)$.
It follows that
\begin{equation}\label{smin}
S\ge S_T\min\left\{\frac{\phi\nu_1}{2C_*(\phi+\nu_1)(\phi+\rho+\nu_1)}
\left(\sum_{j=0}^NC_j\right),\frac12\right\}
\end{equation}
It is then clear that for a given value of $\rho$ and fixed values of the
parameters other than $S_T$ this leads to a contradiction if $S_T$ is
sufficiently large. In other words, given any $\rho>0$ there is a lower bound
for $S_T$ which implies that $\gamma S\ge\rho$. It is convenient to make the
restrictions that $\kappa R\ge 1$ and $L_1/R\le 1$ since then it is
possible to replace $\sum_{j=0}^NC_j$ in (\ref{smin}) by $\frac{3L_1}{4(1+\nu_1)}$
by using (\ref{sumleading}).
From (\ref{rpm}) it can be concluded that
\begin{eqnarray}
&&r_-=\frac{\phi}{b+\gamma S}(1+O(\eta)),\\
&&r_+=1+O(\eta).
\end{eqnarray}
where $\eta=\frac{\phi+\nu_1}{b+\gamma S}$.
This gives approximate expressions for the roots of the characteristic equation
if $\frac{\phi+\nu_1}{b+\gamma S}$ is small. As a consequence of these
equations
\begin{equation}
\frac{r_-}{r_+}=\frac{\phi}{b+\gamma S}(1+O(\eta)).
\end{equation}
Taking the expression for $C_1$ supplied by Lemma 2 and using
(\ref{aplusminus}), (\ref{sumcj}) and (\ref{sumleading}) gives
\begin{equation}
C_1=r_-\frac{\kappa RL_1}{\kappa R+\nu_1}(1+O(\eta)).
\end{equation}
This implies that $C_1=O(\eta)$ and the
expression relating $S$ and $C_1$ then shows that
$\frac{S}{S_T}=O(\eta)$.
In fact
\begin{equation}
C_1=\frac{C_*S}{S_T}(1+O(\eta))
\end{equation}
These relations indicate that in leading order $r_-$ is proportional to $S$.
However it is also the case that
\begin{equation}
r_-=\frac{1}{S}\frac{\phi}{\gamma}\frac{1}{1+b/(\gamma S)}(1+O(\eta))
\end{equation}
which indicates that in leading order $r_-$ is proportional to $S^{-1}$.
Hence
\begin{equation}
r_-=\frac{C_*(\kappa R+\nu_1)}{\kappa RL_1S_T}S(1+O(\eta))
\end{equation}
and
\begin{equation}
r_-=\frac{1}{S}\frac{\phi}{\gamma}(1+O(\eta')).
\end{equation}
where $\eta'=\max\{\eta,b/(\gamma S)\}$.
Combining these two relations gives
\begin{equation}
S=\sqrt{\frac{\phi\kappa RS_TL_1}{C_*\gamma (\kappa R+\nu_1)}}(1+O(\eta')).
\end{equation}
Substituting this back into the equation for $r_-$ gives
\begin{equation}
r_-=\sqrt{\frac{\phi C_*(\kappa R+\nu_1)}
{\gamma S_TL_1\kappa R}}(1+O(\eta')).
\end{equation}
This means that
\begin{eqnarray}
&&C_N=(\sum_{j=0}^N C_j)r_-^N(1+O(\eta''))\nonumber\\
&&=\left(\frac{\kappa R+\nu_1}{\kappa RL_1}\right)^{N/2-1}
\left(\frac{\phi C_*}{\gamma S_T}\right)^{N/2}(1+O(\eta''))\nonumber\\
&&=\left(\frac{\phi \beta}{\alpha\gamma S_T}\right)^{N/2}
\left(\frac{\kappa R+\nu_1}{\kappa R}\right)^{N/2-1}(L_1)^{1-N/2}(1+O(\eta''))
\end{eqnarray}
where $\eta''=\max\{\eta',L_1/R\}$. Choosing $L_1$ small enough makes
$L_1/R$ small. With $L_1$ fixed, making $S_T$ large enough makes $\eta$ small.
Thus $\eta''$ can be made as small as desired by choosing
$L_1$ sufficiently small and $S_T$ sufficiently large.
\noindent
{\bf Theorem 2} Consider the response function $\log C_N=F(\log L_1,\nu_1)$
for the steady states of the system (\ref{phen1})-(\ref{phen4}) with $L_2=0$
and $D_j=0$. Choose fixed values for all parameters in the system except
$L_1$ and $S_T$. Suppose that $\kappa R\ge 1$. Let $\epsilon>0$. Then there
exists a constant $\delta$ with $0<\delta\le R$ such
that the following holds. If $0<L_0<\delta$ there exists $\mu>0$ such that if
$S_T\ge\mu$ the inequality
\begin{equation}
\left|\left(\frac{\phi \beta}{\alpha\gamma S_T}\right)^{-N/2}
\left(\frac{\kappa R+\nu_1}{\kappa R}\right)^{1-N/2}(L_1)^{N/2-1}
F(\log L_1,\nu_1)-1\right|<\epsilon
\end{equation}
holds on the interval $[\log L_0,\delta]$.
\noindent
{\bf Proof} To obtain the conclusion of the theorem it suffices to show
that under the given assumptions $\eta''$ can be made as small as desired.
That this is possible follows from the discussion above.
Note that this theorem implies, in particular, that for $N>2$ and suitable
values of $L_1$ and $S_T$ there exists a range of $L_1$ in which the
response function is decreasing. The theorem also implies that in this
regime the reponse function can be an increasing function of $\nu_1$. This
effect was not captured by the calculations of \cite{francois13} since there
$\frac{\nu_1}{\kappa R}$ was assumed to be so small as to be negligible.
Finally we examine the regime where $L_1/R$ is small but the phosphatase is
close to being completely activated. This means that $S/S_T$ is
close to one. This holds provided $C_1$ is sufficiently large compared to
$C_*$. It remains to check that such a regime actually occurs for some
values of the parameters. It is possible to make $\sum_{j=0}^NC_j$ large
while keeping $L_1/R$ constant. This can be done by making $R$ large.
This makes $a_-$ large without making $r_-$ small. Hence it makes $C_1$ large
and hence $S$ close to $S_T$. In this regime the function of $r_+$ and $r_-$
occurring in the expression for $C_N$ can be replaced by its limit for
$S\to S_T$ and we again get a region where the slope of the graph of
$\log C_N$ as a function $\log L_1$ is one but the line has been shifted
compared to that obtained for $L_1/R$ small.
In \cite{francois13} these types of behaviour were exhibited numerically in
the case $N=5$ with biologically reasonable choices of the parameters. We
found that changing these parameters a little allows similar observations to
be made in the case $N=3$. In the plot shown in Figure~\ref{fig:regimes}
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.5]{rendall_C3_L1_loglog_07aprilrev4_file_but_blue_crop.png}
\end{center}
\caption{%
Log-log plot showing linearity of $\log C_3$ as a function of $\log L_1$ for
small $L_1$, followed by decreasing, increasing, and saturation regimes.
Parameters are
$\alpha = 1$,
$S_T = 6\times10^5$,
$\beta = 5\times10^2$,
$\kappa = 10^{-4}$,
$R = 3\times10^4$,
$b = 4\times10^{-2}$,
$\gamma = 1.2\times10^{-6}$,
$\phi = 9\times10^{-2}$,
$\nu_1 = 10^{-2}$.
}
\label{fig:regimes}
\end{figure}
the three regimes can be seen
together with a fourth regime where $L_1/R$ is no longer small. It is clear
that a regime of this type must exist since the response function is globally
bounded.
We now turn to the dependence of the response function on $\nu_1$.
It has been suggested in \cite{lever14} that the kinetic proofreading model
with negative feedback as studied here is not able to explain the presence of
an optimal dissociation time, a biological effect confirmed by the
experimental work of \cite{lever16}. The plots of the response as a function
of the dissociation time in that type of model in \cite{lever14} show that it
is increasing. Having an optimal dissociation time would require that there be
a region where this function is decreasing. The response function being
increasing as a function of the dissociation time corresponds to its being
decreasing a function of $\nu_1$. Here we have given an analytical proof in
Theorem 2 that there exist parameters for which the response is an increasing
function of $\nu_1$, in contrast to the plots in \cite{lever14}. Since the
theorem is of limited help in finding explicit parameters for which
this happens we also did a numerical search and identified parameters
of this type. The results are displayed in Figure~\ref{fig:c3nu1}, where it is seen that
$F$ has a maximum as a function of $\nu_1$ for fixed $L_1$, which
corresponds to an optimal dissociation time.
\begin{figure}[ht]
\begin{center}
\mbox{%
\includegraphics[scale=0.5]{rendall_c3nu1_a10_st107_be_103_ka_104_R_105_b_1_ga_1eminus2_phi1_L1_103_mult100_crop.png}%
\ \ \
\includegraphics[scale=0.5]{rendall_c3nu1_a10_st107_be_103_ka_104_R_105_b_1_ga_1eminus2_phi1_L1_103_mult100_loglog_crop.png}
\end{center}
\caption{%
$C_3$ as a function of $\nu_1$ in model with $N=3$, showing non-monotonic
behaviour for some values of parameters.
Left: linear scale, right: log-log scale.
Parameters are
$\alpha = 10^{-1}$,
$S_T = 10^7$,
$\beta = 10$,
$\kappa = 10^{-6}$,
$R = 10^5$,
$b = 10^{-2}$,
$\gamma = 10^{-4}$,
$\phi = 10^{-2}$,
$L_1 = 10^3$.
}
\label{fig:c3nu1}
\end{figure}
The conclusion of both the
analytical and the numerical work is as follows. The claim that the kinetic
proofreading model with feedback can only produce a response which is a
decreasing function of the parameter $\nu_1$ is dependent on the parameter
values chosen to do the simulations and not a general property of the model.
\section{Including the antagonist}\label{antagonist}
When the antagonist is included the output variable expressing the degree of
activation of the T cell is $C_N+D_N$. Now asymptotic expressions for this
quantity will be derived. It has already been shown that for a steady state of
the system (\ref{phen1})-(\ref{phen7}) the quantities $\sum_{j=0}^NC_j$ and
$\sum_{j=0}^ND_j$ can be expressed in terms of the parameters. The equation for
$S$ can be solved to give the relation $S=S_T\frac{C_1+D_1}{C_1+D_1+C_*}$. $C_j$
solves the same difference equation as in the agonist-only case and $D_j$
solves the difference equation obtained from that one by replacing $\nu_1$ by
$\nu_2$. The quantities $r_-$, $r_+$, $a_-$ and $a_+$ differ in the two cases. We
can nevertheless proceed as in the former case to see that the solutions
for $C_j$ and $D_j$ allow parametrizations in terms of these quantities as
before. Note that using the equations (\ref{total1}) and (\ref{total2})
it is possible to eliminate the $D_j$ from the equation for $C_0$ and the
$C_j$ from the equation for $D_0$. Thus we have coupled equations for the
$C_j$ and $D_j$ which can be analysed just as in the agonist-only case to
express $C_1$ and $D_1$ as functions of $S$ and the parameters. We can also
write $C_N$ and $D_N$ as functions of $\Sigma_1$ and $\Sigma_2$ respectively.
Proceeding as in the agonist-only case we get an expression for $C_N+D_N$ in
the kinetic proofreading regime. The multiple of $L_1$ obtained there as
leading term is replaced by a linear combination of $L_1$ and $L_2$.
Next the intermediate regime will be considered. For this it is necessary to
define a new parameter $\eta=\max\{\frac{\phi+\nu_1}{b+\gamma S}\}$. There are
asymptotic expressions for $r_-$ and $r_+$ where the leading terms are just as
in the agonist-only case. In particular they are the same for $C_j$ and $D_j$.
Two asymptotic expressions for the quantity $C_1+D_1$ can be obtained.
\begin{eqnarray}
&&C_1+D_1=\frac{C_*S}{S_T}(1+O(\eta)),\\
&&=r_-\left(\frac{\kappa RL_1}{\kappa R+\nu_1}
+\frac{\kappa RL_2}{\kappa R+\nu_2}\right)(1+O(\eta)).
\end{eqnarray}
This gives an expression for $r_-$ in terms of $S$. As in the agonist-only
case this gives an expression for $r_-$ where the dependence on $S$ has been
eliminated in leading order.
\begin{equation}
r_-=\sqrt{\frac{\phi C_*}{\gamma S_T}
\left(\frac{\kappa RL_1}{\kappa R+\nu_1}
+\frac{\kappa RL_2}{\kappa R+\nu_2}\right)^{-1}}(1+O(\eta'))
\end{equation}
where $\eta'$ is defined in terms of $\eta$ as in the agonist-only case.
Following the steps used in the agonist-only case leads to an expression
for $C_N+D_N$ which is the same as that previously obtained for $C_N$ except
that the expression $\frac{\kappa RL_1}{\kappa R+\nu_1}$ is replaced by
$\frac{\kappa RL_1}{\kappa R+\nu_1}+\frac{\kappa RL_2}{\kappa R+\nu_2}$.
This leads in the end to an asymptotic expression for $C_N+D_N$ under a
suitable assumption $L_1$ and $L_2$. The assumption made in the
agonist-only case can naturally be written as an assumption on
$\frac{\kappa RL_1}{\kappa R+\nu_1}$ and in the present case it is replaced
by an assumption on
$\frac{\kappa RL_1}{\kappa R+\nu_1}+\frac{\kappa RL_2}{\kappa R+\nu_2}$.
This implies that under certain circumstances $C_N+D_N$ increases when
$L_2$ increases and $L_1$ is held fixed. An increase in the amount of
self-antigen can lead to a decrease in the response to a foreign antigen.
\section{Conclusions and outlook}
In this paper some properties of the solutions of the model of \cite{francois13}
for T cell activation were proved. A new discovery was that already in the
case of three phosphorylation sites ($N=3$) there can exist more than one
positive steady state for given values of the parameters. Another new
observation is that damped oscillations can occur. It was also proved
that, as suggested by the calculations in \cite{francois13}, the output
variable $C_N$ (concentration of the maximally phosphorylated receptor) is
a decreasing function of the concentration $L_1$ of antigen in some
parts of parameter space. In an analogous way it was proved that under
some circumstances the activation in response to an agonist can be decreased
by increasing the concentration of the antagonist. It was proved that it can
also happen that $C_N$ is an increasing function of the dissociation
constant $\nu_1$. This abstract result was given a concrete illustration
by a plot showing that $C_N$ can have a local maximum as a function of $\nu_1$.
The stability of the steady
states was only determined analytically in the very special cases $N=1$ and
$\alpha$ close to zero. For $N=3$ numerical calculations showed the
occurrence of two stable steady states for certain values of the parameters.
It was proved that damped oscillations occur but can there
also be sustained oscillations (periodic solutions)? It is thus clear
that there remain several aspects of the dynamics of this system which would
profit from further investigations, analytical and numerical.
In immunology it is important to describe diverse situations including the
course of different types of infectious disease, the development of autoimmune
diseases and the destruction of tumour cells by the immune system. It would be
unreasonable to expect that a simple mechanism could be the key to describing
all these situations. One strategy to try to obtain more understanding is to
choose one mechanism and to investigate which types of situations it suffices
to describe. This may be done by combining mathematical models with experimental
data. What are the restrictions under which the type of model studied in this
paper might be appropriate? The first assumption is that in the situation to
be explained the distinction between self and non-self takes place within an
individual T cell. In other words it is assumed that it is not necessary to
consider the population dynamics of the T cells involved or even the interaction
of their population with that of other types of immune cells such as
regulatory T cells or dendritic cells. A quite different type of mathematical
model, where population effects are considered, can be found in
\cite{sontag17}.
In that case, in contrast to the lifetime dogma, the response depends on the
rate of change of the antigen concentration. The second assumption which is
important for the models studied here is that the distinction between self and
non-self takes place on a sufficiently short time scale, say three minutes. On
longer time scales there may be essential effects related to the spatial
distribution of molecules on the T cell surface (formation of the
immunological synapse) so that a description by means of ordinary differential
equations may be insufficient. It may also happen that some T cell receptors
become
inactive on a longer time scale (limiting signalling model, cf. \cite{lever16}).
In this paper we have concentrated on studying the mathematical properties
of a particular model for the biological phenomenon of T cell activation
with arbitrary values of the parameters. A complementary question is to
what extent known experimental data on the parameters may further
constrain the dynamics in this model. In addition it is important to know
whether this model is consistent with all biological data and how it
compares to other possible models for the same biological system. For a
discussion of this we refer to \cite{lever14}, \cite{francois15} and
\cite{lever16}. It was indicated in \cite{lever16} that the situation
where $C_N$ is a decreasing function of $\nu_1$ cannot be reproduced using
the model of \cite{francois13}. Our results indicate that a failure of the
model to reproduce this effect must depend not only on the model itself
but on the choice of parameters used for simulations. At the same time it
may be that this effect only occurs in experiments where the measurements
are done on long time scales (many hours) and not on the time scale of
the initial activation (a few minutes) for which the models of
\cite{altanbonnet05} and \cite{francois13} were primarily intended.
We plan to investigate these questions further elsewhere.
|
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Q: need example shows difference between directed set and full directed set we know that A Set $S$ together with a relation $\ge$ which is both $transitive$ and $reflexive$ such that for any two elements a,b in $S$, there exists another element c in $S$ with c $\ge$ a and c $\ge$ b In this case, the relation $\ge$ is said to "direct" the set.
But can we call A Set $S$ is A FULL directed Set : If Every subset From $S$ is a directed Set?
I need an example of a directed set but it is not A Full directed one.
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Three companies have won places on a new framework contract to supply IT software to GPs across NHS Scotland.
EMIS Health, Microtest and Vision (InPractice Systems) can now develop accredited systems, the functions of which should include increased access for patients to online services.
NHS National Services Scotland (NSS) said the new systems will be more intuitive and user-friendly and will include improved functionality specified in consultation with GPs. Expected benefits include improved online services for patients, better monitoring and reporting, and remote and mobile access for primary care teams.
The contract award follows a procurement process, managed by NSS and overseen by the national GP IT Re-Provisioning Project Board.
Scotland's Cabinet Secretary Jeane Freeman (pictured) said: "I welcome the fact that the new systems will allow people greater access to online services.
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{"url":"https:\/\/www.techwhiff.com\/issue\/what-are-the-ways-to-develop-quality-of-life--577381","text":"# What are the ways to develop quality of life\u200b\n\n###### Question:\n\nwhat are the ways to develop quality of life\u200b\n\n### What does an acid+base make\n\nWhat does an acid+base make...\n\n### Shannon orders two tacos that cost $3.75 each and a juice that costs$2.55. She gives the cashier a twenty-dollar bill. How much change should she receive?\n\nShannon orders two tacos that cost $3.75 each and a juice that costs$2.55. She gives the cashier a twenty-dollar bill. How much change should she receive?...\n\n### Correct answers only! To the nearest cent, how much will he have in 2 years? Use the formula B = p(1 + r)t, where B is the balance (final amount), p is the principal (starting amount), r is the interest rate expressed as a decimal, and t is the time in years.\n\nCorrect answers only! To the nearest cent, how much will he have in 2 years? Use the formula B = p(1 + r)t, where B is the balance (final amount), p is the principal (starting amount), r is the interest rate expressed as a decimal, and t is the time in years....\n\n### 18 36 72 96 What number goes between 36 and 72 to finish the sequence ?\n\n18 36 72 96 What number goes between 36 and 72 to finish the sequence ?...\n\n### Which sentence correctly uses parallel structure? A. The doctor recommends resting for a week and that you eat low-fat food at regular intervals. B. I want to find a job that will provide an adequate context for my skill, and to earn enough money for my family. C. Apart from dabbling in playwriting, he has tried creating a great epic novel and composing a long, tragic epic. D.After Ruth receives her monthly paycheck, the money goes into her savings account, equity funds, and toward her credit ca\n\nWhich sentence correctly uses parallel structure? A. The doctor recommends resting for a week and that you eat low-fat food at regular intervals. 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I need to review text details to see how the author makes complex characters believable.B. I need to focus on characters in the text and whether or not they are believable in a historical...\n\n### Which expression is equivalent to 64 2\/3 PLS HELP\n\nWhich expression is equivalent to 64 2\/3 PLS HELP...\n\n### 1. Characteristics of competitive markets The model of competitive markets relies on these three core assumptions: 1. There must be many buyers and sellers\u2014a few players can't dominate the market. 2. Firms must produce an identical product\u2014buyers must regard all sellers' products as equivalent. 3. Firms and resources must be fully mobile, allowing free entry into and exit from the industry.\n\n1. Characteristics of competitive markets The model of competitive markets relies on these three core assumptions: 1. There must be many buyers and sellers\u2014a few players can't dominate the market. 2. Firms must produce an identical product\u2014buyers must regard all sellers' products as equivalent. ...\n\n### Simplify this and show your work :\n\nSimplify this and show your work :...\n\n### How would thomas hobbes define history?\n\nHow would thomas hobbes define history?...\n\n### Match the literary term with its correct definition. an author's word choice 1) allegory 2) aphorism 3) diction 4) in medias res 5) metaphor\n\nMatch the literary term with its correct definition. an author's word choice 1) allegory 2) aphorism 3) diction 4) in medias res 5) metaphor...\n\n### Which statement describes the graph of function g? f(x) = 2x g(x) = 2x + 3 A. The graph of g is 3 units above the graph of f. B. The graph of gis 3 units to the right of the graph of f. C. The graph of g is 3 units below the graph of f. D. The graph of g is 3 units to the left of the graph of f.\n\nWhich statement describes the graph of function g? f(x) = 2x g(x) = 2x + 3 A. The graph of g is 3 units above the graph of f. B. The graph of gis 3 units to the right of the graph of f. C. The graph of g is 3 units below the graph of f. D. The graph of g is 3 units to the left of the graph of f....\n\n### While observing slides under a microscope, Jack adjusts the microscope's mirror so that a circle of light can be seen. Which microscope is in use? A-Compound microscope B-Electron microscope C-Scanning electron microscope\n\nWhile observing slides under a microscope, Jack adjusts the microscope's mirror so that a circle of light can be seen. Which microscope is in use? A-Compound microscope B-Electron microscope C-Scanning electron microscope...\n\n### You are extremely tired but need to drive somewhere soon. You decide to eat a large meal in hopes that it will wake you up. Is this a good idea?\n\nYou are extremely tired but need to drive somewhere soon. You decide to eat a large meal in hopes that it will wake you up. 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Bradford (Park Avenue) AFC är en engelsk fotbollsklubb i Bradford, grundad 1907. Hemmamatcherna spelas på Horsfall Stadium. Smeknamnet är The Avenue eller BPA. Klubben spelar i National League North.
Namnet Park Avenue kommer av klubbens hemmaarena fram till 1973 som hette så och tillkom för att särskilja klubben från Bradford City.
Historia
Klubben kan spåra sitt ursprung tillbaka till 1800-talet, men då var det en rugbyklubb som även hade ett fotbollslag. 1907 bestämde man sig för att sluta med rugby och enbart ägna sig åt fotboll och samma år ansökte man om inträde i The Football League. När ansökan inte beviljades gick man i stället med i Southern Football Leagues högsta division, Division One. Trots en blygsam 13:e plats ansökte man på nytt om inträde i The Football League, och denna gång beviljades ansökan på bekostnad av Lincoln City.
1908 började BPA därför att spela i Second Division, Englands näst högsta division. Säsongen 1913/14 kom man tvåa och blev uppflyttad till First Division. Första säsongen där blev man så bra som nia, vilket skulle visa sig vara klubbens högsta ligaplacering. Därefter kom första världskriget och ligaspelet återupptogs först säsongen 1919/20. Då blev klubben elva, men säsongen efter kom man sist och åkte ned till Second Division. Väl där kom man näst sist och åkte ned till Third Division North.
Efter två andraplatser 1922/23 och 1925/26 vann klubben Third Division North 1927/28 och gick upp till Second Division. Där spelade man sedan fram till andra världskrigets utbrott med som bäst en tredjeplats 1928/29. Efter kriget fortsatte man i Second Division, men det blev inga framskjutna placeringar och 1949/50 blev klubbens sista säsong på denna nivå. Då kom man sist och åkte ned till Third Division North. Säsongen 1955/56 kom man näst sist där, men blev återinvald i The Football League.
När Fourth Division bildades 1958 placerades BPA där efter att ha blivit tredje sist i Third Division North föregående säsong. Under den tredje säsongen i Fourth Division kom klubben fyra och blev uppflyttad till Third Division, men där hängde man bara kvar i två säsonger. De följande tre säsongerna gjorde man hyggligt ifrån sig i Fourth Division, men sedan följde fyra säsonger då man först blev näst sist och därefter sist tre säsonger i rad. Efter den sista av dessa säsonger, 1969/70, blev klubben utröstad ur The Football League efter att ha spelat i ligan sedan 1908.
BPA gick 1970/71 med i Northern Premier League i stället och spelade där i fyra säsonger innan klubben lades ned efter 1973/74 års säsong på grund av stora skulder. Därefter spelade en klubb med samma namn så kallad Sunday league football på låg amatörnivå.
1988 ombildades klubben och började spela i West Riding County Amateur Football League och säsongen efter i Central Midlands Football League och dess Supreme Division. Ytterligare en säsong senare bytte BPA till North West Counties Football League Division Two och efter en tredjeplats där gick man upp till Division One. Den divisionen vann man sedan säsongen 1994/95 och man gick då upp till Northern Premier League Division One. Klättringen i ligasystemet fortsatte säsongen 2000/01 med seger i Division One och uppflyttning till Premier Division.
Efter 2003/04 års säsong bildades nya divisioner på nivån ovanför Northern Premier League – Conference North och Conference South. BPA fick därför trots en 17:e plats i Premier Division chansen att kvala sig upp. Detta lyckades man med; i kvalfinalen besegrades Burscough med 2–0. Man kom dock sist i Conference North nästföljande säsong och fick återvända till Northern Premier League Premier Division, där man genast kom näst sist och åkte ned till Division One. Efter ett misslyckat kval till Premier Division första säsongen 2006/07 hamnade man 2007/08 i Division One North när ligan omstrukturerades. Denna division vann klubben på första försöket och var därmed tillbaka i Premier Division. Klubben misslyckades med att kvala sig upp 2009/10 och 2010/11, men 2011/12 gick det vägen efter vinst i kvalfinalen över United of Manchester med 1–0 efter förlängning.
Conference North bytte namn till National League North inför 2015/16 års säsong. 2016 bytte klubben ägarstruktur genom att klubbens medlemmar fick överta ägandet från den tidigare huvudägaren Gareth Roberts. Klubben bytte då också namn till Bradford Park Avenue CFC (Community Football Club). Efter bara två år bytte man tillbaka till det gamla namnet Bradford (Park Avenue) AFC och Roberts tog över som ägare igen.
Meriter
Liga
Premier League eller motsvarande (nivå 1): Nia 1914/15 (högsta ligaplacering)
League One eller motsvarande (nivå 3): Mästare 1927/28 (North)
Northern Premier League Division One: Mästare 2000/01, 2007/08 (North)
North West Counties Football League: Mästare 1994/95
Cup
FA-cupen: Kvartsfinal 1912/13, 1919/20, 1945/46
Ligacupen: Tredje omgången 1962/63, 1963/64
Northern Premier League President's Cup: Mästare 2005/06
Referenser
Noter
Webbkällor
Externa länkar
Bradford (Park Avenue) AFC på FCHD
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Members of 3PB's commercial group include specialists with substantial experience in advising and representing clients in disputes relating to intellectual property and related rights. We undertake work in relation to trade marks, passing off, copyright, database rights, design rights, domain names, confidential information and trade secrets.
The team frequently represents clients in the High Court, the Intellectual Property and Enterprise Court, and the United Kingdom and EU Intellectual Property Offices (UKIPO and EUIPO). We are able to respond quickly to obtain urgent Court relief, where necessary. We can advise you on the suitability of urgent relief, and guide you through the tactics and procedure of obtaining it.
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Begin Reading
Table of Contents
About the Author
Copyright Page
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For LAW
Love is a flame; we have beaconed the world's night.
—Rupert Brooke
FROM THE DESK OF SOPHIE CASS
New York
August 28, 2000
Dear Mr. Lambert,
Here's the thing. You can ignore my letters if you wish, but my piece will run in The New York Times whether you, or your lawyers, like it or not. If it weren't for our family connection, I'd sue you in a heartbeat. I don't like threats. I am a professional investigative journalist, Mr. Lambert, and I'm just doing my job. This is not some personal vendetta as you claim, nor have I used underhand tactics. As an artist, I'm sure you appreciate the key is to look, to listen clearly, and see the things that other people miss. I looked, and I saw the truth—plain and simple.
My sources are rock solid, and more importantly I have new evidence for my claims. You see, I have discovered photographs, Mr. Lambert, taken by Alistair Quimby in 1940, the summer before you met Varian Fry in Marseille. Not only are there stunning photos of my great-aunt Vita, there are photos of you at work in the studio at the Château d'Oc. Both of you. I assume I don't need to explain any more?
Your lawyers have done their damnedest to warn me off, but now I am warning you—this piece celebrating your illustrious career and 95th birthday will run. I'm going to give you one last chance to put your side of the story before I file my copy. I will come and see you after Labor Day, at midday on Thursday, September 7. I imagine now you'll be ready to talk.
Yours,
Sophie Cass
ONE
FLYING POINT, LONG ISLAND
September 2000
GABRIEL
There are few things emptier than the space where a Christmas tree used to be. At least, that's what Annie always says. If my lovely girl had her way, we'd have a tree all year long and the white lights strung out on the deck. When I was younger, I'd swim out at sunset and see Venus and the stars above and that rope of lights guiding me home, luminous as pearls on the dark throat of the bay. When we walked the dogs along the beach at night, she'd take my hand or loop her arm around my waist. Gabe, she'd say, look, it's like our own galaxy on the shore.
Plenty are surprised by this home of ours. They expect the great man to have something grander, but they don't see I have everything I ever dreamed of here. I have the clear light of the sea spilling in through the square open doors to the barn where I work, and I have Annie. One summer I paid a bunch of boys to come up from Pennsylvania to raise the barn for me—it's an old red colonial, with a hipped roof and doors that open wide to the north. I need space to work, to spread my canvases, but Annie never wanted a studio. When I offered to build her one, she said, "What do I want with that?" She just set up a little table off the kitchen, facing the wall. She says she likes to be in the middle of things, so the kids can come and go as she sews her beautiful clothes. Her work's in collections across the country, but you wouldn't know it to look at her with her head bent over her embroidery. There's always a dog sitting at her feet and a cat curled up in the warmth of the lamplight beside her as she sews. First it was our own kids, and now there's always one grandchild or another in the high chair beside her or playing with their bricks or dolls on the floor and the radio singing away in the background. Maybe that's why people love her art—each stitch pulses and shines with life. My paintings scare people, intimidate them. They are big and intense, just right for uncompromising white-box galleries and soaring corporate atriums. I know which I'd rather live with. Over time people confused me with my paintings. Here's something I've learned—if you do good work, make it to my age, and keep your trap shut, you become a reclusive genius. I like my reputation, play up to it: the enigmatic old man of abstraction. People are afraid to ask too many questions, which suits me just fine. Only the bravest bother me out here. I glance over at the girl's letter on the table. Only the boldest find me.
My cup of coffee's grown cold as I rock the child in my arms and look out to sea. I'm listening to Billie Holiday singing "It's a Sin to Tell a Lie" on my 1959 Magnavox Imperial, and Lady Day's voice is dancing up among the rafters with the angels, the Calder mobile swinging with the tune as it rises into the air like bubbles in pure water. This song is Annie's favorite, and she always has a mischievous glint in her eye, singing along softly as we dance, our bare feet shuffling on the sandy deck overlooking the sea. My wife knows me well. I've lied about plenty in my life, but I never lied about loving her.
I love this time of year, too, always have. There's nothing like an empty beach for me, with the dazzling white sand, the sea turning to winter swells, and the pure blue sky above you. I can breathe here. Every day I wake feeling like the first man on earth, with his Eve at his side. After Labor Day, once all the fancy cottages along the dunes close down for the season and their topiary is tied up in its dust sheets for the winter, Annie and I kick off our shoes, get our toes in the sand, and take a walk down to our beach. We build a bonfire, grill a fish or two, and toast another summer's passing with a glass of Chardonnay. Annie doesn't drink much, but those nights she curls up under an old plaid blanket, and her cheeks flush, and she talks freely about the past, our life, and the future like the girl I met used to when we walked in the woods at Air-Bel like children in a fairy tale.
I am a contented man. This is all I wish for my children. I gave them each enough to help them start out in life, but not so much that they didn't have to work at it. Too much ease can ruin you; that's what I've always thought when I look at some of the kids of wealthy folks I've known. You lose your edge. I didn't want the kids to struggle as Annie and I did, but the rest I quietly gave away. If some fool banker wants to pay my dealers a million bucks for one of my early abstracts, then let him. Hell, I am the Robin Hood of the art world. There are plenty more folks in the world need the money more than we do, and I have so much to be thankful for in this life that I've had, this good and simple life I don't deserve.
TWO
WILLIAMSBURG, BROOKLYN
Wednesday, September 6, 2000
SOPHIE
A flurry of white wings wakes her, whirling, silhouetted against the bright morning light spilling through the curtainless loft window. Sophie leaps from deep slumber, sits bolt upright on the living room sofa, shielding her face with her arm from the light, from the bird. She squints her eyes, makes out the dove's frantic search for the narrow margin of space it slipped in through, helpless wings battering the high panes of glass.
"How did you get in here?" Sophie throws wide the window. The noise of the waking city spills into the studio on the warm breeze: honking traffic below, the whirr of air conditioners on the roof above, a tinny radio somewhere, playing "Said I Loved You... But I Lied."
Jess's favorite, she thinks instantly. A Pavlovian cocktail of memories swirls in her mind. She remembers teasing him about his taste in music when they met, how his unlikely love of power ballads became a running joke: Seriously? Beneath that Brooks Brothers suit you're all big hair and stonewashed denim? Sophie thinks of the night they slow danced to the song on the deck at some party in East Hampton—his request. Everyone had laughed and groaned as the DJ played the tune, but that gave the moment, the way Jess had walked toward her and held out his hand, a perfect lightness. She remembers his certainty, his focus only on her, the sound of the surf, and the sweet taste of strawberries still on her lips as he kissed her. She flexes her hand at the memory of the ring sliding onto her finger, the glint of the stone in the moonlight. Sophie gathers up the white bedsheet. Of all the songs. Our song... She corrects herself. His song. The tune carries up, up into the morning sky, from an open doorway on Grand Street, taking her thoughts with it.
Sophie speaks softly to the bird, calming its frantic search for freedom. "There you go," she says, dropping the sheet over it the moment it lands in the corner of the studio. Gently, she enfolds the bird in her hands, feels the staccato beat of its heart against her fingers, the fine, cathedral arch of its breastbones.
At the window, she releases it, watches it soar up across Brooklyn into the hazy morning sky. The air is hot, edible, laced with the scent of the streets—gasoline, coffee, ripe melon skins in the Dumpsters behind the grocery store. The dove joins its dull-plumed friends roosting on the pediment of the building opposite, a pale punctuation mark among the Morse code line of birds cooing and shaking the night from their feathers.
Sophie sits on the windowsill in the sunshine, the brick already warm against her aching back through the thin cotton of her white camisole. She closes her eyes, raises her face to the morning sun, and cricks her neck from side to side, loosening the tense ligaments. Her gold-blond hair spills across her shoulders, a glossy halo she scoops up and secures in a loose bun with practiced ease. Beside the window, the sagging red velvet sofa and single pillow still bear the imprint of her restless sleep. The tune ends, and as the radio station jingle releases her from thoughts of Jess, the letter runs through her mind again, as it has on a permanent loop since she woke at four A.M. Was I too hard? she thinks. What if Lambert won't see me? She has read the letter so many times that she knows it by heart. "Professional investigative journalist." She cringes inwardly. Wonder what effect "newly graduated arts writer with zero professional experience" would have had on his hotshot lawyers?
At the sound of paws trotting across the bare concrete floor, Sophie turns and smiles. "Hey, Mutt." The dachshund yawns and stretches, front legs extended, tail wagging high. "C'mon," she says, swinging down from the window. She freshens up, picks out a clean white shirt, and shrugs it on. Sophie reaches for the lead spooled on her suitcase. She rubs her thumb across a curling airline sticker on the case, thinking of the last trip she took with Jess to Mexico. A knot forms in her throat as she remembers it all—the color, the light, the heat. Coming home, she thinks. She remembers dozing on Jess's shoulder in the taxi as they drove through the snowy streets of New York to their cozy apartment in Greenwich Village and wonders if she'll ever be that happy again. It's all gone. I made my choice. She glances across at the dog, who sits waiting for her, his head tilted. "I know, I know," she says, peeling the sticker off and screwing it up. She flicks the ball of paper into a wire basket beside Alisha's drawing board and stops to look at the wedding dress designs her friend is working on. Sophie reaches out and touches the cool bolt of duchess satin. "Could have been me," she says under her breath, and she pads across the studio, sunlight warming the open space in wide parallelograms now. "It's just you and me, now," she says as Mutt follows her, tail wagging in hope. She clicks on the coffee machine, takes out a fresh pack of Zabar's coffee from the mercy parcel of provisions her mother insisted she take, and rips open the seal, inhaling the rich, smoky scent as she tips it into the filter. She glances at the bag of groceries, spots her mother's familiar, looping hand on a note tucked beneath some bagels. Sophie smooths it out: Hang in there. I love you, Mom x. It is pinned to a copy of a Henry James essay, and Sophie sees she has underlined a section.
At the door, she pauses to slip her tanned feet into a pair of white Converse sneakers and loosely buttons the shirt, rolling the sleeves and knotting it at the waist of her wide-legged chinos. Mutt's paws skitter impatiently on the floor, and he nudges her leg with his head.
"Just a second," she says. Sophie tucks a pair of Ray-Bans into her hair and checks her reflection, rubbing away a smudge of mascara from beneath her sea-green eyes. She wipes a trace of toothpaste from the crease in her bottom lip with her thumb. You can do this, she tells herself, her mind racing ahead to her meeting with Lambert, her stomach taut with excitement and nerves. She has imagined what it will be like to meet him, finally, a thousand times. "Be generous, be delicate, and always pursue the prize," she says under her breath, quoting from the James essay. She tucks the papers into her battered leather satchel and grabs her keys and a few dollar bills, then slides back the dead bolts on the heavy metal door. "Let's go."
On the street, her tension eases as they walk around the block to Bedford Avenue, the pulse of a bass line drifting from a pimped-up Chevy on a side street beating in time with her heart as she pauses for Mutt to christen his favorite lamppost near the Kam Sing Restaurant. The metal cellar door is propped open, and the scent of last night's cooking oil and spices drifts up to her. Sophie reaches into her pocket for her phone and dials her mother's number, stepping aside for a group of early gray-faced commuters heading toward the L. She catches her reflection in the window, a white ceramic lucky cat waving at her from the counter. Sophie frowns at the busy signal and slips her phone away.
"Finished? Sure?" she says as the dog walks on. She ties him up outside the grocery store on a wall scrawled with graffiti, and he waits, his unwavering eyes on her as she buys fresh orange juice. Sophie takes a copy of The New York Times from the vending machine. She sees his name instantly, there, beneath the headline. It's as if the print loses focus, leaving two words in dazzling clarity: Jess Wallace. Mutt barks, impatient, and Sophie can't help smiling as she walks over and unties the lead. "Hey, I was only a minute." The dog's good mood is infectious; joyful wags contort his whole body—they are together again, simple as that. Sophie loops the lead and bag over her wrist and flicks on through the paper as they walk, deliberately not reading the front-page article but instead searching for her latest column on a new exhibition that has just opened. She finds it way back in the "Arts" section, tucked among the advertisements. It feels like an afterthought.
In the studio, she hears the shower running, Alisha singing loud and true along to Macy Gray on the radio. Sophie pours a cup of coffee and clears a space among the sketches and bolts of fabric on the dining table, spreads out the photographs and documents from her satchel as if she is dealing cards. Each is labeled with a Post-it note, written clearly in black ink: Gabriel Lambert, 1970? Last known photo. Varian Fry, André Breton, 1940. JC: Gabriel and Annie Lambert, party, Long Island, 1960s. This last photograph Sophie picks up, studies closely. The young woman wears her blond hair fashionably loose, a heavy blunt fringe over dark kohled eyes that gaze, full of love, at the lean, tanned man at her side. His black hair is graying at the temples, worn long enough to brush the collar of his faded denim shirt. His eyes are fierce, the color of the sky. The chemistry between them is palpable, even down the years, no air between their bodies, his arm protectively around her waist, her palm resting flat against his chest. How do you do that? she thinks. How do you keep that passion for a lifetime? Her stomach tightens with nerves at the thought of finally meeting them. Their legendary love affair fascinates her; the idea that sometimes there is a happily ever after is a beacon of hope in the darkness.
"Hey, honey, I didn't hear you come in last night," Alisha says. The red of her sarong flares as she walks through sunlight, beads of water glistening on her freshly oiled skin.
"Didn't want to wake you. How was your break?"
"You know, Labor Day, my family." Alisha purses her lips. "How's your mom?"
"She sends her love."
"D'you manage to store all your stuff in her barn?"
"Just about." Sophie raises an eyebrow as she looks down at Mutt. "Mom suggested we should just move in there. What do you think?" The dog cocks his head, listening.
"Where's she live again?"
"Montauk."
"Could be worse." Alisha shrugs, takes a bright green apple from the bowl on the counter. "If the Times don't take you on full-time, you could get yourself a job in one of those fancy-assed lobster restaurants."
"Thanks for the vote of confidence," Sophie says. "Everywhere's closing for the season, but I'm heading out to Long Island again tomorrow, so I'll see if anyone's hiring, just in case."
"Ha, ha." Alisha bites into the apple. "Man, you spend more time out there than in the city lately. This article you've been working on?"
"Yeah." Sophie rubs the bridge of her nose, rests her lips against her index finger as she stares at the photo of Gabriel and Annie. "I'll be glad when it's done."
"Sleeping any better?"
Sophie looks up from the photo. "So-so. I've just got all these questions running around in my mind."
"Tonight you take the bedroom for a change. You need some sleep if you're going to face off with the great Gabriel Lambert, baby girl." Alisha comes over to her. "Were you up all night working again?"
"It has to be perfect." It's got to be, she thinks. What was it her editor had said? The correction rate on your pieces is too high, Sophie. Accuracy is everything at the Times. You're quick, you're willing, but you're making too many beginners' mistakes. If you want to make it, you've got to check and double-check every lead, every line. I know you've had a tough few months, but if you can't nail this Lambert story, I'm going to have to let you go. "Talking of perfect, see this?" She taps the front page of the paper.
"Jess? Is he back?"
"Who knows? There's some big summit at the UN, maybe he's in town for that." Her stomach turns over at the thought. "I haven't spoken to him in weeks."
"Good." Alisha glances at the clock. "It's the best way. A clean break. You'll be back on top in no time, trust me." She raises her chin. "Say, hadn't you better be going?"
"Is that the time?" Sophie scoops up her papers, swings the satchel across her body. "Thank you." She hugs her friend. "I promise you, this won't be for long."
"Take all the time you need to find somewhere. Me and the Mutt will be just fine, won't we?" she says to the dog. "He's a big hit with all my clients." The dog looks up at her, cocking his head.
Sophie checks her watch and wheels her bicycle toward the door. "Anyone coming in today?"
"Yeah." Alisha gestures toward a mannequin draped with muslin sheets in the corner of the studio, near her drawing board. The shape hints at the full skirts of a wedding dress, and a crystal tiara rests on top, sparkling. "Damn, I didn't—" She hesitates, glances at Sophie.
"It's okay," she says, her voice softening. "I was thinking the same thing. It would have been last weekend." Sophie opens the door. "Kind of ironic, isn't it, bunking down in a wedding dress design studio." She glances back, forces a smile. "Like I always say, the universe has a sense of humor."
Alisha holds open the door. "I sure wish it would share the joke."
In the hall, Sophie drags the elevator cage open and wheels her bike in. "Catch you later?"
"MoMA at five?" Alisha raises her hand as Sophie disappears.
On the street, Sophie pauses as her mobile vibrates in her pocket, the old-phone tone rising above the traffic. She wheels the bike along the pavement with one hand, crooks the phone beneath her jaw. "Mom?"
"Hi, darling, sorry, I was just doing my Pilates. I saw you'd tried to call. Are you okay?" Sophie can hear an old James Taylor track playing in the background.
"Fine...." Sophie pauses. "No, I'm not fine," she says. "I feel like I'm going round and round in circles with this story—"
"But he is going to see you?" Sophie hears her mother click off the radio. "I spoke to Gabe's son, like you asked."
"I feel like such an idiot, asking my mom to call for me. But thank you."
"Hey, that's what I'm here for. And honey, trust me. If I know them, they wouldn't have let you within a mile of Gabe if you weren't family."
"I just want to do it justice. Do Vita justice. I care about this story...." Our story, Sophie realizes. "I miss him."
"Your dad? So do I, darling. So do I. Every single day." She hears the emotion in her mother's voice. "He would be so proud of you. You've sacrificed a lot for this."
"You mean Paris, Jess?" Sophie pauses. Happily ever after? she thinks. "He didn't get it at all, did he? But then I never told him the real reason I want to write this story."
"Well, we all have our secrets, don't we, darling...."
THREE
MANHATTAN
Wednesday, September 6, 2000
SOPHIE
Sophie runs up the stairwell of the New York Times' offices on West Forty-third Street, the exhilaration of her ride across the Williamsburg Bridge into Manhattan still with her, her cheeks flushed with pink. The stale scent of cigarette smoke hangs on the air in the hall as she pushes open the door. "Hey," she says to one of the interns on the way through the dingy, windowless newsroom. "Is the boss in yet?"
"Not yet, Soph." The boy flexes his wrist and winces.
"RSI playing up, still?"
"Yeah, and the culture editor wants me to be his hands today."
She roots around in her bag and tosses him a gold pot. "Tiger Balm helps. Weren't you on the TV guide? Those grids are a killer."
"They're kind of boring...."
"Shh!" Sophie's eyes widen. "For God's sake don't let anyone hear you say that. You've got to suck it up, we've all been there."
Sophie glances around for any sign of her boss as she walks on through the office. "Thank God," she says under her breath as she reaches her desk safely. She slings her satchel onto the back of her chair and swivels around to the computer just as the phone rings. "Sophie Cass," she answers, crooking the receiver against her shoulder as she begins to type. She waves her hand, clearing the cloud of fruit flies hovering above the bowl of red apples by her computer.
"Cass," he says. The hum of the strip lights and tap of keyboards across the room seem to intensify.
"Wallace," she says finally, trying to sound amused, playful.
"Damn, your voice is sexy as ever." She can hear his breath on the phone line. "You're late."
She eases back in her chair. "I prefer to think of it as making an entrance." Sophie pauses. "How do you know?"
"I always liked you in chinos. Very Hepburn, very yar..."
"Are you comparing me to a boat?" She swivels around and stands. Jess is leaning against a desk at the end of the newsroom.
"Why not? Trim. Responsive. Lively handling..." He flicks his cell phone closed and walks toward her as she replaces the receiver with deliberate care.
"What a surprise. I didn't know you were in town." You didn't warn me, she thinks.
"How are you?" He leans down to kiss her cheek, and she smells a new cologne—something citrus and fresh. "You look great."
"Thanks. I've been at the beach with Mom." She can't immediately place what has changed about him. He's wearing the same deep blue suit she loved the best, his red-blond hair is immaculate, as ever. "Are you here for the summit?"
"Yeah, I had some leave, so I flew in from Paris last night. Biggest gathering of world leaders in history, never know what I might pick up."
"You never stop, do you?" Sophie indicates the phone. "I should—"
"Sure." He holds her gaze. "Can we have a drink later?"
"I don't know."
"Old times' sake?" He leans over her, searching for a pen. She imagines for a moment reaching out to him, lacing her fingers in his. How simply it could all begin, again. "Around six P.M.?" Jess scribbles down an address.
"Why not?"
"See you later, Cass." As he turns, he calls back, "Don't be late."
And then, as he walks away, Sophie realizes. It is not Jess who has changed, it is her.
* * *
"I'm sorry, I'm sorry..." Sophie waves, running along West Fifty-third Street, her heels clicking. "The meeting ran over."
"Well, look at you," Alisha says. She shoulders open the door to the museum, and they walk against the crowds leaving for the day. "You shouldn't have gotten all dressed up for me...."
"I didn't, I'm meeting—"
"Jess."
"How did you know?"
"He rang the studio just after you left for work this morning. I hope you know what you're doing."
"Of course I don't." Sophie smooths down her black dress. "Do I look okay?"
"You look beautiful. Hold on a second." Alisha reaches around to the nape of Sophie's neck and snaps off the tag. "Don't want him to think you're making a special effort or anything." She screws it up and tosses it into a bin without breaking her stride.
"Oh God, I didn't?" She touches her neck.
"You did."
"Anyway, it's not for Jess. I bought it for tomorrow. I just thought I might as well wear it now. At least he won't be able to complain I didn't make an effort."
Alisha takes a breath. "Where are you meeting him?"
"Some cigar bar up on First and Forty-eighth."
"Typical. Convenient for his parents' place?"
"Don't start."
"Listen, I have seen you put yourself back together again over the last few months and it has been hard for you. He comes breezing in from Paris and you—"
"It's just a drink. That's all. It's over. I just—I just want to make sure he's doing okay." They walk on through the museum, and Sophie checks her watch. "So what did you want to show me?"
"I want you to know what you're up against." Alisha points the way, the gold bracelets on her wrist jangling. "When I was in here the other day I saw they'd done a rehang. I guess your Mr. Lambert's due for a revival with his birthday coming up and all." She glances across as a young guard walks toward them.
"The museum is closing," he says.
"Okay, honey." Alisha tosses a glance over her shoulder like a silk scarf as she walks on. "Over here," she says to Sophie, stopping in front of a series of huge dark paintings, broad abstract strokes gouging into the canvas. "Gabriel Lambert." They stand side by side in silence. "I know this isn't your period—"
Sophie squints at the Perspex tag on the wall: Mars. "Give me a Matisse or Monet's water lilies any day," she says quietly, stepping backward to take in the full scale of the paintings.
"Lambert makes the rest of them—Pollock, Rothko, all those abstract expressionists of the fifties—well, he makes them look like pussycats." Alisha sweeps her arm around the gallery, and Sophie turns a full circle, returning to Lambert. She stands in silence for a time, dwarfed by the paintings. It feels as though they are pressing down on her.
"We should go," Alisha says, noticing they are the last ones in the gallery.
"The thing is," Sophie says, pointing out canvases as they walk on through the museum, "these guys—the Europeans like Breton and Ernst, all the people Varian Fry helped escape from France, when they arrived in America it was like art exploded. They were the catalyst for all of that." She points back to the room with the abstract expressionists. "But Lambert's work changed so much, I just don't get it."
"Well, if you ask him why, you're a braver woman than me. Your man's like the missing link, isn't he? Born in France and made in America."
"I came into the archives here a while back, and looked at the 'Flight' portfolio of prints Varian Fry curated to raise funds for the refugee organization he worked for back during the war. I don't understand why so few of the artists he helped contributed work."
"Did Lambert?"
"Nope."
"Doesn't surprise me. He's a tough old SOB." Alisha pauses on the pavement, and Sophie hears the museum doors locking behind them.
"Maybe after everything he went through during World War Two he didn't want to look back," Sophie says, thinking it through.
"Perhaps once you start running, it's hard to ever stay still." Alisha puts her hands on her hips. "Look, don't let this old guy run rings round you. From what I've heard he's tricksy. Did I tell you he blanked me at an opening once?"
"No? When was that?"
"Way back, way, way back. I was still at Parsons. It was a friend's show—I saw this old guy in a denim shirt and worn-out jeans, and espadrilles with his toes poking out of a hole at the front, and I thought: I know you. I mean, his hair was white as white, but those eyes... like those snow dogs, you know? Ice blue." Alisha whistles softly. "Man, he must have been fine in his day."
"So what happened?'
"I went over—you know me, I'll talk to anyone, and I started in with: 'Excuse me, Mr. Lambert, I just wanted to let you know what your work has meant to me....'"
"And?"
"He just stared me down. Those goddamn beautiful eyes, cold as fire."
"That doesn't make sense."
"It will when you meet him. Hot and icy, all at the same time. He just walked on by me midsentence, gushing like a damn fool about how much I loved his work." Alisha shakes her head. "Take my advice. You want your story, keep it professional. Don't show a chink of weakness—"
"I won't."
"Sophie, you're a romantic." Alisha cups her face between her hands. "I know you've fallen for the story of Gabe and Annie Lambert, their mythical lifelong love affair—" Sophie rolls her eyes. "Okay, okay." Alisha holds up her hands in surrender. "You sure you want to see Jess? I'm going to see Love and Sex...."
"Again?" Sophie laughs. "You and your thing for Jon Favreau." She kisses Alisha's cheek and walks on. "You're obsessed."
"Takes one to know one," Alisha says under her breath as Sophie walks away.
* * *
Jess is reading a copy of Wonder Boys, a plume of blue smoke rising from the cigar in his right hand. Sophie walks toward him through the dimly lit bar, conversation humming around her like bees in dense grass. Jess throws aside the book and taps his Rolex as he unfolds himself from the dark leather wingback chair to greet her.
"Give me a break," Sophie says, kissing his cheek.
Jess gazes down at her. "I'd forgotten how beautiful you are."
"That didn't take long."
"You cut your hair? I always loved—" He stops, registering her frown.
She remembers how light and free she felt the night Alisha cut it for her a few weeks ago, the heavy hanks of waist-length hair drifting to the floor. "I like it." She touches the nape of her neck.
"Fresh start?"
"I had a broken heart, didn't you hear?"
"You and me both." His gaze travels down. "You should wear this tomorrow, you know. You look professional."
"I was aiming for irresistible."
Jess smiles, his blue eyes creasing. "That too."
"How do you know about tomorrow?"
"I was talking to your editor about this arts story you're writing." He settles back in his seat and beckons over the waiter.
Sophie tosses down her bag and takes the chair beside him. "I feel like a kid with a bad report card."
"Don't sound so defeated. Show them what you're capable of."
"Jess, I'm not like you." Sophie rests her head against the palm of her hand. "I've tried, I really have, but I just feel like a phony—"
"Sweetheart, I've always told you—fake it till you make it—"
"You didn't just 'sweetheart' me?" Sophie raises an eyebrow.
"Sorry, sorry!" Jess holds up his hands in defeat. "Damn, forgetting who I'm talking to." It's an old joke they share. The moment's tinged with loss.
"Maybe I should have stayed on at the university," Sophie says, breaking the tension.
"Ironic, isn't it. If you were still at the Sorbonne, we'd be together in Paris now."
"Really? I hadn't thought of that."
"I see your biting wit hasn't left you." Jess looks up at the waiter. "What'll you have?"
Sophie takes a sip of his drink and pulls a face. "Neat vodka?"
"Whiskey started giving me killer hangovers. I miss it, though—few things finer than late nights and Scotch."
"Hemingway wannabe."
"Pseudo-Sontag." He holds her gaze. "Chablis?"
"Sancerre, thanks." She smiles at the waiter. Jess takes a drag on his cigar and exhales, tilting his face to the ceiling.
"Bravo," he says.
"Sorry?"
"Your taste, it's refining."
"It suits you, this place."
"Sophisticated? Old school?"
"Expensive. Up its own ass."
"I see you've forgiven me."
"And old. Not old school." Sophie glances at him, smiles. "How was your fortieth?"
"Lonely."
"I don't believe that for a moment."
"I missed your birthday, too. What was it? Your twenty-fourth?" Sophie nods. "Well, Happy Birthday to us." He raises his glass to her as the waiter hands her the wine, and they chink a toast. Jess offers her the cigar.
"I gave up."
"Doesn't count."
Sophie takes a drag, exhales slowly. "Happy?"
"Not really. You know me." The intensity of his gaze makes her head spin. "How are you?"
"Just dandy, can't you tell?"
"And Mutt?"
"Doing fine; he's doing just fine." Sophie hands back the cigar. "You know you have visitation rights, anytime."
"Weekends and holidays?" Jess shakes his head, drains his vodka. "No thanks. It was hard enough saying good-bye the first time. Does he still hog the bed?"
"He prefers his basket to Alisha's sofa."
"May I get you a refill, sir?" The waiter clears Jess's glass.
"Yes." He frowns, waits for him to leave. "Soph, why haven't you found anywhere yet?" Jess leans toward her.
"I've been busy. And you know how expensive it is."
"Christ, I feel terrible. Where is it Alisha lives? Williamsburg?"
"It's hardly roughing it." Sophie laughs. "It's up-and-coming, you know."
"Yeah, if you're a meth dealer."
"You watch. It'll be wall-to-wall artisanal cheese shops before you know it." She shrugs. "It's not Greenwich Village, but..."
"I'm sorry about that, leaving you to sort the apartment out."
"It was fine," Sophie says. Fine. It's all fine. How many times has she said that over the last weeks, when it's been anything but? It's been hell, Jess, that's what she wants to say. I missed you, and my heart's still mending. Her throat tightens at the thought of their home, the cozy rooms she had spent months painting and furnishing, all the hopes and dreams that came to nothing.
"I want you to have some of the money, Soph." He glances at the waiter. "Thanks."
"No," she says.
"Everything you did to the place added thousands."
"Your parents bought it for you—"
"For us."
"And we broke off the wedding."
"Only because I insisted on taking the job in Paris." Jess waits for her to look at him. "I've missed you."
"Don't, Jess. We've been over this a thousand times. My life is here." Her stomach flips over at the memory of the arguments toward the end. "What did you expect? That I'd just sacrifice everything and follow you?"
"That's what people in love do."
"Would you do it for me?" She waits, part of her still hoping. "I didn't think so."
Jess swirls his drink. "Your dad would be proud of you."
"Don't, please."
"No, I admire you, I really do." Jess knocks back his vodka. "Hell of a man to live up to, that father of yours."
Sophie closes her eyes, exhales. "Not this again? Jess, you are the one who wanted to go and be a foreign correspondent—"
"You could have come with me." He leans forward. "Don't you remember how it was, when we met? Don't you remember Paris?"
"Of course I do." She can't look at him.
"Soph, I wanted to see you tonight because there's something I need to tell you—" She picks up on his tone immediately. Tell me. Something to tell me, not ask me.
"You've met someone new." It's a statement, not a question. She places her wine carefully on the table. "Is it serious?"
"Nothing's happened, yet." He takes her hand. "If you won't come with me—"
"Of course. I'm happy for you." What did I expect? She rubs her thumb against the side of his, and they rest fingertip to fingertip.
"Jeez, Cass. Aren't we worth more than some job? People do write in Paris, it has been known."
"Not like this, not like the Times—"
"Still going for that Pulitzer, are we? Daddy's little girl...."
"Be happy, Jess." Sophie swallows down her disappointment as she stands to leave.
"I'm sorry. Don't go." He reaches over to stop her. "I'm flying back just as soon as the UN summit ends on Friday. Once you've seen this old guy tomorrow, and knocked out your story—"
"Jess, this isn't just some story." Sophie squares up to him. "I care about this, more than anything—"
"More than me?" His brow furrows. "Come with me."
"I can't—"
"Just think about it. You're due some leave, I know." Sophie tenses as he stands and hugs her. Too damn right I am. It should have been our honeymoon. "Come with me to Paris, not for good, not yet, but just to see if we still—"
"Love one another?" Sophie reaches up and kisses his cheek. Did you ever really love me? Or were you just in love with the idea of me?
"Stay with me tonight." He turns to her, his jawbone brushing her temple. "My folks are out of town."
"I can't—" Her voice breaks. "I just can't... do this anymore, Jess." She steps away and smiles, her eyes glistening. "Besides, Mutt's waiting up for me."
"He watches too much TV. Does he still like Friends?"
"He's into Frasier lately."
"Is he? I guess we all move on." Jess gazes down at her. "Stay. Please."
Sophie shakes her head. "I've got to be at Penn Station early."
"It'd be quicker if—"
"Good-bye, Jess."
"No." He raises her chin with his index finger.
"What do you mean, no?"
"I'm not saying good-bye to you, Cass. Not yet. You've got a couple of days to think about it. Montmartre, the galleries, strolling along the Seine... Wait till you see my apartment near Sacré-Cœur."
"You don't give in, do you?"
"Never, with you." Jess drapes her jacket over her shoulders. "One last chance, is that too much to ask?"
FOUR
SOUTHAMPTON, LONG ISLAND
Thursday, September 7, 2000
SOPHIE
A little after ten, Sophie's train pulls in to Southampton station, and she jumps down to the platform. She slips on her sunglasses as she walks out through the low redbrick building and glances around for a cab. On the flagpole opposite the station, the Stars and Stripes snap in the warm breeze. A man with jet-black hair leans against the side of a dusty blue pickup truck parked beneath the flag. Spotting Sophie, he strides out into the road and calls to her.
"Sophie Cass?"
"Yes," she says. Those eyes. "Who—"
"Harry Lambert." He reaches out and shakes her hand. Sophie's shocked by his directness, his resemblance to Gabriel. Reading her expression, he shrugs. "Yeah, I know. Everyone says it's uncanny." He turns his head from side to side, an amused look on his face. "I haven't seen any photos of Grandpa at my age—"
"I have. It's amazing how alike you are." She thinks of the photos Alistair Quimby took, tucked safely in her bag. "How old are you?"
"Man, you're direct." Harry laughs. "How about getting to know a guy first?"
"I just meant—"
"I'm twenty-six, thank you for your interest."
"Who says I'm interested?" Sophie takes off her sunglasses.
"Say, I know you."
"I don't think so."
"You were at that opening in SoHo a few months back. Red dress." His eyes crinkle. "I tried to get your attention, but you were deep in conversation with some stiff in a suit."
She remembers the night, the argument. Jess had left for Paris the next day. "I don't remember—I mean, I'm sure I'd have noticed you." It was on the tip of her tongue to say: I'm sure I would have noticed someone who looks so like the man I've spent months researching.
"Yeah, well, sometimes you don't see what's right under your nose." Harry smiles. "I looked kind of different then, too—I'd shaved my hair off."
"Bad breakup?"
"No." He runs his hand through his hair. "Though there was that, too. I did it for charity."
"Well, small world."
"Gabe got your letter," he says. "We figured you might be on this train if you were coming in from the city."
"And what? You wanted to stop me getting to see him?"
Harry shrugs. "Just wanted to check you out."
"So?"
Harry searches for the word. "Unexpected."
"Look, I can appreciate you're protective of him." She pauses. "Hey—how did you know it was me?"
"Magic." Harry strolls toward the truck. When Sophie doesn't follow him, he turns to her. "Okay, I did my research, too." A smile flickers over his lips. "On Yahoo. Wanted to see what we're up against."
Sophie walks after him this time, and he puts his arm out, protecting her from the traffic. He opens the passenger door of the truck and helps her in. Sophie glances at the child's car seat strapped to the bench and throws her bag down into the footwell. She's surprised at the jolt of disappointment she feels. "Thanks." The cabin smells of fresh-cut timber and oil. She notices he's wearing heavy work boots. His faded jeans are splashed with white paint, but his crumpled blue-and-white-checked shirt is clean—she catches the fresh scent of detergent as he winds down the window.
Harry starts the engine, and Bear in the Big Blue House blasts from the speakers. "Sorry." He flicks the stereo to the radio station. "There goes my street cred."
"Very smooth," Sophie says, laughing. Harry ejects the CD and flips it out. He reaches for the glove compartment as he watches the traffic, the back of his hand accidentally brushing her knee. Sophie shifts her leg and searches for the case, taking the disc from him.
"I sometimes forget to turn it off—"
"C'mon, admit it. You love Bear."
Harry glances over at her as he pulls out onto Hampton Road. "Who doesn't?" Sophie tucks the disc back in the glove compartment, dappled shadow from the tree-lined road dancing over her arm. "Pick whatever you want." Savage Garden is playing on the radio.
"No, this is fine." She slides her sunglasses down and settles back, her arm resting on the open window. I knew I loved you... Sophie can't help thinking of the photograph of Gabriel and Annie as she looks at Harry.
"How old's your kid?" she says.
"Kid?" Sophie gestures at the car seat. "No, she's my niece. I just help my sister out sometimes, dropping her at nursery."
"Oh." As they drive on past high hedges, white clapboard houses peeking above, she relaxes. Sophie reaches into her bag for some gum, and she spots the Henry James essay, the note. Her mother's voice comes to her: good-looking, sense of humor, good with children... She offers him a stick, and the scent of fresh mint fills the air. "Tell me what you meant by unexpected?"
"Thanks. You've just surprised me, that's all. From those hard-assed letters of yours we were expecting—"
"We?"
"The family." Harry spins the wheel. "My dad spoke with your mom the other day, Miss...? Ms.?"
"Doctor."
"Doctor?"
"Art history." Sophie looks over at him as they stop at a junction. "Though I do know a little first aid."
"I'll bear that in mind if I need mouth-to-mouth."
Those eyes, Sophie thinks, easing the gum against the roof of her mouth with her tongue. She feels the heat rising in her blood, molten, warm.
"Dad's normally so protective of Gabe he won't let any journalists near him." Harry glances at Sophie. "But your mom was pretty persuasive."
"She has a way with words. You may have heard of her—Paige Cass?"
"Vaguely. Poetry's not really my thing."
"What is your 'thing,' Mr. Lambert?"
"Harry. I'm an artist."
"Like your grandfather?"
"And you are a writer, like your father." Sophie looks at him, surprised. "I told you I checked you out, Dr. Cass."
"Sophie."
Harry drives on. "Jack Cass, Pulitzer Prize–winning New York Times journalist and all-round hero...."
"Bravo."
"So what is this? Trying to prove you're as good as Daddy? Going in for the big story?"
"Something like that." Sophie folds her arms.
"What happened with him? I saw he was killed—was he on assignment?"
"Now who's direct?" Sophie shakes her head, a tendril of hair loosening. "No. It was the wrong drugstore, wrong holdup, wrong time to be a hero." She turns to the window, gazing out as a succession of identical brick driveways with impenetrable electric gates sweeps past.
"Damn, I'm sorry." Harry reaches over and touches her arm, waits for her to look at him. "There's never a wrong time to be a hero."
Sophie acknowledges his kindness, a tilt of her head. "Maybe you're right." She felt it, at his touch, a quickening. Desire blooms in her like hunger. She looks up the road, hiding her surprise. "I took you for a contractor."
Harry settles back, one hand on the wheel, his other arm looped easily across the back of the bench seat. "Got to pay the bills somehow."
"I assumed—"
"What? That none of us have to work, because of Gabe?" Harry throws his head back and laughs. "You really haven't met him, have you?" He indicates to turn off the main road. "Listen, I've got to do a couple of chores on the way. Do you mind?"
"Sure, no problem. As long as we get to Gabriel on time."
* * *
After driving for a time, Harry pulls into the yard of an architectural salvage store and switches off the engine. Weathered stone garden statues and a pair of Doric columns line the entrance to the clapboard barn, a brass bedstead leans by the doorway, basking in the sun. "I won't be long."
In the rearview mirror, Sophie watches him walking away. He moves with a lean, agile strength. A blue glass lantern hanging in the tree by the barn door swings in the breeze, catches the light. What am I doing? I haven't felt like this since... Sophie flicks back through her memories. When? Since the first time I saw Jess? No, this is different. She quickly flips down the sun visor and slicks some Carmex across her lips, lets down her hair. Meeting Jess was crazy, and wonderful, and Paris... Her green eyes are dark in the shaded mirror. But it wasn't like this. By the time Harry returns, arms laden with tiles, she is leaning against the truck, idly scrolling through her phone.
"What do you think?" he says to her, dumping the tile boxes in the back. "First impression?"
"Bit wary, but charming in a rugged artist slash builder way...."
"Less of your lip, missy—as Gabe would say." He puts his hands on his hips. "Not me, the tiles."
"Kitchen or bathroom?" She stands beside him, feels him watching her.
"Bathroom floor."
"Those." She runs her fingertips over the warm limestone slabs.
"Yeah, that's what I thought." He walks around and opens the door for her.
"Is this for a project?" she asks as Harry starts the car.
"Nope, I'm renovating a place up the coast."
"Really?" Sophie turns to him, her leg tucked beneath her. "I've always dreamed of doing that." Her imagination runs ahead of her. She pictures an empty house, white rooms full of light, two chairs on the porch, talking late into the night.
"It's not much at the moment," he says. "But it's a start, and I'm going to build a little gallery on the side as soon as I can." He glances at her. "In fact, I could do with a curator—someone who could write the catalogs and so on."
Sophie shrugs. "You never know. If you build it—"
"She will come?" Harry grins. "I loved that film when I was a kid."
"Yeah, so did I." Sophie leans her head against the palm of her hand as she looks at him. It's like looking at Gabriel, but with none of the hardness and anger etched into every photograph she has seen of him at that age.
"What?"
"Nothing," Sophie says. She had been wondering what it would be like to kiss him. "You're just... unexpected, too." She glances at her watch. "Listen, do you mind if we head over to the house now? I don't want to be late."
He hesitates. "Sure." He pulls out onto the main road again.
"Is it far?"
"No." She feels a distance between them, suddenly. "Do you know Flying Point?"
Sophie shakes her head. "Nope."
"City girl, huh?" Harry looks on ahead.
Sophie frowns, doesn't correct him. "Any advice? I mean, anything you can suggest that might make it easier?"
"With Gabe?" Harry puts his foot down, heading toward the coast. "Just be yourself. He doesn't like phonies."
"Okay. Anything else?"
"Take it easy on him, okay?" She sees the concern in his eyes as he turns to her. "Gabe's not... I mean, he's ninety-five."
"About that—"
"Look. You're talking to the wrong person. You have some great theory about Gabe, then have it out with him. I tend to trust people, you know? Treat them how they treat me. And Gabe..." Harry pauses. "He's the best." They fall silent as Harry drives on. Sophie glances at her watch again. Five to twelve. "Not much further." The sudden edge in his voice cuts through her. "Look, in fact, do you mind if I drop you here? It's not far."
"Sure." She hesitates, uncertain whether to ask him what has changed so suddenly. What an idiot, she thinks. I've been out of the game so long, I completely misread him. Disappointment settles quietly on her like mist on a lake. Harry pulls in to the side of the narrow road. "Thanks." Sophie hops down from the cab. "It was good to meet you." She reaches over and takes her bag from him.
"Sophie..."
"Yes?"
"Don't do this, please?" He touches her hand. "I know your mom said you've been working on this for months—"
Sophie laughs uneasily. "Sounds professional, doesn't it?"
"It's just a story—can't you let it go?"
"I can't do that," she says, reluctant to pull away. "It's about my family, too. I want to know the truth about Vita, and Gabriel's the only one who can tell me that."
"But he's an old man, and now..." Harry settles back. He won't look at her. Emotions pass over his face like cloud shadows scudding over hills. He points toward a rough track up ahead. "Head down there. When you get to the end..." He pauses. "Turn left. Keep on walking up the beach, you can't miss it."
"Thanks for the ride. Once you've built that gallery, why don't you give me a call?"
"Hope you're patient." Harry raises his hand in farewell as she closes the door and steps back. "See you around?"
"Sure," Sophie says, and watches as the truck drives away. A line from her research comes to her: I'll see you soon, in New York.
FIVE
FLYING POINT, LONG ISLAND
2000
GABRIEL
This year, for Annie, we had Christmas a little early as a surprise for her. I can see Tom and his brother, Albie, dragging the tree across the yard to the trash, a thin line of green needles trailing them like a file of ants. Some days it still surprises me to see my boys grown men, with sons of their own. I still remember the pair of them running across the beach in the summertime after school, their hair bleached white, their beautiful tan bodies thin and flexible as reeds, lambent, full of sap and new life. The yellow school bus would drop them at the end of the lane, and they'd race straight through the house, flinging their bags down on the way to the beach. In and out, and gone. Damn, it goes by so fast.
A strand of silver tinsel catches the light as the bare branches sweep along the sand. I can see Tom pointing at the studio and saying something to Albie. He's probably telling the boy to keep an eye on me while he finishes up outside, but there's no need for that. Tom always was the stubborn, bossy one, as most eldest kids are, but then so was the man he was named for.
We set up the crèche just the way Annie likes it, too, with the cradle empty until the day. The girls are packing it away in the house now, but I've taken the little Santon figure Annie bought me in Marseille in November 1940 with me. The little clay figure of a shepherd facing the mistral is worn and chipped now, but then so am I. I remember the market, the lamps strung from the stalls, the crush of bodies, and the warmth of Annie's hand in mine. There was little food by then, but they were roasting chestnuts on braziers, and sweet smoke filled the air. She chose him for me, her gloved fingers dancing over the identical heads until she saw just the right one for us, and she wrapped it in her lace-edged handkerchief, pressed it into my palm. "One day," she said, "this will be in our home."
Now, my littlest great-grandchild holds it in her tiny fist. She's curled up asleep on my lap in the studio. The guileless sleep of children moves me. The child's fontanel is pulsing in her sleep, and her perfection, her peace, awes me. I think I've slept with one eye open since 1940.
The girls shooed me out of the way while they tidied up the lunch, told me to stop the little one from crying. Her first teeth are coming in and her cheeks are flushed. I'm not use for much these days, but I can still soothe a baby to sleep, rocking in my old Shaker chair looking out to sea. In her sleep, her fingers extend like a starfish, her perfect little mother-of-pearl nails shining in the autumn light. The shepherd tumbles from her palm to my lap, and I put him away safely in my pocket as I carry her back to the house to tuck her up in her cradle in the kids' old bedroom.
My daughter glances up as I close the screen door behind me, and I raise my finger to my lips. She smiles and bends her golden head to scrub the old wood table clean, her sleeves rolled up, her hands soft and pink from the hot water. A chain of silver stars and shells glimmers on her wrist.
One of the toddlers is napping on the bottom bunk already, and the midday sun is diffuse and warm through the orange curtains. I lay the baby down in her cradle and switch on the little lamp at her side. Her eyes open lazily, register the familiar stars and shells rotating slowly around the room, then close, contented, and I pull the door to, silently.
How many nights did I do that for my kids when the wind raged outside? My throat is tight, suddenly, at the thought of all the days, the thousands of nights, that have gone by unremarked, and I lay my head against the door. The sand is running over the smooth hip of the hourglass. I know the creak of this particular door and the click of the latch by heart. I know every breath and sigh of this house that I built with my own hands.
I wait at the porch for the girl to come and turn the shepherd over in my fingers in my pocket, the old clay smooth and good to the touch. The little figure Le Coup de Mistral, this man battling against the gods, holding on to his hat as he soldiers on, leading his sheep out of the storm, always makes me think of Varian. I find I think of him more often now. I owe him, I owe them, all of this, and I never had the chance to thank him. Because of him, we are here.
Sometimes I think I left my heart at the Villa Air-Bel in Marseille. When I think back to the war, my memories are all of the house of dreams. Our greatest joys and tragedies played out there. Ask Varian, ask any of them, and they'd have said the same. We knew, even then, that life would never be as vivid as this again, felt guilty, even, that we found such unexpected happiness.
Air-Bel was a sanctuary. There, I saw André Breton conjure a court of miracles. Perhaps when the horror of war surrounds you, when everything—life itself—could be taken from you in an instant, this is what men like André and Varian do. They become gods, fight back any way they can. Some of us—artists and writers, lovers and children—well, while others give up and wait to die, some of us fight the only way we can and create something marvelous.
I still see them walking down the driveway of the château for the last time: Varian, the police at his shoulder, and his loyal dog, Clovis, at his side. After the rain, even the cedar trees seemed to be weeping. In a year, Varian said, we lived twenty. I have never been so afraid and yet so alive.
When he wrote his memoir, Varian said he had to exorcise his ghosts. Maybe that's what I have to do today. He reckoned he couldn't lay them to rest until he'd told his story—all of it. Well, this is my story. I don't want anyone to feel sorry for me—the world is full of miserable childhoods, it's incredible any of us survive. It's where you go, not where you're from, that counts in life. You're dealt a bad hand—make something of it. Every day I thank the stars for my luck. I made a living, made a life doing something I love, with someone I love at my side, and I gave my kids the childhood I never had. Maybe I had a lucky star guiding me home to Annie. I've lived with guilt my whole life, but I am a fortunate man.
For fifty years of peaceful tides and stars, I've waited for this day to come, and now it's here. I close my eyes. The girl, Sophie, is near, walking along the road toward me, her slender dark figure growing closer, and I am afraid again. Bring on the ghosts. It's time.
SIX
FLYING POINT, LONG ISLAND
2000
GABRIEL
"Are you Gabriel Lambert?"
"No. Go away." Well, it's worth a try. I'm ready for you, missy. No one has seen me in public for years, and it might just fool her. I squint my eyes against the clear autumn light and try to close the screen door, but the girl slings her fancy suede bag up against the jamb. It thuds on the deck with the weight of a brick, and the door gets stuck on a silver laptop I can see poking out the side like a knife. "I said, go away. This is private property. Didn't you see the sign up on the track?"
"Yes." She folds her arms. What is it with girls today, they try to make themselves so deliberately plain? Not a scrap of makeup. I used to love watching Annie making up her face in the morning. It was like she was joining the dots, clearly defining the beauty in her face. I loved it when our kisses smudged her lips, when her eyelids grew smoky with kohl. Nothing like that with this girl, no sir. Even the heavy black-framed glasses are some kind of statement: Listen, buddy, I know I'm young, and my limbs have the refinement of a gazelle, but I wear this beauty lightly, with a sense of irony. Jeez, the things we take for granted—if I knew how swiftly the power of my body would desert me, I would have spent the first decades of my life on a beach like my kids, just enjoying the grace, the luck of being alive and young. These days, when I wake in the warm, familiar nest of my bed with Annie at my side, there's a weightless moment when I am still as fresh as this girl. Then—I always think of it like a white-haired janitor in brown overalls walking through a dusty warehouse flicking on strip lights—my old body crackles into life and all the aches and pains fire up, one by one. I'm useless now, before my shower in the mornings. The hot water eases my old limbs, and I've always had my best ideas in the shower. Annie gave me some whiteboard markers for my birthday a while back, and now it's the best time of the day for me—I can shower, and think, and sketch out my ideas on the glass of the stall and the tiles. If it's going well, I stay in there for ages, whistling tunelessly, through the gap where my pipe stem has worn down my teeth like an old limestone doorstep over the years.
But look at this girl, with her sharp black suit and her attitude. Just in case that's too sexy, she's got her hair scraped back so hard, it's like she's had a face-lift, and her jacket's buttoned up tighter than a preacher's pants.
"You don't take any notice of signs?" I say.
"You don't take any notice of e-mails. Or your lawyer doesn't." She smiles sweetly, but there's clarity to her speech that makes me feel she doesn't take any nonsense, and a toughness to her eyes that says: Come on, give it your best, big shot. I'm not afraid of you. I'm not impressed by your so-called reputation. I can see her weighing me up, how I've changed from the most recent catalog photograph, which is thirty years old now. I never did like having my picture taken. What does she see? The thick hair in need of a cut turned from black to white. The hollow cheeks tanned and burnished like driftwood, the silver scar above my jaw. Faded blue jeans and espadrilles, a billowing white shirt, untucked. Maybe she notices the cerulean-blue paint under my fingernails and wonders what I'm painting now. "You gave me no choice...."
There have been a lot of girls like this over the years, journalists or students from Parsons or Columbia, come to kiss the hem of the great man or to try to dig up some dirt. I live at the end of an unmarked coast road on Long Island, well away from the chichi villages and "cottages" the size of civic buildings, but still the most determined find me. Sometimes I think it would be easier if they just stuck me in formaldehyde like the sharks and cows the young guns are showing and exhibited me in MoMA under a flashing red arrow with a neat Perspex sign saying, Gabriel Lambert, artist, next to some of the big abstract expressionist pieces that made my name way back in the fifties.
Annie used to field the girls for me, kill them with kindness—cookies and milk at the old pine table in the kitchen or chicken soup in the winter. If they had any romantic illusions, she soon sorted them out. They were out of the house and on the bus back to New York before they realized they hadn't seen me or my studio. Annie never had any cause to worry, there was only one girl for me. Is only one girl. I've loved only two women in my life—not much, I know, in all my years, but enough for me. The first, well, that didn't end so well. Annie was different. We're like swans, she'd say, bobbing along side by side whatever the current throws at us. There were often a few girls sniffing around, but none for a while. The aphrodisiac power of fame and money never ceases to amaze me—they can make a twenty-year-old girl overlook white hair and haunches sagging like an old sofa. But this one isn't looking at me like that. She's different, I can tell. She looks determined, in spite of her lips, lips that remind me of someone. The bottom one is fuller than the top. It has a crease in it like the indentation of a head on a plump pillow.
"E-mails?" I say. "I don't use a computer."
"What about my letters?"
I think guiltily of the endless letters on creamy laid paper, her looping script. Who uses a fountain pen these days? When she kept on writing, I got my lawyer to scare her off, but that didn't work either.
"I assume you got my last letter?" She's talking slower now, like she's thinking maybe I'm deaf or doolally. Well, I can use that to my advantage. I got your letter, sure I got it. Saved your ace till last, didn't you, missy. I hear the sounds of them tidying up the lunch behind me, my great-grandchildren's laughter a bubbling brook, Tom ragging on his brother, the soothing tones of my granddaughter trying to smooth things over.
"Letter?" I make my hand tremble as I rub my brow. "I'm sorry, dear. I get a little confused these days...." It's working. I see her face soften. She was expecting an angry old man of art, but I've got a few surprises of my own.
"This isn't personal, Mr. Lambert, I want you to know that. The editor commissioned me to write the article because of my great-aunt Vita."
"Vita?" I say vaguely.
She bites her lip in impatience, but her voice is steady. "I just want to get my facts one hundred percent straight. I promise, I won't take much of your time."
Facts? Silly girl, there are no facts in a situation like this, only opinions, old memories, and they change over time like a painting fading in sunlight. The girl tries to peer through to the shadows of the cottage, but everything is thrown into darkness by the bright rectangle of light from the open terrace overlooking the sea. Maybe she can make out the Flos Arco lamp curving over the table like an attentive servant or our two old Jacobsen chairs overlooking the sea, the tan leather shaped to our bodies like eggs to chicks. Annie is there. I feel the certain, reassuring weight of her presence behind me, the south to my north like the counterweight of a compass needle.
"You are Gabriel Lambert?"
I hesitate. The trouble with telling lies is you have to remember them. They don't come naturally to you like the truth. You have to hold all the strands together. "Who wants to know?"
"I'm Dr. Cass." She offers me her hand, and grudgingly I take it. It's as slight as a bundle of reeds. "But please, call me Sophie." Of course. Sophia, the wise one. She thinks she's wise, you can see that. She's young enough to think she knows everything still. Life hasn't knocked her edges off yet. Give her time. Pretty soon she'll realize that true wisdom is knowing that the more you learn about life, the less you realize you know. I feel stupider now than I ever did at twenty, and she can't be much more than that.
"Well, 'call me Sophie,' you don't look like a doctor."
"Of art history," she says. "I write arts and culture stories for The New York Times."
"So you're a hack?"
"I prefer investigative journalist."
"Do you have a card? You can't be too careful."
"Of course." I was hoping she'd pick her bag up so I could lock the door in her face, but she doesn't budge. "As you know, the paper plans to run my piece on you..." Her voice trails off as she squats down and shuffles through her bag. She hands me her personal business card—expensive, engraved, naturally. The letters of her name are like Braille beneath my fingertip. As she looks up at me, I notice her knee is poking through a hole in her black stockings, I see smooth white skin and torn flesh.
"What did you do?"
"Sorry?" She hooks a hank of blond hair behind her ear. There's a sharp chrome spike pinning it in a bun at the nape of her neck. "Oh, my knee? It's nothing. I fell over on the road up there."
"You should take them off."
"I beg your pardon?" She squares up to me, full of righteous indignation.
"I only meant you should get some sea air on your skin, wash off the blood." I kick her bag clear and pull the screen door shut. "You've had a wasted journey."
She slams a manila envelope up against the screen. She's written on it: Vita, last paintings. "Like I said. I won't take up much of your time."
Fear chills me, I feel goose bumps as the hair rises at the nape of my neck, on my arms. Vita. One glance at the triumphant look on the girl's face and I know if I shut the door on her, she'll still be there on the porch come twilight, so I edge the screen open and take it from her.
"I just want to ask you what you remember about these photographs," she's saying as I slip the thick, glossy black-and-white images she's probably sneaked out of some art history library from the envelope. Sure enough, there's an official-looking stamp on the back: Do Not Remove. Naughty girl. "All I know for sure is that Vita was with you in the Languedoc during 1939 to '40, and these photos were taken before the fire...." She's clever, this kid, coming at the subject sideways, not saying overtly what she knows and I know.
An old drinking buddy of mine told me a story once about a guy who turned around to him in a bar on Canal Street and punched him in the throat. No rhyme or reason, no warning, just wham. That's how I feel right now as I turn over the photographs and see what Sophie must have seen. Oh, dear God, it's happened. I close my eyes. I've feared this moment my whole life.
"So, I was right." Sophie smiles, her gaze clear and sly. I see it, shimmering there in my mind's eye like the Cheshire Cat's. "Where shall we begin, Mr. Lambert? The Château d'Oc?"
"I can't... I don't want to talk about that, not yet."
"We can come back to Vita." Her words whisper around my mind the way the wind runs its invisible fingers through the trees. The years tumble away like dry leaves on the breeze.
"Why don't we begin with Marseille, then? Tell me about Varian Fry, Gabriel. Tell the story of how you met Annie."
Marseille, of course. It all began in Marseille.
SEVEN
MARSEILLE
October 1940
GABRIEL
It was October 1940 when I arrived. As I close my eyes, I can see it again, the gray hills, the lush green of the palms. I'd paid some guy in Arles for the name and address of the American angel all the refugees were talking about. When I got to Marseille, I discovered I could have found him for free—everyone knew Varian Fry. Someone told me he had moved offices. He said I should head up to the American consulate and ask there. So I did. It looked like a castle. You could tell you were heading in the right direction because the closer you got, the fuller the tram became of refugees. I'd sold Vita's car the morning I arrived, because I didn't want to draw any attention to myself, driving around in some cute little red cabriolet. How I made it from the Languedoc to Marseille without having an accident, heaven knows. Sometimes I think I have a guardian angel watching over me.
I stood up to give my seat to an old woman on the tram. It was packed by the time we reached the consulate, people hanging on to one another. There's nothing more pitiful than seeing good folks trying to keep their standards up. Men and women who haven't had a decent bath or meal for weeks, who still button up filthy shirts and pin down crushed hats on hair that needs a wash. I don't suppose I looked any better.
It was beautiful out there, with the plane trees and the sound of the sea. There was water rushing through the ditches. To be surrounded by such beauty in such desperate circumstances was worse, almost. The consulate was like a castle in a fairy tale, and you could see the hope in people's eyes as they traipsed up the white steps. I stood in the blue shade of a café awning—I think it was called the Pelikan—and let the crowds go ahead. Each face bore the same expression. Everything—all the fear of arrest, of concentration camps and the Gestapo, of every machine-gun bullet dodged on the hundreds of miles they had walked—had led to this. They were paralyzed with fear at the thought of staying in France and terrified by the notion of leaving.
As luck would have it, just as I was asking some hard-looking receptionist at the front desk for information about Fry's organization, the American Relief Center, I noticed a tall, kindly-looking man watching me. The ARC had a near mythical reputation for the way they could help refugees escape to safety, and my desperation to get there must have been written all over my face.
"The consulate has no connection with the ARC," the blonde was saying. She was chewing gum and I caught a glimpse of bubblegum pink between her canine teeth. Her hair was brittle and bleached. Blue streaked her eyelids—she couldn't even be bothered to look me in the eye. She wore a tight sweater, and she had the kind of conical breasts that looked like they could do you an injury if you dared go in for a clinch.
"I'm sorry to bother you," I said, and started to walk away.
"Say..." The man pointed a sheaf of papers at the folding easel strapped to my bag. "Are you an artist?"
"Yes, sir."
He ushered me outside, past the snaking queues of people. "Pay no notice to her," he said. "Miss Delapré does a hard job fielding all these people day after day." He smiled down at me. He had an unusual face for such a big guy—apple cheeked and radiant. Stick a white beard on him and he could have been Santa Claus. I guessed he must be around six feet four, six five. His thick, round glasses caught the sunlight as he turned and pointed back down the hill. "Why don't you head down to rue Grignan? You can't miss the ARC—"
"Just follow the queues?"
He smiled and shook my hand. "You're catching on. The reception is open Monday, Wednesday, and Friday mornings. Tell them the vice-consul, Bingham, sent you. Good luck."
* * *
It's amazing, looking back, how people queued without question. We were all terrified, jittery, and yet everyone waited with the dull resignation of sheep. I've seen animals herded into stalls on the way to a slaughterhouse—one whiff of blood and they panic, scrambling over barriers, crushing the weaker ones beneath their feet. It wasn't like that at all, not yet. People queued to save their lives like they were waiting in line at the baker's. I waited all morning to get an interview at the ARC office, and I can tell you the atmosphere was benign, even calm. There was a young American guy on the door. He had some uniform on, but he looked more like a matinee idol. When there was a bit of pushing and shoving, he'd hold up his hands and say in a southern drawl: "They-ah now. Take it eas-eh, fellah. Evrabody gets his turn." His French was worse than my English at that time, but he made himself understood, and his uniform looked official enough. He was charming and at ease with himself—the French say bien dans sa peau. Once in a while he'd step out onto the pavement for a cigarette or to blow a few bars on his trumpet. Just as I shuffled to the head of the queue, he appeared, the mouth of the instrument swinging into view from the dark doorway like a golden halo. He leaned back against the doorjamb and crossed his right leg at the ankle. I nodded along as he played, and he glanced across at me, his fingers working the valves.
"Cigarette?" I said, offering him one when the tune finished.
"Thanks." He cupped his palm around the flame of my lighter. "You play?"
"Me? No, I just like jazz."
"Man, I've played with some of the greats...." I liked Charlie—that was his name. One of nature's gentlemen. I found out later he'd been a professional wrestler in the States, so when he ended up in Marseille after leaving the American volunteer ambulance corps, Fry took him on to keep order at the ARC. Seemed to me he could just as easily have earned a living as a performer. He told me about jamming with musicians who were like idols to me. "And Louis Armstrong said to me: 'Charlie—'"
"Blow, man, blow," a blond girl cut him off. She stepped out onto the pavement with a clipboard.
"Heck, Mary Jayne." He frowned. "That's the best bit of the story."
"And we've all heard it a hundred times," she said. "Are you next?" Her gaze was direct and unnerving. I noticed her hands—they were surprisingly large and strong for a young woman. I had the feeling she didn't suffer fools.
"Yes," I said, and stubbed out my cigarette on the pavement. I slung the bag over my shoulder and followed her inside. Each desk was full, young men and women talking in hushed voices to refugees. The office was cramped, but through the door at the end I could see a farther room. A young man, taller than the rest, with thick dark hair was pacing up and down. He was one of those people who seem to spark with energy. In a chair at his side, a blond man lounged, his arm resting on the desk, his fingers leisurely tracing the pattern of the wood as he talked.
"Please sit down," the girl said. "Why don't you start by telling me your name?"
"My name?" I blinked. "Gabriel Lambert." She wrote it down. "I need to see Varian Fry."
"So does everyone," she said with an edge to her voice. "May I see your papers?" From my jacket pocket I pulled out my documents. She flicked through them, her gaze traveling from the photograph to my face. I felt the saliva in my throat dry. "Fine," she said, jotting down some notes. She handed them back to me, and I took a breath.
"Now, what makes you think you qualify as a client of the ARC?"
"I am an artist," I said. "Perhaps you have heard of my work?"
She shook her head. "Unfortunately, my colleague Miss Davenport is not here." She leaned toward me. "Miriam has the best eye around here. She was studying art history, you see." She sat back and folded her arms. "Do you have some work you can show us, perhaps?"
"Of course." I had taken the precaution of bringing a few satirical sketches and clippings from an exhibition catalog. I handed these to her.
"These are lovely," she said, looking at the paintings. I saw her smile grimly as she looked at a cartoon of Hitler. "Forgive me, though." I caught the tiredness and impatience in her voice as she tossed the clippings onto the table. "We have rather a lot of people coming through here, pretending to be who they are not. Would you mind terribly doing a quick sketch? The Vieux-Port, perhaps? Miriam always asks people to go and sketch the boats."
"May I?" I said, taking the pen from her. She offered me a sheet of paper. "I am known for my figurative work," I said, glancing at her before deftly sketching her face. My hand was shaking. "I don't do boats."
"Really?" She bridled at my arrogance, but I saw her lift her chin a little as she realized I was sketching her.
"Does everyone have to do this?"
"There are many people pretending to be artists, desperate for our help. We can't assist everyone. I mean, our remit changed somewhat. Originally we were helping only artists and intellectuals. That is still our priority, but my colleague Monsieur Hermant suggested we should give aid to a few of the most deserving ordinary relief cases as well." I liked that about Mary Jayne. I found out later it was she who was helping the so-called ordinary refugees by bankrolling what they called the Gold List, but she never bragged about it, not once. She was a wealthy young woman who had been swanning around Europe in her own plane, so when France fell to the Nazis, Mary Jayne could have left Paris and run on home to the States, just carried on living her charmed life. But she made her way to Marseille and stayed around to help.
"So, which am I?" I asked her, handing over the sketch. In a couple of minutes, I had produced something good enough to fool the girl, anyway.
"An artist, clearly." She flashed those big old blue eyes at me. "Very flattering, Mr. Lambert," she said, glancing from the sketch I handed her to the black-and-white clipping from the exhibition catalog. "May I keep this?"
"Of course," I said, inclining my head. "I really would appreciate your help, Miss..."
"Gold. My name is Gold." Of course it was. Everything about this girl was clearly a cut above. I wouldn't have been at all surprised if she had a hallmark stamped on her somewhere. I shook her hand. "I will discuss your case with my colleagues. Perhaps you would be kind enough to come back on Monday morning?"
"As I said, I have enough money to pay for my passage to America, I just need your assistance with visas...."
"Yes, I've noted it all down. There is one other thing," she said.
My heart fell to my boots. "Yes?"
"We only help people who are known and trusted by us or our clients. We have to be on the alert for Nazi and Vichy agents, you do understand? Do you have anyone who can vouch for you in Marseille?"
Oh God, so close, so close. I thought frantically. "Vice-Consul Bingham," I lied.
She looked up in surprise. "You're a friend of Harry's?"
"More of an acquaintance." I leaned toward her and fixed her with my gaze. "I can't say too much. It really is imperative I leave France immediately," I said. "I've been an outspoken critic of Nazism for many years." I pointed at the political cartoon.
"I do understand, Mr. Lambert," she said, lacing her fingers together. She leaned toward me. "But you must understand, too. Every person who walks through this door is scared for his, or her, life. We have to choose who we help very, very carefully."
* * *
I saw Miss Gold again late Sunday night, in a bar in the Vieux-Port. I was surprised to see her in a joint like that, but she looked as happy as a debutante at the Ritz. A slim, dark guy with round glasses escorted her onto the dance floor. Her face was all lit up with love like Christmas as they danced, but he looked hard, I can tell you, lean and mean, not the kind of fellow you'd expect a rich girl like her to go for. But then, people can be funny like that sometimes. The bar was noisy with voices and laughter, and the band had to fight to be heard. Miss Gold and her man were dancing in the middle of the old parquet dance floor, dark figures moving around them. The dim red light lit up her blond waves, blushed her cheeks like a fallen angel's. When I think of Mary Jayne, that's how I remember her, with a smile on her lips as they turned slowly on the spot, lost in one another.
EIGHT
MARSEILLE
1940
VARIAN
Varian Fry struck his gold Dunhill lighter. The flame licked the soupy night air east of the Vieux-Port, illuminated his horn-rimmed glasses beneath the brim of his dark homburg. The signet ring on his finger glinted. The last warm evenings of a Saint Martin's summer had lingered, but now the mistral swept down from the Vaucluse mountains, a chill whisper in the ear of the gilded Virgin of sinners and sailors atop Notre-Dame de la Garde on its way down to the city and the sea. The wind rustled through palm and plane trees, bringing the metallic tang of the week's catch from the sluiced-off market stalls on the Quai des Belges. That night they were undertaking the third attempt to help a number of the refugees in greatest danger escape to Gibraltar by boat, and it was the biggest risk they had taken yet. After two failures, this has to work. It will work, he told himself again, his stomach knotting with fear at the thought of discovery by Vichy troops or the Gestapo.
"Thanks." Beamish leaned in to the flame, lighting his cigarette. His calm impressed Varian. You would not want to play poker with Beamish. The way he leisurely flicked his cigarette with long, dismissive fingers reminded Varian more of the studied nonchalance of a boy at a dance than of a man arranging one of the most daring escapes of the war. Varian wondered if he was scared, too.
"Are you sure you weren't followed?" Beamish said.
"Of course. I may not be as sharp—"
"Je me débrouille." I look after myself. A grin illuminated his face. That smile was why Varian had nicknamed Albert Hermant—aka Hermant the varmint, as Miriam and Mary Jayne called him—Beamish. A demon of ingenuity with Puck's smile, that was how Varian described his best friend and right-hand man in Marseille. Who would guess that Beamish had been born in Berlin and arrived in Marseille via the Sorbonne, the London School of Economics, and a doctorate at the University of Trieste? He had fought for the Republicans in the Spanish Civil War, but when Spain fell to Franco's fascist forces, Beamish ended up in France, a refugee himself. Now, he was putting his intellect and battle-hardened street smarts to good use helping the ARC. Bound by a common intelligence and drive to save as many refugees as they could, Varian and Beamish had become fast friends.
"I'm not a fool, though. No one was tailing me." But it would have been easy enough. Varian had strolled beneath the blue-painted streetlights of La Canebière with the natural assurance of a Harvard man. His long-legged, confident stride, even his height, marked him out among the teeming crowds of refugees clogging the dark arteries of the city. His anxiety was hidden deep. He wore his learning like armor, a sword and a shield, his nationality like a cloak—I am untouchable. I am the child of a great and neutral country. I am an American.
Now, as he sheltered with Beamish in the shadows on the pier, he had a small valise at his side. Somewhere in the darkness, a ship's foghorn sounded. Until this evening, it had always amused him that the cannons of Fort Saint-Jean pointed inward, as if to keep the inhabitants at bay. Now, he felt the harbor had him at gunpoint, and the dark bulk of the Île d'If made him think of the Count of Monte Cristo, of capture. His gut instinct told him something was wrong. Varian slipped his hand inside his coat and winced.
"You need a holiday," Beamish said. "At least you look like a tourist on vacation."
"Good, that's the idea." Varian checked his Patek Philippe. "The train leaves for Tarascon at midnight."
"Make sure you're on it. The flics can't hear you were anywhere near here tonight." As he spoke, Beamish's gaze followed a group of figures lurching along the dock. "Don't worry, it's nothing."
The door of a shed on the quay near the lighthouse eased open a crack, and Charlie poked his head out. "Hey, boss, we're all set," he called quietly. His warm southern drawl was reassuring, but his fingers worked nervously, as if he were playing his trumpet.
Varian looked around, then strode over and slipped inside, his eyes adjusting to the darkness. He made out the shadowy figures of the refugees, the pale, exhausted faces of British prisoners of war on the run, Spanish and Italian Republicans, writers, artists, and among them the towering figure of Georg Bernhard, his white hair like a cloud on a mountaintop. He stepped forward and shook his hand. "Good luck, Bernhard."
"Thank you, Fry."
"I'll leave you in Charlie and Beamish's capable hands. The boat will be here in an hour. You'll be in Gibraltar before you know it." The man's intelligent, haunted face touched him. For once, the editor of the anti-Nazi newspaper Pariser Tageblatt had lost his natural assurance. Number three on Hitler's Most Wanted list, Varian thought. I'd be worried, too. Two other ARC clients—the German politicians Rudolf Breitscheid and Rudolf Hilferding—were numbers one and two. As leading members of the Social Democratic Party, they were in grave danger. They think they are untouchable, sitting leisurely in the same café every day when they should be in hiding, Varian thought. He had given up trying to warn them to be more careful. These great men believed the so-called unoccupied zone was safe, but Varian knew better. No one is safe. "That couple ah sons ah bitches ah just asking for trouble," that's what Charlie said the other day about the pair of them. But it's up to us to get them out, too, Varian thought.
The hot bovine bulk of the dark figures in their winter coats, the muffled sound of shallow breathing in the claustrophobic shack, unnerved him. "I'll see you soon, in New York," he said, and stepped out onto the dock, closing the door.
"Sometimes," Beamish said quietly, "I think you say that as much to give yourself confidence as them."
* * *
Striding on through the Vieux-Port, Varian imagined the cellars and subterranean smugglers' tunnels beneath the steps of his polished brogues. He sensed the hidden layers of the ancient city, the Phoenician foundations and Roman walls buried below. He glimpsed the gilded Madonna on top of Notre-Dame de la Garde. Once this city worshipped Astarte, Venus, he thought, gazing up at the night sky, trying to locate the planet as he walked on. He loved the age of the place, the sense of history. He had taken a boat out before the weather turned cold to try to look for the cave of Mary Magdalene in the cliffs near the city. "I have found him whom my soul loves," that's what she said.
Varian glanced up at a cry from an alleyway, the heavy thud of a man's body hitting the pavement. He heard footsteps racing away, splashing through dark puddles oily with gasoline. Lines of washing festooned the alleyway, like dark flags in the moonlight. Varian paused, wary of getting involved. It seemed to him that half the people in Marseille were gangsters and the other half wanted to be. The man was already on all fours, shaking his head as he struggled to his feet, silhouetted against the green light seeping from a bar.
Varian slipped the hip flask from the pocket of his overcoat and knocked back a slug of cognac. For a moment he longed to be back home, in New York, with his wife, Eileen, just reading a book by the fire or in the comfort of his own bed. My old poo-dog, he thought. His face softened with sadness at the thought of the letter she had posted to him in Marseille the day before he'd left America. She hadn't the nerve to talk to him face-to-face. Eileen had asked him to bring back a war orphan with him, a child they could adopt and raise as their own. What does that say about us that she couldn't talk to me, and I've been unable to reply? She'll think I've just ignored her. Varian walked on and took another hit of cognac. She doesn't understand, can't possibly understand what I am going through here, how it's changing me. He was so caught up in work, he hadn't even noticed he had missed the day planned for his return flight to New York. He thought of Eileen's latest letter, coolly asking if he was ever planning to come home. I can't leave, he thought. Not yet. They need me. They all need me. He raised his hip flask and hesitated. Or do I need them?
Varian felt as if he had lived a lifetime in Marseille already, but it was only a couple of months since he'd arrived at the Gare Saint-Charles with $3,000 strapped to his calf and a list of two hundred of the greatest creative minds in Europe to rescue. The newly established Emergency Rescue Committee in New York had needed someone on the ground in France to help the refugees to safety. When no one else stepped forward, Varian volunteered. He was an able multilingual journalist and editor, but he had no experience of relief work. Alfred Barr, an old Harvard classmate and now director of the Museum of Modern Art in New York, had helped compile the list of the most important refugees in danger. Varian's was matched almost name for name by the Nazis' blacklist of targets, a deadly mirror image. How many have we ticked off so far? he thought, totting them up as he walked. Not enough. Not nearly enough. He had thought it would take a couple of weeks to spirit his clients out of Marseille, but his work was only just beginning. With no official help from the U.S. consulate, Varian had soon realized the only way out of France for his clients was through illegal and secret routes.
The endless stream of hopeful, desperate faces filled his mind, the refugees queuing outside his cramped hotel room at the Splendide and now at the ARC. I feel like a doctor during an earthquake half the time. The threat of Article 19, which directed the French to "surrender on demand" anyone sought by the Nazis, hung over Marseille like the sword of Damocles. Fry thought of some of the petty crooks and gangsters they had no choice but to work with to keep the ARC afloat. Dealing with people like that, it's only a matter of time till someone betrays us. It felt to Fry like the very air of the city was alive, crackling with fear and suspicion. Thank God for my team, he thought, secure in his trust for them, at least, and that the outwardly respectable relief center's activities giving subsistence allowances and advice about visas to refugees was hiding their more clandestine work, spiriting people out of the country.
"It's essential that we help André Breton and his family," Miriam Davenport had said during the last meeting, leaning forward into the lamplight. "He's more than a poet, and the leader of the surrealists. He's the epicenter of an entire generation of European intellectuals...." Varian listened quietly as she argued André Breton's case with passion, her left hand beating the palm of her right. He was proud of Miriam, how the young art history student had learned to make her voice heard among the men. She had been one of the first to join his staff, only two weeks after he arrived. When the Paris universities had closed their doors after the German invasion, she had made her way to Toulouse. There, a refugee had told her the only chance for them was "to be wrapped in the American flag." Miriam resolved to do just this and hurried to Marseille to help any way she could. Her good nature and intelligence had proved invaluable to Varian, and it was thanks to her introductions that he had recruited several key staff, including Mary Jayne. Miriam had been afraid to speak up at first, until Beamish took her aside one night and said, "Listen, these guys don't know any more about relief work than you do. If you don't argue for your clients, they are going to be shipped off to the concentration camps or killed." After that, she fought for every case. Meeting her was a lucky break, Varian thought. He had needed someone with languages to help with secretarial work and interviewing refugees. Now, she was indispensable, even putting her academic skills to work discerning which of the would-be clients really were artists and which were hoping for a lucky break.
As Miriam read on from Breton's letters of recommendation, Varian looked around the crowded table. Among the many faces, he searched out those he relied on the most. Daniel Bénédite handed the secretary a pile of files as she tidied up for the day; Danny's attention was on Miriam, and Varian could see he was digesting her case with the sharp focus of a man used to working in the prefecture of police in Paris. The Bénédite family were friends of Mary Jayne's, and she looked on Danny as a brother. When the Bénédites had fled to the south of France after the Occupation, Danny, his wife, Théo, and their small son headed to Marseille. Tall, slim, and dark, with intelligent, kind eyes that missed nothing behind his thick glasses, Danny ran the ARC with incredible efficiency. Just as Beamish was Varian's closest friend and ally in the more clandestine work of the organization, Danny was relied on to run the official work of the center, giving aid and advice to refugees. Behind him, Varian could see another of Mary Jayne and Miriam's recruits. Justus Rosenberg, or "Gussie," as they affectionately called him, was the fourteen-year-old office boy the girls "adopted" as they fled south to Marseille. His innocent face had proved useful messengering documents across the city. Now he sat guarding the entrance door with Charlie.
They were all adrift, Varian realized, all on the run themselves, yet they were risking everything to help people whose work had changed the world. Whose work will change the world, he corrected himself. I'm going to get every name on that list out of France if it kills me. Some days it felt as though it just might. But then, with his team, every day miracles seemed possible.
"I have another recommendation," Miriam said, sorting through the files. "Mary Jayne interviewed a chap—Gabriel Lambert. A highly respected painter, more art deco than avant garde...."
* * *
Even Mary Jayne, he thought now, whose sheer privilege and... Varian frowned as he tried to pin down what it was that annoyed him so much about the beautiful young American heiress. Was it her money? Her taste in men? Goddamn that punk Couraud. The thought that Mary Jayne's boyfriend could betray them at any moment, could bring too much police attention to the organization, troubled him. He bit the inside of his lip as he walked on. He had to admit, even Mary Jayne had been invaluable. She's given freely of her fortune and her time. Thanks to her, a second list—the Gold List—had been drawn up, and the $7,000 she had given to the rescue committee had saved hundreds of lives already. She risked her own, too, Varian had to admit. He had been impressed that she managed to get four prisoners of war out of the camp at Vernet. Though I'd rather not think how she persuaded the camp commander.
Now, as he wove through the back streets toward the station, he thought not for the first time that the city was half cesspool, half asylum. Varian marked off in his mind the brothels and fleabag hotels where their refugees were hiding out, waiting for visas and passports. Brothels were safe because the police were bribed not to raid them. Elsewhere in the city, the refugees had to take their chances.
Varian switched his suitcase to his other hand and shouldered his way through the crowd outside a bar. Demobbed soldiers jostled on the pavement—a group of Zouaves in Turkish trousers and Senegalese fighters in bright turbans was arguing. He saw the flash of a knife and skirted around, adrenaline pulsing through him. His stomach lurched with hunger at the sight of a woman spooning bouillabaisse from her bowl. He thought of his empty hip flask and checked his watch. There was just time to get it refilled, so he pushed his way through to the bar.
NINE
MARSEILLE
1940
MARY JAYNE
Mary Jayne Gold and Raymond Couraud danced close, lost in one another on the crowded dance floor. She felt the pressure of his hand at the base of her spine, the warmth of his palm through the thin silk of her blouse. His cheek was smooth against hers, freshly shaven, and she breathed in the scent of him. The small room was dark and busy, figures pressed up against the bar, a hum of conversation, the chink of glasses and bottles punctuating the jazz.
"Don't let's ever get old, Raymond," Mary Jayne said, watching a gray-haired couple sitting in silence at the edge of the dance floor, their slumped backs twin c's of defeat.
Raymond followed her gaze. "We won't ever be like that, bébé." He glared at a young man with black hair and pale blue eyes watching them from the bar, staring at him until the man downed his drink and left. Raymond eased her closer to him, his lips grazing her jaw, her neck. "We'll still laugh, and fight, and make love. There will always be passion. I'll love you forever with all my black heart."
"Forever is a terribly long time," Mary Jayne murmured. "What if we don't have forever, Raymond?"
"Then we have now, we have tonight."
"Sometimes I think that's all I am to you. A good time, a meal ticket—"
"You're my girl, it's as simple as that."
"And you're my bad, bad boy?" Her laugh was rich, throaty.
"I think you like that, Mary Jayne. I think you like it that your friends say I'm no good for you. You like a little danger, no? Something your money can't buy." She felt the lean muscles of his shoulder, his back, flex beneath her hand. "Tell me you love me."
"You underestimate me, Raymond. I see what you are capable of, I see the strength and the bravery hiding in that black heart of yours. You're my 'diamond in the rough,' and I know plenty about diamonds, trust me." Mary Jayne laid her head against his collarbone. "I adore you, darling boy."
"Tell me you love me."
"Not here...."
"Why won't you ever say it?"
She lifted her head, gazing into his eyes. "Darling, it's impossible for us to be together, you know that. Now you've left the Foreign Legion, you shall go off to Britain with de Gaulle and fight with the Free French, and I shall..." She paused, wondering what the future held for her, for them. It was such a short time since she had met Miriam in Toulouse, since they had traveled to Marseille together and met some young Americans, and Raymond. There was a dangerous, masculine energy to him that had drawn her the moment they met, a self-assurance verging on arrogance. She felt the intensity of his gaze now, as he waited for her answer, and a heat rose in her, responding, helpless to resist him.
"You will go back to America, and forget me." Raymond pulled away from her, the dim light gleaming on his dark hair, glinting on his round glasses.
"You can write to me," she said, stepping close in the crush of the dancers. Her lips brushed his ear. "You can tell me about all the battles you have won, and all the hearts you have broken."
"You will marry some rich idiot—" He wouldn't look at her, his hands clenching into fists.
"Why would I marry?" Mary Jayne laughed, a breath against his neck. "I don't need a husband. I have money of my own."
"Then marry me, for love." The raw passion in his gaze caught her off guard, and Mary Jayne felt her stomach free-fall.
"I told you, I don't plan ever to marry," she said lightly, hiding the effect he had on her. "I shall travel the world with Dagobert for company."
"You love that dog more than you love me."
"You dear, sweet boy." She pouted, imitating him. "Don't sulk." He kissed her then, claiming her, his hands in her hair. Mary Jayne broke away, placed her fingertips on his lips as she glanced around, self-conscious. "Darling, stop. Not here."
"I want you," he said, holding her close. "And you want me, I know you do." Raymond took her hand in his, kissed the palm, his gaze not leaving her. He laced their fingers together, and they danced on, cheek to cheek. She felt alive with him, her body coursing with desire. "Will you forget me? Will you forget what we have?"
"Never. How could I?"
"Stay with me tonight." His lips brushed her ear, but Mary Jayne didn't answer. She was looking toward the bar, where the waiter was handing Varian his filled hip flask.
"Good heavens, what's he doing slumming it down here?"
Raymond frowned. "Probably looking for a whore."
"Don't be so beastly." Mary Jayne sauntered over to Varian.
"Mary Jayne?" he said in surprise. "What are you doing here? It's not safe...."
"She is safe with me," Raymond said, slinging his arm over her shoulder.
"Good evening, Couraud," Varian said. The silence strained between them like an overtightened violin string.
"Darling, would you get me a glass of wine?" Mary Jayne said finally to Raymond.
"Sure, bébé." His eyes narrowed. "Dance with my girl, Monsieur Fry, feel free. I will look after your little bag." He gestured toward the dance floor and took Varian's valise and hat, tilting the homburg onto his own head. Varian clenched his jaw.
"Don't," Mary Jayne whispered, pulling him toward the dancers.
"Why do you let him talk to you like that, like he owns you?"
"No one owns me."
"Bet that's the last I'll see of my case," Varian said, taking her in his arms. They moved easily to the music, his hand resting lightly on her waist.
"Oh, stop it." Mary Jayne sighed. She was pleased to see him. At least with Varian she was on home ground. She knew him, knew his type, there was no need for explanation. In spite of their differences, they were both Americans abroad, and his directness was a relief. There are cat and dog people, she thought, glancing over Varian's shoulder to watch that Raymond still had the suitcase. Raymond is a cat, and Varian is a dog, definitely. Confident, direct... She watched Raymond chatting to a pretty young brunette at the bar. Loyal.
"What are you doing in the Vieux-Port?" she said.
"Just felt like a stroll before my train to Tarascon."
"Nonsense. I know you boys are up to something. You think you're all being so cloak-and-dagger, and trying to keep Miriam and me out of it, but we could be useful, Varian. I mean with the clandestine work, not just the ordinary relief cases."
Varian glanced around him to see if anyone was listening. "Keep your voice down. You and Miriam are doing excellent work, but there are some things that are just too dangerous—"
"For a woman? What about Vernet? Wasn't that dangerous?"
"Mary Jayne, I'm grateful for all you are doing for the ARC."
"But?"
"But I can't in good conscience involve you with some of the more delicate work while you are... while you're..."
Mary Jayne realized he was talking about Raymond. "While I'm sleeping with Killer?" She wanted to shock Varian, provoke him. She saw a slight tremor pass over his face, but he didn't rise to the bait.
"What do you see in him?"
"Something that everyone else doesn't."
"He's a hoodlum, Mary Jayne, a petty crook, and God knows I bet he got his nickname for killing more than the English language."
"It's just a joke that Miriam and I came up with. His accent's appalling."
"You could do—"
"I could do better? You really are trotting out all the bourgeois clichés tonight, aren't you, Varian?" She tossed her blond hair back and looked him straight in the eye. "You want me to be with someone suitable, like you?"
"Would that be such a disaster?"
"Careful, Varian, you're a married man. Anyone would think you're making a pass at me."
"I didn't mean me, you silly girl."
"Girl? We're practically the same age."
"Do you ever take anything seriously?"
"Not if I can help it."
"You need to consider your position. Who's buying dinner tonight, Mary Jayne? Who's going to pay for the hotel room?"
"Stop it." Varian's words had shaken her, but she wasn't going to show it. "Just stop it."
"Be careful," he said as Raymond approached.
"I'm a big girl, Varian." She smiled up at Raymond as he handed her a glass of red wine. "Thank you, darling."
"I have a train to catch. I'll bid you good night," Varian said, taking his case and hat. He pushed his way past Raymond and out of the bar.
"I wish you two would get along," Mary Jayne said, settling at a small table with a red lamp. Raymond took out a silver case and lit two cigarettes, passing one to her.
"Play nicely, you mean?"
Mary Jayne kicked him gently under the table. "It would just make things a lot easier. I..." The words died on her lips as three policemen walked into the bar. They spoke to the bartender and showed him some papers. "Raymond," she said quietly. "We should get out of here. Something's going on. There's going to be a raid."
Raymond exhaled a plume of blue smoke, his eyes closing as he tilted his head back. "We will be fine, bébé."
One of the policemen strode over to their table. "Raymond Couraud?"
"Who wants to know?" he said, taking another drag of his cigarette.
"Are you Raymond Couraud?"
"I might be."
"Papers." The policeman snapped his fingers. Raymond leisurely pulled out his wallet and handed over his identity card. "This is him." He beckoned over his colleagues, and they blocked his escape as Raymond made a break for it.
Mary Jayne cried out as the table went flying, the glass of wine smashing on the floor, the red lamp rolling in an arc, light swinging across their shoes as they scuffled. "Raymond!"
"We have a warrant for your arrest. You are absent without leave from the Foreign Legion," the policeman said, snapping the handcuffs closed.
"No! There's a mistake!" Mary Jayne clung to Raymond, who was panting, his arms held firmly by the other two policemen. "He has his demobilization papers, I've seen them."
Raymond leaned down to kiss her and whispered in her ear, "Forged."
The color drained from her face. "What can I do?"
"Money," Raymond said, "and get a good lawyer."
TEN
RUE GRIGNAN, MARSEILLE
1940
GABRIEL
Mary Jayne wasn't at the office the next morning. When they told me to come back on Wednesday to see Fry, who was out of town, I made sure I was first in the queue. I hadn't gotten a whole lot of sleep in my cold, pestilential room anyway. Bedbugs, or fleas, or both, had kept me awake all night, and I was scratching at a bite on my wrist as a kid with sleep-mussed eyes unlocked the door.
"No one's here yet," he said. There was something about his face, an innocence, that touched me. He must have been only a couple of years younger than me, but there was a goodness to him that I'd lost somewhere along the way.
"Hey, Gussie," Charlie said as he squeezed past into the office. "Good night?"
"All quiet, Charlie."
I glanced along the pavement and saw a tall, dark-haired man striding down the road, his overcoat flapping around him. A little black puppy was trotting along at his side, pulling on the leash. "Good morning," he said cheerily, holding on to his hat as they blew into the office. A few people were gathering behind me now, and it wasn't even light yet. It seemed to take hours for the office to open up. I couldn't bear to think what it would mean if they couldn't help me. How else was I going to get away? My mind filled with visions of police cars, cells, the Gestapo. I could be hauled up for manslaughter, murder, even.... I screwed my eyes shut as I thought of the fire. Vita, I thought, my guts twisting with guilt and sorrow.
"Lambert," Charlie called out, reading from his clipboard. There was no answer. "Gabriel Lambert?"
Something inside me registered he was calling my name. "That's me," I said.
"Say, I remember you. The jazz fan. Well, come on in, fellah," he said, and clapped me on the back. The queue shuffled forward behind me as I stepped into the office. "Head right on through," he said, pointing to the room at the end. I smoothed down my hair and knocked.
"Come in," a voice said. I pushed the door open and found the dark-haired man sitting behind the desk.
"I am Lambert," I said. "I am... I..." My faltering grasp of English deserted me.
"We can talk in French," Fry said with perfect fluency. A young woman stood at his side, handing him paper after paper to sign. "I won't be a moment, do take a seat." He glanced at me and smiled. In the silence, my breath seemed loud. My heart was jolting so hard, I could feel it pulsing in my throat. The puppy yawned and turned a couple of circles before flopping down into a wicker basket by the fireplace. "There," he said finally. "Thank you, Lena. Would you mind closing the door?" He stood and offered me his hand. "I'm Varian Fry. Pleased to meet you, Monsieur Lambert."
"Likewise," I said, horribly aware that my palm was sweating. I quickly wiped it on my thigh before shaking his hand. I was so nervous to finally meet the fabled Varian Fry that as I tried to smile my lips trembled, and I saw a flicker of concern on his face. I tried to calm myself. The ARC was my gateway to freedom, and Fry held the key. It wouldn't look good to be too scared.
"Well," he said, scanning quickly through the file on his desk. I could see my name on the tag: Lambert, Gabriel. "You made quite an impression on Miss Gold." He held up the ink portrait. "This may have had something to do with it. Most charming."
"Thank you."
"But, I checked with Bingham." Fry made a steeple with his fingers, pressed them to his lips. I held my nerve, stayed silent. "As you know, we only help people we can trust."
"You can trust me."
Fry stared me down. "Bingham couldn't place you—"
"We met only briefly." Panic chilled my stomach, and sweat trickled down my spine. "I assure you—"
Fry raised his hand to stop me. "Fortunately, a couple of my colleagues had come across your work in Paris, too."
"I had a certain amount of celebrity," I said, glancing down in a way I hoped looked modest.
"Ordinarily, your case would not be a priority."
"But I must leave, at once!" I cried. "I have spoken out against Nazism."
"So has any sane person," Fry said calmly. "The thing is, you are not, I believe, classed as a 'degenerate' artist by the Nazis. In fact, your glamorous art deco work is—forgive me..."
"Decorative?" I said, challenging him.
"Admired. And the authorship of your political cartoons is not known generally. You're not on the Emergency Rescue Committee's lists."
"I don't understand. I thought this was the American Relief Center."
"It is." Varian pressed his fingertips together. "The committee is our parent organization in New York. It was established to help refugees displaced by the war in Europe. My remit in setting up the ARC in Marseille was to help specific clients in grave danger." Fry glanced up at a noise from the outer office. "Our work has expanded somewhat, as you can see from the queues, but the American Relief Center's priority is to sort through the thousands of applicants and help those in immediate danger...."
Decorative, degenerate? Who cares? I thought as he talked on, desperately concocting any number of degenerate artworks in my mind. "Please, you must help me." My throat was tight. "I lost... I lost everything, you see?"
"Yes, yes," Fry said, pulling a clean handkerchief from his pocket and offering it to me. "There, there, old boy. Don't upset yourself." Was I crying? I felt hot, and dizzy with hunger and emotion. "I understand you have funds for your ticket, and the ARC will help you obtain the paperwork and visas—"
"Oh, thank God!" I leapt forward, and clasped his hand. "Thank you!"
"Please, calm yourself, Mr. Lambert," he said. "Listen, all we can do is wrap anti-Nazis like you in the American flag. It's your only chance." He leaned toward me. "Someone said to me the other day that we are in the export business, pure and simple. We are exporting men and women." He made it sound so easy. "Now, I understand from Miss Gold that you have the finances to take care of your ticket," he went on, "which is a big help. All we have to do is guide you through the lengthy process of obtaining the correct papers. I'm going to assign your case to one of my colleagues. You'll like him, he's a good, steady chap and he'll get you sorted out." Fry leaned forward and tapped the side of his nose. "Don't you worry. We have ways of getting you out of France."
I could have embraced him. What it meant, after all the weeks on the road, after everything that had happened to me, to have this American angel talking so confidently to me, I can't explain. There was something, of course, a small voice of conscience in me that said: You have no right to this kindness. But to be honest, my heart was broken and gone. Then, I was thinking only of myself, and it's only now, after decades of guilt for the way that I deceived that good man, that I will make amends.
ELEVEN
RUE GRIGNAN, MARSEILLE
1940
VARIAN
Varian knew the moment Beamish walked into the American Relief Center that the poet Walter Mehring hadn't been exaggerating. "Baby," as they all called the writer who looked more like a vagabond than a man of letters, had appeared at Varian's hotel room the night before, shaken and terrified. Varian had last seen him in the dark shack near the lighthouse, peering around the bulk of Bernhard's shoulder like a bird with its head tucked down behind a cliff against a storm. Baby should have been in Gibraltar by now, but he told Varian that the boat had failed to materialize, and at two A.M. Beamish had sent them back to their hiding places in small groups so they wouldn't attract too much attention.
Varian couldn't let the refugees trailing out of the waiting room see how exhausted he felt. "I'm sorry," he said brightly, "we've just run out of time for today. Please come back tomorrow."
Lena, Varian's secretary, was still hard at work, her head bent over her desk by the white marble fireplace, blown tulips in a blue vase nodding toward the ledger she was writing in as she handed out meal tokens and a subsistence allowance to a couple whose desperation was etched in every line on their gray faces. She talked calmly to them, as you might to a frightened animal. As the man began to babble, to plead, she looked up and tucked a strand of dark hair behind her ear.
"Il ne faut pas exagérer," she said, smiling at them. You must not go too far.
Good old Lena, Varian thought. She was another one who seemed to have arrived like a miracle. How many professional social workers who speak six languages fluently are there around when you need them?
"Everyone's ready, boss," Charlie said quietly. He seemed cheerful enough, but there was something in his gaze that made Varian uncomfortable. Sharlee looked more like a matinee idol than a doorman, he'd always thought, but his old ambulance uniform gave him something of an official air, and he did a good job of keeping the refugees in order and good spirits as they waited on the steps up to the second-floor offices. The ARC had set up camp above a leather and pocketbook store abandoned by the shopkeeper when the October 3 Jewish Statute forced him out of his job and his home. The man had suggested they use it until they found somewhere larger, and in the early days they did their work surrounded by cases of stock, the warm scent of leather and polish filling the air. A six-foot flagpole with the Stars and Stripes stood in the center of the office like a beacon of hope. Varian knew he was the eye of the storm, that for many of these people he was their only chance, and all he could do was try to look as confident as possible. Once the last of the refugees had gone, Varian walked briskly to his office. One glance at the faces of the men ranged around the table confirmed his fears.
He handed Lena a couple of files with handwritten notes to be typed up and closed the door after her. Varian allowed his head to fall forward and rest against the door a moment. He slid his thumb and forefinger beneath his glasses and rubbed the bridge of his nose. "So?"
"The situation's hopeless but not serious," Beamish said, a line, a joke, from the trenches of World War I they often threw around. He balanced a paper knife on the blotter, spun it on its point beneath his index finger.
"What the hell happened?" Varian burst out.
Beamish sat back in his chair and calmly folded his arms. "It looked like it was going to work, right up to the last minute. We met the captain in Snappy's bar like it had been arranged." Fry knew it well, a seedy joint in the Vieux-Port, a favored haunt of British officers hiding out in Marseille. "Even the Brits thought it was a sure thing. There's no way they would have risked trying to get so many of their prisoners out on the boat if they'd thought the deal was crooked." Beamish paused. "The bastard wouldn't budge without the cash up front."
"The captain took the money? All of it?" Varian felt nauseated.
"He told our man he'd go to get the boat—"
"Forty-five hundred dollars. I thought we agreed—no money until the boat was safely at sea."
"It wasn't our contact's fault." Beamish looked impassive, bored, even, but Varian knew him well enough by now to see how shaken he was. "We both tried reasoning with the skipper. He wouldn't budge until we paid up front. He's probably halfway to Morocco by now." He pursed his lips. "Some of the soldiers tracked down the hoods who set this up for us. They won't be doing any more deals for a while, but there's no sign of the money."
"I'm sure that made them feel better, but it hardly helps us."
"We took a gamble."
"And lost." Varian slumped in the chair at his desk. "Where are the rest of the refugees? Mehring is in my hotel room and refusing to budge."
"Back in hiding," Beamish said.
"And Bernhard?"
"He's safe for now. We have him and his wife hiding out in a maison de passe. The owner believes they are a middle-aged couple having an affair."
"Good. Getting him out of Marseille as quickly as possible is our priority. We could try the F route...."
"I agree, Spain is the best plan, but not quite yet."
Miriam knocked on the door and looked in. "Quel pagaille, it was crazy out there today. Are you going to lock up—" Her voice broke off into a hacking cough.
"I wish you'd get that looked at," Varian said.
"Me? Oh, I'm just tickety-boo," she said, catching her breath. "Varian, I've written to the Bretons to say the ARC is going to help them, and asked them to come to Marseille."
"Excellent. I'll make a note to chase up Peggy Guggenheim about helping them with their passage to New York."
"Say, Mary Jayne and I are going house hunting tomorrow. Any of you want to come?"
"You're still going ahead with that harebrained plan?" Varian rolled his sleeves up.
"I just think it would be a good idea for a few of us to find a little cottage somewhere, out in the country."
"This isn't a vacation, Miriam," Beamish said.
"You think I don't know that?" She folded her arms. "Everyone's exhausted, that's all. It would do us good to get out of the city, and I want to get out of my fleabag hotel."
"Hey, it's not so bad," Gussie said as he sidled past loaded down with client files and dumped them on Fry's desk. "Anyway, you gave me the best room. Oh, the view from that garret."
Miriam burst out laughing and dug him in the ribs. "At least the bedbugs can't be bothered to walk up all those flights of stairs." She scratched vaguely at her wrist. "Are you working late?" she said to Varian.
"There are a couple of things to sort out."
"Don't do anything I wouldn't do. See you in the morning." The lightness of her tone belied her shrewd expression.
Once the door closed behind her, Varian beckoned to Beamish. They spoke quietly, heads close together, as the other men chatted. "We need more funds, urgently."
"I'll go and see Kourillo. He always has a few associates who need to shift their francs out of the country."
"That hood? Can we trust him?" Varian frowned as he thought of the White Russian. Five feet tall at a stretch and with a handshake that always reminded him of an empty glove, he made Varian's skin crawl.
"Do you have any better ideas?"
* * *
Varian unplugged the phone cord after his final call of the day to Harry Bingham at the U.S. consulate and poured a large brandy. You couldn't be too careful. If they were connected all the time, their conversations could be listened in to. That was something I hadn't bargained on adding to my résumé, Varian thought. Classicist, journalist, editor, spy. His task seemed Herculean sometimes. For each person, each family, they helped, several more appeared who were equally deserving. Varian ran through the usual litany of paperwork that each client needed: Passport. Transit visa from Vichy. French exit visa. Entry visa for Spain. Transit visa for Spain. Travel visa for Portugal. Transit visa from Portugal. Travel visa for any other country. The list went on and on. Then they needed a ticket paid in full and a firm sailing date. Varian knocked back his drink. Even his dreams were filled with an endless paper chase.
"Are you ready?" Beamish buttoned up his jacket, and Varian scooped up the papers for the last cable of the day to New York with the list of clients' names they had taken on that day for the ERC to request visas from the State Department. The names of any clients too high profile to risk sending by telegram would be concealed in tubes of toothpaste and smuggled out by escaping refugees to send on to New York.
They talked in low voices as they walked along the dark rue des Dominicaines, their footsteps echoing down the empty street.
"What's the latest with Bernhard?" Varian said.
"I sent a message to our people on the French border today."
"Good." They paused as the blue lamps of the police station came into view. A large rat scuttled along the damp pavement into the gloom of the alleyway. "It seemed so easy a couple of months ago. I just assumed it would go on like that. Those boobs in New York have no idea what we are up against."
"We had a few lucky breaks. We've managed to get several groups out over the mountain passes."
"Not counting poor Walter Benjamin."
"None of us knew he'd kill himself rather than risk arrest on the other side." Beamish paused and listened for footsteps, signaled Varian to walk on.
"Christ, it's unbearable," Varian said. "Any one of the names on my list is a man or a woman who can change the whole course of human history for the better."
"We'll get them out. The rules will change again, you watch. The Nazis have everyone right where they want them. France is the biggest man-trap in history."
"As long as they believe all we are doing is giving refugees pin money to live on, and helping them to obtain official visas..." Varian paused as a car swept past the end of the road. "We've just got to do what we can in the time we have. I'm glad that you suggested we take on ordinary relief cases."
Beamish laughed softly. "Ordinary people, ordinary soldiers, ordinary Jews?"
"You know what I mean."
They had the visa de télégramme stamped and took the cable to the post office near the stock exchange. Varian filled out the cable address—Emerescue for New York. He thought of it as the answering echo to their "Amesecours" in Marseille, two tin cans on a very long string.
It's not Hercules I feel like, Varian thought as he walked in weary silence at his friend's side toward his hotel. I feel more like Sisyphus. Each day we push the boulder up the mountain, and each day it rolls back down again. Beamish pointed ahead. "There you go, home safely."
"Thanks, Beamish." Varian shook his hand. "You know you don't have to walk me home every night. I feel like a blushing girl."
Beamish shrugged. "It's safer if there are two of us." He glanced over Varian's shoulder and pulled him into the shadows of the side street. Varian turned to see a dark Mercedes pulling up outside his hotel and a group of five Nazi officers climbing out, the blue streetlights gleaming on the insignia on their caps. "Give them time to get inside and to their rooms before you go in."
Varian felt his stomach drop with fear. He remembered a conversation with Charlie:
Say, Varian...
Yes, Charlie?
I'm scared as hell all the time.
So am I.
Had they come for him or someone else this time? A knock on your hotel door had always meant room service to him. Now he knew it could mean the end of the world. "I think perhaps Miriam is right," he said. "It's time to find a new place to stay."
TWELVE
FLYING POINT, LONG ISLAND
2000
GABRIEL
"Mr. Lambert? Tell me about the photographs." The girl won't give up. I have to be ready for her. Dragging myself back from the past to the present, to her unflinching gaze, feels like swimming against the tide.
"Look," I say. "I've never seen these pictures before." But of course I have. Quimby took them just before he left the Languedoc in the summer of 1940. I thought I'd destroyed the only copies in Marseille. Trust Quimby to have more tucked away somewhere, the blackmailing son of a bitch. "Where did you find them?"
"It was luck, really. A friend of Dad's in London knew about my family connection to Vita, and he spotted them in an archive among Quimby's papers."
"He stole them, for you?"
"Borrowed, Mr. Lambert," she says firmly.
Nonsense, I know those places, you can't turn a page without a pair of white gloves. As if they'd let this girl bring them halfway around the world. Christ, all this time, these photos have been floating around? "It was just a—well, it was a shock to see Vita's face again after all this time. I can't help you...."
"But you must!" The color rises in her cheeks, a gorgeous bloom. "I'm going ahead with the story whether you like it or not."
"Not."
"Please, Mr. Lambert." She tries to control her frustration. "For me it is about family, as much as anything else. You're the only one who was there with Vita—" She stumbles, realizes what she's said.
"I'm the only one alive, you mean?"
"I don't mean to be tactless." She's too young to know how to handle this. "It must have been a terrible time for you. Everyone knows..." She chooses her words carefully now. "Everyone has heard how you lost your son, and Vita. I just hoped you'd be able to help me find the truth."
I close my eyes for a moment and rub the bridge of my nose. The afternoon sun has burned a vivid red-and-gold corona behind my lids. The truth? I don't even know what this is anymore. Vita, Vita, Vita... Christ. Days go by now, weeks, even, when I don't think of that name.
"I can't help you."
"Can't or won't?"
Damn, she's cocky. "Why don't you just clear off back to the city?"
Good, I've shocked her. "That's not very nice."
"Nice?" I rail on her, then lower my voice. "Who the hell said artists were supposed to be nice?" Look at Varian, at everything he did for the artists whose work he loved. How did they repay him? Sure, there was the odd exception—like Lipchitz, he was the best of the lot and a good friend to Varian—but most of them turned their back on him once they didn't need him. Lipchitz never forgot what Fry did for him, but Chagall wouldn't even sign the print Varian practically had to beg him to put into the Flight portfolio. After all Varian had done for him. I stare at the girl, and she flinches. Maybe there's still something in my gaze after all. I've rattled her. "Let's take a walk."
I usher her on around the house toward the beach. I follow the trail of pine needles to the lean-to where we stack split logs every autumn and the boys have left the Christmas tree. Her high heels are sinking in the sand, and as she pauses to slip them off, I see their red soles flash like a warning. My chest is tight, and once she's walked on ahead, I search in my pocket for my inhaler. I've left it in the studio. I start to panic, but Annie's voice comes to me: Easy, Gabe. That's it, try and relax. Easy now. Breathe. Once my lungs have eased, I catch up with the girl and she turns. "It's beautiful here," she says. "I can see why you love it."
I hesitate. She can't sweet-talk me, oh no. "You came out here alone?" What I want to say is: Who knows you're here?
"I'm a big girl." She takes a deep breath of cold sea air, guileless, and shrugs off her jacket. "When I heard you had a cottage down here on the beach, I was expecting something..." Her voice trails off, aware that she's said the wrong thing again.
"Something grander?"
"I just meant," she says carefully, "something different, what with your success, and reputation."
"It's not much, but Annie and I built this place ourselves the summer we moved out here, 1951." I run my fingertips along the flaking white clapboard as we skirt around the side to the beach path. Every time we had a few bucks, or a new baby, we added on a room. It's higgledy-piggledy, as Vita used to say, but we love it. Now, the house and I have both seen better days. I glance up at the terrace where Annie, my Annie, sits gazing out at the sea, a smile on her lips. Our home, the place we chose to plant our flag. Gabe and Annie, a couple of kids playing house. I always expected at some point it would feel like we were grown-ups, but it never did. Still doesn't. Sometimes it still surprises me to look in my shaving mirror and see an old guy staring back. My hip twinges as I climb gingerly down the wooden steps and shuffle onto the beach. I pick up the stick I turned one winter from its place beside the steps, and we walk down toward the hard-packed shore, where it's easier walking. "We lost everything in France. Once that's happened to you, you realize how little things matter. We never needed anything more than one another." I wave my stick at the sea. "Than this."
"How many kids do you have?"
"Kids?" I laugh. "Our babies are old crones now. My youngest grandson Harry's about your age." Come to think of it, this girl would be just his type. He likes these city girls, tough and polished as hazelnuts, sweet and yielding inside. Maybe I'll get him to drive her back to the station, work his magic. Maybe a distraction will make her forget she hasn't found out what she came here for.
"Harry?" She sounds wistful. Perhaps luck's on my side and she's alone or lonely. "You're lucky, having family nearby. Are they all painters too?"
"My kids? No, they took after their mother, far too sensible. I didn't care as long as they did something vaguely creative. I couldn't have borne it if they'd become bankers and lawyers." I throw her a bone. "My grandson's a painter, though." I know you shouldn't have favorites, but I love that kid. He looks just like me at that age. All our children turned out blond and fair, just like Annie, but he has my olive skin and dark hair.
"Is he any good?"
"It's too soon to tell if he has it in him."
"So how did you and Annie end up here, from France, I mean?" She hesitates, wary now, trying to hide how much she wants her answers. "I'm surprised. You don't even sound French."
At that, I laugh. "It was a long time ago, and it's a long story." I turn to her, the surf crashing against the shore, the sunlight glancing off a mirror mobile spinning in the breeze on the porch. The infernal tinkling of Annie's wind chimes drifts across to us. "Listen, I'll cut you a deal. If you leave me and Annie out of the story, I'll tell you anything you want to know about Vita." She waits silently, until I give in. Patience will serve this Sophie well. It's amazing how many people will talk to fill a silence and say more than they intend. "Damn it. You want me to tell you everything, don't you?"
She nods. "Tell me about France. Start at the beginning."
THIRTEEN
MARSEILLE
1940
MARY JAYNE
Mary Jayne and Miriam settled into their seats as the blue-and-cream tram lurched away along La Canebière toward Aubagne, sounding its foghorn. Dagobert, Mary Jayne's large black poodle, circled once, twice, then flopped down in the aisle beside them, his snout on his paws. The landscape opened up as they headed east toward the suburbs; gray limestone hills, palm trees, and the dancing light of autumn sun on the water trundled past. "I do love the smell of these old things," Mary Jayne said, checking her reflection in a platinum compact. The air was rich with the scent of charcoal burning. Sometimes the city seemed to smell like one big fireplace, so many people had fitted charcoal burners to their vehicles now there was little gasoline. She dabbed at her nose, clicked the compact shut, and slipped it into her handbag.
"Boy, what a swell day for a trip to the countryside," Miriam said. There was a fresh breeze from the sea, teasing gold leaves from the trees. The Cimetière Saint-Pierre passed by, and the tram headed for the hills.
"It's just good to get out of that place."
"You're not still smarting about Varian, are you?"
"Varian? I couldn't give a damn about him." Mary Jayne snapped her bag closed, and her husky voice lowered to a growl. "You'd think after I managed to spring those four guys from the Vernet camp he'd treat me with a little respect by now." She bit her lip. I know men like him, she thought. I bet he reinvented himself at Harvard, started eating burgers with a knife and fork and took up smoking just because it looked elegant. To hell with him. Stuck-up dilettante, that's what he thinks I am. Spoiled little rich girl.
"I'm sure he does respect you, he just doesn't show it. I know some of the refugees think he's buttoned up, but it's just a front to give them confidence—"
"Jeez, Miriam. Respect? Don't you get it? Men like him don't know what to do with a woman if you're not in their bed, typing their letters, or keeping their house. The ARC is an all boys' club. I said as much to his face the other night."
"Oh, Mary Jayne, you didn't?"
"Well, why not? They think they are being so clever hiding what they are up to from the girls, but we all know they are doing something crooked."
"I think he's rather wonderful. Don't you think he's attractive?"
Mary Jayne laughed briskly through her nose. "Not at all, my dear. I prefer more macho types."
"Like Killer?"
"Raymond is... He's not what he seems."
"Well, neither are you," Miriam said. "You succeeded where letters from the ERC and the American consulate failed. You still haven't told me exactly how you persuaded the commandant to let those four prisoners out of Vernet."
"A lady never tells." A smile twitched at the corners of Mary Jayne's mouth. "As Beamish said, I have the most innocent face in the world, and let's just say the commandant wasn't immune to my feminine charms." I felt more like the Trojan horse than Helen. Mary Jayne had been their last chance. Emergency U.S. visas had been issued for four of the political prisoners in greatest danger, but all diplomatic requests to bring the men to Marseille under guard to collect them had been refused. Mary Jayne had dressed carefully in her best blue suit with yellow piping and all her grandmother's diamonds. When she'd looked at the reflection in her hotel room mirror, she'd thought, Good, I look exactly how they want me to look—like a pretty, rich American girl. She remembered how, when the commandant offered her a cigarette, Chanel No. 5 wafted from the cuff of her blouse as she leaned in to the flame cupped in his palm. "God, I was glad to get out of that place. They've got the whole camp penned up behind two barbed-wire fences, and the guards are told to shoot to kill." She looked down at her hand and twisted the ring on her finger. She could hardly bear to remember the sight of the shaven-headed men, their emaciated faces. They smiled at me the way poor kids light up at the sight of a Christmas tree.
"I hate it," she said, "it's inhumane seeing people penned up like that. Everyone knows the Gestapo are just going around cherry-picking whoever they want. The Vichy lot are just doing their dirty work for them."
Miriam squeezed her hand. "What you did was very brave."
"The guys did as much as me. If Beamish and the boys hadn't plied the camp guards with wine and women at some brothel when they brought the prisoners into town, they would never have been able to disappear."
"Promise you won't let Varian get to you?"
"Don't give it any more thought. Il est un emmerdeur."
"Mary Jayne!"
"Well, he is. He drives you nuts and he's a pain in the ass. I love that word emmerdeur."
"The only way you get anything done around here is by being a pain in the ass."
"Oh, I can handle him, and if your visa for Yugoslavia comes through, someone's going to have to keep him on his toes."
"Good. You know, if we do find a house..."
"No way." Mary Jayne folded her arms. "I told you, I'd rather go sleep in a maison de passe with the hookers in the Vieux-Port than share a house with him."
"Okay, okay." Miriam laughed. "It's just he's working so hard."
"We all are."
"I still can't figure him out." It was one of their favorite games, trying to decipher their boss. "One minute he's sitting there in his Blackwatch boxer shorts, knocking back the Armagnac in our late-night meetings, and the next he's all buttoned up again."
"A regular sphinx, our Varian," Mary Jayne drawled.
"Don't be like that."
"It's okay for you. You have a role. I just wish he'd let me do more than interviewing the odd client." Mary Jayne gazed out across the sea. "I can do a lot to help, and he just... well, he's just Varian."
"Listen, you've got to realize with Varian that his way is the right way. There's no point in trying to fight against him. He's the reason the ARC has been so effective."
"God, you're loyal, aren't you? He's lucky to have you." After half an hour the tram pulled into La Pomme, and Mary Jayne craned around, pointing out the window. "Look, there's a café back there. Why don't we jump out and see if they know of anywhere to rent?" She rubbed her hands together. "I'm freezing. I could do with a coffee to warm up." The girls held on to the straps as the tram shuddered to a halt. "Come, Dago!" she called, tugging on the dog's lead.
Miriam jumped down and looked around her. "This is great, it's perfect!" As the tram pulled away into the distance, she cupped her ear. "Listen."
"I can't hear a thing," Mary Jayne said, heading to the café.
"Exactly!" Miriam hugged herself. "It's perfect. Peace, quiet..."
"Don't you dare say 'Varian will love it.'" Mary Jayne turned and wagged a finger at her. "Excuse me!" she called, waving at a young girl with long blond hair walking along the opposite side of the road. "How do you do? I'm Mary Jayne Gold of the American Relief Center."
"Bonjour." The girl smiled. "I am Marianne Bouchard."
"Tell me, are there any houses to rent here?"
Marianne shrugged. "You could try our neighbor, old Thumin." She gestured toward a driveway beside the road. "Air-Bel has been empty for years."
"Thank you!" Mary Jayne waved in farewell. "Perhaps we shall be neighbors."
"Look, I can see someone in the grounds over there," Miriam said. "Why don't we ask him if he knows this Thumin fellow?" The girls stopped at the entrance to 63, avenue Jean Lombard, where two redbrick pillars with white stone tops and iron gates marked the entrance to the estate. In the distance, they saw a small man in a black bowler hat raking leaves. His wide black trousers flapped in the wind.
"I'll be damned!" Miriam gazed upward at the engraved white stone capitals on the pillars. "Look at that: Villa Air-Bel. That's the name of my fleapit hotel!"
"Hotel Bel Air?" Mary Jayne peered through the gate.
"It's meant to be. I'm sure of it."
"I don't know, Miriam. This place is huge. It'd be crazy, much too big for us." At the end of the leafy drive, she could just see the corner of a great block of a house, three stories high. Gold and copper leaves fell in slow motion onto the white stones of the driveway, settled on the low slope of the pink-tiled roof. "It's like something from a fairy tale, a sleeping château. We just need a little cottage."
Miriam called and waved to the old man. "Bonjour!" He paused in his raking and limped toward them.
"He won't know anything, he's just the gardener."
"Bonjour." The old man eyed them suspiciously through the bars of the gate. Mary Jayne turned her back as Miriam spoke to him in French, and her breath caught. It's heaven, she thought, captivated by the view framed by plane trees and cedars, sweeping down over terra-cotta rooftops to the sparkling sea beyond.
"He says there's nothing for rent here."
"This house," Mary Jayne said clearly in French to Thumin, and pointed at Air-Bel. "Is this house for rent?"
"Non, non, non," the old man grumbled, and started to walk away.
"We are Americans," Mary Jayne said, and he stopped and turned to her, chewing on his gums as he ambled back. "Americans," she said again, just to make sure he heard her.
"Bon," he said, and took a hoop of keys from his pocket and unlocked the gate. He ushered the girls ahead. "This is my house," he said.
"Your house?" Mary Jayne tried unsuccessfully to keep the surprise out of her voice.
"I am Dr. Thumin. I live with my sister, over there at La Castellane. Air-Bel is too big for us."
"The other neighbors?" Miriam asked. She had a broad smile on her face, watching Dagobert race ahead, kicking through the piles of burnished leaves. The crisp air was perfumed with the smell of bonfires. "We met a charming girl—"
"Marianne? The Bouchards are good people. Quiet, conservative," he said, shuffling along the drive, sorting through the keys. "They wouldn't bother you."
"Tell me, how long has the house been empty?" Mary Jayne's gaze traveled upward as they reached the large terrace overlooking the formal garden with its boxwood hedges and pond. Huge plane and cedar trees marked the boundary of the gardens.
"A while. It is perfectly habitable, though."
"These grounds are lovely...," Mary Jayne said, "but there's too much to do."
"Non," Dr. Thumin said forcefully. "I rent the house, not the grounds. There are eighty-five acres of land that you can enjoy looking at—the magnolias, the olives, the acacias, but everything in them is mine." He stepped toward her. "Including the firewood."
"Okay," she said, and threw a wide-eyed look at Miriam once he turned to the door. The key creaked in the lock, and the door swung open.
"Come," he said, beckoning over his shoulder. "We are a quarter of a mile off the main road, and the house is quiet and peaceful, as you will see."
The house cast its spell on them the moment they walked through the great doors into the black-and-white-tiled entrance hall. Dr. Thumin shuffled ahead, throwing open the metal shutters. Mary Jayne thought of a stately old woman, loosening her stays and sighing with relief. Sunlight poured into the house, waking the rooms, chasing shadows from the high ceilings. She walked in silence from room to room, only vaguely aware of Dr. Thumin telling them about the Louis Quinze tables, the refinement of the Second Empire furniture, the classical frescoes in the library. The atmosphere struck her as unmistakably French—somber and obscure, caught in time. Her reflection in the antique mirror over the marble fireplace in the living room was opaque, the clock beneath stuck permanently at a quarter to twelve.
"It's wonderful," Miriam whispered, taking her arm. "Can you imagine the cozy winter nights in here with a big fire roaring in the hearth?" She ran her fingertips along the keyboard of the old piano. Ornate candlesticks stood ready to illuminate sheet music.
"Just think, if all your plans go well to rescue that handsome fiancé of yours from Yugoslavia, you could be dancing cheek to cheek here by Christmas."
Miriam's smile faltered just for a moment, and her eyes grew soft. "Do you really think so?" She hugged herself with her free arm. "It's too much to hope for, to be together, here."
"Sweetie, after all you've done to save him, he'd better make you the happiest woman alive or he'll have me to answer to."
"I will. I'll make it safely back with Rudolf."
"Attagirl." Mary Jayne leaned in to her friend, nudging her gently. She hated the idea of Miriam leaving and realized at that moment how much she would miss her. She glanced at her, worried suddenly at the risk she was taking for the man she loved. "You will take care, won't you, Davenport?"
Miriam grinned. "You? You are telling me to be careful?" She lifted her arms as if she were dancing with a partner and waltzed away across the echoing room. "Imagine it. Christmas... music, dancing..."
"I don't know. It's too big," Mary Jayne said again, catching Thumin's cunning expression at their excitement. And yet, and yet, she thought as Dr. Thumin pointed out the six-meter-long range in the kitchen and the château's only bath next door. "It's like the one Marat was murdered in," she whispered to Miriam, and the girls stifled their laughter.
They followed Thumin upstairs, through suite after suite of rooms. "There are fifteen rooms up here," he said. "Each bedroom has its own fireplace, so you won't be cold."
"Want to bet," Mary Jayne said under her breath. On the top floor, they gazed out of the windows across the lawns to the sea.
"Look!" Miriam said. "There's a marvelous palm shading the table out there, and acacias and magnolias. Can you imagine how beautiful it will be in the spring?"
"It's the worst kind of dubious, bourgeois elegance." She raised her hand as Miriam began to speak. "Yes, I know. Varian will love it, but he's a snob." Mary Jayne traced her fingertip in the years of dust on the sill. "How much?" she said abruptly, halting Dr. Thumin's monologue. He sucked his teeth.
"It is very expensive."
"How expensive?" she said.
"Thirteen hundred francs."
"A month?" Mary Jayne pursed her lips. "We'll have to think about it." Miriam joined her, and they talked in low voices as Dr. Thumin pretended to check the shutters at the far side of the room.
Miriam did a quick calculation. "That's about thirteen dollars! The smallest hotel room costs fifteen francs a night. Listen, if we were to share this place...," she whispered.
"Like some kind of commune?" Mary Jayne wrinkled her nose.
"It would be a blast, like at college. It would be more like a swank private hotel. We can invite some of my clients like the Bretons to share with us. Split a few ways, we could pay for a cook, and maid, and this place would still cost us less than the ratty hotels we're all staying in." She squeezed Mary Jayne's hand. "Imagine it! Imagine the space, and the freedom..." They heard Dagobert's footsteps echoing down the staircase through the empty house.
"It would be wonderful to have somewhere for us to escape to," Mary Jayne said. "And it would be so much better for the families like Danny's with children." She frowned. "But who else? That's not enough to fill this place. We were only looking for somewhere big enough for us and Danny."
"There's Var—" Miriam began to say.
"No," Mary Jayne said firmly.
"I know he'd love it."
"Then tough luck to him. As I'm bankrolling this whole jolly adventure, maybe he should have been more civil." Mary Jayne stuck her chin out.
"What about Beamish?"
"No, if I know him, he'll want to stay in town." Mary Jayne did a quick head count. "But I reckon we can swing it." She nodded at Dr. Thumin. "We'll take it." She could tell from the confusion on his face that he was expecting to negotiate and now wished he had asked a higher price. "When can we move in?"
FOURTEEN
FLYING POINT, LONG ISLAND
2000
GABRIEL
I watch Sophie bend with the grace of a ballet dancer and scoop up a white oval stone from the shore. What I wouldn't give to be that lithe still. Sometimes when I'm pottering along the shoreline here, I imagine my younger self powering through the surf on my daily run. Day after day for years in all weathers until it suddenly became a chore, not so easy. When was that, 1980-something? Years and decades run into one another now. There was always a dog or two running at my heels. When our last old fella died, we didn't replace him. I won't be long for the happy hunting ground myself, and I didn't want to leave some young dog brokenhearted. It's easier that way, though I miss the clatter of paws on the deck, the sure weight of a dog on the bed at night.
"It sounds like Casablanca," Sophie says.
I glance over at her. "Sorry?"
"I was saying, it sounds like Casablanca."
She's smart, this girl. "That's what some people say, 'the real Casablanca,' or that Varian was the artists' Schindler." I want to say: Life was already over for me when I met him. Then along came this American—tall, kind, talking with quiet confidence like some actor in a gangster B movie: "Don't worry. We have ways to get you out of this mess."
"What was he like as a person?"
"Varian? Have you ever heard something's like a riddle wrapped up in a mystery inside an enigma? That was Varian. None of us could figure him out. But he was an extraordinary man. Courageous, tenacious, and permanently good-humored. He was kind, too, so kind." My eyes prick now, just thinking of it. "After everything I went through, when I landed at his desk and he said he'd help, I cried hot, stupid tears. I couldn't help myself. He handed me a clean red-and-white handkerchief from his breast pocket, and clapped me on the back. I guess he saw scenes like that every day. When people made it as far as Marseille, there was a sense of relief—that you were home free. Then the reality set in—going door-to-door with the hotel concierges all saying, 'Nous sommes complets,' discovering that thousands of other souls had had exactly the same bright idea as you." When I think of Marseille, I remember the smell of wine, and pissoirs, and fish, and ink from the newspaper stalls, and the sea, always, the sea. The place was bedlam. People wandered the streets with all their worldly goods piled up on handcarts. You couldn't get a room anywhere. Then the realization that you were trapped here dawned—that Marseille was like a holding pen, really. You had got this far, but could go no further without your papers being en règle. That's what Varian realized, and calmly set about overcoming it all. What is it they say? Tears may be the path to grace, a way for women to become angels. What of men? "Fry was like an angel of deliverance to us. A regular miracle."
"Well, that's all fascinating, but can we backtrack a bit? I don't want to waste your time when you could be with your family." She smiles, cocks her head, all charm. I'm on my guard now, missy. "Tell me about the first time you saw Vita."
Vita. I have to think for a moment, scroll back through the years like the pages of a photograph album. There she is. I check I have the right page, the right line of the story. There are many ways to tell the same tale. "Yes, now, let me think...," I say, buying some time. The light is dazzling on the white sand, and I shield my eyes, rubbing at my brow.
Sophie butts in, impatient. "The official story is that Gabriel Lambert, enfant terrible of the art deco crowd, met a young British art student, Vita, at a party in Montmartre in 1938. Lambert was quite the catch—rich, talented, and brave. He'd fought with the Republicans in Spain, and entertained his friends by firing off satirical sketches of Hitler, Franco, and Mussolini. He was, in other words, sexy and dangerous and catnip to a girl like Vita."
"Thank you. Go on," I say.
"Most people never knew if she had a second name—she was always only Vita, decades before Madonna or Cher thought of it. It was love at first sight. You were thirty-three, she was younger, closer to twenty, though she never let on her real age. She could have been seventeen or twenty-seven, you never knew with Vita."
"If you know all this, what are you bothering me for?"
"I'm trying to track down her history between leaving England and arriving in Paris, but it's like my great-aunt appeared there out of nowhere. I don't get how someone's history can disappear."
"People disappear all the time, particularly when there's a war on." Or if they don't want to be found.
"Tell me what she was like."
"She was..." I search for the right word to describe her. Passionate. Crazy. A messed-up, beautiful kid. "She was dazzling."
Sophie smiles, reassured. "And her work? Was that dazzling, too?" She labors the word, mimics my drawling pronunciation.
I stop dead. "You want to know the truth? Vita had more of a talent for living than she did for painting."
Her eyes damn near pop out of her head. "How can you say that?"
"Have you even seen her work?"
She flounders. "No, of course I haven't. I mean, a few early sketches attributed to her, but you know that none of her later work survives, only these photos of her studio." She taps her bag. Oh God, the photos. Clever girl, slipping in a reminder that she's just warming up for the knockout blow. I wonder if she's been smart enough to make copies?
"So if this article is about me, why are you writing about her?"
"It's the story, Gabriel." Sophie speaks slowly, as if she's talking to a sulking toddler. "The discourse..."
"Phooey."
"Vita was just starting out when she was killed," she says defensively. "She could have been great."
"Could have been, might have been... who gives a damn about all the what-ifs? It's what you do in life that counts, not what you might have done." I kick at the sand with my espadrille and walk on. "You're running up a blind alley, kid. The truth is, she never progressed."
"I don't believe you."
"Sweetheart, Vita told me herself. She reached a point with her painting, and couldn't get any further. Some people don't. She wasn't satisfied with being second-rate. When she died, she was thinking of going back to acting."
"Acting?" Sophie's voice shoots up an octave.
"You didn't know that about her?" Good. If she didn't know that, hopefully she's missing a few other vital details. "Listen, Vita's life was her art, her best creation."
Sophie is quiet for a moment, processing my revelation. "Let's go back. I'm trying to picture Lambert and Vita on the night they met. In all the biographies they say he—"
"I."
"They say you insulted her dress—a revealing gold-beaded shift, by all accounts, and from what I can make out of the photographs. Apparently you said she hadn't got the figure to wear it. Anyway, she poured an entire bottle of champagne over you. The rest is history. Vita became your most celebrated muse—she had just the right liquid grace for those art deco girls, like a greyhound at full stretch or a chiffon scarf in the breeze."
Vita was beautiful, for sure. She would have been a good actress, I think, her voice was wonderful, too—it poured out of her as naturally as a draft of cold water from a crystal jug. Half the faces are forgotten, but Vita stands out. She had some kind of mirrored band around her forehead, a plume of ostrich feathers. She looked like a queen. Dazzling.
"I've read about the costume party, of course," Sophie says. "It was legendary. I've read how you all danced madly," she says as we walk on, and I imagine the screeching horns of the jazz band.
"Like I say, it was a lifetime ago, I don't remember." But I do. I remember it all.
The question is: How much can I tell her?
FIFTEEN
MARSEILLE
1940
MARY JAYNE
Mary Jayne lay on her narrow bed in the Continental Hotel, her feet resting against Dagobert's stomach. Pink blown roses on the faded wallpaper trailed up toward the ceiling from the brass bedstead, and her golden hair spilled around her on the white pillows. She wore men's blue-striped pajamas, with the legs rolled up around her slender ankles, and she hummed along to the swing tune drifting up from the bar below, tapping her foot to the beat as she read her book. An alarm clock on the nightstand ticked contentedly.
"Last night in this joint, Dago," she said, tossing the book aside and stretching. He raised his head and laid it on her leg, gazing up at her. "You don't care, do you, dear dog?" She ruffled the springy fur on his crown. "You're just glad to be along for the ride." Mary Jayne looked up as someone knocked on the door. "Who is it?" she called.
"It's me, Miriam."
"Just a minute." She swung her legs down and opened the door. "Hello, darling."
"Aren't you packed yet?" Miriam laughed, gesturing at the clothes and lingerie hanging from every surface. "How much stuff did you have in those two suitcases of yours?"
"Oh, there's plenty of time for packing."
"I'm so excited about moving into Air-Bel tomorrow, I'm sure I shan't sleep a wink. The fun we'll have out there! All the artists are coming out this weekend to welcome Breton." Miriam hugged herself. "Danny signed all the papers this afternoon."
"Did Thumin try and up the rent?"
"Don't be such a cynic."
Mary Jayne closed the bedroom door and padded across to the nightstand. "Whiskey?"
"What a treat." Miriam flung herself down in the armchair.
"I've been saving it for a special occasion. We may as well toast our new home." She poured two fingers into each glass. "To new beginnings, and old friends."
"I'll drink to that." Miriam sipped her whiskey. "I thought you might be out with Raymond."
"No, we just felt like a quiet night in, didn't we, Dago?" The dog pricked up his ears at the mention of his name. "It amuses me that Varian thinks I am such a good-time girl. I'm never happier than when I'm by myself." She shrugged and swirled the drink in her glass. "Besides, Raymond has been arrested."
"Arrested?" Miriam sat up in her chair. "What for?"
"Desertion. Forging his demob papers—"
"Oh no, Mary Jayne, I warned you!"
"Please don't say you told me so." She couldn't look Miriam in the eye. "I can't bear it. They have him locked up in the Fort Saint-Nicolas awaiting trial."
"What will happen to him?"
"I can't let him rot in jail."
"Please think clearly, darling—"
"I don't care what any of you think of him," Mary Jayne challenged her. "I don't care that he's done wrong."
Miriam whistled softly. "You're in love with him, aren't you?"
"No. I... It's madness, I know, to fall for someone like Raymond." She sat on the edge of the bed and held her glass to her cheek. "I can't see us settling down with a white picket fence anytime soon, can you?"
"Oh, Mary Jayne..."
"I know what I'm doing."
"No, you don't."
Mary Jayne's mouth twitched, a small smile. "He makes me feel alive. Really alive, for the first time in years." She took a sip of her drink. "Perhaps I've met my match."
"No, you're better than that."
"You are a darling." Mary Jayne looked at her. "I know it's crazy. I feel quite mad, having sleepless nights over some boy...." She stared down at her hands. "I can't abandon him. There's a side to him he never shows, you know. Beneath that tough-guy exterior there's a good and brave heart. I believe in him. When we are alone together..." Her words trailed off as she thought of Raymond. She felt a warmth bloom deep in her, a heat rising. Mary Jayne shook her head and laughed. "Don't you dare tell him any of this."
"Okay, okay," Miriam said, backing down. "You're going to need to hire a damn good lawyer."
"I've done it already."
"Does he know who to bribe?"
"Of course," Mary Jayne said. "Come on. Why don't you give me a hand to squeeze this lot into my suitcases, and then I'll take you out for a late supper."
"Wonderful. I haven't had a chance to stop for a bite all day. The office has been bedlam." She began to neatly fold Mary Jayne's clothes, handing them to her. "Is the chest of drawers empty?"
"Yes, I think so. I never really settled in."
"Do you ever?" Miriam laughed. "You're like a wild bird, following the snow and the sun in your little airplane."
"God, I miss that freedom."
Miriam slid open the top drawer. "Look, all your jewelry, and you've got a new pack of silk stockings in here!" She offered the stockings to Mary Jayne as if she were holding a priceless work of art. Mary Jayne saw the longing in her eyes.
"You take them, darling."
"I couldn't!"
"Take them for your trousseau. Give Rudolf a thrill." She quickly cleared the drawers, tossing jewelry boxes carelessly onto the top of the suitcase.
"If I ever get to Yugoslavia."
"You will," Mary Jayne said, embracing her. "Your visas will come through, and you will rescue that man of yours, and live happily ever after when you escape to America."
"It all seems impossible sometimes." Miriam hugged the packet of stockings to her chest. "Thank you. I love them."
"And I'm sure Rudolf will appreciate your wedding present. Now, what am I going to wear out?" Mary Jayne looked down at her pajamas.
Miriam clipped on Dagobert's lead as Mary Jayne dressed. "Do you feel like heading out to the Pelikan to see who's around?"
"Sure. We might get a bit of news from the U.S. consulate."
Miriam sat on the bed and opened one of the red leather jewelry boxes. A diamond brooch gleamed, light refracting across the wall. "This is lovely."
"My grandmother's."
"Is it safe, darling, leaving all these lying around?"
"You mean with Raymond?"
"I don't mean to pry. I'm just worried about you." Miriam clicked the box shut and tucked it into the suitcase.
"They have sentimental value, more than anything." Mary Jayne looked at her reflection in the mirror. "Daddy gave me most of them, before he died." She adjusted the neckline of the blue wool dress she had chosen, tucking the silk strap of her slip away. "It's funny, isn't it, how unimportant things become when you are running for your life. When we fled Paris after the invasion, I left most of my luggage on the side of the road to Toulouse. I can't even remember now what was in the trunks."
"Some of us didn't have much to leave behind in the first place."
"Just you wait, once all this is over I'm sure you will have a darling home full of beautiful things, and hordes of children."
"So will you."
"Me?" Mary Jayne picked up a hairbrush. "No, I don't think so. I can't imagine choosing a life like that."
"You know you're going to have to make a choice, though, don't you?"
Mary Jayne brushed out her golden hair and put a slick of red lipstick on, rubbing her lips together. "A choice?" she said, spritzing Chanel No. 5 into the air and walking through the mist. "Come, Dago," she said, taking the lead from Miriam.
"Varian and the committee can't risk someone like Raymond having any connection with their work here. It's too dangerous."
"You mean I'll have to choose between Raymond and the ARC?" Mary Jayne locked the door and tossed the key into her clutch bag.
"Darling, it's extraordinary how much you have helped with the Gold List, but if you stay with Raymond, you know the committee will just see you as some kind of gangster's moll. You know the types he's involved with."
"I'd rather that than be at Varian's beck and call." She marched ahead, Dagobert trotting at her heels as they walked out into the corridor and down the sweeping staircase to the lobby.
"Do think about it. The work we are doing here counts. It really counts. I'd just hate to see someone like Raymond taking advantage of you." Miriam took her arm as they reached the lobby. "Do you really love him that much?"
"Perhaps I don't know what love is," Mary Jayne said finally. "But this is turning out to be quite some year for a nice girl from Evanston."
SIXTEEN
FLYING POINT, LONG ISLAND
2000
GABRIEL
"Gabriel," Sophie says. "Gabriel." She tugs at my sleeve and I snap back to now, to the clear blue sky above me and the white sand beneath my feet. "I read that after the New Year's party in Paris you and Vita were inseparable. Is that when your relationship started?"
"I suppose you could say it was." See, it's not a lie, as such.
"You don't look so good. Are you getting tired?" Sophie asks me.
"A little." My hand's shaking as I reach for my breast pocket. Damn, I've forgotten my tablets, too. I'd wanted to enjoy a last lunch with a clear head before we closed up the house for the season, and I didn't want to feel woozy looking after the little one. The kids have decided that we are too old to spend another winter out here. What do they know? I can still walk to Marv's place or heat up a can of soup when we're hungry. Which isn't often these days.
The sun is dazzling on the sea, the white deserted sand. We must look like a couple of chess pieces walking along—the white king facing the black queen. Some days, when the beach is empty like this, the sense of space is so infinite, I swear you can see the curvature of the earth on the horizon, the suggestion of a perfect circle. Maybe that's why my paintings lately have all been arcs and lines inscribed on empty fields of blue. It takes me days to prepare just one canvas, finishing each with a single white dot in a fractionally different place from last time. There is every color in white light, did you know that? To capture this luminosity, that's what I've been chasing, the weightless sensation of being on our deserted beach, the white sand beneath your feet, the limitless cold blue air arcing over you.
Once photography came along, there was no point in rendering real life. I wanted to capture raw experience, sensation, to make people feel. There is nothing more difficult than simplicity. Oh, the critics will have a field day with my new work, coming up with some theory or other, when the reason, the inspiration, is here in front of them. I love this place. It's breaking my heart thinking of leaving, of being a guest in my son's home in the city. This is ours. All of this is ours.
Sophie notices the sign by the steps. "Why don't we head up to the café?"
"Good idea. I'll buy you a soda."
"I'll pay, Mr. Lambert." She folds her arms across her rib cage.
I wave my hand dismissively, that word my great-grandchildren seem to love so much on my lips: Whatever. This girl has destroyed my peace, and now she's getting smart with me? I'm not going to show her that she has me rattled. I'm going to stick with the "angry old man of art" act and hope she won't see past that.
"I'm sorry," she says, at my side again as we walk up the next flight of wooden steps and head across a half-empty car park toward the small café. It hasn't changed in thirty years. Marv's only concession to progress is the satellite dish up on the roof of the old cabin for the cable TV and Internet for all the whiz kids who can't leave their work behind on their summer weekends. I catch a couple of surfers, wet suits peeled down from their torsos, staring at us as we talk, checking her out. Perhaps she's cuter than I first assumed.
"Hey, Marv," I call across to the owner, and settle with relief into the booth by the window, my booth. Most places are already closing for the season around these parts, but Marv stays open all year. He says the locals and the odd surfer or dog walker are enough to keep him ticking over through the winter, and what else would he do? This place is his life.
"Haven't seen you for a few days, Gabe. You okay?"
"Me? I'm fine," I say.
"Okay, okay. Just asking." He sets a jug of ice water down on the chipped yellow Formica. As he leans down, the lamplight gleams on his bald head like polished mahogany. "D'you hear the Knicks have traded Ewing?"
"Yeah. I remember when he came in as the number one pick in the '85 draft," I say, pouring two glasses.
"Fifteen years, man. It's sure a good trade for Seattle," Marv says as he shuffles away, shaking his head.
"You like basketball?" Sophie asks as she sips her water.
"What, you mean just like a regular guy?"
"You're kind of touchy, aren't you?"
"Just had my fill of people who seem to think all artists live on ambrosia." The water is good, I'm thirsty. "It's a mistake to think artists only want that." I lean toward her. "You really want friends who'll talk about the weather, or how they're building their boat, or trade you recipes for beans."
"Don't disillusion me." A smile is playing across her lips. "Next thing you'll be telling me is you do your own tax returns, and the washing up...."
"And diapers. I've changed more diapers than you can imagine." I try not to smile at the disbelief on her face. "Between all the years of kids and grandkids, and puppies and kittens, I seem to have spent half my adult life picking up caca."
She laughs and settles back into the booth. "Maybe people waste a lot of time trying to pin their ideas on artists."
"Are you talking about yourself?"
"I was thinking about a book I'm reading on Duchamp."
"Ah, the master. He taught me that your life can be a work of art, kiddo."
"Do you always talk like a bad gangster movie?"
"I learned English watching Humphrey Bogart and Jimmy Stewart."
"It shows."
Back in the day, I'd have snapped at her for being so bold. Now it doesn't seem to matter. "You hungry?"
"Oh, no. I haven't—"
"My treat." My guess is she's one of those career girls who don't even stop for breakfast, just tank up on coffee. "Do you like pancakes? Marv, get us some of your blueberry specials."
"Whatever you say, Gabe." Marv slings a white cloth over his shoulder and goes through to the kitchen, whistling along to the Shirelles on the jukebox: "Will You Love Me Tomorrow."
"How long have you been coming here?" she says.
"Forever."
"Tell me what you like about it."
"Listen, kid, I like normal. Jeez, if you had to live with my mind..." I rake the heels of my hands across my temples. "If you had to live with that, you'd like normal, too. I like logs piled up for the winter, and Annie's laundry room, and this beach when it's empty. I like a pile of clean white plates on the kitchen table, and the tick of the longcase clock in our hall." I look her in the eye. "I like order, and peace. God knows that's not what I have in my head."
See, I have a theory about artists—their heads are like a basin with the tap running. Once in a while they have to let the plug out and set free a painting or a book, or there'll be a hell of a mess. When you get to my age you are better at regulating the flow, but it hasn't always been so easy. Maybe that's where my reputation comes from.
"Have you always lived here?" she says, tracing a line in the condensation on her glass.
"Always? Pretty much a lifetime. Annie and I shared a studio in Brooklyn for a while, but as soon as we could we headed out to the coast."
"She's an artist? I thought you said—"
"She was... well, she calls herself a crafter. Maybe you've seen some of her textiles. She did beautiful work, but then her hands got bad."
"I'm sorry." She pauses. "I've been wondering. You two never wanted to go back—to France, I mean? A lot of the artists did."
I shake my head. Sure, there are still days when I wake from dreaming about France and I wonder if we should have gone back. The dreams are fragments, really—the color of pale blue shutters changing chromatically with the light. Beneath a slate sky when the mistral is whipping through the olive trees, they seem gray. Beneath a cloudless blue sky they are bleached like the firmament above them, the perfect shade of old denim. I miss stone walls that radiate the heat of the day at night. I miss the smell of rosemary on a bonfire, the taste of a cold pression, and the soft light and shade of a plane tree in a market square. I miss great cheese, and geckos, and those mad, sun-blind dogs they have in the south of France with amber eyes. I miss, I miss... oh, avenues of lime trees like bleached bones along the roadsides, and fat asparagus and peaches. I miss the age of the place—how old villas with lime smudged onto their wall like ocher pastels look at sunset, with their rusted eau de nil gates, and deep indigo convolvulus on glorious mornings. The hardest thing of all, for many, was leaving behind family graves, of course. But I'm not going to tell her that. Too many of us had to do the same thing. Meyerhof said to me once, "You pick up your life and you don't think back." That's the secret. Never look back. Your home, your birthright, is tied to you but lost forever.
"The thing you've got to remember is that some of us dreamed about America, even before the war."
"Really?"
"Sure. At least I did. Jacqueline, André Breton's wife, said it was the Christmas tree of the world. Chagall worried there would be no cows here. But I wanted this my whole life, the possibility of it, this new world. I still get a kick every time I take the train into the city and see the Statue of Liberty, the Empire State."
"But it must have been incredible, your life with Vita?"
"It's another lifetime. I was a different person."
"That's what I want to talk to you about," Sophie says. She slips a tape recorder from her bag of tricks and nudges it on the table. "Do you mind?"
It's on the tip of my tongue to tell her this was a mistake, but then Marv rolls over with a stack of pancakes and the coffee.
"Gabriel...," she says.
"You can call me Mr. Lambert."
"Well, that's kind of formal, considering how long we've been friends," Marv says.
"I wasn't talking to you," I say, the breath rattling in my chest, "I was talking to her." I wave my hand at Sophie. She jumps, knocking the coffee, and I snatch at the napkins in the chrome holder, mopping at the table.
"I can't stand mess," I snap at her.
"I'm sorry." There it is again, that bloom in her cheeks.
"Sure," Marv says, backing away. "Sure thing, Gabe. Enjoy your pancakes." I watch him go out back and pick up the phone.
"Mr. Lambert," Sophie says, "I'm sorry. We got off on the wrong foot. I'm not just a journalist. Think of me as an old family friend, like Vita." Phooey. There's no one on earth like me and Vita, believe me. I can almost see the radiance of Sophie's halo gilding her eyelids as she looks down at the plate in front of her. I don't trust her an inch. While I'm busy watching Marv, I hear Sophie quietly click on her little mumble machine.
"Why are you kids so hung up on the old stories?" I round on her. "You should be making history, not rehashing it."
"How can we learn if we forget the lessons of the past?"
"Listen, kid. There is only now. The past is a fabrication." I wave my hand as if I'm sewing. "It's a patchwork quilt of so-called facts and hearsay. History is up for grabs, a fiction spun by whoever has the strongest voice." I try to catch my breath, bunch my fist under the table. "The future is conjecture. All I want is now. The paint beneath my fingernails, to eat, to sleep, to fuck."
"At your age?" she says without looking up from her notebook.
"Yeah. It had more effect when I used to say it."
She settles back in the booth and folds her arms. "I've read that line in at least three of your biographies."
"That's what happens when you get to my age. You start repeating yourself."
"You're a walking cliché."
"Maybe. But I'm a happy one." I play with the coaster under my water. "So you've read them, then?"
"I need to put my article in context. If I'm honest, I'm more interested in Vita."
How amusing. I'm just the frame, not the picture.
"To lose one wife tragically is heartbreaking," she says quietly. "To lose more smacks of carelessness."
I have to think for a moment or two, it's all so long ago. My life here with Annie has been everything, is everything. All that went before—it's like trying to read a letter that's been left out in the rain. I hope the expression on my face reads as tragic rather than confused. "You're wrong there. I was never married to Vita."
"I know that." Good, she's getting exasperated. With any luck, if I play the old, doddering fool card, she'll get frustrated and start making mistakes soon. "But she was as good as, common law and all that. I'm talking about Rachel. She was your model, too, wasn't she?" She checks her notes. "She died just after you met Vita."
There's her first mistake. Rachel died before Vita was ever on the scene.
"Are you saying that was convenient in some way?"
"You tell me."
"There was a car crash. She was driving."
"Presumably heartbroken...."
If only you knew the truth. "People die for more stupid reasons than love all the time." I run my hand across my brow. "Does it really matter?"
"Yes, it matters. It's part of your story, of Vita's." She leans toward me. "In history, anonymity is the enemy. It reduces everyone to no one. It's my job to put names to the people that we have forgotten."
"You're wrong. It's an artist's job to name."
"Like God?" she says.
"Don't be cute. Besides, he's the greatest artist."
She taps her pen on the table. She's nervous. Good. "You don't want to talk about Vita yet?"
"No." I recognize that voice. Like my sons when they were toddlers, throwing their toys out of the stroller.
"Fine." Her voice is placatory. She's going to humor me. "Then tell me about Air-Bel. Tell me about Annie."
SEVENTEEN
VILLA AIR-BEL, MARSEILLE
November 1940
GABRIEL
"Say, Gabriel, good to see you." Varian shook my hand. He had been deep in thought, staring at the murals in the library, but he turned to me now and smiled warmly. "Have you come to welcome André Breton? Do you know him?"
"Of course, by reputation," I said. "I admire him greatly."
In my pocket I still had Fry's scribbled note with the address: Villa Air-Bel, La Pomme. I fingered the paper nervously. It was too much to hope for, that I should meet Breton, too. It felt like the world had tilted on its axis, and after the horror of the last months I had arrived in a place where your wildest dreams could come true. I wanted desperately to belong here, with this crazy group of nomads and artists. I felt awkward around Fry, still, his confidence awed me, and the way I'd seen him slip easily and fluently between English, French, and German as he chatted with his friends and clients. I felt tongue-tied in comparison.
"How is everything? Do you have money for food? We can give you some meal vouchers if you're having trouble with cash. You have somewhere safe to stay?" I nodded. "Good. Now you stay out of trouble, okay?" Fry shook my hand. "Don't you worry. We'll get you out of here just as quickly as we can. It may take a couple of months, but be sure we'll do our best for you." He smiled reassuringly. "Meanwhile you're welcome at Air-Bel whenever you want. We're planning a little get-together on Sunday." Varian glanced up at the sound of Mary Jayne's voice drifting through from the first-floor landing, the sound of her heels clicking down the staircase to the entrance hall. At her side trotted Dagobert, the poodle an ever-present shadow, his claws tapping on the wooden steps. We followed her downstairs to wait for the Bretons.
"He's a smart dog, old Dago," Varian said. "Have you seen his party trick?"
"No."
"If you say 'Hitler, Hitler,' he'll bark ferociously." Varian laughed. "Maybe I can train Clovis to do the same."
"There you are!" Miriam skipped down the steps beside us. "The Bretons will be here any minute."
"This place is swell, Miriam. I was just enjoying the murals," Varian said, gesturing at the library.
"They're beautiful, aren't they?" Miriam hugged herself in delight. "I just knew you'd love this place. Do you know what they are of?"
"Sure. I'll show you later. There's even one of Aeneas, son of Venus, carrying his father, Anchises, from the flames of Troy."
"Well, that's appropriate."
"The original refugee," Varian said. "When he fled Troy, carrying his father on his back, he set a precedent for us all, didn't he?"
"Listen, Varian, I'm glad I've caught you," Miriam said. I walked on ahead, but I could still hear them.
"Is something up?"
"When I left the office tonight, I found out my visas for Yugoslavia have come through."
"Oh." Varian couldn't hide his shock and disappointment. "Of course, I'm delighted for you and Rudolf. Will you marry now?"
"That's the idea, and then I hope I can get us safely back here before going on to the States."
Varian took his glasses off and polished them with his handkerchief. "It's funny. I mean, I know the deal all along was that each of you would escape yourselves just as soon as you could, but I hate to see you go." He looked at Miriam. "You've done great work, Davenport. Thank you. People like the Bretons are safer because of you."
"Oh, stop it. You're making me blush."
"You will take care, won't you?"
Miriam hugged him. "I'm going to miss you, too, boss."
"When do you leave?"
"Four days."
"So soon?" At the sound of voices in the hallway below, he slung his arm around her, squeezing her shoulder affectionately as they caught up with me. "What a shame you're leaving just as you've found this place. I can't think of a better refuge from the center than this...." His voice trailed off.
"I'm hoping Mary Jayne will come round," she said as they paused on the landing and looked down into the black-and-white hallway. Mary Jayne stood at the open doorway, and Dagobert bounded out onto the terrace at the sound of tires stopping on the gravel drive. "It would do you good to live here, too."
"I wouldn't want to put Miss Gold out," Varian said.
"Don't be like that," she said. "If you think I'm going off to Yugoslavia while you two are still at one another's throats—"
"We'll be fine. She thinks I'm a stubborn ass, and I think she's a highly strung little rich girl."
"Just wait and see," she said. "One day you two'll look back at all of this as the best of friends."
"Like a donkey and a racehorse put out to pasture?" he said doubtfully. "Have you told her yet?"
Miriam shook her head. "I will, later. I just didn't want to spoil everything, not yet."
I glanced downstairs as a tall man with a chestnut mane of hair strode into the hallway. A slender blond woman walked at his side, holding the hand of a young girl. The woman moved with the grace and certainty of a dancer, and her full black-and-white-striped skirt swung as she stepped into the hall. She stood with her hand on her hip, chin raised. A woman who is used to making an entrance, that was the first thing I thought. The lights of the chandelier gilded their hair, danced from the mirrored clips in the woman's hair.
"Monsieur and Madame Breton," Mary Jayne said, stepping forward from the crowd gathering in the hall to greet them. "Welcome to Air-Bel." André Breton shook her hand and introduced his wife, Jacqueline, and daughter, Aube, to the welcoming group. As they talked among themselves, André's gaze traveled around the house. I thought of photographs I had seen of lions gazing out across the plains of Africa, how they always seemed to be part of the world and yet somewhere else, too. Perhaps they see something we don't, I thought. Just then, André looked up at me, his hair a blazing halo above his dark green suit and red tie, and raised his hand in greeting like a blessing.
* * *
If you look really closely at the photos they took that night, you can see me hanging around in the shadows. I can never figure out why we all looked so much older than we were in the photographs taken at Air-Bel. Maybe it's the formal clothes—we all wore a shirt and tie every day back then. Kids of sixteen looked like old men. Perhaps it's not your calendar age but what you've experienced that shows up in a photograph. Annie says I'm an old soul. Me, I just think I went through a lot in 1940.
I was introduced to Breton and his wife but hadn't the guts to talk to him properly. I was still feeling pretty nervous when I headed out to La Pomme on the tram that Sunday. Almost turned back a few times at the thought of this house full of people I had admired my whole life. Sundays at Air-Bel became legendary in Marseille—all the young artists and writers were talking about it, and here I was waiting on La Canebière for a tram to La Pomme. The city looked beautiful in the snow. To talk to a resident of Marseille, you'd think it was never cold, never snowed, but this was the first indication of how bitterly cold that winter would be, and my feet were soaked through already. But the snow covered up the worst of the dross and the filth in the city and made everything feel brand new, including myself.
You wouldn't be going to meet Breton if Fry knew the truth, I told myself. I'd already come to picture my conscience as me as a schoolboy. I imagine the little fellow even now, sitting on my right shoulder, swinging his legs, his shoes shined. You're a bad boy, Gabriel, he says. But it's not my voice. Who said that to me, and when? My mother, perhaps? I don't know whose voice it is, but I know that sense of mortifying shame only too well, the sickening feeling of having been caught.
To hell with them, another voice piped up as I gazed out of the tram window, squinting at the bright light bouncing off the distant waves. This voice is more like my father, lounging around on my left shoulder with a glass in his hand. I have him under control most of the time now, but then... well. I was so messed up and broken inside, most of the time he made his voice heard.
You go out there, and you have a good time, he said. Drink their wine, soak up their ideas, suck them dry. Don't let them give you any nonsense. You can hold your own with them. "I am Gabriel Lambert," I said under my breath.
A few people jumped down from the tram at La Pomme, just before the railway bridge, and I followed them at a distance. I turned up the collar of my overcoat and tucked my head down, the wind whipping through my hair. They chatted among themselves—they were obviously friends, relaxed in one another's company. They turned into a long driveway, and the last of them—a tall, good-looking fellow with a beautiful woman on his arm, held open the iron gate for me.
He had fine features, and his hand on the gate was slender, long fingered, his skin the color of burnished teak.
"Are you here for the salon?" he asked.
"Yes. I'm Gabriel Lambert," I said, offering him my hand.
"Wifredo Lam," he said. "This is Helena." The girl smiled at me and walked on.
"Are you a painter?"
"Yes. I've been studying with Picasso."
"Picasso?" The name of the great man stuck in my throat like a fishbone. I almost ran at that point, I felt so out of my depth.
"Are you managing to work at all?"
"A little." My voice sounded unnaturally high.
"It's the only way," he said, walking up the drive to the house. "I feel my drawings are changing here. I'm illustrating André's new poem."
"André?"
"Breton," Wifredo said, laughing. He chatted on about the poem, Fata Morgana, and introduced me to the others on the terrace, to Óscar Domínguez, and André Masson. I was so relieved that Wifredo had taken me under his wing, I have no recollection of what we talked of at all. I do remember him saying that he had fought with the Republicans in the Spanish Civil War, and thinking how unlikely it was that this tall, gentle man should be caught up again in conflict. I was just happy to listen to them all talk, happy to blend into the group. I could hear music drifting from the house, some wild Count Basie jazz tune with screeching trumpets and thundering drums on the radio. I began to sweat, thinking of that night only a few months before with Vita, the sound of the band and the pounding beat.
"Lambert!" I heard Varian call, but I could not see him. Through the open French window, I saw André Breton dancing with Jacqueline. His head was lowered, his cheek resting tenderly against her temple. They were surrounded by people but seemed lost in each other. "Up here!" Varian called, laughing. I turned and looked back across the terrace to the trees.
"Good heavens," I said, "what on earth are you doing?"
"We're having a little auction," he said, waving from the branches. "Would you mind passing me that last canvas?" I flipped it around and handed it carefully to him. I whistled softly, realizing I had a Max Ernst in my hands. Varian casually swept it up into the branches. He took a length of string and tied the wire on the back of the picture to the tree, holding one end of the string in his teeth as he tightened it. "There," he said, and scrambled down. The paintings spun in the breeze, bright paint glimmering against the trees, the air like flowers in the park.
I could hear the sound of voices behind me, exclamations and greetings as Breton welcomed his friends. I was too nervous to turn around and introduce myself.
"What do you think?" Varian asked me.
"It's incredible...," I started to say.
"It is marvelous!" Breton cried, and clapped his hands. The trees on the terrace were strung with thirty, forty paintings. Looking back now, I realize that it was a collection any fine-art museum would kill to have. "We shall hold the auction later, but first, we play!" I trailed inside, following the young boys I had seen at the ARC office, too shy to introduce myself to Breton.
I hung back, observing the artists from the edge of the room. It was freezing cold—they say 1940–41 was the coldest winter on record—and everyone was done up in their overcoats and scarves, sitting around the huge polished wood table. It was impossible to stay still for more than five minutes without the cold becoming unbearable, and everyone was restless, blowing on their hands. Breton had put magazines, scissors, paper, and glue at the center of the table, and glasses with pencils and crayons. I couldn't make out what was going on. A single sheet of paper was passed from artist to artist, and they folded down the section they had drawn before passing it on to the next person.
"What are they doing?" I whispered to Varian.
"They are playing games," he said. "They call this one cadavre exquis. I think before now, they have used words, made chance sentences. Now they seem to be experimenting with images. Breton calls them les petits personnages. After they have drawn a few pictures, he decides the best."
It's rare that people surpass your expectations, but André was magnificent—every bit as provocative and extraordinary as his writings had led me to hope. He presided over the gathering, stooping occasionally to murmur words of encouragement to the artists. When they had finished, he sorted through a sheaf of papers and raised a drawing of a head in the air. "Stupéfiant!" he cried. "A true collaboration. We shall call this 'The Last Romantic Has Been Buggered by Marshal Pétain'!" The table erupted in cheers and laughter. "Formidable."
"Do you not want to join in?" Varian asked me.
"No, no," I said. "I'm happy to watch."
"You should! A lot of the games they play seem to be collaborative."
Only now do I realize what those games meant. When all around us the world was turning dark, the surrealists were focused on the light. They believed in absolute freedom, and this is what those crazy-looking games were all about. They wanted to free the unconscious mind. They showed us the luminous, random beauty around us and in our dreams, and everything Breton and the others did has changed the way we look at the world forever. Air-Bel became the house of dreams. In that little room in the falling winter light, I watched men and women who, for an afternoon a week, put aside their fear and hunger and created, and at the beating heart of them was Breton. I still wish I'd had the nerve to talk to him, but the greatest lesson Breton taught me was that the most effective course against despair is to preserve your freedom of mind.
"As I have said many times before," Breton said to one of the men, raising his voice above the crowd. "When one ceases to feel, I am of the opinion one should keep quiet, my friend." The group dissolved into laughter, and one of the artists loped off, red faced. It was like seeing a badly behaved cub get a clip around the ear from the head of the pride. I opened my mouth to ask him what he meant, what was this "pure psychic automatism" they kept talking about, but the words tightened in my throat. I was afraid of getting the same treatment.
I felt completely out of my depth around them all, like the new kid at school. I wanted to belong here, in this beautiful house, with these incredible people, but I felt like a phony. I pushed my way out onto the terrace and gulped down the cold air. The hum of voices, the clear notes of the music, fell away as I walked across the lawns. Yet more people were heading up the driveway with their heads bent against the wind like pilgrims. I turned away toward the parkland at the back of the house and walked on.
I heard children laughing up ahead, and I followed the sound. There was a little blond girl with a red ribbon in her hair, and a small boy about the same age, five or six, I guess. They were on the ground, arcing their arms and legs, making snow angels. It looked like fun, so I lay down beside them and started kicking and swinging my arms. The milk sky was heavy with snow above me, and sound was muffled, so when I heard a girl laughing beyond the garden wall, the note of her voice was clear and pure as a bell. "Who's there?" I said, turning my head. The snow was cold against my cheek. "I said, who's there?"
A snowball arced over the wall and hit me squarely on the chest. I blinked, flakes wet in my lashes, on my lips. "Hey!" I cried out, and scrambled up. I could hear footsteps crunching through the snow on the other side of the wall, running away. I chased her, deeper into the woods where the wall began to fall away and a tall, dark hedge marked the boundary. I was breathless, my heart pounding. The grounds weren't tended well this far away from the house, and the hedge was old and patchy. I caught a glimpse of her once or twice—a flash of blond hair between the dark leaves, a pale hand or cheek. I squatted down and breathed deeply, the cold air hurting my lungs, my breath a pale cloud in front of my face. I could see her slender legs between the trunks of the hedge—her dark stockings and boots, the hem of her dress. She had her back up against the leaves, hiding against the trunk of a tree. I crept forward on my belly, silently, pushed my way through a gap in the hedge a little farther up. Everything seemed to slow down. My breath trembled in my throat. Then, just as I poked my head through into the light, my foot must have caught on a dry branch. A twig snapped, and she spun around, startled, her eyes wide and alert like a fawn. You hear people talk about love at first sight in songs, but that was it for me. Looking back over my life, I see that there are a handful of moments like that, which I can recall with perfect clarity. Not all the in-between times we lose along the way, but the moments that matter. The first time I saw Annie, I was absolutely present. Not thinking about the past or worrying about the future, but there, with her. Too often when you're young, you fall in love with your idea of a person. I always reckon people stay happy just as long as their idea of each other fits. Annie's never wavered for me, not once. I saw her for the first time, and I knew her—recognized her, even—and she knew me.
Annie's hair swung after her, pale and luminous. Her face registered fear, then amusement when she saw me, and she scooped up a handful of snow, molded it into a ball as she backed away from me. Just as I scrambled clear of the roots, she swung her arm back. She was grinning now, and I saw she had a little gap between her front teeth. Her lips were unnaturally bright, full and red in the cold, her cheeks flushed pink. She was, she is, the most beautiful girl I had ever seen. I was unmanned, and she knew it. She threw the snowball with the aim of a marksman and it clocked me between the eyes.
"That does it," I yelled, scooping up a big handful of snow. On open ground she was no match for me, and I sprinted after her, stumbling. The snowball hit her on the back of the head. She shrieked, fearful and excited as I grabbed her by the waist and we fell into the snow. "Now, we're even." We lay in silence, face-to-face—aware, I guess, that we had never met. She looked uncertain, suddenly. "My name is Gabriel," I said. Her blue eyes gazed at me. The truth is, it felt as though I'd known her, had been waiting for her, my whole life.
"I'm Marianne," she said.
Like an echo, I heard a woman's voice calling, "Marianne!... Marianne!" She sat up quickly and glanced back over her shoulder at me, smiling. "But you can call me Annie. I have to go." She leapt to her feet, looking down the woods to the little stone house by the road. I could make out a plump, shrewish-looking woman dressed in a black coat bustling through the back garden gate.
"Wait!" I caught at her hand. "Who are you? Where do you live?"
"I'm always here," she said, slipping away.
"Can I see you again?"
She laughed, as if it were the most natural question in the world. She glanced down the hill; the woman—her mother, I guessed—was steaming up the hill, her breath puffing out of her like a train. "Do you live there, with all those crazy people?"
"Me? No. I'm an artist, though. I'm just... I'm visiting."
"Good." She inclined her head toward her mother. "She would not like it if you did. My parents think Air-Bel is full of Communists and sex maniacs," she said. "It's quite the scandal in the village that the old miser Thumin has rented the villa to them. Who is the woman who goes shopping with bracelets around her ankle and a stuffed bird in her hair?"
"That must be Madame Breton."
"People are talking. They do not like things that are different around here."
"Do you care about that?" I asked, pulling myself up from the snow. I was about a head taller than her, and as I looked down at her all I wanted to do was take her hand and keep on running, away from her parents, the village, the war, away from everything.
"Of course not." She glanced quickly over her shoulder. "I love art, in fact I want to study, after school." It was impossible to tell how old she was. Sixteen or seventeen, perhaps. She seemed older. I think sometimes now that young people like that who have grown up in one place, who have only ever known certainty, the sureness of where they belong in the world, have a confidence I'll never possess, even as children. Of course she was beautiful, and exhilarating, but I think that is what drew me to her as surely as north follows south. When people ask how we met, she always says it was love at first sight—and that's true. But there was more to it than desire. I recognized something in her that I needed like air, like water. I coveted her roots—how real she was. Marianne, my Annie, has always only ever been herself. Unlike me.
"I'd be happy to take a look at your work," I said.
She weighed me up with that clear blue stare of hers. "I will meet you in town next weekend. I have my ballet class in a hall near La Vieille Charité at two o'clock on Saturday. There is a little café with red shutters just down the road." She backed away, raised her arms to a graceful fifth position, her pointed foot arcing the snow. "I'm meeting a friend there after the class."
"A boyfriend?"
She grinned. "No, a girl if you must know." She looked back over her shoulder as she walked away. "You'd better go, my mother doesn't like trespassers."
I crawled back through the gap in the hedge and jogged along until I caught up with Annie. She was trailing her hand against the branches again, and I reached up and did the same. I caught her, once or twice, looking at me—a glimpse of her eye, her lips. I could hear her mother coming, a tirade of complaints rattling from her lips like keys in a tin can. As I neared the garden, I could hear the auction beginning on the terrace. "Annie," I whispered to her, and she stopped walking and turned. I reached through the leaves to her, bobbed my head until I found her, her gaze, her smile. I pushed aside the branches and touched her fingertips. Then, as her mother's voice reached me clearly, she turned and was gone, as swift and silent as a bird in flight.
EIGHTEEN
FLYING POINT, LONG ISLAND
2000
GABRIEL
"Annie...," I say quietly. The girl is watching me. I have to be careful. "You made short work of those," I say to her.
"I was starving," Sophie says. She runs her finger through the last of the blueberry juice and sucks it clean. That's the kind of thing that would have got me going a few years back, but now, nothing. "Thank you," she says, dabbing her mouth with her napkin. "They were very, very good." She checks the recorder still has tape left and reaches into her bag for a pen and notebook. As she flips through the pages, words jump out at me: Vita. Gabriel. Why? "Let me just check something," she says.
"Sure."
Sophie flicks through her notes. "According to the research I've done, you moved to the Château d'Oc, which is near..."
"Carcassonne."
"Yes, you were near Carcassonne in 1938, correct?"
"Yes." Which way is this heading?
"And in the summer of 1940, you were living alone there with Vita?"
"Most of the time."
"What do you mean?"
"I mean, people came and went."
"Such as?"
"Quimby, my dealer. Friends," I say vaguely. She's not buying it.
"I heard you were a recluse by then."
I pick up the sugar shaker and stir a slow stream into my coffee before I remember the doctor had told me to cut it out. "Did you now?" I look directly into her eyes as I put it down. "As you can see, I don't much like company, still."
"You seem to have a houseful today."
"Family. That's different."
"Talking of family..."
Oh God, here it comes.
"Tell me about your son." She has the good grace to blush slightly. "I mean, if it's not too painful?" Her eyes flicker down to her page and she unscrews her ink pen.
"Who the hell uses a fountain pen these days?"
"I do. At least for important things."
"Like writing to me?"
"So you did get the letters."
"And about me," I say, reading her notebook upside down. I tap it with my dry old finger. My nails are curved and hard, more like claws these days.
Sophie hesitates, looks from her notes to me. "I thought you're dyslexic."
Clever girl. I'm going to have to be careful. "Maybe I struggled with War and Peace, but I get by." I wait for her confidence to waver and hope she doesn't push that line of inquiry. "I don't give a crap what people say about my work, but you're not concocting some fairy tale about me and Vita just to suit the story you're trying to conjure up. You're wrong about that, I told you. I never stopped Vita doing anything."
"But these photos of Vita's studio...," she says, sliding the black-and-white photos onto the table. Oh God, my heart's racing again. It's there, staring her in the face. Maybe I'm in luck, maybe she's not noticed, not seen? But then, I know in my gut she has. She hinted as much in her letter. Perhaps if I can get her to focus on Vita, she'll forget. "Lambert?"
"Quimby took them."
She checks her notes. "You say he was your dealer?"
"For a time."
"Look, I don't believe you when you say Vita was no good. I mean, it's hard to make out, but these paintings look... well, to use your word about her, dazzling. Why did she never show them?"
My gaze falls to the images. I know them like the lines on my palm. Vita didn't paint them. I did. "They were the beginning of something," I say to the girl. Or maybe the end.
Memory is a funny thing. I spend more time thinking of the past than the present these days. It is more detailed, more full of life, than the days I drift through now. Just like that, I'm back in the Château d'Oc. I am more there than here in the café with the girl. I can still hear myself, my old voice worn out by years of sea air and pipe tobacco, rabbiting on to Sophie, weaving a string of lies, the story she wants to hear, but I'm long gone.
Here is the beat of my heart, racing in my chest as I follow Quimby into the house. Here is the thrum of hooves on the hard earth as Vita rode into the party. Here is the pulse and the rush and the pull of the past. I'm back, I'm back, and I can't bear it. I close my eyes, push on, past the fire. The days, the months, fast-forward, and the memories kaleidoscope until I am safe again at Air-Bel, the night I met Annie and my life changed track forever.
NINETEEN
VILLA AIR-BEL, MARSEILLE
1940
VARIAN
That night, Air-Bel swung to the tinny sound of American jazz, picked up on the shortwave radio. The reception rooms hummed with conversation and laughter, and blazing fires danced in the hearths. From the kitchen drifted the sound of pots clattering on the stove and the voice of the chef, Madame Nouguet, shouting orders to the housemaid, Rose. Above, children's footsteps clattered across the wooden boards, muffling for a moment as they ran across a rug, then racing on, their excited voices pealing like bells through the old villa.
"We shall christen this house Château Espère Visa," Varian said, raising his glass of wine. He stood before the crackling hearth, his hand resting by the stopped clock.
"Perfect!" Mary Jayne cried. "Haven't we been calling this old place the château?" she said to Miriam. They sat beside each other on the sofa, legs curled beneath them. Their cheeks were flushed with wine and the warmth of the fire. "Of course, 'château' is a bit grand."
"Well, I think it's a swell place," Varian said as he joined them. He placed his glass of red wine on the marble fireplace and reached for the bottle to top up the girls' glasses.
"Thanks," Miriam said, smiling up at him.
"So, has Miriam told you the good news?" he said to Mary Jayne. "Her visas have come through. Miriam is getting married." He raised his glass. "We should make a toast. To Miriam and Rudolf!" He chinked his glass against hers.
"Why didn't you tell me?" Mary Jayne said quietly to Miriam.
"I didn't want to spoil tonight." Miriam looked crestfallen. "I told you not to say anything, Varian."
"Typical," Mary Jayne said under her breath. "Damn it, Davenport. I knew you'd get your papers sooner or later, but I'm going to miss you." She squeezed her friend's hand. "And it's just as we've found all this." She pointed across the room. "Tell me again, Varian, who are these people?"
"That chap's a writer. Well... a writer and a revolutionary," Varian said quietly. "He's in great danger because he's spoken out against Stalin. In fact, he was the first writer to call the regime 'totalitarian.' He's been in and out of prison for years, and he's stateless and penniless now. I believe a lot of his family are either imprisoned or in gulags."
"How terrible."
"I hope this old place is a refuge for him."
"Varian certainly loves it—you should have seen him!" Miriam said. "He's been like a child all afternoon, going from room to room, looking in all the drawers." She glanced up as the children raced through, running after Dagobert and Clovis. "I don't know who's had more fun, Varian or that lot."
"I said you'd love it." Mary Jayne sipped her drink.
Madame Nouguet appeared at the door. "Dinner is ready to be served."
* * *
The table was laid for twelve in the heavy, Spanish-decorated dining room. The room had the natural exuberance of a funeral parlor, with faux Córdoba leather walls and heavy mahogany table and chairs. As the group filtered through, Mary Jayne directed everyone to their places. "I'm so glad you could join us for dinner," she said pointedly to Varian.
"Thank you for inviting me," he said.
"I'm not sure what Madame Nou..." Her voice trailed off as she saw André whisk away a red cloth from the center of the table. "What on earth?"
Varian pulled back her chair. "Thank you," she said, craning forward to look at the pile of leaves arranged in front of her.
Varian adjusted his horn-rimmed glasses and squinted at the arrangement. Just then, one of the twigs moved and swiveled a triangular head toward him. "Ha!" he cried in delight. "A praying mantis."
"Two, male and female," André said, settling in the chair opposite Varian. "Just watch."
* * *
The dinner continued late into the night, long after the sparse meal had been eaten and the servants had retired to bed. People danced in the hall, swinging to the jazz tune crackling through the ether from Boston, and the air in the dining room swirled with wood and cigarette smoke, the table littered with empty wine bottles and overflowing ashtrays. For the first time in months, Varian felt truly at ease—he lounged back in his chair, shirtsleeves rolled up, tie loose around his throat. He smiled benevolently as he looked around, at Miriam with her head thrown back, laughing as she danced; at Mary Jayne admiring the tiger's-tooth necklace Jacqueline had strung around her neck; at Danny, who sat with his arm slung over his wife's shoulders as he debated with the men. For the first time in his life, Varian had the feeling that he had come home. It felt almost indecent to be this contented, enjoying a simple dinner with friends in a setting as perfect as this. And yet, he thought, his eye caught by a movement at the center of the table. How long could this last? How long until they were kicked out of the country, arrested, or worse? Varian reached for the bottle of wine as Gussie passed it to him and nodded his thanks. How long will we be safe? he thought as yet again, Varian's attention returned to the two praying mantises stalking each other at the center of the table. It was how he felt every minute of every day—under threat, from the gangsters, the cops, the Gestapo. He glanced at Mary Jayne. Even her relationship with Killer put them at risk.
"Aren't they marvelous?" Breton said, interrupting his thoughts.
"Where did you find them?" Varian asked André.
"In the greenhouse," he said, peering closely at them. "It's almost time." He sat back. "Have you seen the garden yet? Old Dr. Thumin's greenhouse is quite marvelous. I think I shall work in there."
"Are you managing to write, in spite of it all?"
"One must always write." André drained his glass.
"What are you working on?" Varian said, and hesitated. "If I may ask?"
"Something new." He reached into his breast pocket, and Varian glimpsed a folded sheet of paper scored with looping green handwriting. "A new poem." Then he lunged forward, pointing at the center of the table, his eyes glittering with satisfaction. "There!"
It seemed to Varian that the flames, the candles in the dining room, the firelight, flared and glowed in André's eyes the very moment that the praying mantis bit down on the head of its mate.
TWENTY
FLYING POINT, LONG ISLAND
2000
GABRIEL
Whenever I see one of those lumps of amber fashioned into a paperweight in those fancy decorator's stores that Annie likes browsing in, I think of the summer of 1940. I stopped in the street to look at one in the window of a store last time we went to town. Annie had on that old blue dress of hers I love so much, and she was laughing, chatting to an old friend about the grandkids. I could hear her talking, and the thrum of the convertibles driving nose to tail up the main street with their identical cargoes of guys in Ralph Lauren polo shirts and khakis and their expensively blond wives and kids who look like they are out of a catalog. The light was bright and clear, but as I looked at that amber, it seemed to darken and I felt again the heat, the weight of the Château d'Oc, and the taste of the dust in my mouth, and I saw the sulfurous yellow tang of the sky over the hills. I can't remember what the paperweight was sitting on, some flimsy-looking desk or writer's table or something too small to ever be a useful working surface, but it was the quality of the light caught in it that stopped me. It reminded me of looking into a tin of golden syrup. I didn't buy it, of course. What is it they call that style the decorators are all crazy for now? Shabby chic. I don't get that at all, the fakery and faux finishes. Something should be what it is, of its time. It should have heart and authenticity. Everything I have in my life is shabby, but it's beautiful and useful and real, and it's grown shabby because I've loved it and used it. Why is everyone looking backward, instead of creating something new and marvelous now, hey? These decorators just get some kid in a garage to beat up some piece of pine junk, triple the price, and stick it in the window of one of their little chichi shops in East Hampton, artfully scatter a few magazines or papers around, and then hold it all together with a great piece of amber. What do I know—that's probably plastic, too. Anyway, you can guarantee that stuck at the heart of the amber there's some hapless insect. That's how I felt before I met Annie—trapped. Even thinking of it now, my breath tightens.
There have been times when I wasn't sure I'd make it through one of my attacks. Annie's always been good at calming me down, but when I was younger, before the meds got better, there were days when each breath had to be torn out of the air. I had an old mirror in the studio, and I'd stare at myself, trying to still my chest, thinking over and over: I am Gabriel Lambert.
"Say, Gabe, I know who you are, buddy."
"Hm? What?" I say. I look up to find Marv standing over me.
"I'm sorry, man," he says. "I've got to close up early today. Lil wants me to take her into town."
"Ain't that just the way," I say. "Annie's always on at me to take her into town more. I say, 'What do you want to go to town for when we have everything we need here?' She says she just likes to look." I search in my pocket for some change and catch Marv watching me.
"No, you're all right, Gabe," he says, and pats me gently on the shoulder. "It's on the tab. In fact, this is on me."
I laugh out loud. "It must be Christmas."
"Soon enough, Gabe," he says, shuffling away. "Soon enough." He pauses as he flicks out the lights in the café one by one. "You sure you're okay?"
"Never better," I say.
"You want me to drop you back at the house?"
"No, it's a beautiful day for a walk." The sun is sinking already, washing the windows apricot and gold.
"Gabe..."
I can see the way he's looking at me. I'm choked up suddenly. "Now, Marv, don't go getting all sentimental on me." We slap each other on the back awkwardly, half hug, half tussle.
"Don't be a stranger, you hear?"
"I'll be around," I say, and wink. I step aside to let Sophie walk ahead of me, out onto the deck overlooking the beach, and as I turn from Marv my face sets hard. There's no one on the beach now, it's a strip of perfect white sand arcing beneath clear sky. The sunset is seeping up into the blue like rinsing Rose Doré from your brush in a jar of cold water.
I can't see her face, but the wind is whipping the girl's blond hair free as she walks ahead of me down the wooden steps to the beach. I glance behind me. Marv's car is just pulling out of the car park, and the surfers are long gone. The cottages are deserted. We are alone.
The stick is heavy in my hand, the end round like a cudgel. For a split second, I imagine bringing it down on her head. I wouldn't be able to take her down hand on hand anymore, but if I were to catch her unawares... One blow to the temple, I think, would do it, and then I'd just set her loose in the sea. My heart is jolting in my chest, my ribs a taut xylophone beneath the skin. I've killed before, I could do it again, if I had to, to protect Annie, my children.
The guilt has never left me. I've never been able to forgive myself. I've asked myself so many times if leading a good life redeems your soul. Do thousands of ordinary days atone for one deadly act? You'd think you would forget how it feels to kill, but it is always there, tainting everything. I look down at my hands, with their long fingers and wide, full palm. They look innocent enough, but it's always there, beneath all the tenderness, the touches that have created, and healed, and aroused.
She knows too much, this girl. Why now, after all these years, so close to the end, so close to getting away with it? I will do anything to protect my family. If she pushes me... I'm just waiting for her to turn and point her finger at me and say the words I've waited sixty-odd years to hear.
But she doesn't, she just keeps on walking, her steps so light and free they barely leave a trace in the sand, just licks of wind. No matter how fast I walk, she is just out of reach. "So after Vita and your son were killed, you just upped and left for Marseille? You never looked back?"
Never look back. Like I always say, the ones in the myths who look back are the ones they turn to stone or pillars of salt. Always look forward. "Yep." I have to shift my weight and tighten my grip on the carved wooden stick as we hit the soft sand. "Gabriel Lambert left for Marseille."
TWENTY-ONE
MARSEILLE
1940
VARIAN
"Well done, Bill. These are perfect." Varian held the visas up to the window in the artist's studio, then placed them carefully on the drawing board, peering down through a magnifying glass at the papers, one bespectacled eye magnified, blinking in the bright lamplight. Bill Freier pulled the light closer and pointed at the stamp he had forged on the document.
"Not too perfect," he said, pointing at a smudged edge. "That would be suspicious." Part of his skill was turning the brand-new identification cards he could still pick up in tabacs into convincingly battered documents. "A few thumbprints and dog-ears help to pass them off."
Varian stood and reached into his breast pocket for his wallet. "What do I owe you?" He glanced up at the sound of quick footsteps on the wooden stairs leading to the attic room.
"Pfft," Bill said, waving him away. "Fifty cents apiece, call it five bucks."
"Hello, Varian," a slender brunette said as she walked in. She slung an empty-looking basket onto the kitchen counter.
"How are you, Mina?"
"Cold, hungry...," she said. Bill wandered over and kissed her. Varian smiled indulgently—the young couple's love was clear to see. It radiated from them, a heat that even the chill mistral couldn't destroy. When was the last time Eileen and I looked at one another like that? he thought. Mina gazed up at Bill, giggled as he whispered something in her ear. Have we ever looked at one another like that? They were still at the stage where their hands danced around each other like butterflies, never still, full of the novelty and joy of being young and in love.
Mina unpacked her basket as Varian paid Bill. He glanced over and saw a single onion, a half loaf of bread. "Thank you, Bill," he said, slipping an extra note to him. "You are being careful?"
"Of course. Why would they be interested in a little fellow like me?" Bill tossed the roll of money onto his desk, where it landed among the brushes and inks, the piles of blank visas and passports waiting for his attention. "Do you have anything for me today?"
"As always." Varian clicked open his polished brown leather briefcase and slipped the documents Bill had given him inside a copy of Virgil's Aeneid.
"Say, I just saw your friend Hermant outside the café Au Brûleur de Loups," Mina said.
"Beamish? Did you?" Varian said, glancing at his watch. He handed Bill a file of documents, each with black-and-white passport photographs pinned to them. He shook Bill's hand. "Take care now, both of you."
* * *
"There you are, Buster," Beamish said as Varian walked toward his table. "Shall we move inside?" The café Au Brûleur de Loups was quiet at this time of the day, and Varian spotted the man they were meeting immediately. The gangster Kourillo sat at the back of the empty café, hidden from the road by a large pillar. Varian recognized him from his hand, the fluid way he reached again and again, flicking his cigarette impatiently into a rectangular yellow ceramic ashtray stamped Ricard Pastis. He followed Beamish in silence. Varian disliked Kourillo, distrusted him on gut instinct. But yet again, we have little choice but to deal with men like him if we are to fund the ARC.
"Monsieur Fry," Kourillo said, shaking his hand limply. "Monsieur Hermant."
"Kourillo." Varian sat opposite and folded his arms. He nodded as Beamish ordered a carafe of red wine for them.
"How is the relief business?"
Varian folded his arms. "I wouldn't call it a business."
Beamish glanced at him, warning him.
"We are all in business, my friend." Kourillo laughed softly. Varian noticed he had tiny teeth, like a child. "Now, I have a proposal to discuss." He poured water from a carafe into his pastis and watched the glass grow opaque, opalescent.
"Go ahead," Varian said, trying to conceal his impatience.
"No, no, no." Kourillo sipped his drink. "Not here. I just wished to see if you were open to... ideas."
Varian bit down hard on the inside of his lip. "Monsieur Kourillo—"
"Of course we are," Beamish said smoothly. "Shall we meet you at the Dorade tomorrow? I imagine Charles is involved?"
"Naturally. Vinciléoni is involved with everything." Kourillo rose and put his hat on his head. "Until tomorrow."
Varian waited until Kourillo had left the café before he spoke again. "That man," he said, his words clear and quiet. "Wasting our time like this—"
Beamish knew the signs. "Calm down."
"Don't tell me to calm down." Varian glanced up as a couple took a table not far from them. He lowered his voice. "What the hell do you think he is talking about?"
Beamish shrugged. "I've heard rumors about gold."
"Gold?" Varian whispered. "Jesus, Beamish. Laundering francs is one thing, but if we get caught trading in gold, we'll all be locked up."
"We need money, urgently. You said so yourself."
"I know, I know. If only we didn't have to do business with men like him."
"You don't get it yet, do you? Men like him are running this city." Beamish drained his glass and pulled on his knitted woolen hat. "We have no choice but to do business with crooks. It's the only way we are going to get the good guys out of here."
TWENTY-TWO
VILLA AIR-BEL, MARSEILLE
1940
MARY JAYNE
"Well, our little friend has settled in," Mary Jayne drawled. She was sitting with Miriam at the table on the terrace in the evening sun, the slatted shadows cast by the great palms shifting over them like the pelt of a wild animal. Dagobert ran across the terrace, a yapping black poodle puppy at his heels. "I can't believe you twisted my arm about Varian."
"Clovis!" Varian yelled, racing after the dogs, a lead flapping in his hand. "Clovis!"
"Think of it as a parting gift to me," Miriam said, laughing, pulling her coat tighter around her against the cold. "It is good to see Varian looking so relaxed. I think the puppy is good for him. Maybe he's been lonely. I've been worried about him."
"I'm still surprised he chose a poodle," Mary Jayne said.
"I'm not, he adores Dagobert." Miriam nudged her.
"He likes my dog well enough. It's just me he can't stand."
"Phooey," Miriam said, and laughed, watching the children run after the dogs hand in hand, chattering excitedly. "Oh, I'm going to miss this place." She closed her eyes as she turned her face to the sun. In the distance, the Mediterranean shimmered, light sparkling on the rose-gold surface like crystals on a gown. "I do hope I can bring Rudolf back from Ljubljana with me, he'd love it here." She laughed uneasily, and her voice shook. "I don't know what I'm going to do if they won't let us back into France."
"Don't you dare. We promised, remember? No tears," Mary Jayne said firmly, her voice throatier than usual. She took Miriam's hand, rubbing some warmth into it. "If they won't let you back in, then you just get the hell out of Europe some other way, you hear? We all know this isn't forever. The château is just... well, it's just a wonderful adventure, that's all. And you're not leaving yet, there's still tonight," she added, nudging her.
"You should ask one of the artists to take my room," Miriam said. "What about that chap Lambert?"
Mary Jayne narrowed her eyes. "I don't know that I trust him. There's something 'off' about him."
"Do you think so? He's been through a terrible ordeal."
"Really?"
"I found out that his wife and son were killed."
"God, how terrible. He never talks about it."
"My dear, we are all men and women of mystery these days." Miriam sighed wearily. "Too many secrets, too much to hide." She smiled at her friend. "Think about it, at least. He spends most of his time here anyway."
"I think hoping to see Marianne Bouchard has as much to do with that as paying court to Breton and the gang."
"I am glad you agreed to let Varian stay," Miriam said. She smiled as she watched him tossing a red ball for the dogs, the children racing to and fro across the lawn. "He's so happy here."
"I've certainly seen a different side to him." Mary Jayne's gaze followed three men walking up the driveway. "Who are they?"
Miriam shielded her eyes with her hand. "More surrealists, I imagine. Now Breton has arrived, they are flocking here like homing pigeons." Birdsong sparkling across the grounds melded on the air with music drifting from the house. The girls heard someone's fine baritone voice singing a barrack-room song, then laughter before others joined in with the chorus.
"It's more like Breton is a king, or pope and they're all coming to pay court to him."
"That's what this place is." Miriam laughed. "A court of miracles. I'm so glad I've had the chance to see a little of it. I'll never forget meeting men like Masson, and Breton..."
At that moment, André appeared at the French windows leading to the terrace, closely followed by Jacqueline. They were too far away for the girls to hear their conversation, but it was obvious they were arguing.
"I am thirty," she yelled suddenly. "My life is over!" A string of curses fell from Jacqueline's lips, her hands clutching at the air. "You see in me what you want to see, but you don't see me, André, you don't see me." Finally, she tossed her head and stalked away across the garden to where she had strung a trapeze from the branches of one of the trees. André's chin fell to his chest, and he gazed at the ground, his hands in the pockets of his green tweed jacket.
"What do you think is going on?" Miriam whispered.
"I don't know," Mary Jayne said. "I talked a little with her last night. She was happy, it seems, in Martigues. She loved the freedom and how beautiful it is there."
"Is she not happy at Air-Bel?"
"I think she loves the intellectual vibrancy of the place, but..." She hesitated. "It must be hard, sharing your husband with so many people." She slipped a pack of playing cards from their case and began to shuffle them.
Miriam thought of the line from Job: If I beheld the sun when it shined, or the moon moving in brightness. "They both have such presence. Maybe it's inevitable they clash sometimes."
"They are obviously in love. I overheard him calling her his little squirrel yesterday."
"It can't be easy, being married to a man like Breton," Miriam said quietly. "I've heard all about his reputation, the disturbances..."
"Oh, he's a pussycat," Mary Jayne said, laying out the cards for solitaire. "Quite charming, and so charismatic when he's talking at night, or reading from Duchamp's letters..." She paused, the ace of hearts in her hand. "Don't you think he is like the magnetic core of the château? He never shows off, he's just fascinating. I love it when he produces some rare copy of Minotaure or one of his books from those cases of his. It's like he is presenting treasure to us. His words are enchanted...." She smiled, glanced quickly at Miriam. "When he says, 'Alors, on joue...'"
"Don't go getting any ideas!" Miriam laughed. She looked over to where Jacqueline was swinging to and fro on the trapeze with the ease of a gymnast, her slender body flipping upside down, flexing in a smooth curve, arms outstretched to the ground. "I daren't think of the consequences." Jacqueline's seductive presence seemed to permeate the house. Whether she was working on her paintings or sitting in quiet reflection with a halo of cigarette smoke around her head, there was a refinement to her, a magic.
"I'm not interested in him like that. I just wish I understood more—I mean, what are all the games the surrealists are playing? Is it some kind of catharsis? I know humor is a great antidote to fear, but I just don't get the subtleties of what they are doing."
"I don't know. I heard André say the other night that they believe that love is a fundamental principle of moral and cultural progression." She paused.
"Love?" Mary Jayne said, her brow furrowed. "I know Breton was extolling the importance of monogamy and exclusivity the other night. It surprised me. One tends to think artists are all at it like mad."
Miriam laughed. "Why don't you ask him about it? I have no idea either—it all seems rather racy."
"You need to loosen up."
"Unlike you," Miriam said.
"I don't know what you mean."
"Mary Jayne, you will be careful, won't you?" Miriam turned to her.
"I know what I'm doing."
"Do you? What do you really know about Killer?"
"I know that I adore him, I told you," she said, smiling to herself. "I know that he makes me feel alive...."
"He's dangerous," Miriam said firmly. "How old is he? Twenty-eight, twenty-nine?"
"Actually, I found out the other day he's twenty."
"Twenty! And you are?" Miriam insisted.
"Old enough to know better, I know." Mary Jayne gave up on the game and spread the cards in her hand into a fan.
"You're thirty-one, Mary Jayne. You're nuts to be mixed up with someone like him. We don't know why he left the Foreign Legion, or what he's mixed up with here. He's trouble. Varian doesn't trust him...."
"Varian can go to hell."
"If Killer compromises the safety of our clients, of the ARC..."
"Not this again." Mary Jayne threw down the cards, scattering them on the table. "He won't. You have my word. Nothing I do will reflect badly on the ARC. Just as soon as Raymond gets out of jail, I'm going to help him get to England to fight with de Gaulle, and Varian and the precious committee won't have to fret about a thing. Honestly, they're like a bunch of old women."
"Let's not fight," Miriam said, putting her hand on her friend's arm.
"It just drives me crazy. Here I am gladly giving thousands of dollars, and I get treated like some dumb blonde." She scowled for a moment, then smiled as Miriam caught her eye.
"Danny says even when you stayed with his family in Paris, you liked the bad boys."
"Oh, Danny was forever teasing me."
"I think you just like danger. Look at the way you used to tool around Europe piloting your Vega Gull."
"My feelings for Raymond scare me, but the plane never did. I was in control." Mary Jayne's face clouded. "Perhaps I should have hung on to the plane. It would be rather useful down here, but I thought the French forces could make better use of it." She gazed into the distance, imagining how it would feel to pilot herself and Killer to freedom. She pictured her plane swooping low over the dazzling sea, flying free with him at her side. They could go anywhere—west to Lisbon or London, perhaps, then on to New York. He'd hate it, she thought, trying to imagine Killer in America. He'd never come. He says he loves me, but he just wants to get back in the war, to fight. I adore him, but I have to let him go. He'll be a hero one day. That'll show them. That'll show them all. She sensed Miriam's concern and looked at her. "You know, I've always been able to have pretty much anything I wanted in the world," she said. "But I can't have him. At least, not for long. I know that."
"Oh, Mary Jayne." Miriam took her hand.
"No, it's fine. I've quite come to terms with it." Mary Jayne lifted her chin. "I've always thought that no one and nothing can contain you," she said quietly. "Whatever stops you living your best life is your own fault—the bars are inside of ourselves, and I'm going to help set Raymond free."
* * *
The next morning, as the rest of the house slept, Mary Jayne joined Miriam in the dining room. The fire's embers still glowed in the hearth from the night before, and their eyes were red with exhaustion.
"Coffee?" Mary Jayne said, pouring a cup for Miriam. She slipped on the jacket of her pink suit and perched on the edge of the table, her feet in their white ankle socks and sandals swinging above the floor.
"So-called coffee." Her laugh was flat. Miriam gulped down her drink, scalding her mouth. "Listen, Mary Jayne, I—"
"Stop it." Mary Jayne smiled, blinking, unable to speak for a moment. "No grand farewells. This isn't good-bye. If luck's on your side, you'll be back here with Rudolf in a couple of weeks." She checked her watch. "We have to go, we mustn't miss your train. Varian and Beamish are going to meet us at the station to say good-bye."
The friends walked in silence, arm in arm along the driveway. The cold gravel was wet with dew, crunching beneath their feet. Melting snow banked on the verges still, and mist rolled among the cedar trees, birds singing their dawn chorus in the branches. At the gates, Miriam turned and took a last look at the house. Her throat was tight, her eyes welling with tears. "I'm going to miss you. I'm going to miss you all so much."
At the Gare Saint-Charles, they raced along the platform. The train was waiting, steam billowing along the roofs of the carriages. Miriam craned her head above the crowd at the sound of Beamish's whistle and saw him halfway down, holding open the door of the train, with Varian at his side.
"Thank you, Albert," she said to him, catching her breath.
Varian embraced her. "Stay safe, Davenport." He pressed his lips to the top of her head. "You stay safe, you hear?"
Miriam hugged him, tears brimming in her eyes as she turned to Mary Jayne. The girls stared at each other, unable to find the words. They fell into each other's arms as the whistle blew. "This damn war," Mary Jayne said. "What if we never find one another again?"
"We will, we will," Miriam said. She hugged her tight and whispered, "Be careful."
"Of course."
"I mean it." She took Mary Jayne by the arms. "Listen, you look out for yourself, do you hear?" Beamish swung Miriam's suitcase into the packed corridor, and Varian helped her up, slamming the door behind her. Miriam lowered the window and stuck her head out as the train pulled away. "Thank you," she said, raising her voice as the whistle blew and the train chugged out of the station. "I'll never forget any of you, any of this." She waved. "Good-bye! Good-bye! I'll see you soon, in New York!" she cried.
TWENTY-THREE
FLYING POINT, LONG ISLAND
2000
GABRIEL
I'm struggling to keep up with her now. The girl's walking on ahead, her blond hair tumbling down, free and loose. I remember Annie standing on the shore when she was not much older than this girl, her hair blowing in the wind just like that. After the babies were asleep, she'd come down here once in a while and just stand with the surf lapping her feet, staring out at the horizon. I know she was thinking of France and everything we had left behind, but I never did. I never looked back, not until today.
The sun's going down now and the beach is luminous and empty in the half-light, Venus gleaming above us.
"Something I've always wondered...," the girl says, calling over her shoulder.
I stop and try to catch my breath, but it's rattling around my chest like nickels in a tumble dryer. "Yes?"
"Why was there such a huge change in your work between 1940 and the paintings you did in America?"
Here we go. "How could I not have been affected by meeting men like Breton?"
She stops and gazes out to sea, her hand shielding her eyes against the falling sun. "Is that the best you can do?" she says. "I know a whole generation of American artists like Pollock and Rothko were all deeply affected by the exiled artists over here. I know Duchamp, and Breton, and all those people made a huge difference, but you don't make any sense."
"I'm sorry?"
"It's like you became a different person."
My hackles rise. "That's ridiculous...."
"Did the war really affect you that much? I mean," she says, walking on, "your work before the war was lovely, but... well, decorative." She says it like an insult.
"That's the line they always throw at art deco. It was a fashion, that's all. You can't deny the sheer technical skill of my early work."
"After you arrived in America, your painting..." She throws her arms out to the sky, fingers extended. "It exploded. The anger, the clarity—"
"Like I said. It changed, because of the war." It changed, I changed... what's the difference anymore. The blood is singing in my ears now, and I rock slightly on my feet.
"Say, are you okay?" That voice. I...
"I'm fine."
She sits down on an old log that has washed ashore and stares out to sea. I stumble as I join her, my feet tripping in the sand. I end up sitting on the ground, my back against the smooth driftwood. Sophie's sitting behind me, so all I can hear is her voice. I feel the weightlessness of her hand on my shoulder.
"It's time. Tell me everything. What happened in Marseille, Lambert?"
TWENTY-FOUR
RUE GRIGNAN, MARSEILLE
November 13, 1940
GABRIEL
"Still no news after your report?" I heard Beamish ask Danny. He was shuffling through the pile of papers in front of him on the desk. "Your tour of the concentration camps has to have had some effect, surely? Tens of thousands of men, women, and children are penned up behind barbed wire like animals." Beamish shook his head as he scanned Danny's meticulously typed document. "Look at this—dysentery, typhus. It's like the Middle Ages, not a modern European country." Eight men sat around the meeting table in the rue Grignan office, a single lamp illuminating the files in front of them. The main office lights had been switched off, and I sat with Charlie and Gussie in silence, guarding the street door. The truth is, I didn't much enjoy my own company in those days, and when I wasn't at Air-Bel hoping to bump into Annie, I was happier around the ARC crowd. They had grown to trust me, and I made myself useful enough around the place. The hours I spent alone dragged, and I lived in fear of bumping into Alistair Quimby. I knew he was in Marseille somewhere, and he haunted me like a specter. I would be walking through the market, my stomach groaning with hunger at the smell of the food on the stalls, and I would see him walking toward me—or at least I'd think I'd seen him, and I would run, doubling back on myself, trying to lose this ghost from my past. At least at the ARC I felt safe and among good people I could trust.
"Not a word, not a damn word from Vichy. They're not budging an inch." Beamish leaned back in his chair, his arms folded. He had his habitual pout on his lips, and though I couldn't see his face clearly in the dim light of the domed chrome lamp on Varian's desk, I knew something was wrong.
"What's up, Beamish?" Danny said.
He shrugged. "Nothing definite. I just..." He leaned forward into the lamplight and picked up the model airplane on the blotter. He spun its propeller speculatively. "I think it's time to get as many people out as quickly as we can. Things are changing, for the worse."
"What makes you say that?"
"It's just a feeling. Up until now the Gestapo has been happy to let the Vichy lot do their dirty work, but I think—" He was interrupted by a hammering on the front door, the glass rattling in its frame. As one, we all looked up, alarmed, like a herd of deer sensing a predator.
"Are we expecting anyone?" Danny said quietly.
"No. Varian won't be back from Vichy for a day or two." Beamish jumped up. "Stay there," he said under his breath to the men around the meeting table. "Turn out the light." He pulled the office door to and locked it behind him. "Open up," he said to Charlie. He sat quickly at the table beside me and picked up a pack of cards, dealing four hands. My fingers trembled as I fanned the cards he dealt me. Someone knocked again, harder this time.
"Hey, fellah, keep your hair on," Charlie said, noisily working his way through a hoop of keys until he saw that Beamish had settled back, casually picking through his hand of cards. Charlie ran downstairs and unlocked the main door, opening it to the street. A car rumbled by in the darkness, its dim blue lights sweeping up the staircase wall. "Why, good evening Detective Dubois," I heard Charlie say, loud and clear. "Come in out ah the cold." Charlie walked back up to the office, followed by the policeman. I glanced at Beamish, saw him pause, ready to lay a king down on the table. In the silence, I heard the click of the card on the wood.
"Monsieur Fawcett," Dubois said, taking his hat off. "Hermant, Gussie." He glanced at me. I felt the hair at the nape of my neck prickle. "May I come in for a moment?" He went to the window and lifted the blind a crack, watching the street.
"How can we help you?" Beamish said.
"Is Monsieur Fry here?" Dubois glanced at the locked office.
"No," Beamish said. "He left some time ago, for Vichy."
"No matter. It is you I came to see, Monsieur Fawcett." Dubois turned to Charlie. "It's time for you to leave town."
"Me? Why?" Charlie paled.
"You're going to be picked up in the morning."
"In the morning? But why? I haven't done a thing," Charlie said, his voice tight with anxiety.
"How do you know?" Beamish asked Dubois.
"Because you're going to be picked up," Dubois said, "by me." He put his dark trilby back on his head and stepped toward the door. "Unfortunately, Monsieur Fawcett, your gallant—if bigamous—efforts to free Jewish women from the camps has been uncovered. A couple of days ago, two Mrs. Charlie Fawcetts turned up in Lisbon at the same time." He held Charlie's gaze, his eyes crinkling. "I'll be here at six A.M. sharp."
Charlie held it open for him. "Thank you," he said, shaking his hand.
"I hope I won't be seeing you in the morning."
"You can count on it."
"Alors, bon soir, et bonne chance."
We waited until we heard the street door close behind Dubois, and then we exhaled as one.
"Jeez, Charlie," Gussie said. "You kept that one quiet. Just how many girls have you married?"
"Five or six. I kind of lost count." He raked his hand through his hair. "Darn it, of all the rotten luck. Imagine two ah them turning up on the same day. There was never anything in it for me, you know, it just seemed like a good way to get the girls out of the camps."
"Well, good for you, Charlie." Beamish knocked the table as he went to unlock the office, and the playing cards scattered on the floor.
"Was that Dubois?" Danny marched over. "What did he want?"
"They're coming for Charlie," Beamish said.
"When?"
"Tomorrow."
"Then you must leave tonight." Danny clapped his arm around Charlie's shoulders. "Don't worry, my friend, we'll get you out of here."
The meeting broke up, and one by one the men slipped away into the night. I stayed behind with Danny and Gussie to help Charlie get ready. We worked through the night. On a small table in the kitchen, Charlie tucked some documents into the hollow center of some of his sculptures, and I helped him reseal them with plaster of Paris. Next, he rolled translucent paper lined with tiny script into narrow tubes and threaded them into the third valve of his trumpet. I knew better than to ask what the documents were—I'm not sure if Charlie knew, either, to be honest. He just wanted to help Varian get information to the United States about some of the clients in greatest danger.
"What if they ask you to play something?" I said.
"It's okay," Charlie said. "I know a few tunes where I only need a couple ah valves." The smile on his face faltered as he looked at the maps on the table. "It sure looks easy, doesn't it, when you just trace a line on a map via Gibraltar or Casablanca. Freedom seems so close." He cleared his throat. "Danny?"
"Hm?"
"It's been crazy, man, hasn't it...."
Danny punched him on the arm. "This isn't good-bye, Charlie."
They glanced up as Beamish tapped out the code on the front door. Gussie ran down and let him in. "Right," Beamish said. "It's all arranged. You're going out through Spain," he said to Charlie, pulling off his woolen hat. "We're going to get you out through the station, and then in Madrid you look for the red-haired porter."
"He's a good chap," Danny said. "He's helped a lot of our clients and he'll get you to the safe house until your train for Lisbon is ready."
"But what about the border, what if I'm stopped?"
Beamish tossed a dozen packs of Gauloises Bleues and Gitanes Grises and Vertes down on the table. "These are for the guards. Just keep your head and you'll be fine. You're simply an art student, heading south. You'll have to get rid of that," Beamish said, pointing at Charlie's ambulance uniform. "Are you packed?" Charlie nodded. Beamish checked that the top of the suitcase was secure. "Good. The border guards are idiots. They never think to check for a false top to a suitcase, they're only ever interested in false bottoms. Did you manage to get everything in the sculptures? We may as well get as much information out with Charlie as we can."
I tapped the base of one. "Just about dry. If they look too closely, they'll guess."
"I know!" Charlie said, pulling a sketchbook out of his case. He dragged over a stool and quickly sketched out a voluptuous woman straight out of a pinup. "Say, Lambert, give me a hand here." He tore off a couple of sheets and handed them over to me. As Danny and Beamish wrapped the sculptures in Charlie's clothes and tucked them in the case, I sketched a leggy art deco beauty, her body entirely Vita's, her face... I hesitated. I had drawn Vita so many times, but now her image was fading for me. The face was Annie's. "There we go," I said, tossing the nude casually on top.
Charlie whistled in appreciation, slotting his own sketches underneath. "Hopefully, if they stop me, they'll take one look at these and won't bother searching the rest of the case."
"You know, it might just work," Danny said, tilting his head to look at the drawings. Charlie clicked the case shut and picked up his trumpet.
Beamish looked at his watch. "Come on. We need to go."
Gussie opened the street door and shook hands with Charlie. "Good luck," he said.
"You too, kid," Charlie said.
Danny embraced him. "Stay safe, you hear?"
"Will you say good-bye to Varian and the others for me?"
"Sure."
Charlie shook my hand. "Like Varian always says, I'll see you soon, in New York."
TWENTY-FIVE
MARSEILLE
1940
GABRIEL
I never knew for years what happened to Charlie. He made it through the border safely enough, thanks to his cigarettes and our saucy drawings, but it turns out he was arrested in Spain. They brought him back to France to be interrogated by the Gestapo, but the guards can't have been up to much. They left him unattended in the railway waiting room for a moment, and Charlie calmly picked up his trumpet and his little suitcase with all the hidden information and jumped onto a train that was heading down to Madrid. He was one of the golden ones, old Charlie—slipped right through the fingers of death time and time again.
Like I said before, people disappeared all the time in Marseille. It was an easy place to get lost in. The winter days rolled into one another, and I could have been there two days or two months, it felt, as I wandered the streets and waited to see Annie again. I'd taken to hanging around Air-Bel every day in the hope of seeing her, but it had been a week since the Sunday we first met, and I longed to be with her, counting the hours until our arranged meeting after her ballet class.
I was early. Of course I was, I was longing to see her. As the Saturday crowds milled around the pavements, I walked the streets, burning off some of my nervous energy. I happened to see Annie, across the road, jumping down at the tram stop on La Canebière. I raised my hand, about to call to her, but I felt foolish suddenly. I didn't want to come across as too keen. But the sight of her—I couldn't wait another hour to be with her, so I followed her to her ballet lesson like a stray puppy; she was talking and giggling with her friends, completely oblivious. The other girls were like children to me, mousy and unformed, but Annie—she was radiant, the winter sun on her blond ponytail, the sure carriage of her head. They ran up the steps to a hall near La Vieille Charité, and I snuck around the side of the building into the alley. My heart was beating fast as I climbed the old metal fire escape, the treads creaking under my boots. I could hear the piano player warming up in the hall, scales and arpeggios drifting up from an old upright piano. At the top of the fire escape, I settled down on the low wall and leaned across, peering through the clear glass corner of one of the large skylights. The girls must have still been changing, because all I could see was the gray head of the piano player, bent over the keys as his hands swept up and down. Up there, I could see across the terra-cotta tile roofs to the old baroque chapel. The delicate dome and the stonework were falling apart in those days, before Le Corbusier took an interest in the place after the war. The music stopped, and I saw the class lining up at the barre, and the gilded crown of Annie's head. Her arms unfurled like the black stamens of a flower. The girls all wore black leotards, black tights, and shoes. None were as lovely as she. I laid my cheek against the cool glass and watched her for an hour, imagining how it would feel to place my hands around that waist, so slender I was sure my fingers would touch. I think I knew I was in love with her then, as I watched, and I grew self-conscious. I slipped away and walked for a while to give her time to reach the café.
I strolled up, ten minutes late. The red awning flapped and snapped in the breeze. She was sitting outside on a bentwood café chair, her ankles crossed elegantly in front of her. She has beautiful feet still—gorgeous toes, a ballet dancer's arch, slender ankles. She was wearing a gray dress with a full skirt and a white collar. She looked a little like a novice nun, with her hair piled up on her head, but there was something in her gaze as she looked at me that sent my guts falling away like the drop of a roller coaster. Then she smiled. That little gap between her front teeth.
"Mr. Lambert," Annie said. I took her hand and pressed her fingers to my lips, our gazes meeting. From the flush in her cheeks, she felt it, too, the energy flowing between us the way the sky becomes alive before a storm. She introduced me to her friends, but I barely registered them. I felt myself expand, looking at Annie, the world shrink around us to a fine point, to her. I was polite enough, made small talk with her friends—even then I was smart enough to realize their approval would count. I ordered a coffee and cognac, and cordials for them. Annie asked for a tisane of chamomile. Once the waiter had served our table and the other girls were chatting happily over their drinks, Annie cradled the steaming white cup in her hands and blew gently, raising her gaze to mine.
"So," I said, leaning toward her, the old wicker chair creaking beneath me. The table was narrow. I could feel my leg brush lightly against hers, and our elbows were just a hand span apart.
"So," she said, her eyes creasing with amusement. "What did you think of our ballet class?"
"You saw me?"
"Of course." She glanced down the table, making sure her friends weren't listening, a smile playing on her lips. "I saw you crossing the road just after the tram stopped."
"Why didn't you say something?"
"I wanted to see what you would do." She leaned closer. "I don't think you would make a good spy, you stand out in a crowd, Mr. Lambert—"
"Gabriel." I was so close to her, I could smell the sweet herb scent of her tea. The cognac warmed my stomach. I could feel a golden heat thawing the coldness from my torso, my limbs, my fingertips. Or was it Annie made me feel like that?
"I felt you, watching from the skylight like a naughty black cat."
"But you didn't look up once."
Annie shrugged. "Didn't want to spoil your fun."
I pressed my mouth to my knuckles to stop myself laughing in my embarrassment. "I'm sorry. I couldn't resist. You looked so beautiful." She lowered her eyes. "Are you angry with me?"
"No, I'm touched you wanted to see me again so badly." She blinked. "I did, too."
"I've thought of little else all week," I murmured. The space between us seemed to contract. I felt light-headed with longing for her. A raucous shriek from one of the other girls broke the moment. A couple of boys they knew dragged over a table to join them. Annie introduced me to them, and we waited for the table to settle down. She looked from the boys to me.
"How old are you?" she said quietly, leaning toward me again.
"How old do you think I am?"
She cupped her chin on her hand and studied my face. I wondered what she saw. The flecks of gray at my temples, the dark olive skin?
"Too old," she said, a smile playing on her lips.
I folded my arms and leaned in to her. "Too old for what?"
"For me." She laughed. "My father would never allow me to step out with a man like you."
I lowered my voice. "What if I told you I was eighteen years old?"
She threw back her head and laughed. "I'd say you were the biggest liar I had ever met." See, Annie knew me well even then.
"Well, the answer is easy," I said, tracing my finger on the table, achingly close to her hand. "We don't tell him."
"A secret?"
I glanced at the kids at the table with us. They were deep in some conversation about the war, not paying any attention to us. I was dizzy with wanting her, quite sick with desire. My thumb stroked the inside of her wrist, the lightest touch. She didn't take her hand away. "I have to see you alone." I could see from the rise and fall of the small gold crucifix strung around her neck that she was breathing quickly.
"Tomorrow," she whispered. "Meet me at three o'clock on Sunday, under the railway bridge at La Pomme."
TWENTY-SIX
MARSEILLE
1940
GABRIEL
I skipped some of Breton's meeting for her, and we met every Sunday from then on in the same place. It was a narrow little bridge, barely a car wide, with elegant stone walls sweeping up the steep embankment to the railway track above. It always felt like entering a grotto as I stepped into the darkness, like something from a fairy tale. I don't know where her parents thought she was, but she brought a little dog with her, and we walked in the woods. It was a greasy-looking terrier that growled every time I went near her and nipped at my ankles. In the end, I had to tie it to a tree so I could kiss her. That first kiss I remember in perfect detail. She leaned back against a bare winter tree and waited for me. The leaves were soft and wet on the ground, my footsteps muffled. The clouds were low, enfolding the trees like a blanket. It was so quiet, I could hear my own breath and the slow drip of water from the branches.
"I saw Madame Breton in the village the other day," she said. "She was like an exotic bird, with feathers in her hair."
"You know she is the wife of André Breton, the famous surrealist?"
"The writer? Even I've heard of his poems." She tilted her head as I put my arms either side of her, trapping her against the tree. "She must have seen us together. She asked me if I was seeing 'that handsome Gabriel Lambert.'"
"Did she now? What did you say?"
"I told her I knew you." Annie shrugged. Her eyes were lowered. "We walked back to the house together."
"What did you talk about?"
"She told me about meeting her husband. She said a love like that is an illness from which you never recover. Do you think she is right?"
"Even geniuses are idiots when they talk about love." I sensed her disappointment and backtracked quickly. My head was swimming with desire for her. "I hope that I never recover from you."
Annie raised her gaze to mine. "She said to me just being pretty isn't enough for a woman to become and remain a part of a great man's life."
"Oh, Annie..."
"What if I'm not enough for you? What if you get tired of me? I'm not a great artist, like you, I haven't traveled, or learned things, not like you—"
"Annie," I said, and cupped her jaw in my hand. "If only you knew..." I hesitated. "If only you knew how I've loved you from the moment I saw you. My love is..." I looked up at the sky, searching for a way to express how I felt about her. "My love is more than the stars. It is like Venus, the morning star. It is there, all the time, night and day, burning brightest of all." I felt her relax; her head fell back. I knew Annie was ready then, waiting. There had been no hurry with her, no urgency, not like Vita, just a slow, deep longing for home. There was an inevitability to us. I remember the roughness of the bark beneath my hands as I leaned in to her, the moment just before my lips touched hers, how the world seemed to dissolve and fall away, a burning light around us as my eyes closed.
* * *
Annie may have hidden me from her parents, but they would have been proud of the way she held on to her virginity. I tried everything short of downright begging, but I fell for the only girl not making love during the war. Every Sunday I traveled to La Pomme and walked miles with that damn little terrier snarling at me, all for a kiss and the hope of more. She was beautiful, and mercurial, and I wanted her, so I came each week. By December, Marseille lay under a blanket of slush. My feet were permanently wet or cold, I remember. It's very true: if your feet are comfortable, you can cope with a lot in life. What is it they say, an army marches on its stomach? Stupid saying, it marches on decent boots. Anyway, I was starving hungry and my feet were bad. They have been, ever since the war. I endured chilblains and tormented nights for Annie, but it was no more than she deserved.
"I like this," I said one Sunday, touching the embroidered gold scarf she was wearing.
"Do you? I made it." She slipped it from her neck and wrapped it around mine. "Here," she said. "Keep it."
"I couldn't."
"Please, it would make me happy to think of you wearing it when we aren't together." She tucked it into the lapel of my coat. "Besides, I think you need it more than me. Your lips are blue." She kissed me. "Better?"
"I'll treasure it. Thank you." I rubbed the pale gold wool between my fingers. "It's beautifully warm."
"It's cashmere." She had embroidered it with hundreds of stars. Later, when I read Yeats's "Aedh Wishes for the Cloths of Heaven," I thought he had written it for us. That's what we did on those walks—we spread our dreams before each other like an offering.
We talked for hours about everything and nothing. She asked me about Paris, about my family, about art school, and I told her. I lied as little as I could, yet I kept the truth about America from her. I think I was afraid I would lose her. Now that I had found her, it made the knowledge that I would leave soon harder to bear.
* * *
One evening—it must have been at the beginning of December—I was heading back to my lonely bed in town after walking her home. We had been to the Santon fair together, and Annie had bought me the little shepherd. It had been the most wonderful afternoon, and I felt a rare contentment as the tram trundled back into town. There was mist in the valley, only the tops of the roofs and umbrella pines poking out as the tram trundled along its route to La Canebière. Everyone huddled up on the tram had the sunken look of cold and hunger. I was no different, but I was lit up inside with love. You can endure anything when you feel like that. I decided to buy her something for Christmas and remembered an old jewelry store I had passed with her on the way to the Vieux-Port.
I stopped in the café Au Brûleur de Loups for a cognac on the way.
"Non, monsieur," the bartender said, pointing at a sign tacked above the bar: Jour sans alcool. "I can offer you a glass of champagne instead, perhaps?"
"It doesn't feel like a day for champagne," I said, and ordered an espresso instead. I love that about the French. Once in a while they banned booze, but you could still get a restorative glass of champagne. I hail from a civilized nation. As the bartender cranked the machine for my coffee, steam poured out, hissing and gurgling. I leaned against the zinc bar and looked in the mirror behind the bottles. Above the noise of the coffee machine, I could hear Varian's voice. I sipped at the scalding coffee and listened. I could just see in the mirror that he was sitting at the back of the café with Beamish, talking. I wandered over and greeted them.
"May I join you?" I said.
"Hello, Lambert. Of course." Varian moved aside his overcoat, and I pulled out the wooden chair. "How are things at the hotel?"
I shrugged. "Not bad. It is the waiting that is getting to me."
"You seem to be passing your time constructively," Beamish said, a smile playing across his lips.
"What do you mean?" I said.
"Your girlfriend is very beautiful."
"She's not... I mean...," I said, blushing furiously.
"Leave the poor fellow alone, Beamish," Varian said, laughing. "Besides, you are one to talk." He leaned conspiratorially toward me. "This chap has a new girl every time I see him."
"And you don't have an eye for a good-looking woman, Buster?" Beamish held Varian's gaze until he looked away, smiling. If Varian had affairs, he was certainly discreet. I never saw him with anyone.
"How are you getting on with the receptionist from the consulate?" he said to Beamish.
"Camille?" He cocked his head. "It is a sacrifice I must make to help our cause." I remembered the tough-looking blonde at the desk the day I met Bingham. If he was seeing her, he was a stronger man than me.
"Does she help you with the visas?" I said, leaning toward him. Sure, I knew that the ARC was the legitimate front for more clandestine work, but they were still careful with the amount of information they let me in on. Beamish looked uncomfortable.
"The ARC needs all the help it can get," Varian said carefully, "and Camille helps."
"A little," Beamish added. "She has an expensive cocaine habit, and I suspect she is selling information to both sides to fund her habit."
"It must be dangerous, though, the work you do?"
Beamish stared me down. "Not at all. We simply help refugees with visas, give them money for food and hotels. What could be dangerous about that?" I realized I had overstepped the line.
"Of course, I understand." I drained my cup and stood. "Good to see you both."
"Are you leaving? Breton and the others will be here soon. I thought you'd come for the meeting," Varian said.
"I am sorry, I have an appointment," I lied. The truth was, I was still so in awe of Breton, the thought of being in this small café with him intimidated me. He was the magus around which an ever-changing cast of writers and artists fluttered, moths to a flame. He knew—that's what his eyes said, every time he looked at me. He knew. I felt, somehow, that if my gaze met his, I'd turn to stone.
"Will you be up at the château on Sunday?" Varian asked.
"I wouldn't miss it." I glanced at Beamish. "How come I never see you up there?"
"Beamish prefers to stay in town," Varian said.
"One of us needs to." As Beamish looked at me, I felt my cheeks burn. "There are too many people playing games around here."
* * *
I walked on through the narrow streets, deep in thought. I had no idea, then, how remarkable Varian, Beamish, and all of them were, of the risks they were taking. On the surface, Varian seemed like the archetypal preppy—oh sure, if you saw him up at Air-Bel letting his hair down, there was a different side to him, but you'd still say he was straight as a die. Beamish, I couldn't figure out. He was smart, real smart, no hiding that, but it wasn't obvious in the way it was with Varian. He was, as the French say, un peu dans la lune. But there was another side to him—when I was up at the château one day, I heard Danny saying that Beamish had fought with the Republicans down in Spain. I admired him, I guess. I wish I'd known him better.
I found myself at the jewelry store before I knew where I was. The old woman was just closing up, turning over the sign on the door, but she let me in. The old brass bell on a hooped spring over the door tinkled. It was warm inside, a little stove behind the counter glowing in the dim light.
"There, that bracelet, please." I pointed to a silver charm bracelet, with stars and shells, in the window. The woman wrapped it in tissue paper and put it in a small red box for me. I walked out of the shop on air, buoyed up at the thought of giving it to Annie the next day. I rounded the corner of the street with a spring in my step, and then someone grabbed my arm.
"I thought I'd seen a ghost," he said.
"Quimby?" My bowels went slack. I would have run, but he had me by the arm, and I didn't want to make a scene with so many people around. Quimby was the one man who could destroy me, and I didn't want him spouting off to the police if we started a fight. He pushed me into a dark alleyway. I could see in the half-light the expressions rolling across his face as he put two and two together.
"Well, well. I was right." He tightened his grip. "What on earth do you think you are doing? I went back to the house to get you and the paintings and found it all burnt to hell. I had the buyer all lined up."
"It's not what you think," I said quickly. My heart was jumping around in my rib cage like a cricket. "I didn't—"
"Kill them?"
"How can you say that?" I pulled my arm free. "You owe—"
"I owe you nothing, you shit," Quimby said, spittle flecking my face. "The thing is, the paintings I have left are selling like hotcakes...."
"My paintings," I said, squaring up to him.
"I'm going to sell the lot before I get the hell out of this dump." He pushed me back against the alley wall, the heel of his hand pressing into my sternum. "Speaking of which, how much have you got on you?"
"I'm not giving you any money."
"You will give me exactly what I want, if you know what's good for you."
"I need time."
"That is one thing we have plenty of," Quimby said. "We're all trapped like rats here. Just remember, one word to the authorities—"
I searched in my pocket, felt the red jewelry box under my fingertips. I pulled out the notes I had left and handed them to him. Luckily I had hidden most of the money I'd rescued from the Château d'Oc under a loose board in my hotel room, and I had already put down the money for my ticket to America once boats became available. "That's all I have."
"All you have, or all you have on you?" He sneered. "Don't try and play games with me. I know how much you took from the château. There was always plenty of cash in the desk. I want another five hundred francs next Friday. I'll meet you outside the Notre-Dame de la Garde."
"I don't have it."
"Then find it." Quimby tucked the money into his breast pocket. "I want you to keep a low profile. Not that you would be hanging out in the kind of hotels my clients are staying in." He pulled on a pair of black kid leather gloves and turned up his collar. "I wouldn't want them to bump into you, that's all. And in case you get any ideas about not showing up, you should know I've been following you for a couple of days. I know all about the ARC, and that pretty little blonde." My mouth went dry. "You should also know that I have certain photographs of you in Vita's studio...."
"Rubbish, you're bluffing."
Quimby calmly pulled out his wallet and slipped out a black-and-white photograph folded in four. I fought the urge to retch. The night returned to me. The flash of Quimby's camera reflected in the mirror near Vita's easel, the pop and hiss of the bulb.
"Don't you dare threaten me," I said quietly. The blood rushed and sang in my ears. Could I thump him, steal his wallet? But Quimby had said "photographs"—he had more, somewhere? I racked my brain, trying to remember that night.
"I'll do what I damn well like." He tucked the photo away. "And don't think about disappearing in this cesspool of a city, or I'll be straight round to nice Mr. Fry with the rest of the charming photos I took at the Château d'Oc."
"You wouldn't dare. I'll... I'll..." I was right. Panic washed through me, chill and sickening.
"What are you going to do? Bump me off?" He laughed, a quick exhalation through his nose. "Or did you do it?" I saw his teeth gleam in the blue streetlight as he turned away, smiling. "Who cares? For what I'll get for the paintings I'll risk dealing with a murderer."
Quimby was clever, that's for sure. He wormed his way straight to the heart of the thing that kept me awake at night. Did I kill them? Sometimes, in the sleepless hours, I still wonder if I pulled that door to and sealed their fate. I stood in the alleyway for a few minutes after Quimby left, shaking with shock and cold.
TWENTY-SEVEN
FLYING POINT, LONG ISLAND
2000
SOPHIE
The light is dazzling, flaring on the water, on the white sand. Sophie tastes salt on her lips as she bites the bottom one, searching in her bag for her phone. She is some way from the Lamberts' house, alone. She has ignored the insistent ring of the old-phone tone twice already, and she goes to switch it off but then sees it is Jess.
"Where are you?" he says the moment she answers.
"Flying Point." Sophie sinks down on her haunches and looks out to sea. "I can't talk now." She rakes her hand through her hair. "It feels like I've walked miles. I'm going round and round in circles...."
"Listen, be careful, okay?"
"What do you mean, be careful?"
"I've been doing some snooping around for you." She hears the slight slur to his words, the background noise of a bar somewhere.
"I can look after myself."
"I know, I know." The phone muffles as he tucks it beneath his jaw. "But I've something that might make a difference."
"To the story?"
"To how much you care about the story. To us."
Sophie exhales and glances over her shoulder. She settles down on the sand. "Go on, then."
"Tell me what you know about your dad's family."
"What? Where's this going, Jess?" Sophie frowns with frustration. "My great-grandfather was from London. He married an American girl, Sophie, who I'm named for. They had two children, my great-aunt Vita and my grandfather Sam. There was some kind of argument, and Vita went off to France before World War Two. When my great-grandfather was killed, Sophie moved back to the U.S. with Sam—I think she met a GI, or something, in London?"
"Right so far." Sophie's eyes narrow at his cocky tone. "Go on."
"Vita, as we know, died in France." Sophie squeezes the bridge of her nose. "I'm going to get to the bottom of that."
"And what about your grandfather Sam?"
"I... I don't know much about his past." Sophie hears Jess exhale a short, soft laugh.
"Go on," he says.
"Sam met a girl in New York sometime in the early fifties, and they had a kid—my father, Jack Cass."
"Nope, not necessarily."
"What do you mean, 'nope'?"
"I mean, what if Sam Cass isn't your grandfather?" Jess sounds triumphant. "I can't believe you didn't check this out—"
Sophie leaps up and paces along the shore. "What are you talking about?"
"It's basic Journalism 101. Take everything you believe to be true that you are basing your story on, and make damn sure it is before you start work. I knew you would have taken this for granted. You trust people too much, Soph." She hears him take a hit of his drink. "Old Sam must have been a stand-up guy, because he married a girl with a small baby." Sophie hears the rattle of ice cubes. "Who knows, maybe it was a lavender marriage and he had something to hide himself; they didn't have any more kids—"
"You're drunk." Sophie closes her eyes, and the bright corona of the sun flares red and orange behind her eyelids.
"I may be drunk, but I'm right. There's a blank on your daddy's birth certificate under 'Father.'"
"Why are you doing this?"
"So that you'll give up on this dumb story, and come to Paris with me. The great Jack Cass was a bastard. There's no family connection now, no magical free-spirited Great-aunt Vita that you clearly wish you were more like—"
"That's low." Sophie shakes her head. "Dumb story?"
"Hey, don't shoot the messenger," Jess says. "It's not your story, Cass. Let the past go."
"Do you really think I care any less about it because of this? You're right about one thing, Sam was a stand-up guy. Whatever the deal was with their relationship, he and my grandmother loved one another, Jess—really loved one another, and they loved Dad."
"I didn't mean—"
"I grew up listening to Sam's stories about Vita, and if you don't get how inspiring hearing about a woman who was creative, and smart, and not afraid of anything was to a kid, then that's your loss." It feels like something has broken free inside her. "If you thought shattering my dreams would make me settle for you, then you were wrong."
"Hey, Cass, hold on—"
"Do you think I care for one moment if Vita's not a blood relation? She's family, Jess. We loved her, and she died. She was younger than me, and I just want to find out the truth, for me and for Dad."
"Yeah, it's all about Daddy...."
"What was I, Jess? Your idea of the perfect girlfriend for the great American writer?"
"Soph, you were—you are, perfect for me—"
"Perfect pedigree? You loved the idea of that, at first, didn't you? Jack Cass's daughter. My dad may have been a bastard as you so charmingly put it, but he was still a better writer than you'll ever be."
"What happened to you?"
"I grew up." Sophie raises her chin. "I owe it to him to write this story, and then I'm done."
"Paris?"
"No, Jess. You've always been so hung up on proving that you are better than my dad, and you've just shown me you are nowhere near the man he was. He was human, and fallible, and wonderful, and he loved me and my mom. You've never loved anyone but yourself, Jess. I deserve more than that."
"You've met someone, haven't you," he says. Sophie thinks of their conversation in the bar the night before. "Is it serious?"
She glances over her shoulder, feels her stomach tighten at the thought of Harry. She's angry, but there's a fierce inevitability to it. "Nothing's happened." She realizes it's not true as she says it. Nothing's happened—yet. "It's over, Jess. It was over with us when you left me, and now I'm over you."
"Good luck to him, whoever the hell he is." Jess's words roll together, angry and hurt. "He's going to have to be one hell of a man to shape up to your daddy. I mean, what was it about him, eh? What did Jack Cass ever do for you that I couldn't? I know you. I know there's something you haven't told me."
"Thanks, Jess," she says calmly. "Thank you for making one decision a whole lot easier." Sophie turns and gazes out at the endless blue sea. "Take care of yourself. Good-bye."
Sophie cups the phone in both hands, fights the urge to throw it out to sea. She wants Jess, and his jealousy, gone, gone for good. What did Jack Cass ever do for you? Her chest rises and falls, her heart a staccato beat. You'll never know now, Jess.
* * *
"How was ballet?"
Sophie can remember her father's voice, still. She remembers the feel of her mittened hand in his, how she leaned in to the warmth of his overcoat, her head resting against his hip, the soft leather of his jacket—the smell of tobacco and motorbike oil. Steam rose from the pavement vent at their side as they waited to cross West Seventy-ninth Street. She remembers it all.
"Ballet was good. I'm ready for the exam next week."
"Good girl. Always give one hundred and ten percent, remember that." She feels him tighten his grip on her hand as they walk on up Broadway.
"I'm starving. Can we get a milkshake?"
Jack glances at his watch. "Your mom needs some groceries from Zabar's, and I've got to get back to work, honey. We'll be at the apartment in a few minutes, can't you wait?"
"I'll be quick, I promise." Sophie points at an old drugstore on the side street, its neon sign and chrome silver front blurred by the first snow falling. "Come on...." Her hand slips from his, and she backs away, daring him. "We can share a chocolate one."
"You're on," Jack says. He scoops her up into his arms and runs toward the drugstore.
Sophie remembers pushing open the door of the drugstore, the old bell ringing high above her, the warm draft of air. They sit at the lunch counter, her feet swinging above the chrome bar of the stool. The windows are steamed up, the streetlights and headlights beyond a pastel blur. It is busy. They have to wait for their order. She remembers the anticipation, swinging impatiently on the stool, round and round as Jack flicks through The New York Times. Finally, the waitress hands over the metal mixing cup and two glasses, two red-and-white-striped waxed paper straws.
Sophie has thought of this moment so often, it has a hyperreal quality. The red of the Formica counter is vivid. She sees her small hand on the counter, reaching for her glass, the smooth flow of the milkshake, hears the hiss of the coffee machine. Just as Jack begins to pour, the shop bell rings again.
Sophie's stomach lurches now with fear.
TWENTY-EIGHT
VILLA AIR-BEL, MARSEILLE
1940
VARIAN
"Timeo Danaos et dona ferentes," Varian murmured, gazing up at the murals in the library. "Beware of Greeks bearing gifts." From the hall below, voices drifted up and among them the clear sound of a young American woman, saying, "And this is for the children."
"You shouldn't have," he heard Danny saying. "Thank you, Miss Guggenheim."
"Peggy," Varian said, jogging downstairs. He was struck as usual by her warm and anxious gaze. It always seemed out of place, somehow, in such an angular, confident woman. Her hesitation, her nervousness, made him want to protect her. Even her slightly bulbous nose added to the charm of her face.
"What a darling place you've found!" Peggy Guggenheim stepped across the black-and-white tiles of the hall and turned in a slow circle, taking the house in. "How clever of you, Varian."
"It was nothing to do with me." Varian glanced at Mary Jayne. She leaned against the doorway to the living room, smoking a cigarette, one arm crossed over her rib cage. "Miriam and Mary Jayne masterminded the whole thing." He registered her slow approval. "We're very happy here," Varian said. "May I take your coat?"
"Oh, no, my dear. It's frightfully cold." She pulled the collar of her dark sable coat tighter, burying her head farther down. "Aren't you freezing?"
"You get used to it." Varian ushered her into the living room.
"I'm so pleased to be here, at last. The artists talk of little else except Breton's Sunday salons." She cupped the curl of her hair in her palm and buoyed it up. "Air-Bel and Countess Lily Pastré's estate at Montredon are like sanctuaries in a dark night, the calm eyes of the storm...."
Varian raised his eyebrows at Mary Jayne as Peggy walked on.
"Lily's a remarkable woman," Peggy said. "I believe Masson is in hiding with her, is he not? And Pablo Casals, Josephine Baker. Under any other circumstances it would be a marvelous party, wouldn't it?"
"Come and warm up, Peggy," Mary Jayne drawled. "Can I get you a drink?"
"A cognac, perhaps? Just to warm up." Peggy curled up on the rug before the fire, her thin legs tucked beneath her body. "Well, who are you?" she said to Clovis, who rolled over on his back, paws cycling the air. "I've met old Dagobert, but you are a handsome young fellow, aren't you?"
"Thank you," Varian said.
"Not you, my dear, I meant the dog."
Varian laughed. "I know you did, Peggy. He's mine, for my sins, aren't you? He's a scoundrel. Can't get him to do a damn thing I want him to."
"Poodles are clever dogs," she said, smoothing Clovis's belly. "You have to make him think it's his idea." She glanced up at Mary Jayne and took a balloon glass of cognac from her. "Thank you, darling."
"Your earrings are adorable, Peggy."
She touched the tiny paintings set in gold dangling from her ears. "Tanguy painted them for me. Do you like them? Now, you must tell me everything. How are you? Will you be going back to the States soon?"
Mary Jayne shrugged. "Perhaps."
"You must, it really is time for us to hightail it out of here, don't you think?" She sipped her drink. "Why, I've spent the last couple of months packing up my art collection single-handedly." She inspected her broken, torn nails. "At least the paintings are safely on their way to Grenoble now."
"The museum agreed to store them for you?"
Peggy nodded. "Now, I just need to get myself to safety." She looked up at Mary Jayne. "If you are leaving for New York soon, may I take your room here?"
She frowned. "I'm not leaving." She glanced at Varian. "Yet."
"I've been wanting to ask you about something," Varian said to Peggy. He watched as Mary Jayne settled on a sofa on the other side of the room. The sound of a woman's laughter drifted through from the open door.
Peggy scowled. "I suppose all those ghastly fake surrealist wives have taken roost here, too? I—" She broke off and flung her arms up in greeting. "Jacqueline!" Peggy put her glass on the hearth and jumped up.
Jacqueline strode into the room, her full coat swinging. She brought the fresh scent of cold air and the woods with her. "Peggy." She opened her arms in an embrace. Varian noticed that both women kept their eyes open, their faces strained as they kissed once, twice. "What a marvelous surprise. Are you staying?" The edge to her voice made him think of paper cuts.
"A couple of days. I have some business to finish up, and then I'll be back again. I adore it here, it's just like Paris, don't you think, with all the surrealists wandering around in a vague way?"
"How delightful. We must..." Jacqueline followed the direction of Peggy's gaze, and her voice trailed off as she realized Peggy wasn't paying the slightest attention. Peggy's face lit up as André walked into the hall, nudging the main door closed with his shoulder. His arms were laden with branches and kindling. She rushed forward, beckoning to Danny and Gabriel.
"Boys!" she cried. "Come and help me. Monsieur Breton should not be carrying logs. It's like seeing Leonardo da Vinci take out the trash."
André carried the wood through. "Nonsense, Peggy," he said. "We are all doing our bit to help run the château, and this is a delightful chore." He tossed the wood into the fire basket and turned to his wife, taking Jacqueline's coat from her shoulders. Aube ran into the room and flung her arms around her father's leg. She glared at Peggy. "You remember our daughter?"
"Of course! Charming," Peggy said, taking up her spot in front of the fire by Varian again. As the Bretons busied themselves with unwrapping scarves and gloves, Peggy leaned in to Varian. "The child's a pest in cafés. I mean, he's clearly besotted with her, terribly indulgent. Don't you think that's what happens if you have children after forty?" She took a cigarette from Varian's silver case and waited for him to light it. "Thank you." Peggy rested her head back against the marble pillar of the fireplace and exhaled. "He must have mellowed. I remember a story about an outburst in Paris when a woman with a baby carriage bumped into him on the street. He rounded on the poor woman and said, 'If you must shit out a child, keep it away from me.' But look at him now, with the girl, he adores her, don't you think?" She stretched and sighed. "Do you have children, Varian?"
"My wife is keen. In fact, she's hoping I'll bring home a war orphan."
"Instead you're bringing her a poodle?" Peggy roared with laughter. "Varian, you're incorrigible. Oh, it is good to sit still for a while, isn't it? I feel like I have been running for months."
"How did you get out of Paris in the end?"
"My dear, it was a close-run thing. Can you believe the idiotic life we were living there? Everyone was determined to party that summer. Do you remember the ridiculous craze for yo-yos? Why, Cartier even made a gold one. Such nonsense." Peggy shook her head. "Why, right up until the last minute we were sitting in cafés drinking champagne while trains were pouring in with machine-gunned refugees. The day Hitler stormed Norway, I walked into Léger's studio and bought a 1919 painting."
"I heard you have been buying a lot of work," Varian said carefully.
Peggy laughed throatily. "People accuse me of profiteering—well, let them. The artists need money. Picasso, if you can believe it, had the nerve to turn me away at the door of his studio, and said, 'Lingerie is on the next floor.' Hateful man. But then, I loathe myself sometimes." She sipped her drink. "I got out on June twelfth, two days before the Nazis marched in. I'd been stockpiling bidons of gasoline on my terrace."
"A good move."
"Well, I just bundled my maid and my two Persian cats into the Talbot and hightailed it out of there. I tell you, it is marvelous to be here, just marvelous."
"Do you know the Bretons well?"
Peggy pursed her lips. "I wouldn't say I was part of the surrealist circle in an orthodox sense, but I am a great admirer of Breton." Her eyes dilated as she looked at André. "Have you not noticed the effect he has on people? No one is immune to his charm."
Varian shifted his arm and spoke quietly to her. "Listen, Peggy, if you really want to do something to help these artists, I realize it must be tempting to head to the U.S. as quickly as possible, but I wondered if you had paid any thought to my suggestion that you take over the ARC from me. Those blithering idiots in New York have no idea what's really going on here, and we need someone good if the office is to carry on its work."
"Of course I have thought about it, in fact I talked to the American consul about it only this morning." Varian's heart sank. "It's preposterous. What do I know about refugee work?"
"About as much as I did when I took over this job," Varian said. "You've got to remember, Peggy. I was an editor at Headline Books. I am an editor... that is, if they'll keep my job open any longer. I was only meant to be here a couple of weeks. I need to get home, to work and to my wife."
"I simply can't help. The consul advised me to have nothing to do with you all." Peggy glanced at him. "Though of course I don't pay attention to everything he says."
"The thing is, Peggy, this work is taking its toll on my health." He hesitated, thinking of Eileen's latest letter. "And my marriage."
Peggy placed her hand on his. "You are doing marvelous, wonderful work here. But I'll be honest, the black market atmosphere in Marseille terrifies me. The very thought of arrest..." She shivered. "I daren't think what you are really up to, behind the scenes, to get all these people out safely. I'm not cut out for espionage. I'm sorry, Varian, I just can't. You do understand?" She picked up her glass. "I am happy to give you money, though, and I guarantee I'll pay for the passages of the Bretons and Max Ernst to America....
"André," she called. "Does the thought of America excite you? I was thinking when I establish my gallery, you could use it as the court for your surrealist gatherings."
"Thank you, Peggy." His face grew somber. "America is... necessary. I cannot say I like exiles a great deal."
"Oh, my dear, it will be splendid—"
"Have you met Max?" Varian interrupted Peggy, quickly changing the subject when he saw André's discomfort.
"Not yet. I find his work quite dazzling," she said.
"I believe he has a similar effect on women, too."
"Varian, you are naughty." She cocked her head. "Though I am quite seduced by the idea of him, and his work."
* * *
André paced at the center of the living room, reading from a letter from Marcel Duchamp. Varian looked around the room—people had squeezed onto sofas, shared chairs carried through from the dining room. Jacqueline sat near the fire, with Aube curled asleep in her arms. His voice is mesmerizing, like a spell, Varian thought. He smiled to himself at the expression of awe on Gabriel Lambert's face. What is it with men like Breton? This power they carry? He's more like a shaman than a poet.
As Breton held court, Varian thought of that morning. He had gone out early to walk in the grounds while it was silent. They had all stayed up late the night before, as usual, so no one was up yet, or so he thought. He walked quietly downstairs, his socks slipping on the wooden stairs. At the landing he paused and listened. Behind the closed doors he could sense breathing. Varian padded down to the hall and sat on the bottom step to pull his shoes on. He buttoned up his coat and slung his binoculars around his neck.
The crisp air was like a tonic to him, clearing the pain of his hangover. I'm drinking too much, he thought, marking off the dull ache of his kidneys, his head, his ulcer. He breathed deeply as he walked, one gloved hand balled in the palm of the other. His thoughts fell into the rhythm of his steps as he ran through the urgent cases they had coming that week. Of the clients most in danger, the German politicians Hilferding and Breitscheid were still in hiding, and they were no closer to getting the editor Bernhard or "Baby" Mehring, the poet, out.
The sound of an ax chopping wood caught his attention, and Varian raised his binoculars, looked ahead through the forest. A thin, worried-looking man with gray, sunken cheeks was swinging halfheartedly at the trunk of a birch in the woods. "Bonjour, monsieur," Varian called. Someone else can't sleep. The man turned quickly in surprise.
"I am not stealing."
"It is none of my business. We don't own these woods."
"Thumin said it would be all right. I am paying him." His eyes darted. Varian thought of a thrush he had held in his hands once, the quick-fire beat of its heart, how fragile its bones felt. He saw the man staring at his binoculars.
"I was hoping to see some birds." Varian suddenly realized the man was frightened. "Do you live in the village?"
"I am Bouchard."
"Varian Fry. How do you do." He stepped forward, his hand extended. Bouchard made no move to shake his hand, so he awkwardly folded his arms. "You live over by old Thumin? I've seen your daughter, Marianne, around." He said no more, seeing the sudden flash of concern on the man's face.
"You are at Air-Bel?"
"I hope we don't disturb you?"
Bouchard chewed at the inside of his cheek. "We hear the music, at night. We see the people coming and going from the parties."
"Perhaps you and your family would like to come over one evening, to meet everyone?"
"No. I don't think so." He swung the ax. "We are quiet people, Monsieur Fry. Your friends would do well not to ruffle feathers here. We do not like 'artists' and..."
Go on, say it. Degenerates. Say it. "Is that a threat?"
"A threat? No. Just some neighborly advice." The ax hit the trunk of the tree, splintering the wood. The elm's bare branches shook and rattled above them.
* * *
Varian cursed under his breath as he stalked back to the house. Ignorant bloody peasants.... In his heart, though, he knew Bouchard was right. Air-Bel was attracting too much attention with the nonstop parade of flamboyant characters making their pilgrimage to the Sunday salons. Perhaps it would be safer to keep a lower profile. The glimmer of a candle in the greenhouse distracted him. Holding his breath, he stepped a little closer and peered through the misted glass. Among the dusty leaves, he saw André sitting at a small table, sheaves of paper with green script littering the books in front of him, scattering to the cracked terra-cotta floor. A candle flickered at his elbow. He wore a heavy green wool dressing gown, and he had his head in his hands, his fingers raking his thick chestnut hair. His eyes were squeezed closed in absolute despair. Varian felt the hot prickle of shame. He felt like a voyeur, a gawping tourist outside the cage of a lion.
* * *
Varian studied André's face now as he paced magisterially within the circle of artists and writers. André had said to him one night that he had no time for empty moments of depression, though he seemed plagued by them. Not a trace of that anguish now, Varian thought as he sipped his wine. A shriek from the hall stopped Breton's soliloquy. He pushed his way through the crowd, Varian on his heels.
"What is it?" Varian said. Breton was kneeling over the prone form of the maid, Rose. She had lost a scuffed shoe and had a hole in the bottom of one stocking. Gabriel appeared at Varian's shoulder and inhaled sharply.
"Is she all right?" he said. Varian heard the anxiety in his voice. Rose lay at the bottom of the stairs, vomit staining her white apron red.
André scooped her into his arms. "It's red wine, not blood," he said. "But I will examine her just to make sure."
"Monsieur Breton!" Madame Nouguet rushed forward. "You must not soil yourself. I'll fetch the gardener...."
"Nonsense, madame," he said, carrying the girl upstairs.
"Such a marvelous man," she said, wringing her hands.
"She'll be fine," Varian said to Gabriel. "Come on, have another drink, old chap." He clapped him on the back.
"I'm sorry," Gabriel said. "When I saw her lying there, it just... it reminded me of something." Of someone, Varian thought.
"Why is André dealing with the wretched girl?" Peggy said. "A great man like him. The girl was covered in vomit." She looked as though she were chewing wasps.
"André is a doctor, did you not know that?" Jacqueline said smoothly. She squeezed past with Aube in her arms, the little girl's head curled peacefully in the curve of her mother's neck. "It's late. I must get the little one to bed." She glanced at Peggy. "Sleep well."
"A doctor?" Peggy said.
"I had no idea," Gabriel said to Varian as he handed him a full glass of wine. "I still find it remarkable to see him... I mean, he's a legend. To see him collecting wood, or tending to that girl..."
"She's an alcoholic, clearly." Peggy clicked her tongue. "I could smell it on her as she served the soup."
"I think she may have drunk a little too much today because she was feeling guilty," Varian said.
"Guilty, why?"
"She forgot to feed the cow this morning, and the beast's bellowing was heard by one of the villagers. The authorities came and took it away this afternoon."
"You have a cow?"
"Had, Peggy. We had a cow. We gave its milk to the children."
"Goodness, how bucolic."
"You know it is all Chagall talks of. Will there be cows in America."
Peggy roared with laughter. "No, my dear, you misunderstand him. Chagall adores cows, identifies with the stupid beasts himself, I believe. Why, didn't he plan to have a cow on his business card? What he means is will they allow silly creatures like him into America."
"Well, I never." Varian glanced up. "Peggy, have you met Gabriel Lambert?" He saw a look of alarm cross Gabriel's face. "Peggy Guggenheim, a great patron of the arts."
"I know." He shook her hand. "How do you do, Miss Guggenheim."
"Perhaps you know Gabriel's work?" Varian said to Peggy.
"Some agent—Quimby, dreadful man—tried selling me some paintings in the summer. They were quite charming, but not my thing." She waved her hand. "Of course they are very accomplished, my dear, but decorative, wouldn't you say?"
"Peggy makes me laugh," Varian said quietly to Gabriel as she bustled away. "Have you noticed how she always answers everything with a question?" He raised his glass, puzzled by the flushed look of relief on Gabriel's face. "Your good health."
"He hasn't been here, has he? The agent she was talking about?"
"An agent? No, I don't think so." Someone turned up the radio, and a few couples began to dance in front of the fire. Varian raised his voice above the hum of conversation and the beat of the jazz tune. "Why, are you looking for him?"
"No," Gabriel said. "He's looking for me."
TWENTY-NINE
FLYING POINT, LONG ISLAND
2000
GABRIEL
The thought of Quimby has me on edge again. Still, after all this time. I screw my eyes closed against the memory of him, just for a moment.
"Gabriel?" The girl's voice is close by, insistent. "We're not done yet. Come on. What happened next?"
"It was that night I finally started to relax around the place." My heart is fluttering in the cage of my ribs, but I'm not going to let on to her, oh no.
"Even if Quimby had been sniffing after Peggy Guggenheim and her fat wallet?"
"Yeah, well, if he'd tried her in the summer, I just had to hope he wouldn't bother chasing her up again now she was in Marseille."
"Go on, then. Tell me what happened next." The girl sits beside me again.
* * *
Breton—I never felt comfortable enough to call him André, he was always Monsieur Breton to me—well, he came back downstairs after half an hour or so.
"Rose will be fine," he said, rolling down his sleeves and fastening the heavy links in his cuffs. "A sore head and a few bruises, but nothing more serious." Peggy rushed forward and took his arm.
"We had no idea you are a doctor, my dear! What other secrets do you keep?"
"Plenty," he said, bowing that great head of his slightly. "I was a medic in the Great War, and recently," he said. "Like my dear friend Dr. Mabille." I saw her bridle a bit at that. Everyone knew she'd refused to cough up for Mabille's ticket to America. Peggy argued that he was just the surrealists' doctor, not an artist of note. Breton was gracious enough not to push the point further. I guess he felt, they all felt, beholden to her.
I slept at Air-Bel that night, for the first time. All the rooms were full, so I curled up on one of the sofas beside the dying fire and had the first good night's sleep I'd had for months. I felt safe, I think it is as simple as that, and I knew that Annie slept in her house nearby. It is a terrible thing to be afraid all the time. Everyone was terrified, because of the Gestapo, just waiting for that knock in the night. And me, I was petrified at the thought that Quimby might expose me. It felt like I was living on borrowed time, that I could lose everything, and Annie, at any moment.
Around seven A.M., people began shuffling downstairs. Varian and his group set off to catch the tram into town to open the office, and as I sipped my coffee at the window I watched him walking abreast with Danny and the others, Clovis racing ahead of them. They reminded me of gunslingers in an American western.
"Morning," Mary Jayne said as she poured herself a steaming cup of coffee. "Are you coming into town?"
"Perhaps later. I was hoping to see someone."
"Marianne?" She flashed me a quick smile. Aube rushed into the room and ran to me. I picked her up and swung her onto the crook of my hip. "You have quite the way with little girls," Mary Jayne said. "How old is Annie?..."
"Sixteen."
"Cradle snatcher," she said, winking at me. "Then again, I'm not one to talk. Listen, if you're going to hang around for a while, why don't you make yourself useful? When Raymond and I were walking in the woods yesterday, we saw a few last mushrooms."
"Has the cold not killed them off?"
"Dear boy, you are asking the wrong person; all I know is the season runs late here. I'm amazed Thumin missed them, but you could ask your little girlfriend to help you see if there are any more. Watch out for Thumin, though, don't let him catch you thieving."
* * *
I dared not ask for Annie at her front door, so I crept into the back garden of the house and tossed a few pebbles at her bedroom window. I saw her face appear behind the glass, and I signaled to her to meet me in the woods. As I waited for her, I gathered the few mushrooms I could find, then sat beneath an oak tree and scraped at the earth with my fingers.
After half an hour, I heard her footsteps running toward me, twigs cracking beneath her boots. She waved when she saw me.
"What are you doing? You look like Little Red Riding Hood with that basket." She leaned down and kissed me. "Or maybe the Wolf."
"I thought there might be some truffles. Don't they grow under oaks?"
Annie laughed. "You need a dog, or a pig for that. Do you know nothing about the countryside?"
"Perhaps Varian could train Clovis."
"A poodle hunting truffles? You are funny," she said, laughing. "I'm sorry, I couldn't get away. Maman wanted me to go into the market. I don't know what's wrong with her, she just won't leave the house these days. I can't stay long. She thinks I'm on the way to the tram."
"Well, we shall go into town together." I scrambled to my feet, and we walked hand in hand into the woods.
"So what have you found?"
I lifted the red-and-white gingham cloth and showed her the mushrooms. "Mary Jayne asked me to find what I could for Madame Nouguet."
"The cook?"
"Do you know her?"
Annie shook her head. "I overheard a couple of women from the village trying to get information about the house out of her at the tram stop." She smiled up at me. "She wouldn't talk, by the way. Your friends are safe."
"I wish you'd come and meet them one night, you'd love them."
"I can't, you know that. My parents practically lock me in my bedroom at night." She leaned against me. "I think they are afraid some handsome artist is going to scale the wall and carry me off."
"That sounds like a fine idea." I bent down to pick another mushroom. As we walked on, Annie offered me a folded shopping bag.
"It's been raining, so you should look for snails, too," she said, and I wrinkled my nose.
"I've never liked them."
"Gabriel, it's not about what we like, it's making do with what we have." She kicked aside a tree stump near the tumbledown wall and lifted an old rock. "There," she said, pulling snails from the wall. "Put some herbs in the bag and leave them to digest them for a few days."
"I don't know how Madame Nouguet is feeding everyone," I said. "You know, she had taken to locking the half-pound bread rations in the pantry overnight, but someone figured out how to take the door off its hinges." I laughed. "We all chip in some rations, but everyone is starving on stewed carrots and rutabagas."
"Stop it, you're making my mouth water."
"You know, yesterday, Varian caught the goldfish from the pond to add to the stew?"
Annie laughed. "Is that the American?"
"Yes, have you met him?"
She shook her head. "My father was moaning about some man spying on him with eyeglasses." She brushed her blond fringe away from her eyes. "He should be careful. People are suspicious. If they start thinking spies are hiding out in Air-Bel..."
"Oh, don't be ridiculous.'
"Really, you must watch out. It's not like Paris, Gabriel. This is a small, provincial village—people talk. One stray bit of gossip—"
"Then let's give them something to talk about." I pulled her into my arms and kissed her. For once, she didn't resist. I could smell the cold earth on my fingertips resting against her neck. "I have something for you," I said, and reached into my pocket. "I was going to wait until Christmas, but—"
"But you can't wait?" Her eyes widened as I handed her the red box. "Oh, Gabriel, you shouldn't have!" She eased back the lid, and her lips parted, smiling. "It's beautiful!" She held up the bracelet, looped over her fingertip. "I've never had anything so lovely."
I took the chain from her and fastened it around her slender wrist. The silver stars glinted in the morning light. I held her hand and raised her wrist to my lips.
"I'll wear it always," she said.
"You know, one of these stars is Venus," I said, touching the bracelet. "It shines night and day, like my love for you...."
"Marianne!" a man shouted. I looked up to see her father striding through the forest toward us. He was pale with rage, a shotgun slung over his arm and a rabbit dangling by its feet from his fist. "Who the hell are you?"
"Papa," she said, running to him. She held him back as he pushed his way toward me.
"Who are you?" he yelled.
"Sir, my name is Gabriel Lambert."
"You stay away from my daughter, do you hear? I'm warning you, if you so much as look at her—"
"You'll what?" I stepped toward him, my fist clenched. The blood sang in my ears. Annie shook her head, pleading with me not to do anything.
"Don't you threaten me." He dragged her after him. "You are to have nothing to do with those people, do you hear?" I heard him telling her.
"Let me go!" She struggled, but he held on to her. She craned her head back around to me. "Gabriel!" she shouted.
"Leave him," old man Bouchard said. "This is too much. I warned you. They will bring nothing but trouble to us, do you hear? It's time they left us in peace."
THIRTY
VILLA AIR-BEL, MARSEILLE
1940
VARIAN
Varian cleared the steam from the mirror on the marble stand. He was exhausted, and his hand trembled as he ran it over his jaw. He glimpsed movement at the edge of the garden through the window and stepped closer, clearing a space in the condensation with his fingertips. Just some birds. Varian exhaled. He had been on edge for days. Steam rose from the bowl of water as he lathered up the shaving soap with the brush and rubbed it into his face in even circles.
"Okay, Lena, the next letter is to Alfred H. Barr, director of the Museum of Modern Art, New York," he said, forcing himself to concentrate. "Dear Alfred...," he began. His secretary sat at the small table by the window, her shorthand notebook balanced on her lap. It was a crisp morning, and beyond her he could see the pines emerging from the mist. The gaunt face of a stranger looked back at him from the mirror. In the greenish light of the lamp, he looked sallow, and deep purple shadows ringed his eyes. The razor rasped against his skin as he dictated the letter, each methodical stroke revealing pink, vulnerable skin. "Best regards, etc., etc." He rinsed the razor, thinking of the letter he had received from Eileen that week. Think of me, she said. I do, he replied, but your husband is a changed man. Don't think of trying to change me back.
"Varian!" he heard Mary Jayne shout, hammering on the door.
"Just a minute," he said. He quickly finished the last couple of strokes, wincing as the razor clipped his jawbone. He splashed water on his face and wiped the last of the soap away on the towel. "What's the matter?" He flung open the door.
"I'm sorry," she said, her gaze darting over his dressing gown, unsure where to look. "It's the Sûreté." She walked him over to the window overlooking the driveway. Outside, two police cars and a panier à salade wagon were pulling up.
"Damn," he said, and dabbed at the cut on his jaw, thinking, This can't be good. He grabbed his address book and threw it into the fire. "Listen, Lena—go down with Mary Jayne."
"What will we say?" She fingered the brooch at the throat of her blouse.
"Nothing, say nothing at all. I'm going to get dressed. I'm sure it's just routine inquiries."
"These Sunday parties have drawn too much attention to us," Mary Jayne said. "We need them to keep a lower profile."
You're a fine one to talk, he thought, hanging out with that punk.
"Stall them," he said, striding across the room. "Is everyone else up?"
"They're having breakfast."
"Good," he said, grabbing his suit trousers. "Make sure everyone looks as relaxed and normal as possible."
"Normal?" Mary Jayne laughed. "In this joint?"
* * *
By the time Varian raced down to the hall, Mary Jayne had opened the door to the police, and the inhabitants of the Villa Air-Bel were drifting into the hall. A burly policeman barred the front door. Varian caught the look of curiosity on André's face turn to alarm and then carefully disguised calm.
"Good morning, Detective." Varian strode toward him, his chin high and shoulders thrust back. "How may we help you?"
"Monsieur Fry? I have a warrant to search the premises."
"What's the meaning of this? We protest and reserve all rights."
The detective calmly began to pick his nose.
"Formidable," André murmured from the corner of the hall.
"We can make this easy, or hard, Monsieur Fry," the policeman said, inspecting his index finger as he stepped closer. "We have a warrant." Potent, stale cognac breath emanated from him like sea fog. He reached into his pocket and waved the papers in front of Varian with the same fingers that had recently been up his nose. Varian saw a carbon copy of the chief of police's warrant to automatically search any premises suspected of Communist activity.
"As I said, we protest and reserve all rights," Varian said slowly and clearly. "We have nothing to hide. Do you work with Detective Dubois? He's well aware of our humanitarian work at the ARC. In fact—"
"Detective Dubois will not be serving on the Marseille force much longer."
"Since when?"
He shrugged. "He leaves soon. Dubois is being posted to Rabat."
Poor devil, Varian thought. No doubt someone blew the whistle on him.
"Now, is everyone here, including the servants?"
Varian glanced around and saw Rose, Madame Nouguet, and the young Spanish nanny, Maria, huddled by the kitchen door. The Bretons and Jacqueline's sister stood together in the corner. "Yes, we're all here." All except Danny and the others from the ARC, he thought. Let's hope they stay out of the way.
"We intend to search the house from top to bottom," the detective said.
"In that case, perhaps you would like some coffee? Madame Nouguet, would you mind?" He watched as the relieved servants raced to the kitchen. "May I ask why you are searching the house?"
"We have had reports of suspicious goings-on."
"Such as?"
He checked his papers. "I believe young Mademoiselle Breton has been telling her classmates about how sad her parents were at the death of their friend vieux Trotsky."
Jacqueline held tight to André's arm. "Inspector, she is just a child, she doesn't understand."
"Still, it is suspicious. We cannot be too careful. Why, only yesterday we received a call about a young woman carrying a heavy suitcase to the house."
"That was me." Jacqueline's sister stepped forward defiantly. "I am visiting. The suitcase contained clothes for Jacqueline and the child, that is all. I'll show you."
"We'll see." The detective signaled to one of the policemen. "Go with her. Make a thorough search of the room, and the suitcase." He looked at each person in turn. "Now, all firearms must be surrendered...."
Varian joined Mary Jayne by the green-tiled stove as the detective droned on. "Do you have anything incriminating on you?" he said under his breath once the police weren't looking. "I have to get up to my room. I left a fake passport on the dresser."
"You should be more careful." She glanced at the young policeman nearby. "Fine. I'll distract him. See if you can persuade them to let you up."
"Excuse me..." Varian strode over to the detective. "I need the lavatory."
The man looked at him in irritation, then signaled to one of the policemen. "Go with him."
Varian thought quickly as they walked upstairs. "Have you been on the force long?" he said.
"A couple of years."
"How do you find it?" They walked on up to the next floor.
"Pas mal." The boy shrugged.
"Here we are." Varian closed the bathroom door and waited, his head leaning against the door as he caught his breath. He pulled the chain and ran the taps, wetting his hands before he opened the door. "Thank you," he said to the policeman. He brushed his nose with his index finger. "I could do with a fresh handkerchief, would you mind? I'll just pop into my room and fetch it," Varian said, hoping he wouldn't follow him in.
"Fine." The boy slumped against the wall and crossed his ankle over his leg.
Varian grabbed the passport and looked around him frantically, darting backward and forward. Oh God, oh God, he thought. There was no time to lever up a floorboard or loosen the mirror, his preferred hiding place in his old room at the Hotel Splendide. He threw the passport up on top of the wardrobe and caught his breath. Calmly, he pulled open the top drawer of the dresser and slipped out a neatly folded handkerchief. As he walked out of the bedroom, he blew his nose. "Thank you."
They arrived back downstairs just as André was placing his service revolver on the table in front of the policemen. A couple of typewriters and guns were there already. "Good," the detective said, checking the Breton family's papers. "Did you find anything else up there?"
"There are a lot of books, and papers," a young policeman said. "I don't know what most of them are, but we found this." A portrait of Marshal Pétain as a Gallic cock dangled from his hand.
"Ce sacré crétin de Pétain?" the detective roared. "You are calling Pétain an idiot? This is revolutionary propaganda!"
"Non, monsieur," André said calmly. "You misread. P-u-t-a-i-n," he spelled out. "Whore."
Varian hid a smile. He saw Mary Jayne signaling to him, and he wandered over, pretending to warm his hands on the stove. "Did you get it?" she said.
"Let's hope they don't look too close."
"Good. Listen, cover me, will you?" She opened the door of the stove, and as she tossed on a fresh log, Varian saw her slip in a ball of paper from her pocket.
"Will this take long?" he asked the policeman walking toward them. He blocked his view of Mary Jayne.
"That depends," the man said. Varian glanced at Mary Jayne and saw her face fall. He followed the track of her gaze and saw Raymond being marched at gunpoint across the terrace by another policeman.
"That's all we need. What's he doing here?" Varian hissed to her. "I thought he was in jail?"
"He was. I helped get him out," she whispered.
"Chéri!" Mary Jayne cried, throwing her arms around Raymond before the policeman could stop her. "Mon amour, mon amour!" He kissed her, and Varian saw him slip an envelope into her pocket.
"Come on," the policeman said, dragging him away toward the detective. "I found him hanging around outside."
Varian watched as Mary Jayne stuck her hands in her pockets and walked over to Jacqueline. He saw her speak quickly to her and nod her head toward the young blond policeman by the terrace window. Jacqueline raised her chin and sauntered over, swinging her hips seductively.
"Do you have a light?" she said, leaning against the window frame.
"Of course, madame." He fumbled in his pocket. Jacqueline inhaled and licked her lips.
"Thank you. What a gorgeous lighter. Did you buy it here?" She ran her red fingernail down the engraved metal.
Behind the man's back, Varian saw Mary Jayne lift the edge of a Chagall painting of a flying cow they had hung on the wall. She slipped the envelope behind it and walked away.
"Bon," the detective said after several hours. "All the rooms have been checked...." His voice trailed off as two more policemen walked in from the terrace, Danny between them. All three were panting.
"He tried to make a run for it," one of the policemen said. Danny's face was bright with anger.
"Ah, another joins the party. Bien. We go."
"Go where?" Varian said. "You assured us this would not take long. We are a busy, and legitimate, relief organization—"
"We have a few more questions, Monsieur Fry, and I would like you, out of the goodness in your heart, you understand, to accompany me to the station."
"We are not under arrest?"
"Bien sûr." The detective opened his palms. "We have nothing against you. You will be back in an hour, you have my word. The servants and the mothers with young children may remain here. We are not animals, Monsieur Fry."
Jacqueline turned to André, flung her arms around his neck. Aube nestled between their legs, glaring at the policemen. Varian caught Jacqueline's whispered words: "Courage, mon cher, courage...." She turned her furious gaze on the policemen, the tiger's teeth around her neck jangling.
Varian squatted down and rubbed Clovis's ears. "Do we need to bring anything?" he said to the detective.
"No, no. It will not take long, I assure you."
As Varian followed André out of the house, he heard him laugh and say, "Not long?" as he slipped a book from the hall table into his pocket.
"Stay, Dago," Mary Jayne said, beckoning to Rose to hold the dogs back. "Keep them on their leashes until we've gone." She buried her face in the warm, springy fur on his head. "Now, you be good for Rose and Madame Nouguet, you hear?" She glanced up to see Raymond struggling as the police pushed him into the back of the wagon. "If anyone comes for the kids, you give them hell," she whispered, and kissed Dagobert's nose.
"Come on," Varian said, offering her his arm. "Let's show these damn fools how to do it."
"Varian, I'm afraid," she whispered. He looked down at her, saw the terror in her eyes. Compassion loosened the tight coil of anger in him, like sunlight unfurling new leaves.
"Don't worry, my dear. They can't touch us." He put a protective arm around her and glared at the detective as he swept out of Air-Bel with Mary Jayne at his side. "Come on, chin up," he whispered.
"I can't," she said, her voice shaking. "Varian, what if—"
"You can, and you will, Mary Jayne. We're Americans." He helped her into the back of the wagon. André, Danny, and Raymond already sat in the shadows on the narrow bench opposite, straight backed and pale faced. We are safe as Americans, but what of the others? The thought of his friends in Vernet, or some other camp ringed with barbed wire, sickened him. Varian felt his guts weaken with fear.
* * *
"How long have we been here?" Mary Jayne whispered, glancing around the packed police station.
"Hours. I don't know." Varian checked his watch. "Jeez, it's six o'clock. They can't keep us much longer, surely?" The room was crammed with people waiting to be interviewed. "Why are they taking so long with André?"
"Don't you know, my dear? He has a list of convictions as long as your arm."
Varian paled. "You're kidding me."
"Back in the day, he was quite the bad boy. Jacqueline tells me there are at least twenty-five counts on his charge sheet."
"Christ! Why did no one tell me?" They sat squeezed together on hard wooden benches, the fug of hot bodies and wet-dog smell of tweedy winter clothes hanging in the air. "Hey, kid!" he called, signaling to a boy selling newspapers on the street. He wriggled his wallet free from his pocket and peeled off a few notes. "Go to the restaurant next door and bring us a bottle of wine and some sandwiches, okay? You can keep the change." The boy ran off. "Are you okay? What do you think this is all about?" he asked André as he joined them.
He shrugged and pulled the book out of his pocket. "Who knows? They want to round everyone up in a raid, they can do, with no reason."
"I wish I'd had the presence of mind to bring a book," Varian said.
André turned the pages and sighed. "But of all the books in the château to pick from, I end up bringing one I wrote."
"Ah, good!" Varian said, spotting the boy weaving his way through the crowd with their food. Mary Jayne handed out paper cups and tore the sandwiches into pieces until there was enough for the group. Varian sighed with relief and chewed his cheese baguette with his eyes closed. "Damn, that's good. We haven't had a thing since breakfast."
At nine o'clock, a door opened and a file of policemen strode into the packed room. "Make ready," one of them shouted. The noise grew, a hubbub of cries.
"Where are you taking us? We are Americans!" Varian shouted, taking Mary Jayne's arm and protecting her from the crush of bodies.
"Fry, Bénédite, Breton, Gold..." One by one their names were called, and they followed in line through the back of the police station to where the trucks were waiting. Is this how it goes? Varian thought, his breath pluming in the cold night air as he stepped into the gaping darkness of the truck. He heard others breathing there, nervous, short gasps in the darkness. Perhaps at the end people do as they are told, meekly like animals? Not all of us fight, resist. He felt Mary Jayne's hand on his arm as the doors slammed shut.
"Where are they taking us, Varian?" Her voice shook. "I don't know what they have done with Raymond...."
"Killer can look after himself." He put his arm around her, steadied her as the truck lurched away. In the darkness, he tried to follow the path the truck was taking. We're heading toward the port, he thought. That makes no sense. If they were going to ship us off to one of the camps, we'd head to the station. The truck bounced along the cobbles, and Varian caught the tang of the Quai des Belges. "We're stopping."
The truck doors flung open, and a policeman signaled to them: "Out."
One by one they jumped down, and Varian blinked in the blue streetlights. Hundreds of people were being herded out of trucks just like theirs. "What's going on?" he said.
"They are forcing everyone onto the ship," André said, pointing behind them. Varian turned to see the huge hull towering over them.
"My God, it's the Sinaia," Varian said. The masts soared above the deck, and two vast chimneys with white bands running around them rose into the night. He felt vertigo sweep over him as he gazed upward. "I traveled around Greece on this a few years ago."
"Now that is a marvelous coincidence, my friend."
"Make sure we stay together," Danny cried out as the crowd jostled them. Varian grabbed Mary Jayne's arm and kept her close as they were forced up the gangplanks. He felt the rock and sway of the boat beneath them, the creak of the boards. Far below, the black sea sloshed against the pier.
"If I'd known they were taking us on a cruise, I would have packed my bathing costume." Mary Jayne's smile faltered.
"Don't worry," Varian said. "They can't do a damn thing to us...." His voice trailed off as they stepped into the hold of the boat. His eyes adjusted to the darkness. "Christ, how many people have they got on board?" The hot, animal smell reminded him of cattle pens, the tang of sweat and fear. Above the stinking pallets of straw, a perfect square of stars hung, the only air from a hatch in the deck. Deep in the bowels of the boat, someone played a Spanish flamenco tune, melancholy and plaintive.
"Over there," a policeman said, pointing to the far corner of the deck.
"What about bathrooms? Where will we sleep?"
The policeman indicated the floor. Squalid straw pallets had been hastily thrown down. "Pipi?" He pointed up to the top deck.
"And food? Water?" Varian demanded.
He shrugged and pointed at a tin bowl on the floor nearby. "Tonight it is too late. Perhaps tomorrow you will be lucky." He swung his gun over one shoulder. "Choose one person from the group to collect food for you all. There will be a loaf of black bread and a pail of coffee to share in the morning."
"Why, it's a regular five-star resort," Mary Jayne said. She shivered in spite of the press of bodies. Varian shrugged off his jacket and draped it around her shoulders. "Thank you." She looked up at Varian. "I'm afraid. Where do you think they are taking us?"
"I don't know." Varian looked at Danny.
"You two will be fine," he said. "We have to try and get word to the American consulate, somehow. As for us..." His voice trailed off as he gazed around the ship.
"They can't hold us here, can they?" Mary Jayne's voice shook. "What—what will they do to us? I'm worried about Killer."
"I imagine Monsieur Couraud can take care of himself," André said. He took off his coat and spread it out on the nearest straw pallet, gesturing for Mary Jayne to sit down. "Courage, my friends. This is going to be a long night."
THIRTY-ONE
FLYING POINT, LONG ISLAND
2000
GABRIEL
It's something that kids like Sophie will never understand, or at least I hope they won't. This was a so-called civilized European country, and they were just rounding up people from their homes, from the streets, and slinging them in holes barely fit for animals. They held them in three boats, four forts, and some of the local cinemas without explanation. In all, about twenty thousand people were rounded up, just to tidy the streets for Pétain's visit.
"So where were you when all this was going on?" she says.
I sway on my feet as I stand, the breath rasping in my throat. The sun seems to flare and arc above the horizon, luminous as phosphorus. I can hardly see her, the light's a halo around her, and I cover my eyes.
"Gabriel, I said, where were you?" She walks at my side as I stagger along the beach.
"With Annie, of course. When you are young, and in love, you steal time together whenever and however you can."
* * *
I had arranged a rendezvous with Annie at the public library in the Palais des Arts on place Auguste-Carli. Her parents could hardly object to her studying, and it was somewhere warm and quiet to meet. It thrills me, even now, thinking of the anticipation of seeing her. The silence, the peace of the library, the quiet breathing of the people bent over the old desks, studying. One day I saw Breton there, his head in his hands as he studied a pile of leather-bound books. I didn't dare disturb him, of course.
It was one of our favorite places to meet—we changed the stacks each time, just in case Annie's mother followed her. This time I was waiting in ornithology. Varian was a big bird-watcher, and he had taken me out in the grounds of Air-Bel, let me try his binoculars. He was a member of some fancy American club—the Audubon Society—and he knew all the Latin names of the birds, their calls, everything. I went early to the library, spent hours flicking through heavy leather books with marble endpapers. The illustrations were dazzling—the clarity and colors. I remember a "meet the artist" lecture in the city thirty years back, when some chinless art historian was pontificating about why one of my most celebrated works was called "Bird." "Oh, it must be because of this and that," he said, "freedom, peace, it's a homage to Brâncuşi or Picasso's doves, clearly...." I was bored by then, so I just let him prattle on. I wasn't about to tell him it was named for an afternoon of erotic anticipation spent flicking through bird books while my friends were being rounded up like cattle. The shame of that makes my cheeks burn even now. The streets were strangely quiet by the time I walked back to my hotel. Then, when I went out to Air-Bel on Sunday, I found out what had happened.
"Bit late by then," Sophie says, her voice floating on the breeze.
"I did everything I could." I ball my fist up on my chest. I can feel my heart jumping around beneath my ribs. "It took me three days to find out where they were holding everyone."
"Wasn't that a risk for you, to go down to the docks?"
"I didn't give a damn. I owed Varian, all of them, everything." The guilt is bitter as lemon juice, still. "Of course, when I was on my way to the Sinaia, who did I bump into but Quimby?" He appeared out of nowhere from a narrow side road like the sudden smell of gas.
"Well, what a fortunate surprise," he said. "I was just on my way to your hotel."
"No," I said. "We agreed you wouldn't come there."
"Would you rather I came out to the château?" He stepped so close, I could smell his sweat and cologne. He repulsed me, but I wouldn't back away. I pulled out a pack of cigarettes and lit one, the flame dancing in his spectacles. He moved back a pace, leaned against the damp stone wall. "I'm out of cash."
"I'm not giving you any more money, I can't."
"Oh, I know you haven't got any on you," he said, inspecting his nails. "Or in your room."
"You have been to my hotel!" I said. Thank God I had entrusted Annie with the money. If I couldn't get to her, there's no way Quimby could get past Monsieur and Madame Bouchard.
"You really are peculiar," he said, still admiring his buffed fingernails. "For one left with so little, you are remarkably picky about your things." He looked up at me then. "I don't care where you've hidden the cash, but go and get it. I want another five hundred francs by tomorrow."
"Or what?"
"Or I'll march right into the ARC and tell them everything." He dusted off the shoulder of his overcoat. "Oh, and in case you are thinking of bumping me off or anything silly like that, I've taken... precautions. And I've left a letter with my concierge to be passed to Monsieur Fry should anything happen to me."
"I'll get it," I lied. I just wanted to get to the docks. I pushed past him and kept running, doubling back on myself again and again through the winding streets to make sure he wasn't following me.
At last I reached the Sinaia. Maybe you've seen that photograph of Marcel Duchamp standing on the prow of the boat that eventually liberated him from Marseille in 1942? He was heading out to Casablanca, and he stood up there like a ship's figurehead, waving, full of joy. People think of boats as great symbols of liberty, but the Sinaia did not seem like that to me. I joined the crowds of women and children on the docks, shouting up to their imprisoned men. The huge anchors tethered the boat, chains plunging into the choppy ocean. All I could see were cranes and rigging, ropes straining, birds shrieking in silhouette. It terrified me. I shielded my eyes, looked along the row of faces. It took me a moment to make them out, but then I spotted Fry, head and shoulders above everyone else.
"Hey!" I yelled, waving my arms above my head. "Hey! Varian!" They were too high, too far away, for me to hear what he was shouting to me. I saw him talking to someone at his side—I couldn't work out who, but then I saw him swing his arm back and throw something.
I pushed through and shoved some kid aside who was reaching for whatever Varian had thrown. It was a note, wrapped around a ten-franc piece. I gave the coin to the boy and waved the note in the air to show Fry I had it.
* * *
I went straight to the U.S. consulate, just as Varian asked. I knew better than to trust the receptionist by now. I just asked to see Harry Bingham and waited quietly for him. It can only have been a few hours, but it seemed to take forever. Bingham finally appeared at the door of his office, his kind, gentle face breaking into a smile.
"Ah, Monsieur Lambert. We met before?"
"Briefly."
"I'm sorry to keep you waiting. It's been a hell of a day with the aftermath of all these raids." He ushered me into his office. "How can I help you?" I explained what was going on as quickly as possible and watched his face set hard with anger. We stood by the window, looking down on the queue of people snaking along the pavement two abreast to the consulate. The café tables and chairs were piled up for the day, stacked between the plane trees.
"From up here, everyone looks the same," he said quietly. "Look—hat after hat, faceless, anonymous." He paused. "The trick is to make people care, you know, to name them to the world. Men like Fry are making a real difference." He laughed dryly. "You know what the U.S. officials call all the paperwork, all the hundreds of letters Varian keeps them tied in knots with? 'Fryana.'" He paused. "I've seen too many people's grief and anger turn to disillusionment. Not with Fry. He and his little tribe of amateurs are outwitting the Vichy stooges with sheer intellect, and the drive that comes from knowing they are serving justice." Bingham sighed. "I envy them. My hands are tied here. The consul general, Hurley, wants nothing to do with the ARC, and Consul Fullerton isn't a bad man, but he's cautious. Often the best I can do is to give an affidavit in place of a passport." Of course, that wasn't true. I found out after the war that Bingham was immensely brave. He helped many, many people and rescued Lion Feuchtwanger, the prominent German Jewish literary figure, from a Vichy detention center. Feuchtwanger had been one of the first to denounce the brutality of the Nazis, so of course they came after him. Harry sheltered him and his wife, Marta, in his own house and helped them escape to the United States. Bingham was a good man, a righteous man, just like Varian.
Ever the diplomat, he kept his thoughts to himself as we stood at the window, and he simply picked up his coat and hat and told the girl on the reception desk that we were going out on official business. I had to laugh at the look on her hard, painted face as outraged people swarmed around her desk, gesticulating and shouting.
The first thing we did when we reached the center of town was stop at a bakery, and Bingham bought a tray of sandwiches. The car drove onto the dock and pulled up beside the armed guards at the foot of the gangplank.
Bingham stepped out of the car and took the plate of food. He was a tall man, like Fry, and imposing, and I saw the crowd part before him like Moses.
"Good day," he said to the guards. "My name is Hiram Bingham and I am the American vice-consul. You are holding American citizens on that boat, and I will be seeking their immediate release." He handed the plate to one of the guards and pulled out his business card. He wrote on the back, To Fry, with compliments, HB, and tucked it beneath the waxed paper. "Now, perhaps one of your colleagues would be good enough to deliver that to Mr. Varian Fry and his colleagues." The guard opened his mouth to protest, but Harry's look stopped him in his tracks. "Meantime, I'd like to have a word with your boss, if you'd be kind enough to point the way."
Less than an hour later, Fry and André walked down the gangplank, Mary Jayne between them. I watched from the shadows as Bingham clapped him on the shoulder. It shook me to see Varian in that state, filthy and unshaven.
"Thank you, Harry," he said. "What a relief. We'd managed to get up to see the captain, but if it wasn't for you, God knows how long we would be stuck on the boat."
"Is that all of you?"
Varian shook his head. "They've held Danny."
"Do you know why?"
"He's done nothing," Mary Jayne said, taking the blanket Harry passed her from the car and pulling it around her shoulders. "Those animals." André stood at her side, pale and dignified, his face dark with stubble. "You know we've had nothing but stale bread and water for days? We've been sleeping on lice-infested straw, and pissing in—"
"Mary Jayne," Varian said, "it's over now."
"Jesus!" she said. "It isn't over, it's just beginning, don't you get it? They're holding Danny to get at you, Varian. They're trying to scare you into quitting. They can't hold you because they don't want to anger Harry's lot."
"Is that right?" Varian said to Bingham.
Harry opened the car door, ushering Mary Jayne in. "They won't rest until the ARC is closed down, you know that, Fry." He took off his hat and ran his hand through his hair. "Don't worry about Danny, I'll have a word with Dubois."
"He's being sent to Rabat."
"Maybe he'll be able to pull some strings one last time." His face fell as he turned. "I don't rate Danny's chances without his help."
THIRTY-TWO
VILLA AIR-BEL, MARSEILLE
1940
VARIAN
"I did warn you," Beamish said, handing Varian a large glass of Armagnac. "I make it a rule to clear out when a fascist head of state is coming to town."
Varian raised his glass. "I'll bear that in mind, next time."
Beamish squatted down and raked the fire in Varian's room. "They've put an effigy of Pétain on La Canebière that's eight meters high. The bloody fascists have been parading in front of it all week." When Fry didn't respond, he looked up. "Are you okay?"
"Me? I'm fine." Varian leaned his head against his hand. "I just... I thought..."
"What? That you were untouchable?" Beamish settled back in the chair opposite Varian. "You thought that as an American you were a superior being, beyond the local rules?"
"No," he protested. "Not—I don't know."
"No one is safe, not now." Beamish swirled his glass, watching the flames of the fire through the amber liquid. "This was just a general râfle to clear up undesirables for Pétain's visit to Marseille. Worse will come."
"But why did the police come all the way out here? I can understand clearing up the Vieux-Port and La Canebière, but—"
"I told you it was a bad idea to move out here, but again you wouldn't listen. La Pomme is provincial. Air-Bel's neighbors don't like the motley band of bohemians and artists coming and going—especially the girls. Other women envy people like Jacqueline at the best of times. Last time I was on a tram back into town I heard a couple of old women at the stop bitching about her. You're too conspicuous out here. You'd all be safer in town."
"I can't ask a man like Breton to hide out in a fleabag hotel or a brothel."
"I'm sure if you asked him nicely, he'd rather be in a maison de passe than a concentration camp."
"Don't be facetious. Someone's betrayed us to the authorities, and I'm going to find out who." Varian bit his lip.
"Do you think Couraud has something to do with this?"
Varian thought for a moment. "No. He's a crook, but he wouldn't put Mary Jayne through that." He pinched the bridge of his nose. "I wish she'd listen to sense. That damn punk is going to drag her down with him, mark my words. You know he made her hide an envelope stuffed with forty thousand francs during the raid?"
Beamish whistled softly. "Where the hell did he get that kind of cash?"
"Where do you think? Stolen, no doubt." Varian swirled the drink in his glass. "No, I reckon the person who betrayed us is a local. I'd bet my last dollar on it being that Bouchard fellow next door. He warned me as much."
"Maybe he doesn't want his precious daughter corrupted by 'degenerates'?" Beamish's head rolled back and he flexed his shoulders wearily.
"I'll talk to Lambert."
"Does it really matter? It could have been anyone. People are squealing on their innocent next-door neighbors because they looked at them the wrong way in 1929. It's the perfect excuse for old prejudices and slights to get an airing."
Varian checked his watch. "Come on. André said he had some kind of announcement."
* * *
Varian and Beamish strolled into the crowded living room, and Danny nodded at them from the far corner. His wife sat at his side, and their little son hung around his neck. The poor kid is probably worried his dad will go away again, Varian thought. He turned as Jacqueline's sister walked in, talking to Mary Jayne.
"Those fools thought I had a suitcase full of dynamite, can you believe it? Just because Air-Bel is near the railway bridge, they thought I was going to blow up Pétain's train!" She laughed. It's not funny, Varian thought. Who saw you with your suitcase? Who told the police? He walked across to the darkening windows and looked out over the deserted grounds. The branch of a tree scraped against the glass. It felt to him like a thousand pairs of eyes were watching the house.
"Varian," Gabriel said, joining him.
"Do you ever feel like you're being watched?"
"All the time. Do you think someone's following you?"
Varian shook his head. "I don't know. I think someone is spying on us, on the house. I don't know why, or who." He rubbed his eyes with his thumb and index finger beneath his glasses. "Ignore me, I'm just being paranoid."
"No, I don't think you are."
"Look, Lambert, it's none of my business who you see, or—"
"Are you talking about Annie?"
"We have to keep a low profile, old man. I've heard talk, that's all. We're ruffling feathers in the village, and now someone has betrayed us."
"I promise, I'll deal with it—" Gabriel started to say.
"Thank you, my friends, for joining us," André said. Varian clapped Gabriel on the back and turned to listen to André. "These are dark days," he went on, pacing in front of the fire. Like a lion, Varian thought, the flickering flames illuminating his mane of hair. Breton's voice washed over him as he talked of the war and of everything to come. "I propose," he said finally, "that this festive season we play a game, the greatest game we have ever played." He stopped pacing and looked slowly around the room from artist to artist. "I have studied the Jeu de Marseille in the library on place Carli. The original tarot deck was named for this city." He pulled a pack of cards from his breast pocket and fanned them out like a magician. He dealt a card on the table and four cards ranged around. "As you know, I often used to consult the cards myself." He placed his finger on the central card. "We are poised in a void, my friends. We will redo the symbols of fate and we will create a new deck that will answer our questions—what is certain, can harm, is hovering, and has been overcome." He shuffled the cards back into the pack. "I propose that we create our own Jeu de Marseille, a collaborative deck, a work of art that will burn away our days of anxiety and waiting...." The group burst into applause and excited chatter.
"I heard W. B. Yeats drew his own tarot trumps," someone shouted from the back.
Breton raised his voice. "We will reimagine the suits. No simple hearts, diamonds, clubs, and spades. I don't want to change the rules of the game, I want to change the game!" he cried, throwing the cards high into the air. The men and women gathered in the room erupted in cheers and clapping. Varian watched as the cards tumbled down in the firelight, and André cried: "To Love, Dream, Revolution, Knowledge! To the Jeu de Marseille!"
THIRTY-THREE
VILLA AIR-BEL, MARSEILLE
1940
GABRIEL
I heard Breton say once that what he loved, whether he kept it or not, he would love forever. I've always been like that, passionate and true, but I've mellowed, just like everyone does with age. When I think back to the night I went storming across to the Bouchards' house, I hardly recognize myself. I was so angry, I had lights dancing in front of my eyes. It was them, the Bouchards, it had to be. Who else lived close enough to Air-Bel to spy on what was going on? And Annie's parents had made it very clear they wanted her to have nothing to do with me. Well, I wasn't about to give up so easily. I loved her—I would love her forever.
I flung the old gate back on its hinges and marched up the pathway. It was late, but I could still see a glimmer of light through the downstairs curtains. I hammered on that door so hard, my knuckles ached.
"Monsieur Bouchard," I shouted.
"Attendez, attendez!" I heard footsteps on the other side of the door. "Who is it?" He didn't open the door even a crack. I heard a window open above me, and I looked up. Annie looked like an angel, her blond hair falling around her shoulders, the white of her nightgown brilliant against the ink-blue star-filled sky.
"What on earth are you doing?" she whispered. "Are you drunk? Go away. This will only make things worse—"
"No." I saw her frown at the anger in my voice. "Someone has betrayed everyone at Air-Bel to the police. Varian, André, Mary Jayne, all of them have been locked up on some goddamned prison ship for days."
"Oh God," she said, reaching to pull the window closed. "Stay there. I'm coming down." I paced the frozen path, my footsteps scuffing the ice. I could hear her now, arguing with her father on the other side of the door. Then I heard her mother talking low and fast.
"Let him in," I heard Annie say clearly.
Her mother sounded desperate. "No good will come of this, I told you. If people see him hanging around here..." The little dog chimed in, yapping and scratching at the door, trying to get at me.
"So let him in before someone sees!" Annie was exasperated. "I know Gabriel, if you don't open the door, he'll stay out there until the whole village wakes up." At that I heard the bolts slide back, and the door opened. The damn dog shot out and growled at me, its white fangs bared. I started to walk to the door, and it went for me, biting my ankle.
"Coco!" Annie shouted as I hopped around swearing, trying to shake it off my leg. "Coco!" She grabbed the dog and pulled me into the house. I had never been inside the Bouchards' home, and I was surprised by how empty it was. From the outside it looked like a gracious old stone house, but inside the living room the furnishings were little more than a battered dark wood table and chairs and an uncomfortable-looking horsehair couch. Annie gestured that I should sit down, and sure enough, it was as lumpy as a sack of rocks. "I'll put her in the kitchen," she said, stroking the trembling dog in her arms. I don't know if it was jealousy or my throbbing ankle, but I hated that dog at that moment.
"What do you want?" Old Bouchard stood in front of the grate and took out his pipe. "I thought I made it clear, you are to have nothing to do with our daughter, or I'll—"
"Or you'll what?" I said, cutting him off just as he started packing tobacco into the bowl of the pipe. I sprang to my feet. "Betray us? How dare you? Because of me, you betrayed all of them?" I realized at that moment that I wasn't just angry—I was guilty, and afraid, too.
"Gabriel," Annie said. "Don't talk to my father like that."
"Don't you see?" I rounded on her. "He threatened me, said he'd get everyone kicked out of Air-Bel." I glared at Bouchard. "Thanks to you they were all locked up. Thank God the American consul got them released—"
"We did nothing." Annie's mother sat in one of the wooden chairs, her back straight. "It wasn't us."
"Gabriel, honestly, it could have been any one of the villagers. It doesn't take much for people to turn on strangers." Annie cupped my hand in hers. "I promise, it wasn't my parents who turned them in."
"You should hear the gossip," Madame Bouchard said. "They are having drunken orgies there. That blond American girl nearly slapped a woman in the café, I heard, and that other one, the one with the crazy outfits and the teeth strung around her neck like some voodoo priestess, well, everyone is talking about her." She rocked slightly as she talked. "Everyone is talking. No one is safe."
Monsieur Bouchard walked across and laid his hand gently on his wife's shoulder, and she stilled. I felt the anger in me ebbing away. "Go," he said. "If you care even a little for Annie, go away and never see her again."
"I can't do that." My fingers interlaced with hers. "I love your daughter. I intend to marry her." I felt her squeeze my hand. When I looked at her, her eyes shone with love.
"No good will come of it. We have lived quietly here, safely...." Bouchard was afraid, I realized. What was he hiding?
"Papa," Annie said, "we can't hide like animals. They will find us soon enough anyway."
"What are you talking about? Who will find you?" I said.
Bouchard gazed at me, clearly thinking whether to talk or not. "I am only telling you this because I want you to go, do you understand? If you breathe a word of this to anyone, I swear..." His wife reached up to touch his hand.
"Marianne is of Jewish descent," she said. "I am Catholic, but my husband's family is Jewish." I felt the truth come together in me like a smooth stone falling in a well of cold water. I looked at them and tried to imagine them young and in love. What must it have taken for them to defy their parents, to marry outside their religion? I could see no trace of that passion. Their eyes were ashes.
"That is why I cannot work anymore," he said, his voice bitter and angry. "This so-called Statut des Juifs forces my people from our jobs, our livelihood. They take our money, our public voice. What will be next, I ask you? Our lives?"
"We have lived here for years, quietly," she said, her hand on her husband's arm. "We do not want to draw attention to ourselves—that is why we want Marianne to have nothing to do with you and that crowd at Air-Bel."
"If we keep our heads down, we will be safe."
"Do you think so, Papa?" Annie's voice broke with frustration. "No one knows what the Nazis are capable of."
"I vomited when I heard about the statute," Bouchard said. "It is the beginning, only the beginning."
That was the first time I saw him as a human being, as a man, rather than an obstacle to overcome. Who was it said "Something horrible happens when you claim certainty"? I will never, as long as I live, be able to understand how that self-appointed so-called master race believed they had the God-given right to destroy the Jews. I saw tears glint in old Bouchard's eyes as he looked down at his wife. He turned to me and blinked quickly. "Don't you see? No one pays any attention to us here."
"Not yet," Annie said.
"You've been seen," Madame Bouchard said quietly. She glared at me. "Oh, you think you've been so clever, the two of you, meeting up for your walks, and in town, but don't you know there are eyes everywhere? That nosy old bitch from the farm up the road cornered me in the market and said, 'I see your Marianne is courting, then?'" Annie looked at her feet. "Imagine, hearing it from her how you have been sneaking around in the woods and the fields."
"I am sorry, Maman." Annie tightened her grip on my hand and looked up at her parents. "I love Gabriel, and nothing you can do or say will change that. I want to marry him."
"No, I forbid it," Bouchard said. "What you must understand, Mr. Lambert, is that not only am I Jewish, if you marry her, Marianne will be Jewish in their eyes too."
"I don't understand?" I thought of the gold crucifix she always wore.
"You see, the Vichy government has made the law against the Jews even harsher here," Madame Bouchard said, going to her daughter. "The Nazis said that anyone with more than two Jewish grandparents is considered Jewish."
"In France," Monsieur Bouchard said, "you need only two Jewish grandparents and a Jewish spouse." His eyes brimmed with tears. "Your mother, I believe, was Jewish. Marianne told me, because she thought I would be pleased."
I thought quickly. I had made a grave mistake in telling Annie. "You misunderstood, darling," I said, skipping back a generation. "My grandmother was Jewish, but she married outside the faith. The Lamberts, my parents, were both Catholic." I saw old Bouchard exhale, but I dared not breathe easy myself, not yet. If we married and the truth came out, it would be a death sentence for Annie.
"My wife, I hope, will be safe," he said. "I don't care what they do to me, but my wife and daughter must be saved. A Jewish father is no great recommendation these days." He hung his head as Annie went to him.
"I love Gabriel," Annie said. "Please, give us your blessing."
He shook his head. "No. This man has Jewish blood, he is part of that... that degenerate crowd at Air-Bel. No good will come of it. If you love my daughter, Monsieur Lambert," he said, unable to look me in the eye, "you must never see her again."
THIRTY-FOUR
VILLA AIR-BEL, MARSEILLE
1940
GABRIEL
I stumbled my way back through the darkness to Air-Bel, looking down at my pale hands. It was like I could see it there, the blood—of Vita, of them both. The thought of adding Annie to the list made the bile rise in my throat. I turned my anger on Quimby, and my fists clenched. I punched the gate as it swung closed behind me. The pain was keen, but it calmed me down.
"Clovis!" I heard Varian's voice somewhere behind me. "Clovis! Where are you?" I could hear him marching through the undergrowth, twigs snapping beneath his feet. "Who's there?" He swung the beam of the flashlight toward me. "Is that you, Lambert?"
I squinted and covered my eyes. "Has he run off?"
"No." Varian strode toward me, the flashlight beam swinging across the dead grass and wintry ferns. "That bastard Raymond Couraud has kidnapped him. Can you believe it? I thought maybe he tied the dog up out here somewhere. When I get my hands on that little punk, I'll—" He broke off at the sound of a car racing up the driveway, a shower of gravel on the terrace. Varian broke into a run, and I raced after him. Just as we rounded the corner of the house, we saw Killer stumble drunkenly from an old black Citroën 11 cabriolet. Mary Jayne ran out and took hold of him. The Bretons and Danny's family trailed behind her, alarmed by the disturbance.
"You idiot," she said to Raymond. "What on earth do you think you are doing?"
"He had fun, didn't he, Mathieu?" he protested, pointing to where Clovis sat on the backseat between two of Killer's gang, panting contentedly.
"He sure did," said an olive-skinned man with slicked-back hair. "Hello, Mary Jayne. You look lovely this evening."
"You asshole," Varian said, grabbing Killer by the collar of his jacket. He pushed him back against the hood of the car. "I swear, if you ever touch him again, I'll—"
"You'll what, you American pansy?" Killer flicked away his hands, squared up to him. "You prig. I'd slit your throat in a heartbeat."
"You don't frighten me." Varian glared down at him. "You leave him alone. And you leave us alone. I heard all about the stash of money you hid here. Don't you get it?" he said, turning to Mary Jayne. "You stupid girl. If they had found that, it would have been enough to have us sent back to the U.S. It would have destroyed everything, just because of him." Varian put his hands on his hips. "You have to make a choice, Mary Jayne. It's him or us."
"What?" She turned slowly to him. "How dare you?"
"You are jeopardizing the entire operation."
"I choose Raymond," she said, taking Killer's arm.
"Then you are making the wrong decision."
"We'll see about that."
"I think you should leave."
Mary Jayne stepped closer and murmured, "You pompous ass."
"And you, my dear, have proven yet again that you are nothing but a spoiled little rich girl." He lowered his head and whispered in her ear, "Your charm, and ease, and beauty will no doubt get you through life, but you need to grow up and get some backbone. Life isn't like a story from F. Scott Fitzgerald for most of the world, Mary Jayne, not all of us tool around Europe in our private plane, following the snow and the sun when the whim takes us. I know you. I know what you are. You're not a kid anymore. All the planes, and jewels, and elegant dresses can't fill the hole in your heart...."
Her eyes blazed, glistening with tears, but her voice was steady. "You're wrong. You are wrong about me. Ask them, ask any of them. I am brave, and kind to my friends, and generous—"
"Oh, sure, I can't argue with that one. But then it is easy to give away that which you've never had to work for."
"You know nothing about me, and you know nothing about Raymond."
"Killer?" Varian sneered. "He's a punk, Mary Jayne. The kid is sore with me because of what I represent, he's outclassed by Breton, so out of spite what does he do? Kidnap a puppy. Some great man you've got yourself there."
Mary Jayne raised her chin. "You have no idea—"
"If you want to be really kind to your friends, do the decent thing and leave."
"This is my house. I found it."
"If you continue to stay here, and see Raymond, you are putting the lives of your friends, of Danny and his family, in danger." That seemed to hit home. I saw her face crumple as she glanced over at her friends.
"All right! You don't have to go on." She sauntered over to Killer, swinging her hips. "You've got yourself a lodger for the night. Okay?"
"Okay, chérie." He put his fingers in his mouth and whistled for Dagobert. The dog bounded down the steps to the car and leapt into the front seat. "Looks like Dago is driving us back to town."
"At least he isn't drunk." Mary Jayne pushed the dog aside and settled behind the wheel. "I'll be back to collect my things in the morning," she said without looking at Varian.
Danny rushed forward and passed her a sweater. "Take this," he said, "you'll freeze." Mary Jayne reached up and squeezed his hand.
"Thank you, darling. Now, you keep this lot in order, eh?"
"Are you sure you know what you are doing, Naynee? You will be careful?"
"Naynee? You haven't called me that in ages." There was a slight tremble to her smile. "You were always cautious enough for the two of us." Mary Jayne started the engine. "Of course I don't know what I'm doing, but what can I do? I'm not going to let Varian boss me around like some stuck-up schoolteacher."
"I could knock your heads together sometimes." Danny looked uncertainly at Raymond. "You know I'm here, if you need me," he whispered. "Please be careful. We've all heard the rumors that he and his buddy Mathieu bumped off some hood in the Vieux-Port. Raymond has dangerous friends."
"I know." Mary Jayne slipped the car into gear. "But I believe in him," she said, looking at Varian, "and nothing you, or anyone else can say will change that."
Varian grabbed Clovis's lead and lifted him from the car. Killer vaulted over the door and slid down into the space. "So long," he said, flipping a salute at them as Mary Jayne turned the car and sped away, the taillights disappearing into the night.
THIRTY-FIVE
VILLA AIR-BEL, MARSEILLE
1940
VARIAN
Varian tied a red ribbon to the fir cone and passed it to Aube. "Now, why don't you make it beautiful," he said. "What color shall you paint it?"
"Red! Like a magic tree!" Aube clapped her hands. She sat at Jacqueline's feet in the dining room. Aube propped the china doll her parents had given her that morning for her birthday against the fireplace and fanned out her printed cotton apron. The doll's slim white legs and tiny black shoes stuck straight out from the white underskirt as though they were emerging from a shell.
"There," she said to the doll, "you must stay warm, little one."
Jacqueline dragged across her sketchbook and sighed. She tucked a pencil behind her ear and flicked through her drawings.
"They are wonderful," André said. He was leaning against the door, watching them.
"Hello, my darling." She closed her eyes as he leaned down and kissed the top of her head.
"That is what I was saying to Jacqueline," Varian said. "She must paint."
"It's hopeless," she said. "I can't work at the moment. I mean, I love it here. With all our friends coming and going, it feels like old times, but..." She frowned and looked down at Aube's golden head, bent in concentration. André squatted down beside his daughter and helped her dab red paint on with his fingertips. Varian felt like he was intruding, so he stood and went over to the fire, raking the embers. He saw Breton look reflectively at his hands, smear the red between his thumb and forefinger.
"There, Papa!" Aube said, lifting the bright red pinecone.
"It's beautiful," he said.
"What do you think, Varian?" she said.
"Marvelous," he said.
"Come, let's hang it on the tree with the others." Breton lifted her into his arms, and they walked out into the hallway. Danny and Varian had dug up a fir tree from the forest and planted it in a terra-cotta pot as a surprise for the children. The artists had painted the branches white and everyone had joined in, painting stars and butterflies, abstract shapes, and ribbons of colored paper chains. Now, in the firelight, it looked magical. Aube reached up and placed the little red fir cone on one of the branches.
Varian looked up as the front door flung open, and the decorations danced in the cold wind. "Thank God, Beamish," he said as his friend strode into the hall. He took a deep breath, felt some of the tension in him ebb away.
Beamish pulled off his woolen hat and ran his hand through his hair, nodding a greeting to them all. He was out of breath and gestured to Varian to follow him.
"Where have you been?" Varian said quietly.
"Not here," Beamish said.
"Come on upstairs. We won't be disturbed there."
* * *
Clovis lay sleeping in a pool of lamplight on Varian's bed beside a half-read copy of Virgil, his paws working as he chased rabbits in his dreams.
"Come on in," Varian said, holding the door open for Beamish. Clovis bounded from the bed, welcoming him with delight.
"Thank God, Beamish. The cops have been at the office looking for you. I thought they might have got you, too."
"Do they have an arrest warrant?"
Varian nodded. "They were looking for 'the one who calls himself Hermant.'" Beamish squatted down and rubbed the dog's ears. "I told them you had resigned some weeks ago and I had no idea where you are."
"Thank you."
"How was Banyuls?"
"Good. Everything is running smoothly at the border."
Varian got to his feet. "We had a bit of a blow while you were away. Little Bill Freier has been picked up."
"Damn. Really?"
"Someone must have ratted on him. The cops walked in and found him surrounded by all the paraphernalia of his forging operation." Varian blinked quickly. "I daren't think what will happen to him." He noticed the look on Beamish's face suddenly. "What is it? Something's wrong."
"Not really, considering the cops are after me. I've, well, I've had some news. A post has come up for me at the University of California at Berkeley."
"I didn't know you'd applied."
"I hadn't. An old professor of mine arranged it through the Rockefeller Foundation." Beamish lowered his eyes. "He's a good fellow. I think he looks on me as something of a wayward child. He knew I'd made it through the fighting in Spain. I think he's worried my luck might run out."
"You? Never."
"He may have a point. I've had a good run."
"Congratulations." Varian shook his hand. "Of course, you must take the job."
"But what about our work here?"
"You think you're irreplaceable?" Varian couldn't hold on to his smile. "It's not safe for you anymore, my friend."
"Was it ever?"
"Damn, we had some times, didn't we?"
"The best of times."
"And you, my friend... you are the best of all of them." He cleared his throat. "It's probably just as well you are leaving now the cops have you in their sights. When will you go?"
"Tonight."
"So soon?" Varian felt like a mountain climber watching his rope snake free from its anchor.
"I won't risk going back to the hotel for my things, but I'll take whatever you need me to, from here, papers and so on." Both men felt the charged atmosphere. "I'll head straight back down and on through Spain and Portugal, then to New York."
Fry reached for his briefcase and sorted through the papers. From an envelope he pulled a torn slip of colored paper. "There, this matches up with the half our people have at the border. They know you, of course, but we may as well keep the sequence and take the next slip they are expecting."
"Of course."
"I can't imagine you'll have any difficulties. They've helped nearly a hundred people over the mountain routes in just six months. You can stay in their rooms for as long as you need, and work with them in the vineyards near the border until it is safe for you to cross over."
"Won't people be suspicious?"
Varian shook his head. "If anyone notices, they always say our clients are friends who have been unable to rejoindre leurs foyers. I do love that phrase, the idea of returning to your hearth, your home...." He fell silent. "You have your visas and passport ready?"
"You know me, I'm always en règle."
"I've never seen a man with so many IDs." Varian poured them two drinks. The bottle was almost empty, and he finished it off between them. On the nightstand he saw Eileen's latest letter. The words much love—if you're interested leapt out at him. Beneath it, he glimpsed the dog-eared corner of the telegram from the committee in New York: Replacement arriving. Return to US soonest. He knew another fitful night's sleep lay ahead. Every night was the same—he awoke at four A.M., worrying about the refugees, his marriage, his health. Now, it no longer helped to go to bed half-cut. He always woke stone-cold sober at four, unable to rest and sleep, succumbing finally just before his alarm went off. I was never qualified for this job, he thought. I only volunteered because there was no one else. It was only supposed to be a few weeks. I've been demanding New York send over someone with experience of a situation like this for months, but now... Varian's heart stilled as he realized nobody had experienced anything like this before. There was no precedent, no one better qualified for the job than him. He had no option but to stay to the end, whatever that meant. These people needed him. He blinked quickly. "We're going to miss you around here." He turned and smiled, handed his dearest friend in Marseille a drink. "To Buster and Beamish," he said, chinking the glasses.
"May we always be one step ahead of them," Beamish said, and downed his Armagnac. "Right, what can I take out with me?"
Varian took his copy of Virgil from the bed and slipped out a couple of thin sheets of onionskin paper, filled with tiny writing. "I've been waiting for the next client to leave, but you may as well take these."
Beamish reached for the tube of toothpaste by the basin and squeezed some out until it was half-empty.
"I don't suppose you have a condom?" Varian said, rolling the papers tightly. Beamish flashed him a quick smile and reached into his pocket. He slipped the documents inside the condom and slid them into the opened end of the tube. He refilled the toothpaste and rolled down the end.
"What will you do now?" Beamish said, wiping his hands. "Will you leave for New York?"
"How can I? Someone has to hold this place together." He glanced at Beamish, a faltering smile on his lips. "I sent them a telegram. Do you know what their response was? The committee has stopped my salary."
"They're trying to force you out?"
"And Headline won't keep my editing job open any longer."
"What does your wife say?"
"Eileen wants me on the next boat out of here." Varian laughed. "But I never did like being told what to do. I think I might move back into town for a while. I'll take a room at the Hotel Beauvau. Like I say, with you gone, we need someone on hand in town." Varian shivered. "It is kind of cold here, too."
"I didn't want to say anything, but this place is like an icebox."
Varian's stomach gurgled. "Don't say that, you'll make me think of food. I dream of steak and ice cream."
"When I get to America, I shall eat a steak in your honor."
"Are you sure you won't wait and come by the office in the morning?"
"I don't think so," he said, unable to look Varian in the eye. "I'm not good at long good-byes."
"Me neither." Varian offered Beamish his hand. "Take care, old boy."
Beamish glanced up at him. "I'll see you soon, in New York?"
"Do I always say that? It seems to give them confidence."
"Like I said, maybe it is you who needs reassuring."
Varian's throat was tight. "You take care, you hear? And I will see you soon, in New York, or California, Dr. Hermant, or whatever your real name is."
"Hirschman," Beamish said, slipping his hat onto his head. "Albert O. Hirschman, but don't tell a soul." He gazed out the window toward the sea in silence for a moment, and Varian saw him smile one last time in his reflection in the dark window, before he turned and was gone.
THIRTY-SIX
FLYING POINT, LONG ISLAND
2000
GABRIEL
"Is it true," Sophie says to me, "that everyone was making love like crazy during the war?"
"What?" I stop and lean over, my hands resting on my thighs. "Jeez, woman. Is that all you kids think about? Everything was different then. Normal rules didn't apply. People changed their partners, their lives, their names. Wartime... peacetime... we're all making it up as we go along."
"I kind of admire people who have the guts to invent their own lives," she says. My head hangs down as I try to catch my breath, but I can see her legs up ahead.
"You're nuts if you think you're going swimming. You'll freeze your ass off."
"I just want to put my feet in the water, wash off the blood on my leg," she says. She peers back at me over her shoulder as she peels off her stockings, her hair blowing long and wild in the wind. I can't make out her face, just those blue, blue eyes looking at me defiantly. Blue? I could swear they were green earlier. Must be the light. "Come on, Gabriel, live a little."
"Not so much of your cheek, missy. I told you 'Mr. Lambert' will do just fine."
"Stop avoiding the question, Mr. Lambert," she says, tossing her stockings down on the sand like a shed skin.
"You young people think you invented sex," I say. "The thing is, when death's all around you it gives living an edge, makes you take risks you wouldn't otherwise."
"Is that how it was with Annie?"
"None of your damn business." God, I hate journalists. All these years poking and prying like the fame of my public work gives them rights to my private life. I haven't spoken to any of them since 1970-something, and this was a mistake, today. I'd never have agreed if the girl's mother, Paige, hadn't called my son. My head snaps up and I've a good mind to tell her to get lost, but she's run off down ahead into the shimmering water, and there's such a grace and lightness to her that I'm lost for words. "Annie was a good girl," I yell after her, my words snatched away by the wind.
* * *
Annie, Annie, Annie... As I close my eyes, I think of that first time. I'd spent the day helping Varian and the boys move the ARC into bigger offices at 18, boulevard Garibaldi. The place they were taking over had been a hairdresser's salon. I guess they had split in a hurry because they had left all their equipment—mirrors and brushes, bottles of potions, all kinds of junk.
"Is that the last of the boxes?" Varian asked, looking up from his desk. "Thanks for your help, Gabriel."
"What do you want us to do with all this?" Gussie said, pointing at the pile of hairdresser's equipment.
"Chuck it out, would you? We need all the space we can get in here." He gazed over to where the team were unpacking files from brown boxes. Roses bloomed across the wallpaper behind Lena's desk, framing a small white marble fireplace. He seemed miles away, in some kind of detached reverie. Varian had seemed distant for the last week or so, ever since Beamish had left.
"Here," I said to Gussie, "let me help you." I grabbed one end of a big mirror and helped him out the back door to the alleyway and the bins.
"It's good of you to help," Gussie said.
"The least I can do," I said. "I wish I could do more." The truth is, I felt like a fraud. The guilt's as fresh now as then. Oh sure, some people are pathological liars—untruths are as natural to them as breathing—but not me. I just wanted to do any little thing that was good, and true, to help them. We carried box after box of junk out into the alleyway, making a great pile. "What do you think will happen to this stuff?"
"It won't last long out here," Gussie said. "We're probably being watched as it is. You wait and see, as soon as it's dark someone will be down here and away with the best of it to sell it on."
I thought of Annie. "Say, do you think it would be okay to take a couple of things for my girl?"
"Sure," Gussie said, pulling the door closed behind him. "Take whatever you want."
* * *
I headed out to La Pomme that afternoon with my pockets stuffed with hairbrushes, clips, shampoo. On my lap I balanced a little gilt-framed mirror. It made me think of Snow White: Who is the fairest of them all?
Annie and I had grown bolder—the confrontation with her parents had brought everything to a head, and now I had figured out how to sneak up the back stairs of the Bouchards' house through the old hay barn and along the roofline to her bedroom window. If we weren't allowed to see each other in public, then we would do it under their noses.
Annie opened the French windows of her room. They swung back from the wrought-iron balcony, and she stood aside, waiting for me to jump in. She had been working on the loom by the window, and she held a hank of blue silk in her hand.
"Wait," I said, "let me pass this to you."
"What on earth have you got there?" She put down the thread and took the mirror from me.
"It's a present," I said, vaulting over the balcony. I closed the window after me, drawing the drapes.
"I swear you were a cat in a past life," she said, kissing me on the cheek. She was like the heroines of courtly love to me, worthy and chaste. Devastatingly chaste. "You never make a sound."
"Are they here?"
Annie nodded. "Maman is in the kitchen, and Papa is asleep in front of the fire downstairs. I said I had a headache, so I've gone to bed."
"Poor darling," I said, pulling her to me. "Let's see if we can do something to help." She let the mirror fall back on the bed and wrapped her arms around my neck.
"I missed you," she said, taking first my lower lip and then the top one between hers. I felt her tongue glide against mine, quick as a fish among the reeds. Oh, those afternoons were an exquisite torture, the silence, the fear of being discovered. They were the most erotic hours of my life, without doubt.
Her hands slid beneath my coat. "What have you got in here?" she said, reaching into my pockets. "Have you held up a beauty salon?"
"Not exactly." I emptied my coat of brushes and bottles, setting them down on the kidney-shaped glass-top dressing table. Annie's bedroom was the only properly furnished room in the house, it seemed. The chintz drapes on the dressing table were faded but good quality, and her bed had a wonderful, deep down quilt that I longed to curl up under every time I saw it. Now, with children of my own, I can see that Annie was the Bouchards' world. They wanted her to have everything they did not. Then it seemed only right that someone so beautiful should have the best of everything.
"Are these really for me?" Her eyes sparkled. "You promise you didn't steal them?"
"I promise." I set the mirror up against the dressing table and guided her to the stool.
"What will I say to Mother? She'll notice."
"Hide them in your wardrobe, or under your bed," I said, letting down her hair. I ran my fingers through it, felt the heavy weight of it. "It can be a secret." I pushed aside her hair and brushed my lips over the nape of her neck, felt the fine strands against my cheek. "Let's play hairdressers."
"Oh Gabriel, you're ridiculous! We're not children."
"Okay, you do your hair, and I'll just watch." I lay back on her bed, the springs creaking and sighing.
"Shall I put it up?" Annie turned this way and that in front of the mirror. She reminded me of Velázquez's Venus. I could almost picture Cupid hovering over the dressing table as she looked at herself.
"Why not?" I pulled my sketchbook from my rucksack. That was the first drawing I did of her, the first of thousands, as it turned out. It was quick and full of longing for her. I caught her, pinning her hair up into a chaste chignon. No, that's not quite right. There's nothing chaste about that drawing. I was weak with desire for her, my head swimming with exhaustion, the sketchbook barely concealing how much I wanted her. "Annie," I said without looking up from the drawing.
"Mm?" she said, hairpins in her mouth.
"Darling, would you take your blouse off?" I could see the shock in her eyes. "Just so I can see the line of your body better." I wondered if I had blown it, if she would throw me out. Night was closing in, the gray light falling outside, and a storm wind rattled the window. Annie's bedroom was small and cozy, up in the eaves of the house with its sloping ceiling. She took the pins out of her mouth one by one, holding my gaze. It was agony. Then, her fingers drifted to the little mother-of-pearl buttons on her white cotton shirt. One by one she undid them, and then, finally, she eased off the blouse. It slipped silently to the floor. She looked at herself in the mirror, her fingertips running across her collarbone, between her breasts. The thin silk camisole she wore outlined the curve of her rib cage. I was afraid to move, but I had to touch her. I was breathing hard as I marked in the final strokes of the drawing, and then I crept forward, padded silently across the room to her. I knelt behind her, slid my hand across the flat of her stomach as I kissed her neck. "Annie," I said, my voice little more than a whisper. She turned on the stool and allowed me to nudge open her thighs. She told me later she had decided that morning that she was ready, that I was the one, but I expected her to stop me. Instead, I felt her hand on my belt, the soft white skin of her thigh above the edge of her wool stocking. And then, and then... I thought I might faint at any moment. She led me to the bed, and we slipped beneath the quilt. We dared not undress in case her parents discovered us. The thought that we might get caught... well, that only added to the illicit joy of being with her. Everything fell away—there was no war, no house, nothing beyond our mouths and our hands on each other. We were the world, and the world was in us.
THIRTY-SEVEN
BOULEVARD GARIBALDI, MARSEILLE
1941
VARIAN
"Happy New Year," Danny said, closing the door to Varian's office. "How was it in Cannes?"
"Not good." Varian sat back in his chair. "And now the Thyssens have been arrested. Not that I care much for Nazi financiers. The net's tightening and it's bad news for our operation. If they've taken the Thyssens, what hope is there for clients of ours like Breitscheid and Hilferding? They were prominent German statesmen, but Hitler is gunning for them now."
"It's the first arrest of many, I am sure of it," Danny said. "Now we know the Nazis intend to use Article Nineteen."
Surrender on demand. "Is there any news of what has become of the Thyssens?"
Danny shook his head. "There's nothing in the press."
"When I was up in Vichy, the first place I went was the U.S. Press Bureau. They were beside themselves—if this story broke, there would be a worldwide scandal, but the censors have stamped right down on it. I don't hold out any great hopes for them, I have to say...."
The main door of the boulevard Garibaldi office flew open, and a young man stepped inside before Gussie could stop him. He put his hands on his hips and surveyed the ARC like a conquering hero. A stooped, gray-haired woman clasping a hard alligator handbag in both hands slipped through the door behind him and stood, waiting.
"Here we are, Miss Palmer," the man said. He marched across to Lena's desk and, ignoring the young refugee family talking with her, thrust out his hand. "Jay Allen of the Chicago Tribune, and now the ERC."
Lena pursed her lips. "How may we help you, Monsieur Allen?"
"Why, New York has sent me to take over this place," he said. "Haven't they cabled?"
"They said something about sending a replacement," Varian said coolly. He strode across the office. "How do you do?" His handshake was firmer than strictly necessary. "Why don't you come through where it's quieter?"
"Excellent idea, old boy." Allen turned, searching for the woman who was still cowering by the doorway. "Miss Palmer," he bellowed, beckoning to her.
"Everything okay?" Danny said as he slipped by.
"It will be," Varian said quietly, ushering Allen and Miss Palmer through.
He closed the door behind them and gestured to the chairs in front of his desk. He opened his mouth to speak, but Allen interrupted.
"I was expecting more of a welcoming committee."
"There's a war on. Banners and balloons are hard to come by." The men glared at each other. "Let me introduce you to everyone—"
"That won't be necessary." Allen leaned back in his chair and laced his hands behind his head. "This is how it's going to work. I will not be coming into the office myself. I will be continuing my work for the North American Newspaper Alliance, and Miss Palmer will be taking orders directly from me, running the day-to-day goings-on in the office."
Varian glared at him in silence, unable to believe what he was hearing.
"If I may, Mr. Fry," Miss Palmer said, her voice little more than a whisper. "New York has asked that we tail you for a few days, see how the office operates...."
"A few days? It will take more than a few days."
"Well, that's no good," Allen said. "I've got to get back up the country. I have a story to—"
"Hold on a minute." Varian leaned toward him. "You think you can just waltz in here—"
"Listen, I'm the foreign correspondent for the Chicago Tribune, and—"
"I know perfectly well who you are, but you surely don't mean you intend to continue working for them and running the ARC?"
"Why not? I'm not afraid of hard work or tough situations."
Jumped-up, would-be Hemingway, Varian thought, and chewed his lip.
"Besides," Allen went on, "as I said, Miss Palmer is going to be my eyes and ears here when I'm not around."
"You fool. You have no idea about the work we are doing here." He leaned closer and spoke clearly, quietly. "We are working eighteen-hour days, often seven days a week—you think you can do that part-time?... Well, do you?"
"Yeah, well, we'll see about that." Allen stood, his wooden chair scraping back on the boards.
"This is ridiculous. I'm not leaving."
A slow smile formed on Allen's lips. "Varian the contrarian. You've been asking for someone to take over this joint for months."
"That was early on."
"Changed your mind, have you?"
"The situation has changed, not me."
"Mr. Fry," Miss Palmer cut in, "you asked for someone to relieve you."
"Someone capable."
"You stuck-up son of a bitch." Allen thumped his fist on the table. "I'll show you. Whether you like it or not, Fry, I'm here to stay and you are heading out on the next boat or plane home."
* * *
"Who was that?" Danny said to Varian once Allen and Miss Palmer had gone.
Varian reached for the bottle of cognac on his desk. "My replacement."
"I thought you wanted to go back to America? Haven't you been saying for weeks how worried you are about your job, and Eileen?"
At the mention of his wife's name, Varian paused pouring his drink, then doubled the measure. "It's not that simple anymore. With Beamish gone..." He hesitated. He'd always hidden much of the clandestine work from Danny and the others. "I'm damned if I'm going to leave the office in the hands of that fool and see all we've worked for go to hell."
Danny stared at him, held his gaze. "Boss, if you ever need help, you just have to ask. We all know how much Beamish did."
Varian sensed that he knew what was going on. "Allen has no idea of the gravity of the situation. He thinks he can just handle all this work on the side as he continues to send reports home." He knocked back his drink. "Well, let's just see how much he can handle, eh?" He flipped through his diary. "I'm going to Nice tomorrow to see Gide, Malraux, and Matisse. Those idiots in New York think it's just a question of saying: 'Well, hello, Mr. Matisse, would you like a one-way ticket to New York? Oh, you would, splendid. I'll just chat with the nice visa people, and we'll put you on the next boat out of here.'" Varian swirled the drink in the bottom of his glass. They have no idea, he thought. No idea about the fake passports, and visas, of the constant fear everyone is living in. "They think everyone will just jump at the chance of going to the promised land. Half the artists of Picasso and Matisse's stature think they are untouchable, and the other half would rather die on French soil than leave."
"Boss, you seem awfully tired. Are you sure it's not time for you to go home?"
"Hell, no." Varian raised his dark-ringed eyes and looked through to the main office at the queue of refugees snaking out the door and onto the pavement. "The ERC wants Matisse and his like to leave France immediately? Let's just show Mr. Allen how difficult this job is."
* * *
Miss Palmer flinched as a parakeet swept across the palms. Water dripped somewhere in the conservatory, the sound muffled by the green leaves blocking the light from the vast glass windows. Up ahead, Matisse shuffled along the tiled path, leaning on a cane. At his side walked Varian and a doctor. He wore a red velvet dressing gown and a purple paisley turban. His orange leather slippers flapped gently as he walked.
"Are you sure we can't convince you?" Varian pleaded. "I am most concerned about your health."
"Thank you, young man," Matisse said. "Monsieur Fry, it was most thoughtful of you to bring a medic with you."
Varian took Matisse's arm and helped him back into the wicker bath chair set up beside his easel. Specially adapted brushes on long bamboo sticks sat in a jar at his side, and an unfinished nude in india ink was pinned to the board on the easel. Around the legs of Matisse's chair, Varian saw multicolored offcuts of paper strewn across the floor like confetti. A large pair of silver scissors sat on the table beside him, resting on a stack of cerise and orange paper.
Matisse caught his breath. "I am not a well man, as you say. I have no interest in leaving my home, my birds." He pointed up at a pair of yellow cockatiels. "Who would feed my birds?"
"You may not be able to feed your birds yourself, soon," Varian said.
Matisse laughed, a light gasp of air. "The Nazis cannot scare me. Why would they be interested in an old man who paints dreams?" His head rolled to the side, and he looked at the empty champagne bottles on his desk filled with wildflowers. "They call me a degenerate, but all I have ever done is paint the beauty in the world. Is that a crime?"
"No, far from it. It's why you have so many friends in America who wish to see you safe."
Matisse reached up and patted his hand. "Tell my friends I thank them from the bottom of my heart, but I will not leave my home."
* * *
"Stubborn old coot," Varian said, pulling his hat down over his eyes as they stepped onto the bustling Nice street.
"For intellectuals they are either brave or naïve. Gide, Malraux, Matisse—we couldn't make any of them see sense." Miss Palmer shook her head. "I don't know what to report back to Mr. Allen."
Told you so, Varian thought to himself. "It's a tough job, Miss Palmer, but I'm sure you're up to it." He glanced over his shoulder at her as he tipped his hat. "I'm sure you're both up to it."
"Of course we are." Miss Palmer's eye twitched as she buttoned her coat against the biting January wind. She pulled up her collar. "Have you booked your ticket yet?"
"Lena's taking care of it."
"Good." Miss Palmer's face betrayed her suspicion. "I'll see you in the office on Monday, Mr. Fry."
Varian strode away through the town. He headed toward the Promenade des Anglais in silence, his thoughts rolling round and round in time with his footsteps. Varian jogged down the steps to the beach and sat down to take off his shoes and socks. He rolled up his trouser legs and walked on across the cool beach. He stood at the shoreline and gazed out to sea, curling his toes against the sand. So, he thought, the Foreign Policy Association won't keep my job open editing at Headline in New York any longer, the Emergency Rescue Committee has stopped my salary here because of Allen, and my wife is threatening to leave me. He took a deep breath and sighed. But I have this. I have all this, and there is much to do. The winter sun broke through the clouds, glittering on the horizon. He realized, in spite of everything, he had never felt more alive.
THIRTY-EIGHT
MARSEILLE
1941
GABRIEL
There's something about the threat of imminent demise that can send the most godless man hurrying on his knees to church. Annie's always had her faith, and I've envied her, truth be told. All the years she's worn that same little gold cross she had when I met her—I guess her mother must have given it to her at her confirmation. We raised our kids to respect all faiths—Annie still lights a candle for her father and my mother on a Friday night and so on. If there's one damn thing I can't bear, it's intolerance and the stifling insistence that someone's right and the other guy isn't. The moment you're certain about something, the game is up.
Quimby and I had argued about religion one night at the Château d'Oc, and I'd said I was agnostic or words to that effect. I imagine Quimby found it amusing to make me meet him up at Notre-Dame de la Garde to hand over his hush money. Not that I minded. I'd rather have met him far away from where we might bump into anyone from Air-Bel, or Annie. Once I'd caught my breath, I liked it up there. The view across Marseille, and out to sea, you wouldn't believe. Everything looks better from a distance—lives, lovers, cities. You miss the grit and boredom of everyday living, just get to see the good bits.
It was cold up there, with the wind blowing around the hill, and I felt the chill of the stone through the seat of my pants as I sat on a wall looking out toward the Fort Saint-Jean and the big old lighthouse. It was peaceful, though, and somehow I forgot about my wet shoes and chilled feet, and I felt myself still. Maybe that's happened to you? It's like a glimpse behind the curtain, when the chatter and the nonsense falls away, and you hear yourself clear and true? Well, I held on to that moment, and I took my chance. I made myself a pact with that little gold Virgin up on top of the church.
I said to her: I know I've done bad things, the worst. I know I've not been to church, or prayed for years, but if you save my Annie, I swear to you I will be a good man. I will put all of this behind me, and I will live a good life. I will do good work, and raise fine children, and I will leave this world a better place than I found it.
It wasn't much as prayers go, but I meant every word. I put my hand into my pocket and felt the smooth paper of the envelope holding my exit visas. Varian had told me that morning as he'd handed them to me that there was a flood of refugees leaving France now that they were authorizing visas again. He reckoned that the Gestapo had most of the refugees they wanted trapped and at their mercy.
I saw my own tormentor sashaying toward me along the path. "Gabriel," Quimby said, tugging off his leather gloves. "Do you have it?" I slid a wad of notes toward him on the wall. "You're very quiet today."
"What is there to say? You're blackmailing me."
"No need to sulk." He flipped through the notes. "You don't mind if I count it?" I caught a couple of rough-looking sailor types looking at him as they walked out of the church and hoped he might get mugged on the way back down the hill.
"Good, all there," he said, slipping the money into his breast pocket. "Same time, same place next week?" He cocked his head. "Or perhaps you'll be gone then?" I said nothing. "I saw you going into the ARC this morning. I imagine you have your exit visas, as they seem to be handing them out like gobstoppers at the moment."
"If it wasn't so tiresome, it would be flattering the way you follow me around."
Quimby put his fists on the wall and leaned toward me. "I'm just protecting my investment, dear boy." He licked his lips. "I don't know how you do it, day after day."
"Do what?"
He moved closer, I could feel his stinking breath on my cheek, but I was determined not to recoil. "Aren't you consumed by guilt, when you see all those hopeful, desperate faces lined up outside the ARC?"
"Go to hell, Quimby."
"I'll see you there first, Gabriel." He pulled his gloves on. "Tell me something, did you do it? Did you kill them?"
I closed my eyes and breathed deeply. Bright flashes of color and light swirled behind the lids as I thought of the fire, of the hand reaching through the grille on the door. I jumped down from the wall and squared up to him, my face close to his. "Why, are you afraid I might come after you next?"
"Not that it matters, either way. I don't care. You've done me a favor as long as I get my money." He pursed his lips. "Do give my regards to that luscious little cutie you're dating. When are you going to tell her you're leaving? Time's running out."
The thought of admitting the horrible truth to Annie made me feel sick. That morning, I had confirmed my passage on a boat heading via Martinique to New York. I just hoped, somehow, I could persuade Annie's parents to let her come with me. I hadn't enough cash to pay for another ticket, not with everything I had been giving Quimby. I hoped they would have the money. Papers were another thing entirely. There had to be a way to get her out of France safely. My own visas crinkled in my pocket like a guilty secret as I stood.
"Oh, and just in case you are thinking of skipping the country without saying good-bye, remember, I'll find you." The sunlight flared on Quimby's spectacles as he turned away. "I think there might be a very good market in the U.S. for artworks by the illustrious Gabriel Lambert."
"No, you wouldn't!"
"You can run to America if you wish, but you can't hide from me, Gabriel. Remember the photographs. I can destroy you any time I feel like it. I own you. I created you." A smile curled across his pale lips. "And we have a great future ahead of us."
THIRTY-NINE
VILLA AIR-BEL, MARSEILLE
1941
MARY JAYNE
Mary Jayne shouldered open the heavy main door at Air-Bel, the wind rustling the heavy velvet drapes in the hall.
"Hello," she called, "anyone home?" She looked up as the maid, Rose, crossed the hall from the dining room, carrying the remains of breakfast on a tray. "Where is everyone?"
"Good morning, mademoiselle." She shifted the weight of the tray. "The Bretons have traveled to town together to try and secure their tickets."
"And Monsieur Fry?" Rose nodded toward the drawing room, and Mary Jayne tossed her coat on the chair. Her heels clicked across the tiles as she followed the sound of men talking. "Hello, Danny," she said. "What are you boys up to?"
Varian and a couple of the men looked up from the papers they were talking about but said nothing and turned their backs on her. Typical of Varian to sulk, she thought.
"It's..." Danny hesitated. "Things are rather difficult at the office. Miss Palmer has just upped and left for the States."
"I'm not surprised. I ran into her and that chap Allen at the consulate. The woman looked like she'd keel over from a heart attack the first time someone said 'boo.'"
Danny shouldered on his overcoat, and Varian picked up his homburg from the table. Danny scooped the papers into a file and tucked them into a canvas knapsack.
Well, I'm not going to talk to him if he's going to be that rude, Mary Jayne thought, glaring at Varian as he swept by without acknowledging her.
Danny saw the expression on her face. "I'm sorry, it's a bad time. Allen is giving Varian hell, and Breitscheid and Hilferding are being as difficult as usual."
"Those old sons ah bitches," Mary Jayne said, imitating Charlie's elegant southern drawl. "What have they done now?"
"We sent a car to Arles to pick them up, at great expense, which would drive them safely to Lisbon. Can you believe it, they turned it down flat?"
"I can believe anything of them. They think they are untouchable."
"Well, they are under house arrest now."
"Danny!" Varian yelled from the hall.
"Are you okay?" Danny asked Mary Jayne.
"Me? I'm fine and dandy," she lied. As fine as a girl who has just found out her lover has stolen all her jewelry can be. Mary Jayne chewed at her lip as she remembered returning to her hotel room to find the place had been turned upside down. When she challenged the concierge, he had described in perfect detail the man who had gone to her room. How could he? How could Raymond betray me, after all I have done for him? She had wept as she'd folded away her clothes and retrieved the empty jewelry boxes scattered around the room. I'm not having it, she had decided. I'm going to find him, and I'm going to get every last jewel back. She thought of her father, her grandmother, overcome with regret. Varian was right, damn it, he was right all along. How could I be so stupid? Well, this is my own dumb fault for trusting him. Love? Killer doesn't know the meaning of the word.
FORTY
BOULEVARD GARIBALDI, MARSEILLE
1941
VARIAN
Varian paced the office, waiting for Danny to return. "Were we followed?" he said as Danny appeared from the alleyway and locked the back door.
"No, I couldn't see anyone."
"There have been people on our tail for weeks, ever since the Sinaia." Varian ushered him into his office and locked the door. "Listen, I need to talk to you before the others arrive." He picked up his telephone and listened, before replacing the handset and pulling the wire from the wall.
Danny leaned against the fireplace. "Now Beamish has gone, perhaps we will all have to do more. You can't do this by yourself, however much you love playing at espionage."
"It's not a game, you're right," Varian said. "But the truth is there's a hell of a lot of fun to be had in rescue work and you have to find it whenever you can, or we would all break down. Regular depression, ennui, has no place here, and I for one am glad of that." He cleared his throat. "Even if our less official work is too much for some to handle."
"You're talking about Miss Palmer, aren't you?" Danny said. "You get a kick out of riling people, don't you? Is that what happened? You scared her away?"
He held Danny's gaze. "What we are doing is a kind of miracle day after day."
"One gets used to miracles. However they are being performed."
Varian leaned on the desk. "This is what I need to talk to you about. As you may know, or suspect, we have been helping people with fake visas and passports." Danny came and sat quietly by the desk as he talked. "There is more to the covert work that the ARC has been doing, and you're right, I need your help more than ever now that Beamish has gone." He folded his arms. "For the time being we are just going to carry on regardless, and pretend Jay Allen doesn't exist. We receive income from many people, not just the committee in New York, and I can't in good conscience simply hand everything over."
"Mary Jayne alone has given thousands," Danny said, "and this office does work far beyond the remit from New York." He paused. "Just tell me what we can do, boss."
"It's dangerous work. Are you sure you want to be involved?" Varian sat back in his chair and placed his fingertips together. "Very well. I think the best, the safest, option is to divide the work that Beamish and I have been doing between a few of you."
"That way if one of us is picked up, they won't be able to get all the information out of us?" The Adam's apple in Danny's throat jumped as he swallowed.
"Let's hope none of us will be picked up. If they do get one of us, then at least the others can carry on." He paused. "We have three problems. How to get people out, how to get false papers, and how to get dollars from our patrons into France from the U.S." He looked directly at Danny. "From now on, I want one of you to be in charge of land routes. He'll liaise directly with the guides at the border to get our clients out over the mountain and sea routes." Varian leaned forward. "I'd like another of you to take care of the fake passports and documents. Bill Freier has been picked up, but we have found a new supplier, and Gussie will continue to courier them across the city for us." He smiled, thinking of young Justus Rosenberg. "The boy has the face of an angel. No one would suspect him of running forged papers."
"And what about me?"
"I'd like you to take over some of the most challenging work Beamish was doing. As you know, he had good contacts with some of the..."
"Shadier elements?"
"Precisely," Varian said. "I want you to be in charge of laundering money."
"Hold on a minute—"
"Listen, how else do you think we have been bankrolling all this? With the donations from the committee?" Varian laughed. "We've been moving money in and out of the country for the refugees via Corsican gangsters. It would be far too suspicious if the authorities knew of the huge sums we are clearing through the office. Camille, the receptionist at the consulate, introduced Beamish to some gangsters, who in turn introduced us to Charles Vinciléoni, who owns the restaurant Dorade. Beamish came up with a way to launder funds. When one of our clients headed for America needs to get out, they give us their francs. Instead of running it through the books, Beamish clears the funds through Vinciléoni. A fellow called Kourillo is in on the deal. He realized that this would be a good way for some of his associates to get their money out of the country, too. The middlemen take a commission, of course—"
"Even Beamish?"
"Naturally. The gangsters would be suspicious otherwise. It's a three-way split between Kourillo, Vinciléoni, and Beamish. What they didn't know was that Beamish gave his commission straight back to the ARC. Once we have the funds here, we tell the office in New York how much to deposit in the client's dollar account. The client then picks up the cash once they are home free in the States." Varian paused. "It also means that any funds donated in New York can be picked up as francs here without going through the official channels. The police have no idea what is going on."
"And you trust these guys?"
"Trust is the wrong word. These guys have a finger in every pie—white slaves, the black market, dope." He thought of them all sitting around the stone-topped table in Vinciléoni's restaurant. Everyone was drinking cognac except Vinciléoni, who drank a glass of bicarbonate of soda. "There's little choice but to deal with gangsters in Marseille. I know this must go against the grain for you, but we need help however we can get it." He held Danny's gaze. "Will you do it, and take care of the accounts?"
"Will I cook the books, you mean? Is that what he was doing all this time?" Danny tilted his head back and nodded. "Sure. After all, who would suspect an ex–police official?"
"Good," Varian said. "I'll take you with me to the next meeting. Kourillo has offered to sell us some gold worth fifteen thousand dollars for eight thousand."
"That sounds too good to be true."
"I know. We're screwed if they catch us with gold, but we have to look at every opportunity, every crazy scheme to get our clients out of here." Varian looked at Danny. "It's up to us, my friend, and we are running out of time."
FORTY-ONE
FLYING POINT, LONG ISLAND
2000
GABRIEL
Where is she? The damn fool girl has dived into the sea. She must be crazy. I can see her, just, doing a lazy crawl out toward the horizon. "She's mad," I say, and slump back onto the sand. There's no fat on me these days, and my bony haunches ache as they hit the beach. I reach into my pocket and pull out my battered old brown leather wallet. It's worn and smooth as a stone, and over the years it's shaped itself to my hip. I open it up and flip through the photographs of all the kids, back through the years like watching the leaves of a calendar fall away. There it is: March 1941. The photographer Ylla turned up at Air-Bel, and she took photographs of the artists and the Bretons—maybe you've seen them in books about the surrealists. There's a gorgeous one of André and Jacqueline sitting with their little girl, Aube, beneath that huge old palm tree on the terrace. They look so happy and contented. It's heartbreaking when you think what was to come. Maybe the photo of them play fighting had more in it. They're standing beside a tall range of windows, opaque and dusty like the corroded silver of an antique mirror. They've both taken the stance of a boxer, fists raised, weight balanced back on their legs. Jacqueline is wearing a pair of wide-legged pants and a plaid shirt. If you glance at the photo quickly, it looks like they are dancing.
Air-Bel was a refuge for us all, and that's the feeling you get when you look at Ylla's photographs, how happy we were there. It was a place apart from time. Ylla took this photograph I have in my hand of a kid called Gabriel and his girl, Annie. There's so much love in our eyes, the photograph is radiant, even after all these years. I flip it over, and there's Annie's writing: I love you, this much, always.
It wasn't like real life didn't touch us there. I remember one Sunday in March, the little nanny Maria was hysterical. The Vichy government had suddenly rounded up all Spanish men and deported them to the Sahara to build the railroad. Just like that, no warning. They had taken her father, and I never did manage to find out whether she ever heard from him again. Imagine that, when the people you love most in the world could be taken from you in a heartbeat, not because they are good or bad, but just because of the chance of their blood.
I sometimes wonder why it didn't make me more bitter, all of this. The thing is if it changes you for the worse, then they've won, the fascist bastards have dehumanized you, and they've won. I never gave up, not once, not even when it looked like everything was lost.
I rub my thumb across the photograph. I remember we went into town the night that Maria's father was taken. Annie was pretty shaken up by it, and we wanted to do something defiant, however small. We heard that a bunch of the artists were meeting at the café Au Brûleur de Loups. They were planning something, and we wanted to be part of it. I remember her walking through the packed café to our table, to me, and the head of every man in there turned to watch her walk by, but she saw only me. She sat on my lap and put her arm around my neck.
"So? What are we going to do tonight?"
"Wait and see," I said. I was playing with the silk rose the café owner had put in a vase on our table, and I offered it to her.
"We can't take that," she said, laughing.
"When we get married, I will fill a church with roses for you."
"Gabriel..."
"I mean it. I don't care what your parents say. I love you, Annie, and I want to spend the rest of my life with you."
FORTY-TWO
BOULEVARD GARIBALDI, MARSEILLE
1941
VARIAN
"Is it done?" Varian said, looking up from the papers on his desk as Danny walked in.
"Yes, I buried the cases under the pine trees at the back of Air-Bel." He slumped down on the chair in front of the table. "That artist, what's his name? Gabriel Lambert? He almost caught me. Scared the life out of me, I thought it was the cops."
Varian smiled. "Was he out walking with a pretty blonde as usual?"
"That's right." Danny took the glass of cognac Varian offered him. "What's been happening here?"
"Well, the good news is we have our friend Mehring safely away."
"Baby? Thank God for that. How did you manage it?"
"As you know, our friends Breitscheid and Hilferding felt bunking down in the bowels of a ship was beneath them."
"The fools. Even if the cabins were all sold out, they should have jumped at the chance of getting out, particularly Hilferding—he's Jewish, isn't he?" He crossed his arms. "Well, they're in trouble now. Their exit visas have been withdrawn. It's a bad sign." He glanced at Varian. "Have you had any luck with your passport, boss?"
Varian shook his head. "I'm in exactly the same boat as many of our clients, now. Instead of renewing it, the idiots at the consulate have confiscated it until I leave."
"It's ridiculous! They are putting you in grave danger."
"Since Miss Palmer's hightailed it home, I have no choice but to stay, whether the U.S. government likes it or not."
The men turned as one at the sound of fists hammering on the front door. Varian felt his heart leap to his throat. He checked his watch. "It'll just be the others," he said.
Danny pushed back his chair and unlocked the front door. Gussie ran into the room, gasping for breath. "They got them," he said. "They got them—"
"Slow down, Gussie," Danny said.
"They've arrested Breitscheid and Hilferding—the Gestapo have them."
"Oh God, I was afraid of this," Varian said. "When?"
"I overheard two Vichy cops in the Vieux-Port crowing about it in a bar: 'Two of Hitler's greatest opponents have been rounded up....'"
"Damn, we were too late." Varian took his glasses off and threw them down on his desk. Charlie had it right about them: "Those couple ah sons ah bitches are asking for trouble."
"It's not good." Gussie looked down at the floor. "Hilferding is dead."
"Oh God, no." Varian slumped back in his chair.
"The Nazis found the poison he always carried on him, but he managed to hang himself in his cell."
"And Breitscheid?"
"They are sending him to Buchenwald."
Varian took off his glasses and rubbed his face with his hands. "Thank you, Gussie," he said. "Keep your ear to the ground. We have to get as many of our people as we can out immediately. If great statesmen like them are being arrested, what hope does the ordinary man or woman have?"
FORTY-THREE
MARSEILLE
1941
GABRIEL
I remember that night like yesterday. The artists who met in the café Au Brûleur de Loups decided to take part in the "battle of the walls," painting graffiti on the streets of Marseille. The BBC were putting out broadcasts, telling people to paint VH—Vive l'honneur ("Long live honor") on the walls. Or sometimes it was VV—Victoire et Vengeance. Either way, everywhere you looked there were red and black Vs. It was a small defiance, but it annoyed the hell out of the fascists. Of course, they were clever and eventually pretended like it was their idea to paint V everywhere, V for Viktoria, some old Teutonic word they dug up. They started printing posters with big white Vs on a red background, even stuck a big one on the Eiffel Tower for a while. But in the early days, it was ours. Everyone was fired up. We agreed to meet the next night, at eleven o'clock in the Vieux-Port.
Annie wasn't allowed out, of course, so we had to wait until her parents were asleep. I'd half dozed off myself, waiting by the garden gate for her. "Wake up," she said, sliding down onto my lap. She showered my face with kisses, and my hands slipped down to her hips. "Oh no, you don't," she said, springing up. She grabbed the pot of paint at my side and hid it in her basket, then ran off across Air-Bel's lawn. "We have work to do!"
You had to be careful, of course. There were Kundt Commission and Vichy cops everywhere, but we managed to do about ten walls that night, before we ran out of paint. Just as I was doing the last V, I heard Annie gasp: "The flics!" She was keeping watch at the end of the road, and she ran toward me, her feet splashing in the gutter.
I slung the empty paint tin into a bin and grabbed Annie. By the time they walked past, we were in each other's arms, kissing. I heard them whistle appreciatively and flicked open an eye, watching them until they turned the corner. "That was close," I said, giving Annie my full attention now. We hadn't managed to be together for a week or more, and my whole body hummed with desire. Every kiss, every touch, every look, lit up the night.
"Come with me, to my hotel."
She shook her head. "What if someone sees me? My mother would ship me off to a convent if she thought we had been to a hotel together."
"Perhaps that would be the safest place for you."
Annie shook her head and laughed. "You're crazy."
"Crazy with wanting you."
"Let's go home," she said. "We have a couple of hours before Papa wakes up." She slipped her arm through mine and leaned into me. "I wish we didn't have to hide, Gabriel. Wouldn't it be wonderful, to be together, always?"
My stomach twisted with guilt. How could I tell her I was leaving? I looked down at her face, at the love and trust there. I'd realized something else sitting up on the hill saying my little prayer. I was going to marry her, and I had to find some way for us to be together. We rounded the street onto La Canebière to catch our tram back to La Pomme, and I summoned up all my courage. "Listen, Annie, we need to talk...."
Her face fell, and she pulled away. "I don't believe it. Mother warned me it would be like this, that if you gave a man what he wanted, then he'd use you up and throw you away. How could I have been so stupid?" She began to run toward the tram stop.
"Wait!" I shouted, running after her. People were turning to look at us. I grabbed her arm and stopped her. "I love you," I said, holding her tight. I kissed her then, and her spine arced back. "I love you, but I need to tell you something...."
"Well, well, my favorite lovebirds." Quimby strolled toward us out of the darkness. "I was just having a late supper in the café over there, when I saw you run past. Not a lover's tiff, is it?"
Annie looked at him askance. "Gabriel, who is this?"
"Has Gabriel not told you about me? I'm a dear friend of the family."
I pleaded with him, with my gaze. She couldn't find out like this, not like this. "Quimby is an old business associate," I said. "We'll be concluding our work together tomorrow."
"Tomorrow?" He tossed his scarf over his shoulder. "Excellent. Shall we meet at the usual place?"
I couldn't get away from him quick enough. Quimby was one of those people who taint everything around him just by his presence, and I wanted Annie away from him as quickly as possible. I spotted a tram coming and bustled her onto the car.
"Who is that man?" she said.
"No one, it doesn't matter."
"You're shaking." She put her hand on my arm. "He's not no one. Why are you afraid of him?... Gabriel?" She refused to budge. It's something I've always loved about Annie. She's straight as a die. My whole life has been built on so many lies, I don't know what's truth anymore, but Annie always cut straight to the heart of something. "Are you in some kind of trouble?" I felt her lips against my ear. "I don't care," she whispered, aware of all the people around us. "You can tell me."
"I will," I said. "But not here." I put my arm around her, and my head sank to hers. I closed my eyes. I felt scared, suddenly, for the first time, after staggering blindly through the horrors of the last months. More than anything, I couldn't bear the thought that what I was about to tell Annie might drive her away forever.
* * *
We jumped down off the tram and began to walk toward Air-Bel. Annie stopped me beneath the railway bridge, just as a train thundered overhead. "Gabriel," she said. "I'm not walking another step until you tell me what's going on." She settled her back against the tunnel wall, and I put my hands either side of her. The train passed by, and the vibration through the brickwork stilled.
"I'm sorry," I said hoarsely.
"Sorry? For what?" She reached up and touched my face. "Gabriel, are you crying? What's wrong?"
"No, I'm not crying," I said, embarrassed, wiping at my eyes with the back of my hand. I was, of course; I've always been a weeper. Never make it through an old movie without a tissue or two, not like Annie, she's the tough one.
"Stop it this instant. You're scaring me. Whatever it is can't be anywhere near as bad as you think."
"It is. I've done something terrible."
FORTY-FOUR
FLYING POINT, LONG ISLAND
2000
GABRIEL
"What did you do, Gabriel?" The girl is there, suddenly, her face close to mine. I feel her breath, the breeze on my cheek. "What did you tell Annie?"
"The truth, missy, what you've been waiting for." I close my eyes as she exhales. "Annie said to me: 'Gabriel Lambert, nothing you can say will stop me loving you.' I said: 'That's just it. I'm not Gabriel Lambert.'"
I sense the girl sitting just in front of me, the fading sun a halo of light behind her head. "At last," she says. "Tell me the truth, just like you told Annie."
* * *
The truth? It's all so long ago now, I can hardly remember where one life ends and the other begins. I was in Paris on June 14, 1940, when the Nazi tanks shook the pavements and their vile black boots goose-stepped down the streets I loved. That day, a boy arrived home from art school to find his mother had killed herself. I was eighteen, but still more boy than man. You could smell the gas as you walked upstairs to our studio. Or could you? Maybe we didn't even have gas, I don't remember. She'd barred the door and by the time the concierge helped me take it off its hinges, it was too late. Our neighbors rushed in and screamed. I remember their bent figures crowding around her, my mother's foot as small as a child's extended beyond their swirling skirts, a hole in the bottom of her shoe patched with newspaper. That's what spooked me the night the maid fell downstairs at Air-Bel—I think I'd blocked the image of my mother's body until that moment, seeing her sad little foot with its hole in the stocking and everyone crowding around.
The funny thing about life is it's not consistent. You can go through years, even decades, without aging, then bam!—something happens, and you wake up older. Loss, war, disease... these are the things that ravage you and burn the lines on your palms and your face. I reckon this is why time moves slowly when you're a kid, why summers are never as endless and sunlit again. When the knocks are coming thick and fast, as they did the summer of 1940, you grow up quickly. It's not the length of years, it's the weight of them—that's what makes people old.
There was no note, no good-bye. I have no photographs of my mother, and my memories of her are confused. All but the image of her small foot, the battered shoe. Helene was fifteen when she fell pregnant with me, to a boy a couple of years older. She never stopped loving him, even though he had abandoned her the moment he found out she was having his child. She named me after him, encouraged me to be an artist, like him. She raised me, groomed me, even, to be just like my father. Sometimes I wonder if she's looking down and regrets quite how well her pupil learned his lessons.
My mother was an artists' model, and more, I don't doubt. That's how she met my father. We lived in the eaves of an old house in the Marais, in a single room with a roof that leaked. She slept in an old cot, I on the cushions from our single chair. The room I remember more clearly than my mother—the slant of the light on a winter's afternoon, the smell of cabbage, the almost edible fug of must, and dust that settled on the place like a shroud.
It wasn't the chichi district it is now. When I went to a show at the Picasso Museum a few years back, I walked past our old building. I've tried to find her, of course, but they just took my mother's body and dumped her in some unmarked pauper's grave. It's like she never existed. But she was a good woman, and she did her best with the little she had. I guess she lives on, in my work. It's because of her I am who I am.
One of her clients must have taken pity on us when I was thirteen or fourteen, and when I showed some talent he paid for my art lessons out of charity. Or guilt, who knows. I remember wondering at the time if he was my father. My mother was nearly thirty-four when she died, but she looked decades older. When I look at thirty-year-olds now, they are soft like children. Not my mother. When the Nazis invaded Paris, she snapped like a brittle twig. As I cradled her body in my arms, she weighed little more. In my memory, a copy of Stravinsky's Firebird is looping on a record player near the window, the needle sliding, bumping across the smooth end of the record, a soft hiss filling the air. But maybe I invented that. I don't see how we could have afforded a gramophone, and she would have pawned it by then if we had. My mother had given up on happiness years before the Nazis marched down the Champs-Élysées, but her dream, her fantasy of art and freedom and Paris, kept her going. When that died, her spirit went with it, but she gave me a parting gift. I learned early on how easy it is for people to disappear.
* * *
The girl, Sophie, can't be much older than I was when my mother died and I joined the great exodus of people fleeing Paris, heading south through France away from the gray Nazi tide sweeping across the country like slops from a mop bucket. Everyone was terrified. All many of us had left was our name. Some of us didn't even have a piece of paper to remind us who we were. Stateless, apatride, we were trapped, fleeing who knows where, with no hope of escape without paperwork or visas. The roads from Paris sparked with fear in the night. If the troops caught up with us, what would happen? People talked in choked voices of concentration camps. I could tell you about the nights I spent sleeping in ditches, or barns if I was lucky. Whenever I see a herd of deer now, and one senses something, a predator, and raises its head, and all the others follow suit before they flee, I think of the nights I spent huddled in dark spaces with men and women, and children. I see their faces.
I could tell you about sheltering under cars as enemy planes strafed the columns of refugees snaking south. I could tell you how it feels to squeeze yourself between the earth and the hot metal chassis of a car, whose faint blue-painted sidelights illuminate the long grass near your hand. Can you imagine how it feels when a bomb explodes nearby, and every cell in your body reverberates, and you don't know for a moment if you are alive or dead? Your ears are shrill with blood, but you can just hear the moaning of those hit at point-blank range, behind you. Never look back. Like I said, the ones who look back are the ones who turn to stone, or salt. You have to keep moving forward to survive. I could tell you how it feels to walk hundreds of miles and find yourself cowering, your face in the dirt, and see bullets ricocheting like hailstones inches from your head. I could tell you how sick you feel when the four-year-old child hiding up ahead isn't so lucky.
That moment has stayed with me my whole life. What was left of the good, of the boy in me, broke then. Now, when I picture myself lying in the ocher dirt, trying not to cry, it makes me think of that line of Mehring's: "Hope cracks and crumbles." Even now when I cradle my sleeping great-granddaughter in my arms, I think of that little girl. I was surrounded by people, but completely and utterly alone. I left everything that was pure, and soft and true, on the side of the road, and the shell of a man staggered up and walked on past the grieving mother.
* * *
But do you want to know about that? The world has compassion fatigue. You've heard stories like my sorry loss of innocence before, haven't you? I was a refugee, just one more face among the nameless thousands lucky to make it out alive. I can see from the look on Sophie's face that all she sees is the sorry old husk of the great man of art. She wants Vita, to conjure her back like a spirit at a séance.
"Okay, okay," I say to her. Flick back in time. Imagine you are this boy who has lost his mother and the only home he has ever known. Your heart is broken—you just want to look your father in the eye and tell him she's gone. Of course, part of you hopes for a reconciliation, that he will accept you, love you, even. Night after night you wonder what it will be like to meet this man. You have been on the road for weeks, scavenging food from hedgerows, drinking water from streams and puddles. You are tall and broad and strong, so you can look after yourself, but you look like a bum, your unshaven face is dark with dirt, your hair is thick and long. The look in your eyes has the weariness and knowledge of a man. All you know for sure is that you have to get away. All you have is an address written in your mother's hand, an unknown village in the Languedoc, a pin in the map gleaming like a beacon. On the road south, the companions you found peel off group by group, to Toulouse, to Spain, to safety and escape, they hope. The refugee telegraph crackles, word of mouth, sightings of friends and family in distant towns. By the time you stumble into the nearest town to this village you have pinned your last hopes on, just as the light is falling, the beauty of the sunset makes no impression on your broken senses. Birds settle on the terra-cotta tiles of the church like five o'clock shadow, but you don't see them. You are shattered, heartsore, all you want to do is sleep. In the nearest bar's cloakroom you wash yourself as well as you can, and you order a beer, a pression, with your last centimes, and it is the best, the coldest, beer you have ever tasted in your life before or since. After this, you think, who cares what happens. Then someone slaps you on the back.
"Lambert!" he says. You turn your head slowly, his voice and face swimming in and out of your senses like you are a prizefighter on his last legs, cornered. "Where the hell have you been, you old dog? I haven't seen you for years. Are you still with Vita? No one's heard from her in an age."
"Vita?" You have no idea who this man is, you realize, but he knows you, it seems. It takes a moment for you to click that he has mistaken you for your father.
"He's squiffy," he says to his companions. "Thank God some things don't change. Lambert's been on one of his benders." You look at the group and wonder if you are hallucinating. There is Pierrot and Pierrette, a jester, a Roman legionnaire, a bear. "Are you coming to the party, at least? You must! One last fling before we all get the hell out of here." They drag you away from your drink and bundle you into an open-top car. Why not, you think. Why not play along? Perhaps there will be food at this party. What harm is there in pretending to be your father, just for a while?
"He doesn't have a costume," Pierrette says. A mask is handed to you, a Venetian carnival mask, gold, with a long nose, and a black cape is draped over your shoulders. Your head sways as the car swings around hairpin bends, up, up into the green mountains. Your eyes are heavy, lead weighted. When you awake, they are climbing out of the car and walking along a gravel path lit by torches, through the great wooden gates of a château. You follow them.
You wonder if you are dreaming. "What a darling idea to hold one last party," someone says as you knock back a drink at the bar. You pour another. "A last cry of freedom," someone dressed as Nero declares, holding a slopping glass of red wine aloft. "Damn them all to hell." Then the band quiets, and all you can hear is a girl, laughing in the shadows of the plane trees. The drummer pounds a beat, one-two, one-two, thumping faster and faster, your heart can't help but keep pace. The trombonists stand, swinging out a tune, and all faces turn to the gate like sunflowers following the course of the sun. And then, she is there.
* * *
Vita rode into my life—literally. She galloped through that gate on a horse—I'd like to say a white horse, but really it was a crazy-eyed bay. Did I say she was naked? The torches in the night scorched the earth between me and her. It was as if the noise of the party, the people, fell away around us. There was only her and the flames dancing in the night. Then there was a flash of light, a pop. Some guy dressed up as a faun with green body paint and shaggy thighs was taking a photograph of her. His yellow horns poked out of his hair just above the camera.
Then the party erupted into cheers, and the horn section of the band rolled into "Sing, Sing, Sing." Vita pulled the horse up short, gravel scattering under its hooves. It reared up on its back legs, turned a circle on its heels. Everyone thought it was part of the show, but she told me later she was scared to death. She jumped down soon enough, and the host cut through the crowd of wild dancers and draped a sheet over her like a toga.
"Your best yet, Vita," he said.
"Happy birthday," she said, kissing him on the cheek. "I never could resist a bet." Then she pushed her way through the crowd toward the bar and picked up my drink.
"What are you staring at, big nose?" she said to me, glancing over the rim of the glass of wine.
It took a moment for me to realize what she was talking about, and then I remembered the mask. "That's my drink."
"It's a free country." She paused. "Or was."
"I like your crown," I said.
She touched her hair, the delicate circlet of gold leaves. "It's a coronet of myrtle leaves. I made it myself, modeled on the Macedonians." She swayed in perfect rhythm to the music, her hips undulating with the liquid grace of a cat. "Dance with me."
"I don't know how...."
Vita dragged me into the heart of the crowd. "Just feel the music," she yelled in my ear, and swung away.
I remember now the drums, the thump of the drums reverberating in my chest, how the swirling masked faces swam before my eyes, and at the heart of it all was her, jiving and swinging, so terribly alive. That is what I remember most about her, how much her name suited her. I had never met anyone like her. "Who are you?"
She stepped closer to me, ran the nail of her index finger over my lips. "I am the poisoned chalice. I am Helen astride the Trojan horse."
"You're certainly melodramatic."
"No, this is the bit where you are supposed to say: 'Of course you are, Vita, quite as lovely as Helen,'" she said, waving her fingers in the air. "You're not terribly good at this, are you?" She tilted her head. "Have I seen you before?"
"I don't think so. I'm just passing through."
"Oh goody," she said, and took my hand, pulling me toward the garden. We were breathless by the time we found a quiet dark spot beneath the trees.
"No, really, who are you?" I said again, catching her by her waist.
She pulled off the cascading blond wig and ran her fingers through her dark bobbed hair. "Does it matter?" she said. She backed away from me and slipped the toga from her shoulder. She was naked and lovely, moonlight dancing through the leaves, across her skin. When you look at all the art deco paintings and sculptures of her now, they are alive with that grace.
I'd like to say I seduced her then, but the truth is I was clueless. Vita took charge. She placed my hand on her breast and kissed me. "We could die tomorrow," she whispered, her lips tracing my jaw beneath the mask, my neck. "I don't want to die without having made love tonight." I pulled the mask from my face, and in the darkness, my blood coursed with the distant beat of the jazz band, the hypnotic buzz and hum of the cicadas in the grass around us. I felt my whole being contract to my lips as she kissed me, like running the film of a firework exploding in reverse, and then, predictably, in sudden, wonderful release I came the moment she touched me.
"Oh God, I'm sorry, I—"
"Don't apologize." She laughed softly. "We have all night." She curled herself around me, skin on skin. "Forget everything. The future doesn't exist tonight, nor the past...."
"Only memories?"
"They change." She slid her hand down my stomach. "All that matters now is you, here with me, tonight."
War is like that for some, you see. It heightens everything, makes you more of what you are, makes you want to do something life affirming, the most natural thing in the world. We were young, and crazy with fear, I think. She was the first woman I had ever been with, and dazzling is the only word for her. When I woke at dawn, I was alone, sleeping on the dew-wet grass with a canopy of leaves above me and a fragile gold coronet placed over my heart.
FORTY-FIVE
FLYING POINT, LONG ISLAND
2000
GABRIEL
"Gabriel," Sophie says, "we're running out of time. Tell me about the house where you lived with Vita. Tell me about what happened at the Château d'Oc."
"It was a magical place, but of course I sold it after the war," I say to her, and begin to rattle out the same old lies I've told over the years to anyone who was curious enough to delve back that far.
"Stop it. Tell me the truth, Gabriel."
"I..." Oh God, the truth. The blood is singing hot in my ears, a volcano surging.
* * *
The truth is I hated the Château d'Oc. I was terrified the first time I walked up the dirt track to my father's house. There was no paved road in those days, just an earth track cut through the trees, climbing up the mountainside. Someone at the party gave me directions, and at dawn I stumbled off along the driveway lined with burned-out torches. I heard them laugh and say, "My God, Lambert must be pissed."
It didn't take long to walk there, an hour or so, in spite of my lungs. You have to remember I was an eighteen-year-old boy and all sinew and muscle then. It would take a month of Sundays for me to do that walk now. Each step brought me closer to the village, to the address my mother had written so many times on letters begging for help that I knew it by heart.
All I knew about the house, about my father, was hearsay and gossip. He had run away the moment my mother became pregnant. They were little more than children themselves, so perhaps it's no surprise he ran like a scalded cat. He became a fashionable artist, grew wealthy in the early 1930s, and married some young heiress, Rachel West. She died in a car crash, and he took up with one of his models. His society clients couldn't get enough of his paintings of languid nudes and his flattering portraits of society women with their strings of pearls and greyhounds. Then, he disappeared.
My mother was relentless, though, and finally tracked him down to the Château d'Oc. We never visited, of course. Ignoring her letters was one thing, but I think she couldn't have borne it if he had rejected her face-to-face. I could see its turrets and pink-tiled roofs above the tree line now as I walked, snaking round and round its fortified base, the road coiling in like the track of a labyrinth. I passed rough cottages built into its base, where scrappy, boss-eyed dogs stared at me with their yellow eyes, yawned, and stretched. Beyond a small chapel, I caught my breath. I stood for a while, looking out across the open hills just beginning to tremble with the dawn chorus of birds and crickets, a red sun flaring over distant mountains. I'd imagined this for years, his home, his land. Finally, I was going to meet my father. Sometimes I pictured myself embracing him, sometimes thumping him on my mother's behalf. I'd never expected to feel so uncertain. My footsteps crunched across the stones littering the road. I remember thinking clearly that I had come to the end of my journey. More fool me. It was only just beginning.
* * *
A peeling piece of paper was tacked to the gate with a rusted thumbtack. I could just make out the name "Lambert." I thought the blue gate looked rotten, its paint flaking at my touch. The château wall seemed to grow organically from the hillside, and the gate was flanked by ferns and ivy that brushed my arm. The swollen gate swung open, dragging on the earth beyond, and I hesitated.
That was my moment. The split second that time seemed suspended and gut instinct or my guardian angel told me to walk away. To keep going. To head to Spain, or Marseille, or some other way out. But, what did I do? I shouldered it open, walked straight in, and the trap sprang shut.
Across a courtyard littered with broken chairs and a dried-up fountain, the first person I saw was Vita. She was sitting on the kitchen doorstep, eating a baguette with strawberry jam smeared on her lips. Wasps muzzed around the open jar at her side, and a bowl of black coffee steamed patiently by her ankle. I remember the shock on her face, how her silk flowered robe gaped open and she knocked the coffee over as she ran toward me. The empty white bowl spun on its side.
"What the hell are you doing here?" she whispered, glancing back over her shoulder. She pulled the robe tight and belted it with a scarlet sash. "I couldn't believe it when I woke up and saw your face. Christ, what a mess. What a ghastly, hideous mess. He'll kill you if—"
"Can I help you?" A man leaned in the kitchen doorway, a burgundy paisley robe doing little to conceal his nakedness. His hair was wet from the bath, and he looked at me with pale, myopic eyes. I recognized him from the party—the faun with the camera. A smear of green still edged his jawline. Was this him? The man I had hated and loved my whole life?
"I'm looking for Lambert," I said. There was a strange expression on Vita's face, something I couldn't place.
"Are you indeed?" The man sauntered over. "Well, I'm Alistair Quimby, his dealer. Any business you have with Monsieur Lambert, you can discuss with me." He pulled a lorgnette from his pocket and grimaced as he settled it on his nose. He stopped short. "Good God," he said, and stepped closer to me. I recoiled as he paced around me, felt his gaze snaking from my ankles up to my face. He pursed his lips in amusement. "Let me guess."
"I believe he is my father."
"Do you? Well, you wouldn't be the first one, lovey, but we've never seen one quite as good as you before, have we, Vita? This divine creature is Lambert's muse," he said, waving a pale hand in her direction. He looked me over one more time as if he were inspecting a prize bull, then beckoned me to follow him into the house and walked away.
"Go. Go now," Vita said under her breath, and pushed me away. "I never would have spent the night with you if I'd known who you are."
"I don't care," I said, my heart already full of desperate longing for her. "I don't care if you're married to this Quimby fellow."
Vita gasped out a short laugh, little more than a breath. "I'm not married to him. I'm with Lambert, you fool." She looked up at me in the dawn light as the sun rose over the dark bulk of the château's walls and struck me. She raised her hand, shielded her eyes. "It's incredible."
"What's your name?" Quimby called.
"Gabriel Lambert," I said.
"Ha. Of course it is." He turned to me. "And your mother is...?"
I told him, and that she had died recently.
"Condolences," he said without much conviction, and clapped his hands. "Well, Gabriel Lambert, prepare to meet your father."
* * *
The first thing I noticed about the house was the smell. Even now, when the calibration of mothballs, dust, cigars, and garlic is just so, I go time traveling back to that old château on the hill. The second thing was how dark the house was. The windows were little more than glazed arrow slits, cut into the walls of the château, so deep a man's arm could barely reach the glass. Clearly there was no housekeeper. The frames of the engravings hung in the entrance lobby were thick with dust, untouched for years.
The kitchen was clean enough—whitewashed stone walls, a long scrubbed pine table with benches either side, and a high-backed settle hard up against the open fire, which was lit and dancing even at this hour. Above the table, a Bec Auer gaslight shed its greenish glow. "I'll make some more coffee," Vita said, and stalked off toward the scullery, swinging a metal kettle in her hand to pump some water.
"Follow me, young man," Quimby said, his pale finger dancing over his burgundy shoulder like a maggot on a steak. Out in the hallway was where it started to get a little crazy. Kilims littered the large flagstones, slipping beneath my feet as I walked after him. Every wall was hung with African masks: beady, glassy eyes and gaping mouths with ivory teeth everywhere I looked. Every tabletop was piled with yellowed journals and papers. "Lambert is quite the collector, as you can see," he said airily. "Primitive art has been a great inspiration to him." I glanced down a curving flight of stone steps.
"What's down there? The dungeons?" I thought of the old woman who babysat me at night when my mother was working. She told me once that in the old stories, the monster is always driven out. Our Minotaurs and Calibans live alone in their labyrinths on the edge of town.
"Once upon a time. Now Vita has her studio down there. She says it's 'womblike.'" He turned to me. "Frankly it gives me the willies." He brushed some dust from the shoulder of my jacket and flattened down my collar. Close to, he smelled sour. His breath was stale with cigarettes, and his lips were chapped and stained from too much red wine the night before. His eyes were cold as a pike's as he smiled, and his yellow pointed teeth reminded me of the horns he had worn to the party. "Smarten yourself up a bit, old boy. Lambert still cares about appearances in spite of everything."
In spite of everything? I wondered what he meant, but as we passed an old gilded mirror glimmering darkly above the open fire in the hallway, I paused and smoothed down my hair. I hadn't shaved for weeks, and the dark beard made me a stranger to myself. The silvering on the old mirror had worn away at the back, and I had to weave my head around to find a patch that did not show only a partial reflection.
"Come on, you'll do."
I followed him upstairs and craned my neck to look up at the wide spiral of mahogany banisters disappearing into the darkness of the house like a nautilus shell.
"Most of the château is closed up," Quimby said, padding noiselessly up the stone steps ahead of me in his purple Moroccan slippers. "It's hard to keep these old places going without staff." We reached the first landing. "Right. Wait here. I'll go and wake him." He knocked softly on the double doors at the heart of the landing and opened the door. A draft of warm air breathed across the landing like a sigh, sickly and sweet.
I remember pacing the landing for some time. I could hear Vita knocking around in the kitchen downstairs, the crackling of the fire in the hallway below, and, from within the room, the rise and fall of male voices. I remember thinking it strange that you couldn't hear any clocks ticking. Houses like that always have a few longcase clocks knocking around, marking off the hours. It was like time stopped. I don't know what I was expecting—sure, part of me hoped my father would walk out of his bedroom, bleary with sleep but delighted to discover he had a son. For a moment, I let myself imagine some guy embracing me, holding me at arm's length, exclaiming, "Well, I never!" The bedroom door opened again, and I heard Quimby say:
"Just wait. He's perfect, I promise you." He beckoned to me. "Come on, Lambert will see you now."
* * *
I stepped into my father's room sick with dread. The heavy red velvet drapes were still closed, and the only light came from the smoldering fire in the grate. I blinked, my eyes adjusting to the darkness. My hearing seemed more acute, and I could make out the wheezing breath of a man. With the dust and the smoke, I could feel my own lungs tightening up, I can tell you. I turned to the heavy four-poster, and against the white sheets I began to make out the dark shape of someone. Of my father. The coal of a fat cigar glowed as he inhaled.
"Gabriel Lambert," Quimby said with a flourish, "meet Gabriel Lambert." I felt like he was laughing at me, at us, and my cheeks burned with humiliation.
"Of course your mother named you for me." My father's voice was breathless and harsh. "The woman never got over me. All those letters, Christ." He leaned forward in the bed and stubbed out his cigar in a Chinese ashtray on the bedside table. The firelight flickered over the purple silk drapes of the bed. Still, I could not see his face. "I'm not at all surprised she stole my name for you, too." He laughed bitterly. "All those years, calling herself 'Madame Lambert.'" I blinked. Perhaps it was the smoke from the fire or the dust, but my eyes pricked with tears. "Well, step closer. I can't see you." I felt as wary as Red Riding Hood.
Vita barged open the door and dumped the tray into Quimby's hands, coffee cups rattling. "Good grief, no wonder. You can't see a thing in here with the drapes closed all day." She flung open the curtains, and my father recoiled in the bed, his hand shielding his eyes.
"Not all the way, Vita." She closed them slightly, so that a thin shaft of sunlight cut through the room, illuminating the tangled roses of the faded rug. "Step forward, boy." I moved to the line and raised my chin. I heard him gasp. "Well, well." He threw back the bedclothes, and again, that sweet, heavy scent caught at the back of my throat. Vita chattered on as she set out the coffee on the table beside the fire, but I watched him in silence as he struggled out of the bed. He walked toward me in darkness, a white nightshirt falling to his knees. We were the same height, I realized, the same build, shadows of past and future meeting. "It's remarkable," he murmured. "It's like looking in a mirror."
"It's like you've been reborn," Quimby added.
My father circled me. Unnerved, I kept my gaze ahead, waiting for him to stop in front of me. This he did, and as he stepped into the light, I tried not to cry out in shock.
* * *
By that evening, I had grown a little more accustomed to my father. In the dim gaslight of the drawing room, he sat in a red velvet wingback chair, talking to Quimby beside the fire. The heat was stifling, and sweat stuck my shirt to my back as I laid white plates on the mahogany dining table next door. They gleamed in the shadows like four full moons.
"Syphilis," Vita whispered to me as I helped her set the table. "In case you were wondering."
"Oh God," I said under my breath.
"You're fine," Vita said tetchily.
"I didn't mean that."
"We're both fine." Vita threw down the last of the napkins and stalked toward a large Chinese gong. She plucked up the carved beater and thumped the gong—the note vibrated through the house, through my chest like a heartbeat. I was so nervous, eating was the last thing I felt like doing, and the smell of the mushroom soup turned my stomach. "Dinner is served," she called.
Lambert and Quimby joined us, still deep in conversation. My father leaned heavily on an ebony cane as he walked and barely glanced at me as he sat at the head of the table. Only when he raised his glass in a toast did he look directly at me.
"As Plautus said: 'I wined, I dined, I concubined—'" He broke off into a hacking cough.
"Perhaps good food, good friends, good health, would be more appropriate, my darling," Vita said, handing him a napkin.
The children of depressives are alert to the subtlest shifts in mood—they are very good at reading people. Living with my mother all those years had trained me well, and I learned a lot about my father during that first meal. He hid his bitterness well, but I saw something flicker over his face as he looked at me. Jealousy. I came to the conclusion that the corruption of his skin was the final manifestation of whatever poison twisted in his heart. He must have been quite something when he was my age, all that vanity and greed cloaked in beauty. No wonder my mother with her desperate unhappiness fell for him—she must have realized no one could make her more exquisitely miserable than him. Now, he lolled at the head of the table, his bow tie loose around his neck, like some young buck at a party. I wondered if he forgot sometimes, if he thought he was still as he had always been, forever young. Perhaps that was why all the mirrors in the house reflected darkly or had been hidden away.
My father drained his glass. "To hell with this, we should be having champagne, celebrate the return of my prodigal son." He clicked his fingers at Quimby. "Fetch a bottle, will you?"
Quimby hid his flash of annoyance rather well. I guessed he must be used to it. He dabbed his lips with the heavy linen napkin and scraped back his chair. I heard the sound of his footsteps going down to the cellar echo through the silent house, and the fire crackling in the grate beside the table. My cheeks were burning under the intensity of my father's gaze, but I was determined not to let him intimidate me. Quimby returned with a dusty bottle of Dom Pérignon in his hand. "Shall I?" He began to peel away the foil and loosen the wire cage.
"Do it like a man." Lambert lurched from the table and grabbed the bottle. "Watch, boy," he said to me, lifting a tarnished saber from the mantelpiece. "Always slide the blade along the seam," he said, warming up with a couple of short slides, the metal scraping on glass. "There!" he cried, slicing the neck of the bottle neatly. The champagne gushed, and Vita casually leaned forward with a glass to catch the flow.
"Gabriel?" she said, passing me a glass.
"In one!" Lambert cried. It was the first time I had tried champagne, and the bubbles caught in my throat, making me cough. "Come on, boy, drink!" Red faced, ashamed, I knocked back the wine. "That's more like it." He filled my glass again. "Drink!"
"Darling, stop it," Vita said, reaching out her arm.
"Go to hell." He pushed her away and concentrated on me. "Drink." So I did. I drank the whole damn bottle and I've hated the stuff ever since, but I proved something to him that night.
Vita cleared the plates in silence, her gold shift glimmering in the candlelight.
My humiliation over, Lambert turned his attention to her. "Why are you so tarted up? I can't believe you still have that ghastly dress," he said.
"It's a special occasion."
"I told you when I met you, you don't have the tits for it."
"Why do you always have to be such a shit, Lambert? Why can't you be nice for once?"
"Whoever said artists had to be nice," he yelled after her as she carried the plates to the kitchen. "We're all heartless, selfish bastards," he said to me, "don't think you are any different, dear boy," and drained his glass, catching a dribble of red wine with the back of his hand. He gestured at Quimby to fill his glass and turned to me. "Oh dear. Have I embarrassed you?"
My cheeks were burning, the champagne lurching in my stomach, but I held his gaze. "Not at all."
"It's quite remarkable, isn't it, Quimby?" He staggered to his feet and walked around to my chair, lowered his ravaged face beside mine. "He's identical." I tried to keep my head steady and not to recoil. He was, what—only seventeen years older than me, but close to, I could see clearly the corruption of his face.
"Identical? In your dreams." Vita thumped down a board of sweating cheeses on the table. "How old are you now, Lambert? Thirty-four, thirty-five?"
"I'm thirty, if I'm a day."
"As I said, in your dreams. The boy's mature looking for eighteen, but you...," she said, her voice low. "Do you know, Gabriel, your father is so vain that his new passport and papers still carry a picture from over ten years ago?" She raised her chin defiantly. "And a false birth date by the sounds of it—"
"Go to hell," Lambert yelled. He heaved himself up from his chair and limped outside to the terrace.
* * *
I don't like conflict, never have, but they thrived on it. Half an hour later, they were dancing in each other's arms on the terrace like Beauty and the Beast, moonlight gilding the beads of her gown. Their argument upset me, though, and that night I couldn't sleep. I was jealous, too, I admit it. The memory of his hands on her ate away at me, and I tossed and turned, fitfully, in my bed, the sheets tangled around me. No breeze came through the open window, and the heat and weight of the empty rooms soaring above mine weighed on me like a tombstone. The house seemed alive at night, full of sighs and creaks. The scream of a fox jolted me awake, and I sat bolt upright in the bed, my heart racing. It was hopeless trying to sleep, so I padded downstairs, feeling my way through the dark house. The cool stone wall was rough to the touch. I remembered someone had told me if you were lost in a maze, all you had to do was hold your left hand to the hedge and follow it around. Is that true? In the hallway I paused. A faint, reddish light glimmered from the stairs to Vita's studio. I began to feel more confident at the thought of catching her alone. Perhaps there was still a chance she might want me as I wanted her. Yes, I know, did I have no scruples? She was my father's girlfriend and all that, but I was eighteen, for goodness' sake. I may have looked mature, but I can tell you I wasn't exactly thinking with my brain in those days.
My footsteps on the stone stairs were silent, and I followed the spiral down. I frowned as I made out Lambert's and then Quimby's voice.
"Do you think he'll go for it?" Quimby was saying.
"Of course he will!" Lambert's words were thick and slurred.
"I think you should just do the decent thing and let the boy go on his way," Vita said. That annoyed me—"the boy." Vita wasn't much older than me. It's funny, that throwaway line stuck with me. It always makes me think of that Noël Coward song "Mad About the Boy." You could picture Vita saying it somehow: "My dear, I'm simply mad about the boy." Well, blame it on her or him—maybe that's why I've spent my whole life thinking I'm Peter Pan. In my head I never aged past 1940. A lot of moons, a lot of years, have come and gone since then, but part of me burned as bright as phosphorus that year.
Vita turned at that moment and must have spotted my foot on the stairs. "Gabriel, is that you?"
"I'm sorry, I didn't mean to intrude." I pushed the heavy, iron-braced door fully open.
"Well, don't just stand there, come in," she said. "Whatever you do, don't slam that door, though. The handle's broken."
I wedged the iron bar propping the door open and rattled the handle. "I can fix this for you, if you like. I'm good with things like that."
"See, Vita, the boy's good with his hands." Lambert raised his chin. "Takes after his old man."
"Could you, Gabriel? The whole damn place is falling apart." She cast a surly look at Lambert.
"What do you expect me to do? We're out of cash, dear heart."
"Which is what we were just talking about, Gabriel," Quimby said. There was a sibilance to his French pronunciation that made my skin crawl. When Annie and I took our first grandchild to see The Jungle Book at the cinema, Kaa, the snake, reminded me of Quimby. "Your father has a business proposal for you."
"We have a proposal for you," Lambert corrected. "You'll make enough cash to get out of France, too, Quimby, if Gabriel will help us."
"Help you? With what?" I said.
"Here's the deal." Quimby settled back on the high wooden stool beside Vita's easel and made a bridge with his hands, pointing at me with his index fingers. The way his thumbs were cocked, it looked like a gun. "Lambert will house you, teach you all he knows about art..."
"Which is no more than a father should do for his son," Vita muttered.
"And in return?" I held Quimby's gaze steadily.
"You must agree to impersonate your father."
"That's ridiculous!" I cried. "He's seventeen years older than me."
"Hear us out," Quimby said. "Before the unfortunate effects of Lambert's condition, he was a handsome man—youthful, tall, olive-skinned like you. He wore a black beard, like you, had longish dark hair..."
"Like you," Lambert said. He had the glazed expression of a bored cat toying with a mouse. I noticed when he looked up at me as he leaned in to the candle flame to light his cigarette, his eyes remained black, the pupils fully dilated.
"All we need you to do is meet some rich American clients of mine who are thinking of buying up everything your father has ready. They collect art deco, and already have several of Lambert's best pieces, but they realize this is a buyer's market."
"Fuckers. People like Peggy Guggenheim are just profiteering," Lambert said.
Quimby coughed delicately. "Like I said. It is a buyer's market. Artists are selling at rock-bottom prices, and they want to clean up before they get out of France."
"The thing is, Gabriel," Lambert said, "they want to 'meet the artist.'" He glanced at Quimby. "Clearly they think they will get a better price if they deal with me rather than with Quimby alone. Of course I can't meet them like this." He waved vaguely at his face. "No one would buy a beautiful dream from a monster."
"Hush." Vita reached out and touched his arm, but Lambert pushed her away. I realized then from the look on her face that she loved him, in spite of everything.
"They'll never fall for it," I said.
"I've looked at your sketchbook." Lambert tapped at the dog-eared book on the table beside him.
I grabbed it, held it to my chest. "How dare you go through my things?"
"Your things?" He roared with laughter. "A few pairs of darned socks and a sketchbook?"
"You have no right..."
He sneered, waved me away with a hand that I recognized as my own. "I don't know what you're bothered about. It's not as if you have much."
"But you will." Quimby stood and walked over to me.
"And if you're worrying about being busted by the clients," Vita said, "I used to be an actress—"
"That's one word for it," Lambert said under his breath, lurching back into an old armchair in the corner of the studio. A cloud of dust rose up as he sat, legs spread straight ahead of him. I noticed there weren't any holes in the bottom of his hand-tooled, supple shoes, not like my mother's.
Vita ignored him. She put her hands on my shoulders and led me to the huge mirror she had placed by her easel. "I can help you age your features a little, put some gray in your hair to match Lambert's." My hackles rose as she ran her fingers through my hair, brushed the skin at my temple. It was less the thought of art lessons from my father than the idea of being alone with Vita, her face close to mine like this, that made me agree to their plan.
As I left, I loitered on the stairs and listened to their conversation.
"Do you think he can pull it off?" Lambert said.
"Of course he will." Vita's voice came and went as she paced the studio.
"We have to get to the States," Lambert said. "I've heard of an American 'angel' in Marseille who is spiriting artists to the U.S. We just go and find this guy, and we're out of here."
"Everyone knows you've always been an outspoken critic of Nazism," Quimby said. "Naturally they will help an artist of your stature."
I heard Lambert cough. "Frankly I'm scared for my life."
"Don't worry. The sale of these paintings will pay for your passage to New York," Quimby said.
"And mine," Vita added.
"What about the boy?" Quimby said.
"What of him?" Lambert laughed, a cold little laugh that made my stomach curl up and harden. "It's a little late in the day to be paternal. After we pull this off, he can stay here if he wants, or he can just disappear again. He's nothing, a nobody. The Nazis will leave him alone."
When someone tells a boy he's worthless, that he doesn't count, it can go one of two ways. Either you believe them, resign yourself to second best, to being a nobody, or you decide to prove them wrong. Perhaps that is what I have been doing my whole damn life, trying to prove my father wrong. I wanted to show Gabriel Lambert that I was as good as him—hell, I wanted to prove I was better than him, who am I kidding. Vanity, greed, talent—they were just some of the gifts my father gave me. I have no illusions about my personality. I never did a thing I didn't want to, I was just as driven and selfish as my father. Maybe that's why I did so good a job, how I fooled so many people. There's the same uncompromising clarity in our work—even after all the people who had known him personally died, his work lived on, and not once did anyone question Gabriel Lambert's progression from decadent art deco nudes to big old angry abstracts, not once... well, not till today, this girl. The critics all put it down to the influence of Breton, of Duchamp, to the torch that these European greats passed on to the new generation of American artists that I became part of. They never doubted for a moment, because there is a perfect, true note in our work, like the way a soprano hitting a high C makes your hackles rise and glasses shatter. I became so adept at imitating my father's art and life, now even I don't know where he ends and I begin.
Like him, I have devoted my life to art, have done what I wanted to, needed to, without compromise. This work has, if you like, left the world a better place. Does that make mine a good life? Has everything I have done in the last sixty-odd years redeemed one unforgivable act? Looking back at that boy on the stairs now, I think it's kind of ironic. For once in my life, I did exactly what someone asked me to do. I disappeared.
FORTY-SIX
FLYING POINT, LONG ISLAND
2000
GABRIEL
Sophie stretches out on the sand beside me. "So how did it work, Gabriel? How did this transformation come about?"
"I was a quick learner, it's as simple as that. And I was motivated—boy, was I motivated. I wanted to show that bastard just what I was made of."
* * *
Is it fanciful to say that my father enjoyed my company? I wonder. He had many faults, but he was true to his word, and when I wasn't out fooling people with more money than sense into buying Quimby's stash of my father's—or should I say "our"—paintings, we worked in the studio from dawn till dusk. Lambert's studio was out across the courtyard in an old barn, where they had opened up the north-facing roof for skylights. Week after week, he had me practicing my drawing, turning out little oil sketches in the manner of his work. He spent the days sitting on the sofa, smoking, teaching me everything he knew about art. Around the house he was listless and apathetic most of the time, but when we were working I saw the fire in his eyes flare up, briefly. I saw what he once was and could imagine him as the toast of Paris.
There wasn't any question of asking me what I wanted to paint—when I showed him some of my modern pieces, he tossed the canvases into the corner of the studio. He said: "You are the son of a great figurative artist, and that is what I will teach you to paint." Anyway, I must have done all right because one day toward the end of October I found him pacing in front of the piles of drawings and paintings I had done. He had them spread out on the big oak table he had by the barn door. I watched him for a while, saw him pause beside a painting I was particularly proud of, a copy of one of his early works, so good that even Quimby couldn't tell I'd done it.
"Not bad," he said without turning to me.
"What do you mean, not bad?" I said, laughing. "Quimby said it was perfect."
"Quimby's a fool," he muttered, and pointed upward. "There's only one source of perfection in the world, don't you forget it." He turned to me, but he was looking past me toward the courtyard. "Sometimes he gets it just right." I followed his gaze and saw Vita doing her calisthenics, turning cartwheels across the yard. "Right, you're ready," he said to me as he limped to the door. "Vita!" he yelled. "Vita." Moments later she ran into the studio.
"What is it? Are you all right?"
He tossed aside his cane and gathered his purple dressing gown around him as he fell back onto the sofa. I saw him wince with pain. "It's time to teach this boy how to paint properly from life. I don't know what kind of nonsense they've been filling his head with in Paris, but we shall sort him out." Vita glanced at me apologetically. "You will model for him."
"Oh." I saw her cheeks flush. "I couldn't, I mean, I—"
"My dear girl," he said, "you're surely not going to play the blushing ingenue at this late stage." He seized his cane and lifted the edge of her blouse with it. "Come along, off with it." Vita looked horrified.
"That is no way to speak to your muse, dear heart," she said, trying to make light of it.
"I'm sure we could find another model, p-perhaps a girl from the village...," I stuttered.
"N-n-nonsense!" Lambert said, imitating me. "A girl from the village? That's like asking a donkey to run the Grand National." He glared at me with those black eyes of his. "If we are to complete one or two of my unfinished paintings, you need to work with the woman who inspired my finest creations."
"It's fine, really." Vita glanced at me and shook her head as if to say, Stop complaining. I realized if we made too much of a fuss, my father would know there was something between us.
Was there? I've often wondered that over the years, whether Vita felt anything for me, anything at all. I know she loved Lambert, but there was something in her face, when I caught her looking at me sometimes. I don't know if she wanted me or just wished Lambert were still whole and healthy like me. Perhaps I was just a reminder of what might have been. All I know is my heart was in my mouth as she disrobed behind the screen in Lambert's studio.
"Put some music on, boy," he said, dragging at a cigarette.
"What did you mean, about the unfinished paintings?" I asked him as I flipped through the stack of albums beside the gramophone. I selected Debussy's orchestration of Satie's Gymnopédies. I'd heard Vita listening to it one night as she sat alone at sunset, looking out over the hills. It suited her somehow—gentle and beautiful. It suited the melancholy sadness in her that she tried to hide.
"You didn't think I was teaching you everything I know out of the kindness of my heart, did you?" My father's eyes glimmered. "Quimby's nearly sold all my finished work, and all the last editions of the prints. I haven't had the energy to work for some time, and he reckons he can sell anything I can give him at the moment. We'll start with that." He gestured toward a large canvas leaning face-to against the wall.
"No."
"What do you mean, no?"
"Passing myself off as you is one thing," I said, struggling, "but to forge your work as well?"
"What on earth do you think I've been teaching you for? Stop blathering and listen. You have to talk convincingly to the buyers about the work. It's not enough that you look like me, you have to be me." He leaned forward and whispered, "You will do exactly what Quimby and I tell you to. What if we were to tell the police that some young con artist has been impersonating me, selling off stolen work..."
"You wouldn't."
"Try me." I carried the painting he pointed at to the easel and pulled off the dust sheet. My lips parted as I stood back to inspect it. "I've done the hands and face, as you can see. The rest should be easy enough for you." Vita's face gazed out of the painting. Her life-size body, arms raised sinuously above her head, her legs at full stretch, on tiptoe, had been sketched in.
"It shouldn't take too long for you to finish, a week, perhaps." A week? A week of looking at Vita, in this pose? I didn't know whether to laugh or cry. It was exquisite torture. "Any fool could finish the background," he said. There was the suggestion of a familiar Lambert motif in the background, an art deco design like a metallic mosaic. His work had all the glamour of Tamara de Lempicka, but there was something else to it, a raw sexuality. "It's the line of her that you must be careful with." He reached out and slapped Vita on the backside like a racehorse as she walked past. "That is the true signature of a Lambert painting."
"Oh God, not this one," Vita said. She paused in front of the canvas and wrapped a faded floral kimono around herself. "I'm sure he's a sadist," she said. "I had the most awful cramps in this position, but would he stop? Would he hell." She stalked away and unwound a rope tied to the beam at the side of the podium in the studio. "And it's bloody freezing in here. If you're going to have me pose all morning, at least light the sodding stove." I rushed forward and clumsily stuffed balls of newspaper and kindling into the wood burner. I struck a match, and as the flame caught, I looked through the dancing light to where Vita stood, winching down a small noose.
"A little higher," Lambert said. His voice was low, and I felt my cheeks burning. I tried to busy myself stoking the fire, but I glanced at him. His gaze followed Vita as she climbed onto the podium. She slipped the kimono from her shoulder, her gaze locked on his. It was as if she were stripping for him alone, and I felt like a voyeur. I could hear him breathing as I walked around to the easel and sorted through the brushes. When I looked up, she was quite naked, her hands looped into the noose for support as she took on the pose in the painting. The heat of the crackling fire, the buzzing of the cicadas in the grass, and the bees muzzing the lavender outside made my senses vibrate. The smell of turpentine and oil paint inflamed me. I turned to her, my brush in my hand, and tried to forget it was Vita. Tried to forget that I had kissed that rib cage slick with sweat, felt her move beneath me. "Perfect," Lambert said softly. "Perfect."
And she was. The truth is that painting had all of us in it—my father, Vita, and me. That's its complexity and beauty. Lambert finished it—two tiny dots of white on her pupils breathed life into the whole thing, like the touch of a god. That painting turned up at Sotheby's in New York not so long ago. It had stayed in the family of the old guy who bought it in 1940 all these years. I don't know how anyone could have let it go. I went into town to see it one last time. The director is a friend, so he let me in after hours to see my painting—our painting. They had it spotlighted at the center of the gallery, up against a black velvet wall, the heart of the collection. I recognized the work of the framer—he'd done a good job, a deco key design picked out in silver against the gilding, echoing Lambert's motif. The painting had "it," that something that makes a work of art unforgettable. I brought my old Sony Walkman with me and listened to the cassette of the Gymnopédies, rewinding it over and over again as I sat in the dark gallery alone with her. The music, the plaintive oboe and soft strings, worked its magic. It was like having Vita in the room with me. The line, the pigments, shimmered with Lambert's genius, my teenage lust, but more than anything with her, with Vita. Looking at her in the peace of that gallery, I wasn't an old wreck of a man, I was a young boy again, and I wanted it, I wanted the magic. I could have bought the painting anonymously, of course. I won't pretend I wasn't tempted, but I couldn't have it in the house I share with Annie. It would have felt unfaithful.
* * *
The last time I'd seen it was when Quimby returned to the Château d'Oc at the end of October to collect the paintings. I don't think I've ever hated anyone more than Alistair Quimby, I tell you. I stayed out of his way, as much as I could. That is, when he hadn't got me dressing up as Gabriel Lambert to sweet-talk some old fool of an art collector into handing over more than they should have for one of the paintings.
"Shouldn't these be signed 'studio of'?" I said to him as we loaded the last of the canvases into the back of his old gray Citroën van. I tucked a blanket around the end of Vita's painting as if I were swaddling a child.
"Nonsense, dear boy," he said, tightening the straps around the blankets so the paintings stayed upright as he drove. Quimby slammed the door shut. "Think of Gainsborough, Rubens, do you really think any of the great prolific artists really painted all their own work?"
"It's forgery."
"No, it's not."
"Well, I bet Gainsborough never had his son impersonate him."
"Hush, dear boy," Quimby said, patting my hand. He stepped a little closer. "Don't pout. You know, you should have some fun. All work and no play—"
"Quimby." Lambert's voice was harsh. Quimby stepped away and slid a pair of round rose-colored sunglasses down from his hair.
"I was just thanking young Gabriel for all his work."
"Hadn't you better get a move on? It will be dark before you reach Marseille."
"Yes, yes." He shook Lambert's hand. "I'll send for Gabriel once I've secured buyers."
"This will be the last time, won't it?" I said.
"I said so, didn't I? I heard Peggy Guggenheim is buying anything she can get her hands on for a good price."
"She's never bought any of my work," Lambert said.
"There's always a first time." Quimby turned to me. "You've done a good job. No doubt now you want to get back to your own work?" I caught the smirk on his face and thought uncomfortably of how Quimby had surprised me in Vita's studio the night before, how the flash of his camera had cut through the darkness of the cellar. Vita was sitting on the stool beside her easel, watching me paint one of the abstracts I was working on. They were never Vita's paintings. They were mine. Lambert pushed past Quimby as he took the photograph, lurched toward my work. His ghastly face, twisted with laughter, reflected in the mirror side by side with my own. Click, the camera went. I remember the pop of the flash, the smell of it. Even to this day, the memory of their laughter, their ridicule, makes the bile rise in my throat. That's what I saw in the photograph Sophie showed me—not Vita, not the paintings, but the truth about us, about Gabriel Lambert. The two of us, side by side. Past and future. Both of us. And the girl knows, I'm sure of it.
* * *
I found Vita curled up asleep in the armchair in her studio that evening. I didn't want to wake her and began to step away.
"Lambert?" she mumbled sleepily. "Is that you?"
"No, it's me."
"Oh." She rubbed her eyes and yawned. "It's getting harder to tell you two apart. Even your footsteps sound the same now. Or at least, they sound how his used to. If you know what I mean."
"Have you been working?"
Vita laughed. "Trying to. It's hopeless. I've come to the conclusion my destiny is to be a muse rather than an artist in my own right." She gestured languidly at a pile of torn sketches littering the floor. "Lambert says I have failed to progress, and he's right, damn him. Perhaps I shall return to the stage...."
"I thought you might like to go down to the village, for a drink?" I said quickly, before I lost my nerve. She looked so beautiful, curled up on the chair. I longed to kiss her again. It was like some kind of divine joke that I had spent my days, weeks, forced to look but not touch, to study every inch of her in detail under the watchful eye of my father. I could draw her still, if I wanted, in absolute perfection.
"No, I don't think so," she said. She stretched as she stood and looked into my eyes, blinking once, twice, like a cat. "That's not a good idea."
"It's just a drink."
"We both know you are asking me for more than a drink."
"Vita," I said, desperately reaching for her. Before I knew what was happening, my arm was around her waist, my mouth searching for hers.
"Gabriel, stop it. Stop it!"
I backed away, my heart thumping. "I'm sorry. Please forgive me, I... I've never felt like this about anyone before." Frustration balled in my chest, I could hardly breathe. "It's just being near you, all the time—I can't, I can't bear it."
She pulled her embroidered shawl around her shoulders and crossed her arms. "You dear, sweet boy."
"I'm not a boy—" I said, my voice breaking.
"Shh! Lambert will hear you." She stepped closer and kissed me on the cheek. "Gabriel, I'm very flattered, and it's understandable that you feel confused, after what happened at the party, but the thing is, I love Lambert. He's a grade-one shit in many ways, but I love him, and I can't leave him, not when he is so ill." She took my arm. "I don't know how long he has left, but I want to be there."
"I understand." I didn't, of course. I was eighteen and randy as hell. I couldn't for the life of me see how a woman like Vita could choose Lambert over me.
"I'm not feeling so great myself," she said. I noticed for the first time the dark circles under her eyes.
"You don't think...?"
"No, of course not. It's just some bug or rotten food, or something." She shook her head and sighed. "Of course it's not syphilis. Lambert and I have always been careful. We've never been lovers, you do know that?"
"No?"
"He was already ill, when we met." She glanced at me as we started to climb the cellar steps. "His wife, Rachel, infected him." She pursed her lips. "She must have been a piece of work, spoiled bitch. Imagine being unfaithful to a man like Lambert? No wonder she killed herself when she found out what she had done."
"She did?"
Vita nodded. "Lambert was in pieces when I met him. She'd just died, and he couldn't work. He knew what was coming, too, how ill he was going to become."
"And then you came along."
"Then I came along." She smiled sadly. "I asked him to marry me, you know, but he won't. He says he doesn't want me to be tied to a monster."
"I'm sorry," I said, and meant it. Now I knew why she was so unhappy.
"Your father was... is a remarkable man. Never forget that."
"But isn't it difficult, for a woman like you, I mean?" My chest was tight with jealousy. "What do you get out of this? Don't you ever—"
"Need someone? It was part of our agreement, that every so often I could have a little 'adventure' if I felt I needed it."
"He knew?"
"Of course he does. I'd never go behind his back." She squeezed my hand as she turned to go upstairs to bed. "He knew where I was that night, but not who I was with." She looked into my eyes. "You turned out to be more of an adventure than most, Gabriel. Lambert must never know about us. It would break his heart."
* * *
I've never been a good sleeper. Overactive imagination, that's what Annie always says, but I've never slept worse than I did in my father's house. I remember lying in bed one night in October, thrashing around like a sprat in the bottom of a bait bucket trying to get comfortable.
I couldn't stand it anymore, thought perhaps I'd be better after some fresh air, so I pulled back the blankets and struggled into my clothes. I leaned against the wall of the staircase as I stumbled downstairs. The fire in the hall cast long shadows, steps zigzagging up the curving wall like teeth. I was heading for the kitchen, but then I heard Vita and Lambert arguing down in the cellar.
I know, I know. I should have gone on, ignored them, but I could hear her crying, and I was worried he was drunk and might hurt her. None of it would have happened if I had just gone on walking out into the night and caught my breath under the cold sky of a thousand stars. But I padded downstairs.
"Whose is it?" he was yelling.
"I don't know." Vita was sobbing now.
"Tell me." I heard Lambert throw something across the room, the mirror above her desk shattering. "This was never part of the deal, Vita. I don't want to be a father. Children suck the bloody life out of you, and leave you a husk. I can't paint around a screaming infant...."
"You can't paint anyway!"
"Go to hell, Vita."
"You don't have to do anything," she said. "You'll never see it."
"You have no idea! The noise, the squalor. You'll get fat and..."
"And what, Lambert? Hideous to look at?"
"That's not fair."
"Do you think I love you less, for what you are?"
"I just want one part of my life to remain perfect. One part. Is that too much to ask?"
"I'm not a sculpture, or a painting. I'm real, and I'm alive, and I just want a child so that when..."
"When I die? Is that it?"
"I don't want to be alone, Lambert. It would be like having a part of..."
"What?"
"Nothing."
The silence pushed me back against the wall of the cellar stairs.
"Oh, Christ, no... Him?" Lambert said. "Not him."
"Lambert..."
"In my own house, with my son, how could you?"
"It's not what you think!"
"I'll kill him!"
"Lambert, no, please—" I heard the sound of his hand hitting her, the thud of Vita hitting the floor.
"Leave her alone!" I yelled, my hand clutching at the wall.
Lambert knelt at her side. "Vita... Vita..." He cradled her head. Blood was already trickling from her ear.
"What have you done?" I cried.
"She fell, hit her head on the table." I glanced across. As she had fallen, a candle had knocked over. Flames were beginning to lick the red velvet drapes behind her easel. As usual, the place was littered with cloths smeared with oil paint and meths. An arsonist couldn't have done a better job. I winced, the smell of smoke filling my nose.
"Lambert, you have to get—"
"You, you little bastard," he said, struggling to his feet. "After all I've done for you."
"It wasn't like that—I didn't know," I said. "Vita didn't—"
"You thankless, heartless piece of shit." He shoved me hard, back against the door frame. I felt the metal rod Vita had propped the door open with fall away as I stumbled. I looked beyond my father, to where the flames were leaping up the drapes. Vita lay on the floor, her pale arm extended toward me. I felt like I was about to pass out.
"Lambert, you must—" He thumped me then, and I fell back onto the stairs. I saw him reaching for the door.
"Go to hell."
"Lambert!" I shook my head, the blood shrill in my ears from the blow. I jumped up as he went to slam the door, grabbing the handle on this side. The handle that I had said I'd fix but hadn't. I pushed against the door with all my strength, keeping it open.
"Get out of here," he said. "I swear to God, I'll kill you if I see you again."
I felt my feet slipping, skidding on the stone floor as he pushed against me. My arms were shaking as I clung on. "The fire," I gasped.
"Get out!" With a final shove, the door slammed shut, and I fell backward.
For a moment there was silence. I imagine he must have turned and seen the fire rising, licking the ceiling. I heard him cry out, then the handle rattled.
"Fire!" He thumped on the door. "Fire! For God's sake, the fire! Get us out... get us out of here!" I tried the door. The lock was jammed fast.
"I can't... I can't open..." Bright lights danced before my eyes.
"Oh God, oh God," I heard him cry out. Smoke snuck beneath the door. I threw myself against the heavy wood again and again, but the door wouldn't budge. "Help us!" he cried, coughing and choking on the smoke. I pulled open the tiny metal grille in the dungeon door and looked in, terrified by what I might see. His eye appeared there, briefly, blocking the light of the wall of flame, and then his fingers snaked through the grille. I reached up and touched him, repelled and afraid.
"I'll go and get help," I said, but we both knew it was hopeless. There was no fire brigade here, no way to pump water down, no locksmith who could cut them free in time.
That image has stayed with me my whole life, his fingers touching mine—again, I think it was the only time we touched. Like God giving life to Adam in the Sistine Chapel. The deaths of Vita, of our child, of my father, marked the end and the beginning of my life. I've asked myself if I could have saved them somehow, whether Vita knew or if she was already dead. For sixty years I have blamed myself. If only I had fixed the damn door. It's the smallest things blindside us and change our lives forever.
"Go," my father said. He took away his hand and stepped back to meet his fate. I peered through the grille, saw him take Vita in his arms, cradling her. "Go!"
FORTY-SEVEN
FLYING POINT, LONG ISLAND
2000
GABRIEL
I lie back on the sand beside the girl. "If only..."
"What happened next, Gabriel?" Her voice seems far away.
"I don't... I don't know. I must have collapsed, outside the château somewhere. The next thing I knew, I woke up in bed."
* * *
"Monsieur Lambert?" I could hear the woman's voice, but it seemed to be floating toward me out of a thick fog. "Monsieur Lambert?" I sat bolt upright in the bed, gasping for breath.
"Hush," she said. Dry old hands pushed me back against the soft pillows. My shoulder was throbbing, and it was agony every time I breathed in. "The doctor thinks you have cracked a couple of ribs," she said. "Try and lie still."
"Vita...," I said, my voice a thin rasp.
"Oh dear," she said. I tried to follow her dark figure in the shadows of the lamplit room. I was in a single bed, the old brass frame gleaming in the firelight. The room smelled of violets—the stale, sweet air laced with the tang of cat pee.
"Where is Vita?"
"Oh dear, oh dear..." She tucked in the lace-trimmed sheet. I recognized her as the old widow who lived in the gatehouse near the entrance to the Château d'Oc. She had been a sculptor, I believe, and several figure studies were ranged along the mantelpiece, their shadows flickering up the wall like a Greek chorus in the firelight. A ginger tabby slipped unnoticed into the room and curled up on the pink chintz chair by the hearth.
"Please, I must know." I took her hand and saw that mine was bandaged. Dark blood had dried on the ridge of my knuckles, and my nails were torn and dark. Her hand was gnarled with arthritis, twisted and bulbous as an old tree root. I remembered then that Vita told me this was how the woman had ended up in the Languedoc, unable to work any longer. She said that my father let the widow live there rent free because she had been married to one of his old teachers.
"Monsieur Lambert. I am so sorry, your wife..."
She had mistaken me for my father. "You don't understand, I am not—"
"I do wish there was someone I could call, to be with you?" I shook my head. "There are friends, perhaps? I don't feel you should be hearing this from a stranger... well, a neighbor, though we've never met." She looked away again, blinking. "I recognize you, of course, from the newspapers. You haven't changed at all, you are just as my dear Philippe described. He always said you were a remarkable boy, such talent—his star pupil...." I stared at the ceiling as she chattered on, trying to collect myself. "I am so grateful for your kindness," she said finally. "After Philippe died, I didn't know what to do."
"This house is your home for as long as you need."
"Thank you. I am so glad to meet you at last, to say that face-to-face. Of course, I understand that artists need solitude, and I never wanted to intrude. Your wife was always quite insistent about that. I assume Vita was your wife? And your son..." She looked away. "I don't mean to pry. Vita told me your son had come to visit. I hoped to have the chance to meet him, but I know you value your privacy at the château, and I didn't want to pry. Oh dear, oh dear..."
I flinched. "Are they dead?"
"Oh dear," she said again and again, fussing over the blanket. "My condolences, Monsieur Lambert. They took away two bodies this morning. They were too... They asked me who they were. I knew you, immediately, of course, and told the police the only other people at the château were your son and your wife. I saw Monsieur Quimby leave some time ago, that's what I told them. I knew it was just the three of you at the château. Not that I pry." I turned my head away, tears pooling in my eyes. "By the time the fire brigade arrived from the town, the fire had swept through the lower floors of the château. They found you on the steps to the courtyard, half-dead from the smoke. Of course, I offered to take you in. The doctor said you must have complete rest—"
I pushed back the blankets and staggered to my feet. My right arm was in a sling, and my ribs were bandaged. I caught my reflection in the oval mirror over the dressing table and saw my right side was a livid purple, bruises seeping beneath the white bandages. "I have to go back," I said.
"Are you sure that it's safe, Monsieur Lambert? The authorities will have to have reported the fire." She lowered her voice. "If the Nazis were to hear that you have been hiding here..."
"I haven't been hiding," I said, shrugging on my father's identity like my old jacket. "This is my home."
"Of course, of course." I could see she was thinking as I dressed. "Monsieur Lambert, I have heard rumors." My breath caught in my throat. What was she talking about? Had she glimpsed my father? She knew the truth, she had just been humoring me.
"Rumors?"
"Your satirical cartoons are well known in certain circles. Why, Philippe had several framed in his study in Paris," she said. "France is no longer safe for people who have spoken out against fascism. Of course, they were not published in your name, but people gossip. Philippe worried that you had lost your way, artistically. Success can do that, sometimes."
"I never stopped working, he needn't have been concerned," I said, struggling into my shirt. The woman untied the sling, and I winced as I slid my arm into the sleeve.
"You are a prominent artist, Monsieur Lambert," she said, doing up the buttons. "It may be wise for you to leave now, if you feel you can travel."
I was intrigued. This was a side to my father I hadn't seen. All the time he had been hidden away in his castle, he had been penning anti-Nazi cartoons. No wonder he was worried.
"I've heard many of the artists are flocking to Marseille," she whispered, tying the bandage at the nape of my neck. "They say there is a man there, an angel from America, who is spiriting people out of the country. There is a man in Arles you can pay fifty francs to find out his name and address." She found a scrap of paper and wrote down the name of a café. "He is there each day at twelve P.M., apparently."
"Then you must come, too, madame." I thought of the bidons of gasoline Quimby and Lambert had been stockpiling in the barn and Vita's pretty little red Peugeot 202 cabriolet. "I shall drive us to Marseille," I said with more confidence than I felt. The most I had done was drive the car around the courtyard with Vita yelling instructions.
"No, no. They will not bother me and Artus." She beckoned to the old cat. "We have done nothing of importance."
"Thank you, madame," I said, and walked toward the door as weak and gangly as a newborn foal.
"Be careful!" she called. "Promise me you will rest once you are in Marseille!"
* * *
The house was silent in the twilight. I realized I must have slept through a whole day. The smell of burning hit me the moment I walked through the gate in the wall. Furniture had been dragged out into the courtyard, where it lay dark and sodden. Lambert's papers and journals caught at my feet, blowing limply in the wind like wet autumn leaves. I bent down, wincing, to pick up one yellowed journal, and peeled back the pages, stopping at a cartoon of Hitler as the Grim Reaper, presiding over a marching army of skeletons. The fluid line of the drawing was unmistakably my father's, but it was signed only with a cartoon of a feather—an Egyptian glyph. Years later, when I remembered this and looked it up, I discovered it was the symbol of Maat—of truth, justice, and balance.
I hadn't the maturity then to look beyond the surface of what my father had become. I saw a ruined man rather than a loyal husband, a good friend, an artist of conscience. Once in a while his drawings come up at auction, and I add them to my collection. If I had stopped then to look at the piles of journals and magazines littering my father's house, I might have seen a different side to him, but my chance had gone.
I staggered as I climbed the kitchen steps. Weak light leaked through the narrow windows of the house. I paused at the steps to the basement and listened to the silence, the slow drip of water from the ceiling. There was nothing for me here, now, I realized. In the hall mirror, I looked at myself, at my dark beard trimmed to match my father's, at the gray hair Vita had bleached at my temples.
"I am Gabriel Lambert," I said aloud, my voice echoing through the house. It was as if by naming it, by naming the monster that stood before me, it would come alive. I padded up the staircase, wheezing. My lungs were never the same after that fire. Aside from the stench of smoke, the fire had barely touched the first floor. I went first to my own room and lit a fire in the grate. Into it I threw every last scrap of my old life—my clothes, papers, my sketchbooks, even. Gabriel Lambert Jr. was dead. The only reminder of his life was the name on a death register somewhere, together with Vita's, forever. Next, I searched my father's bedroom, found the key to his desk. I pulled down a soft leather Gladstone bag from the top of the wardrobe and tossed it onto the bed. It was filthy—my fingers were coated with dust. I remember turning them over in front of my face, the bandages black with ashes. Whose hands were they? From the desk I took everything I could find of value—cash, the deeds to the château, bankbooks, his papers and passport. Sure enough, just as Vita had said, the photograph was at least ten years old. The face that looked back at me was my own.
FORTY-EIGHT
MARSEILLE
1941
GABRIEL
"Who are you?" Annie said.
"I'm sorry," I said, over and over. I'd imagined, sometimes, it would be a relief to be unmasked, for the whole story to come out, but God, the guilt was unbearable; it still is. What if it was my fault? I've asked myself that time and time again over the years since Quimby put the seed in my mind. It's funny how a single comment from someone can burrow down inside your mind like a parasite and blight your whole life.
All I cared about now was Annie. "You see?" I said to her. "I can't marry you, not while we are in France. If I did, and Quimby told the authorities who I really am, then you would be classed as Jewish, and I'm terrified what they would do to you. My father's parents were Catholic, but my mother was Jewish—if we marry, then the statute will apply to you, too."
Annie rocked on her heels, biting her lip. "Who are you?" she said. "Who are you?"
"Please, don't leave me," I said, hanging my head in shame. That's when she slapped me, hard.
"You lied to me."
"I know."
"Look at me, Gabriel." I could see the fire in her eyes. "Give me one good reason why I should forgive you."
"I love you, and I'll never lie to you about anything again as long as I live." I've been true to my word, that I can tell you.
Annie pulled herself up to her full five feet two inches. She smoothed down the collar of my jacket. "Let me get this straight. I thought I had been seduced by a thirty-something-year-old successful artist, but I've really been made love to by his penniless eighteen-year-old son?"
"It sounds terrible when you say it like that."
"At least Papa will be relieved about one thing. He said you were too old for me."
"You mustn't tell him."
"Why on earth not?"
"Varian and everyone at Air-Bel will know."
"But why should they care? They are your friends. It's wartime, Gabriel, thousands of people are living under assumed identities." She thought for a moment, and I saw her face cloud as she realized. "Oh God, you're leaving, aren't you?" She began to back away from me. "That man Fry's helping you get your papers... your father's papers. You've been planning to leave, all this time?" Her voice rose in despair. "How could you?"
I grabbed at her wrist. "You don't understand, I want you to come with me. They mustn't know who I am. They can help me get papers for you, and we can marry in America, I'm sure of it. Quimby has taken all my money, but I'm going to ask your parents to pay for your passage—"
"To America? Are you mad? As if they'd let me go, let alone marry you. And anyway, they have no money. Haven't you seen our house? Papa hasn't worked for months. They had to sell everything. If it wasn't for the pittance we make from sewing, we'd have starved."
"I didn't know," I said, and my head dropped. "I'm sorry."
Annie covered her ears as another train thundered overhead. Once it had passed, she took my face in her hands. "I love you," she said. "I will always love you, but the odds are I will be rounded up with all the other Jewish families, and who knows how long I have."
"No," I said, my throat thick with tears.
"It is what it is, Gabriel. I'll do everything I can to fight back, but you..." She paused and kissed my eyes, my cheeks. "You have a chance," she said, her voice little more than a whisper. "I want you to go, and live a wonderful life, Gabriel, for both of us. I want you to go to America."
"I won't leave you."
"Yes, you will," she said firmly. She stared at me in the way she's looked at all our children since when she's going to make damn sure they do what she wants. "You are Gabriel Lambert. You are going to America."
We walked up to Air-Bel hand in hand, in silence. I felt like that fellow Varian told me about—Sisyphus, struggling under the weight of all my guilt and sorrow. The lights were on in the house, in spite of the early hour.
"Looks like they've been having a party as usual," Annie said, and we stepped through the door. A man in a dazzling white sheepskin coat stood next to a pile of canvases strapped together. He reminded me of a bird of prey with his flash of white hair and those sparkling blue eyes of his.
"How do you do?" he said, stepping forward to shake my hand. I saw him eye Annie appreciatively as he kissed her hand. "Max Ernst."
"Ernst?" I said. "How marvelous to meet you. I am a huge admirer of your paintings. I'm Gabriel Lambert, and this is Marianne Bouchard."
"Ah, you are Marianne," he said. "Yes, I can see why your father might be worried."
"There she is!" Monsieur Bouchard marched through from the kitchen. "Where have you been hiding her? I searched the grounds, the house." He grabbed her arm, but Annie struggled free. "Did you think I wouldn't notice you were gone? You left your window on the latch and it was banging in the wind. Your mother and I have been worried sick."
"I'm sorry, Papa."
"I suppose you have been with him," he said, pointing at me. "We forbade you to have anything to do with him."
"I assure you, Monsieur Bouchard, Marianne has been perfectly safe," I said.
"Papa, I'm here now," she said, and glanced at me. "Besides, Gabriel is leaving for America soon. You will have nothing to worry about then." She ran toward the door. "Good-bye, Gabriel. Good luck."
"Annie, wait—" I ran after her and caught up with her on the drive. I took her in my arms, but she wouldn't look at me. Moonlight gilded her hair, her skin silver. "I love you," I said. "This isn't the end." I heard footsteps behind us.
"Gabriel." Varian reached out and took hold of my arm firmly. "Let her go." He glanced back at old Bouchard. "This will go badly for us all if you make a scene."
I stepped away. "Please, Annie—" Bouchard shot me an angry look as he dragged her away, but it was tinged with relief.
No, no, no, I thought. It would never be good-bye, not with Annie and me.
"Courage, mon frère." Max clapped me on the shoulder and steered me back toward the house. "One must sacrifice everything for love."
"That's just what I was thinking," I said, looking back over my shoulder, watching Annie and her father disappear into the dark night.
FORTY-NINE
VILLA AIR-BEL, MARSEILLE
1941
VARIAN
Spring came early that year at Air-Bel. Varian sat high in one of the plane trees on the terrace, his back against the trunk. The plum trees were in blossom, and irises poked through the sodden leaves beneath them. From the pond, where he had caught the last of the fish a few months ago, toads croaked, spawning. He could see a female lying belly-up, dead, after releasing her strings of eggs.
Varian glanced up into the tree as he heard a lark sing. It was as though the house were awakening from the winter. Lizards basked on the warm stone walls again, and a magpie rooted through the leaves of a yellow fuchsia on the terrace. He laid his head back against the bark and imagined the sap rising through the tree like a pulse.
He stretched out his hand and spun the small framed ink drawing by Wifredo Lam that he had just suspended from the branch, ready for the auction. The colors glinted in the morning sun like jewels. Through half-closed eyes, he gazed out to the Mediterranean. This is the kind of place you could live forever, he had written to his father the night before.
He glanced up as he heard the kitchen door open. Madame Nouguet was pushing the gardener out of the door, looking nervously around her. He was tucking his shirttails into his trousers and pulling up his braces. Madame Nouguet's cheeks were flushed, her hair worked loose. Varian smiled as the man leaned in for one more kiss, and, swept away, she threw her arms around him. They broke apart, and the gardener stalked off toward the greenhouse, his shoulders hunched. The passionate affair between the cook and the gardener was the talk of the château. Mary Jayne was scandalized—she could not understand what the normally prim cook saw in the man. "He smells of compost," Varian overheard her say one night. He thought of Cupid, bow taut, arrow swinging blindly. Who else will he hit? he wondered as he scrambled down from the tree. It's like spring sickness. Even the two rabbits they had brought back from the country for breeding for the pot had just had their first litter.
Varian watched the gardener amble off toward the vegetable patches, where they had planted string beans, radishes, tomatoes, and lettuces. The thought of the vegetables made his mouth water. I must have lost twenty pounds, he thought, feeling his waistband sag as he put his hands in his pockets.
A group of figures made their way up the driveway, ready for the Sunday salon. The doors and windows of the château were flung open to the balmy air that morning, and the celebrations moved outdoors. Already, it was impossible to imagine the biting cold of the winter. The breeze on his face, on the skin at the open neck of his shirt, was like a caress. He was glad to be back, to be home, he thought. It felt like these gatherings were running on borrowed time. The artists would be leaving soon, Gabriel Lambert and the Bretons among them, he hoped. His face clouded as he thought of Breitscheid and Hilferding. After the news of their arrests was confirmed, he had grown more determined to get everyone else out safely.
At least Monsieur and Madame Bernhard are safely on their way now. Varian rubbed the bridge of his nose as he thought of all the plans they had tried to get them out of the country. Now, at last, another contact had come good and the Bernhards were being taken via underground routes and hiding places through Spain to Portugal.
Varian raised his hand as he saw Gussie walking across the terrace. He jumped down from the tree. "Morning, what a fine day."
"I think I have some news that may make it an even finer day for you," Gussie said. They stepped to one side to let a crowd of artists past, and Varian leaned against the ivy-clad wall.
"So?"
"You know our friend Mr. Allen was heading up-country to get an interview with Pétain?"
"In his dreams. No one can get close to Pétain."
"Well, according to some of the journalists I just bumped into in town, our friend crossed over into the Nazi-occupied zone without permission."
Varian raised his eyebrows. "Did he now? Not a terribly good example for a man who is supposed to be heading up a relief center. What have they done with him?"
"He's been arrested. The best he can hope for is that they'll trade him for one of the Nazi journalists later on." Gussie winked at Varian and walked on. "He's out of your hair, at least."
Varian clenched his fist and pumped his arm in the air. The sun seemed brighter suddenly, the colors of the garden more vivid. Men whose names he had known only from books on art, literature, politics, greeted him as an old friend on their way to pay homage to Breton. Varian felt like the ringmaster of a circus, keeping the house in check. I wish Miriam could have got back here to see all this, he thought. She'd have got such a kick out of it. If only their visas had panned out. I hope they get down to Lisbon some other way. He thought guiltily of Mary Jayne, how she was missing this, too. But she'd made her choice. If ever Cupid was blind— His thoughts cut off as Aube wove around his legs and ran toward the lawn, chasing Clovis. Rose and Maria raised their arms high, the glasses and bottles tinkling on the trays they were carrying. They set them down on the trestle tables set up on the terrace.
"Thank you, girls," Varian said, reaching for the corkscrew.
"Will that be all, sir?"
"Thanks, we'll be fine." It was the servants' day off, and Varian always had the impression they couldn't get away fast enough from the Sunday parties. Maybe they're afraid. He'd never been able to get over the sense of being watched, not since the Sinaia arrests. The sun gleamed on the glasses, the bottles glowing ruby in the morning light. Let them watch, he thought as the cork slid out of the bottle with a mellow pop, and he raised the bottle of red wine to his lips.
"Santé," Jacqueline said. She walked with her arm around André's waist, his hand upon her shoulder. She took two glasses from the tray and poured a glass for them.
"Do you think the auction will be a success today?" André asked Varian.
"There are some remarkable pieces. I'd love to buy them all myself, if the ERC hadn't stopped my salary." At least that should change now Allen is out of the picture. Varian gazed up at the canvases strung from the trees. "There should be a good turnout for the Ernst exhibition alone."
"Is Peggy coming?"
"I'm not sure. She said she'd be back soon."
"I have a feeling her interest in Max is not entirely professional," Jacqueline said. "Once she sets her sights on a prize she won't give up until she's won it, or bought it. She'll be back, if I know Peggy."
* * *
There was no food that day, but there was wine and an infectious sense of joy and optimism. Varian orchestrated the auction, waving a wooden hammer in the air like the conductor of the philharmonic.
"What will you give me for this Masson drawing?" He pointed to the last gilded frame in the trees, and Danny scrambled up to turn the picture toward the crowd. Varian took the bids, calling them out with the rolling tone he'd heard an auctioneer use at a house sale in Connecticut when he was a child. The bids rose steadily. "Sold!" he cried.
As the auction wound down and the crowd dispersed, following André inside to see the sketches for the Jeu de Marseille, Fry settled back in a wicker chair. The midday sun beat down on him. A group of Spaniards was clearing the pond for swimming. He felt lazy, fug-headed with contentment. Clovis lolled over, his tongue hanging from the corner of his mouth. The dog flopped down at his side, and Varian distractedly ran his fingers through the springy fur on his head, felt the hard, narrow skull beneath.
"Varian!" Danny called, running along the terrace. He was gasping for breath and clutched his side as he ran toward him. His pale face was flushed, alarmed.
"What is it?" His moment of peace evaporated. The wineglass slopped as he put it on the table, and red seeped into the white cloth. There were people milling around them still, so Danny nodded his head toward the garden. They walked on in silence until they were out of earshot, Clovis weaving among the box hedges ahead of them. "Well?"
"It's the Bernhards."
Varian stopped walking. "No, not them, too?"
"I just heard. They were picked up in Madrid."
Varian slumped onto a stone bench beside the pool. "How did they find them?"
"It was their papers. The new transit visas are obvious fakes, if you know what you are looking for."
"This is terrible. Which other clients have we got going out?"
"I don't know, but I'm going straight to the office so we can recall them all."
"Good. Is there anything we can do for the Bernhards?"
"I'm sorry, boss." He was shaking. "We can't seem to do a damn thing right at the moment."
"Christ, it's not your fault. We're all up against it. You mustn't blame yourselves, you hear?" Varian rubbed his thumb against his lip, thinking quickly. "Our days of grace are over."
"And you?" he said. "It's not safe here for you now."
"I don't give a damn. The U.S. government won't help us, the French want us out, so to hell with all of them. I'm going to stay and fight it out just as long as I can." Varian thought for a moment. "Listen, we're going to need more funds. Kourillo reckons he can sell half the gold for us at a favorable exchange rate. I was going to meet him in town tomorrow, but I've got to go up to Gordes to see Chagall with Bingham. Will you meet him at midday?"
"Sure, boss. I'll dig the gold up tonight." Danny squeezed his shoulder. "I'll need to take it in two cases, it's too heavy for me to carry in one lot."
"Good man."
"Boss, are you sure we can trust him? It was mighty suspicious that the cops turned up at Air-Bel with a warrant to search for gold and foreign currency a couple of days after Kourillo sold it to us."
"Trust him?" He raised his head and focused on the distant sea. "I wouldn't trust him not to sell his own grandmother if he could get a good enough price, but what choice do we have?"
FIFTY
MARSEILLE
1941
MARY JAYNE
"Mary Jayne!" Raymond grabbed at her arm as she walked out of her hotel.
"Go away, I have nothing left to say to you." She struggled free and hurried on quickly toward the tram stop. The morning crowds of people on their way to work milled around the pavement, and people stared at them as they pushed by.
"Wait, hear me out," he called after Mary Jayne.
"I can't. I'm late for work."
"For that stuck-up ass Fry?"
"Don't you dare," she said, swinging around. Mary Jayne slapped him, hard, the force of the blow stinging her palm.
"I deserved that," Raymond said, cupping his cheek. "I've been trying to see you, but the doorman wouldn't let me into the hotel."
"Good." Mary Jayne strode on, keeping her gaze straight ahead. "I told him if you came anywhere near me to call the cops."
"Please, I beg you."
"The great gangster, begging now?" She rounded on him. "How could you?" She glanced around, self-conscious, aware of the people staring, a crowd forming around them. She was standing by the door to a church and motioned at Raymond to follow her inside, away from prying eyes.
The church was empty, only the scent of incense from the morning mass lingering, rows of votive candles glowing on a tiered stand near the altar.
"I'm sorry," he said quietly. "I love—"
"Love?" she murmured, her voice low and husky. She gazed up at the stained-glass window nearby, a cloak of shifting color moving over them as the sun emerged from behind a cloud. "You don't know the meaning of the word. I feel like such a fool for telling everyone they were wrong about you."
"Mathieu threatened me. He said the gang would kill me if I didn't take your jewels."
"Killer was afraid of being killed?" She laughed. "I don't believe you. I think you were in it together. I think all you ever wanted from me was money."
"No," he said quietly, shaking his head.
"I never want to see you again. Do you understand?" He reached for her. "I mean it," she said, turning away. She wrapped her arms around herself. "I don't know what to believe anymore. That hoodlum Mathieu offered to bump you off, you know that? He said he wanted me for himself, can you believe it?"
"He always wanted you." Raymond's face crumpled. "That's the only reason he gave your jewels back. He'll kill me now, for sure."
"Well, that's your problem, not mine. I've done all I can."
"It can't end like this, bébé."
"Don't call me that. You have no right."
Raymond took hold of her arms. "Come with me. We can run away from this place, start again...."
"And what? Keep on running? Always looking over our shoulder in case Mathieu and his gang have found us? God only knows what you have done to cross them, but they want you dead."
"I love you."
Mary Jayne turned her head away. He pressed his lips to her hairline. "It's over," she said.
"No, never. Not with us."
"It's over," she said again, her voice hoarse, breaking.
"I'll always love you."
Mary Jayne unclipped her handbag, pulling out her wallet. "Write to me once in a while. Tell me about all the battles you have won, and all the women you've destroyed with that black heart of yours." She handed him a roll of banknotes.
"I don't want your money."
"Take it," she said, forcing it into his hands. "Take it, and get yourself to England somehow, like you always said you would. Take it, and make something good of your life, Raymond. Prove them wrong." Mary Jayne embraced him, her throat tight with emotion. She screwed her eyes closed, trying to stem the tears she was too proud to let him see. "Prove me right."
FIFTY-ONE
GORDES
1941
GABRIEL
"I assure you, Monsieur Chagall, we love 'cows' in New York," Varian said for what must have been the hundredth time. I couldn't figure it out. Here we were with one of the world's greatest living artists, and they were talking about cows? I'd managed to hitch a ride with Varian and Harry Bingham out to old Chagall's place in Gordes. When I miss France, I think of that day. Chagall lived in an old girls' school, a huge, beautiful old place. That day was perfection—the almond blossom was out, and the air was drenched with perfume. I'll never forget those hills—the gray green and sage, the dark flames of cypress. I think it is what heaven will look like, and I could understand why Chagall was reluctant to leave. I'd always been a big fan of his work, but the man himself was leaving me cold.
"Meh," he said, and shuffled toward an easel in the corner of the studio, draped with a white cloth. "I do not think I can work in America."
"My God," I heard Varian whisper to Harry, "it's like dealing with a recalcitrant child."
"Stick with it, Fry." Harry walked over to Chagall. "The thing is, monsieur, our sources tell us that soon people of Jewish descent will be rounded up."
"The anti-Jewish laws disgust me, but I am safe. I am an artist, a celebrated artist, and a French citizen. They would not dare to touch me." I could see from the expression on Varian's face that he thought they wouldn't think twice.
"Monsieur Chagall," Bingham said in his measured, pleasant tones, "you must listen to Mr. Fry."
"We're running out of time," Varian said. "I implore you, let us help you. We can arrange everything—papers, visas, tickets."
Chagall ignored him. He threw back the dust sheet from a painting of a young girl, flying free in space. "This is Three Candles."
"Let me take a photograph," I said, ushering Fry, Bingham, and Chagall toward the canvas. The painting was sublime. The girl reminded me of Annie.
"Please, at least come and see us at the ARC office, so we can talk through exactly how we can help you," Varian said to Chagall. I was half listening, caught up in the beauty of the girl, flying free.
Of course, by the time Chagall came to town in April, they had started rounding up the Jews in earnest. Varian had to go storming into police headquarters and say to them, "You do realize you have just arrested Monsieur Marc Chagall, one of the greatest artists in all of France, in the world? If I make one call to The New York Times, you will feel the full force of the U.S. government." Varian fought for Chagall and look how he repaid him. I can't figure out human nature sometimes. But then again, like I said to Sophie, why do people assume artists are going to behave like nice, normal people?
Anyway, I owe Chagall. That painting changed my life. I knew now what I must do. On the drive back down to Marseille, all of Provence unfurled beneath us, rolling green hills spreading down to the sea. Through the open window, I could hear cicadas humming. The sunlight through the window of the car was warm, and the leather seat was comfortable. I stretched out on the backseat, dozing as Varian and Harry talked in the front. Maybe their goodness was catching. On that drive I decided to do the first selfless act of my life, and there was a peace in that. I reconciled myself to my fate, and perhaps that made me appreciate the beauty of the countryside, the company of two good men, even more, because I knew it was all about to end. I'll never forget that drive—I felt a rare contentment the like of which I was not to feel again for years.
Varian dropped Harry at the American consulate, and we drove on in companionable silence to Air-Bel.
"Is everything okay, Gabriel?" he said, parking up on the drive. "You haven't said a word on the way back."
"I was just thinking about Chagall's painting."
"Beautiful, wasn't it?" He turned the engine off and stretched out his arms. "You know, some people said to me in New York, 'Why on earth are you risking your life to save artists? Why are they more important than ordinary men and women?' When you look at a painting like that, you just know it's the manifestation of everything good in a civilized society. If we don't keep the flame of culture burning bright, then what will we have left when the fighting is done?" He took off his glasses and rubbed at the bridge of his nose. "My friend Alfred Barr says that art exemplifies freedom. A painting like that is a symbol of freedom."
That was exactly how I felt when I looked at Chagall's painting—free.
FIFTY-TWO
BOULEVARD GARIBALDI, MARSEILLE
1941
VARIAN
The outer door of the boulevard Garibaldi offices was splintered and dented. "Thank God we changed the locks recently," Varian said to Gabriel as they let themselves in to the ARC. "If they'd have got through this door, they could have torched all the files."
"Who do you think did it?" Gabriel said.
"Probably those Vichy fascist kids Gussie noticed hanging around the last few days."
"They're just trying to scare you."
Yeah, or their bosses are, Varian thought, and it's working. The frequent searches, the suggestions that they would not be at all surprised to find Monsieur Varian Fry floating in the harbor with his throat slit, were all beginning to take their toll. "To hell with them, we've got work to do." Varian flicked on the lights as they walked through. "Morning, Gussie," he called.
"Morning, boss." Gussie swung his legs around from the cot in the kitchen and stretched, yawning.
"Quiet night?" Varian glanced at him and noticed the iron bar beneath the camp bed.
"Not exactly." He slid on his shoes. "I didn't think it was worth waking you all unless they made it through the street door. I don't think they'll bother trying again."
"Good chap." Varian peeled off a couple of notes from his money clip and gave it to him. "Listen, go get yourself some breakfast and head up to Air-Bel for a rest. We can hold down everything here."
Varian flicked on the gas stove and ran a kettle of water. "Right, we've got a busy day ahead," he said as the staff began to file into the office. He spoke to one or two of them and handed over the most urgent files. Finally, he turned to Gabriel. "Okay. Why don't you tell me what the hell is going on? I thought everything is in order? You're sailing on the Paul Lemerle with the Bretons."
"No. There's been a change of plan."
Bloody artists, Varian thought. It's terrible to say it, but when some of the most troublesome ones get picked up, it's almost a relief not to have to deal with them. The kettle whistled on the stove, and Varian frowned as he poured the steaming water over coffee grounds in the jug.
"Look. We have little time left, and much to do." He poured them each a cup of coffee. At least there's some good news. Who knows why, but the Bernhards have been released and they are on their way to Lisbon. Thank God, Varian thought, offering up a silent prayer of thanks. Out of Hilferding, Breitscheid, and Bernhard, at least we managed to get one of the three out. Varian slumped down in the chair at his desk and sipped at the scalding coffee. His mouth tasted metallic and his head was thumping. The thought of the day ahead exhausted him. They had stayed up until the early hours celebrating the news that the Bretons' French exit visas had come through, and Varian had drowned his guilt about Danny's arrest in drink after drink.
Just as it seems there's a glimmer of hope, Varian thought. My poor, poor Danny. Kourillo's to blame. He paled at the thought of the lost gold. The crook's an agent for the police and the Gestapo, I'm sure of it.
"Danny's lawyer came in yesterday," Gussie said, shrugging on his coat.
Varian beckoned to Gussie to follow him to a quiet corner, away from Gabriel. "Is he formally under arrest?"
"They've booked him for trading in gold, and transferred him to the Prison Chave."
"Oh God, no," Varian said, raking his hand through his hair. "Did anyone from here manage to speak to Danny?"
Gussie shook his head. "His lawyer saw him for ten minutes. He told me that Danny took the first case of gold, and Kourillo was waiting for him at his hotel on rue Thubaneau for the pickup, just like you arranged. Kourillo took the two thousand dollars in gold, and handed over the francs. When he came with the second case, Danny thought he was being followed, so he tried to walk on by, but Kourillo rushed down the steps and shook his hand. That's when the flics picked Danny up. Three guys jumped him."
Judas, Varian thought. "You're sure he has been charged?"
Gussie nodded. "They've thrown the book at him."
"Damn, damn, damn," Varian said under his breath. "Right. This is what we are going to do. I will go down to the station and take responsibility for it myself."
"You can't do that, boss," Gussie said quietly. "The office depends on you. Danny knew the risks."
"Damn it, we have to do something."
"Danny cooked up a story," Gussie said. "He's told the cops that some of our grateful, wealthy clients have paid us in gold."
"Did they believe him?"
"I doubt it."
"Damn them to hell," Varian said, and balled his hand into a fist, pressing it against his forehead. "I pass that prison twice a day walking to and from this place. I hate to think of Danny in there." He fell silent, his shoulders hunched. "Right. I'm going to see Vinciléoni. I'm going to make it known that we want a hit put on Kourillo."
"Are you sure, boss? Do you really want his blood on your hands?"
"They won't kill him. I just want to scare him out of town. Kourillo is one of their own. Vinciléoni will just tell him to get lost. I simply want to send a message that they are not going to mess around with us again." Varian strode past Gabriel to his desk and reached for his hat. "People disappear in Marseille all the time—only last night some English guy, Quimby, turned up in an alley with his head bashed in. It was in the papers this morning."
"An English guy?" Gabriel interrupted.
"Yeah," Gussie said. "I doubt they would have bothered to cover the story if it was just the usual Marseille ruffian, but they are speculating he was some kind of spy."
"Look, Gabriel, we'll have to do this later. Can you book a time with Lena?... Say, are you okay?" Varian looked properly at Gabriel for the first time. "What happened to your face?"
Gabriel touched the gauze taped to his cheek. "Nothing. I looked at a guy the wrong way in Snappy's bar."
Varian shook his head. "I tell you, you'll be glad to get out of this place. Nobody's safe anymore."
FIFTY-THREE
VILLA AIR-BEL, MARSEILLE
1941
GABRIEL
"Did you kill him?" Annie whispered to me, closing the Bouchards' back door behind her. I could hear her parents talking in the house—it wasn't safe for me to stay long.
"Of course not." I hoped my face didn't show a thing. "Quimby had it coming. I bet I wasn't the only person he was swindling. They're saying he might even have been some kind of agent, or spy."
"At least you won't have to spend your whole life looking over your shoulder, now." When she looked up at me, her expression reminded me of an old stray dog who lived on our street in the Marais. He must have been thirteen or fourteen, a grizzled, charming old thing. We'd give him scraps if we could, which he accepted with all the grace of a down-on-his-luck nobleman. I came back from school one day to find him locked up in the back of the warden's van. We all knew what that meant. The man wouldn't budge, wouldn't let him go. I railed against it, banged the bars, but the old dog just stuck his nose through and nudged my hand. He looked at me with exactly the same brave expression Annie had.
"Annie, I'm sorry—"
"Don't, Gabriel, please." She pressed her fingers to my lips. "Don't make this any harder than it is."
"You will come, tomorrow, to see the boat off?"
"Of course. Even my parents are coming. I think they want to make sure you've left the country." She turned my hand over in hers, traced the long, firm arc of the lifeline with her index finger. "There, you've years and years ahead of you...."
"I don't want to spend them without you."
She shook her head, unable to look me in the eye. "You have to go, now. It's too dangerous here." She touched the bandage on my cheek. "My poor darling," she said. "You have a chance to get away from here. You can start a new life. No one knows you there. You'll just be Gabriel Lambert, not the father or the son. You can just be yourself, the very best self you can be." She raised her gaze, her eyes glistening. "Promise me, Gabriel. Promise you'll never forget me." I held her close, her head against my chest, my lips in her hair. "Live a good life, for both of us, Gabriel. Make it count."
FIFTY-FOUR
MARSEILLE
Tuesday, March 25, 1941
GABRIEL
Dawn's silver light washed the horizon, crept wave by wave across the shore. The Quai de la Joliette was crammed with people by the time I arrived, carrying only my rucksack. I was leaving as I had arrived, with nothing more than the clothes I stood up in. I had even sold my easel. Once again, I had no money, no home. I had skipped out of the hotel owing the last of my bill.
I could see armed guards keeping the crowd separate from the passengers. People were so desperate to get out of the country, they would try anything. I saw an old man on his hands and knees, trying to creep past the guards, but they caught him and cracked his ribs with the butts of their rifles for his trouble.
"Where are they?" I said under my breath. I saw Varian then, shepherding his clients toward the gangplanks. I saw Breton, his hair gilded by the morning sun, carrying Aube in his arms. He was hand in hand with Jacqueline, and they pushed their way through. I've always hated crowds, and the noise, the shouts, the hot jostling figures, terrified me. My palms were sweating, my skin prickling with anxiety. All the Bouchards had to do was bring Annie to the docks, that was all. What if I never saw her again? I checked my watch. They were ten minutes late. The Capitaine Paul Lemerle was making ready to sail, sounding its great horns, steam pluming up into the air from the huge chimneys. Where is she? I looked frantically around and saw a lamppost to one side. I clambered up and looked around the bobbing sea of heads and hats.
"Annie!" I yelled above the noise of the crowd and the ship's horns. "Annie!"
"Gabriel!" I could hear her, followed the sound of her voice. There! I saw her pale hand waving like a drowning woman's, reaching out of the dark sea of hats. I jumped down and pushed my way through toward her, my heart thundering, my breath coming in short, tight gasps. I knocked over a man, tripping on the legs of someone who had fallen. Then she was there, in front of me.
"Annie," I whispered, burying my face in her hair and holding her tight to me, tears in my eyes. Her parents shoved their way through. I saw my anxiety painted on their pale, drawn faces. I held Monsieur Bouchard's gaze, and he nodded. "Come," I said. "It's time." I held on to Annie's hand and shielded the Bouchards with my arm. We fought our way through to the front of the crowd, until I found a rifle pointed at my chest.
"No further," the guard barked, "passengers only."
"He is a passenger," Annie said. She turned to me. "I may never see you again. It's just hit me, that you are really leaving."
"No," I said. "I'm not leaving. You are."
I saw the confusion on her face, and it felt like the noise, the crowd, fell away. It was just me and her. "Annie, it's all arranged. The ARC has helped me get your papers in order, and I'm giving you my ticket." I put the strap of my bag over her head, tucked it under her arm safely. "I knew you'd refuse if I told you before. It's all in here. Tickets, papers—"
"No," she whispered, tears brimming in her eyes. "I can't let you sacrifice your crossing... yourself, for me."
I took her in my arms, held her tight, my lips against her cheek. "Annie, after everything I have done, please, let me do one good thing."
I felt her shake her head, she pushed me away. "No, I won't let you. I won't go. I can't leave you...," she said, turning to her parents.
"Marianne," Monsieur Bouchard said, taking her face in his hands, tears pouring down his wrinkled cheeks. "You are getting on that boat."
"No, Papa," she said, a sob catching in her throat. She looked at her mother. Madame Bouchard reached into the carpetbag she was carrying and forced a bundle of money and clothes into her hands.
"My child, my child..." Her face crumpled as she held her daughter one last time, rocking her in her arms, her eyes screwed closed as she whispered to her. "Go," she said finally, breaking away.
"Mama, no!" Annie screamed, reaching out to her, but her mother pushed her toward the boat, weeping.
"Go now. God bless you, my darling girl. Go!"
The ship's horn blew, and I dragged Annie, crying for her parents, to the bottom of the gangplank. I handed over the ticket and papers for inspection. The official stamped them and unclipped the chain to let her climb aboard.
"Listen to me..." I soothed her like a child, like I have all my own children and grandchildren down the years. "I love you," I said, embracing her. I breathed in her clear scent of sunlight and lavender one last time. "I love you. Wait for me. When I get to America, we'll be married, I promise...."
"Gabriel," she cried out as the guards pushed me back with their guns. "I love you!" The official took her arm, hurried her up the gangplank. "I'll wait for you. However long it takes, I'll wait."
I reached toward her, fought to stay at the front of the crowd for one last glimpse of her. "I'll see you," I cried, "I'll see you in New York!"
FIFTY-FIVE
MARSEILLE
1941
VARIAN
The great ropes loosed from their mooring posts, snaked free into the water. The Capitaine Paul Lemerle heaved, like a black cliff breaking free above the crowd. Varian waved, craned his neck to see the Bretons up on deck.
"Bon voyage," he cried, echoing the shouts around him. It worked, he thought, his heart soaring with elation. It worked. Thank God, they are away. Next to them, he saw Marianne. The poor girl looked terrified. He followed the direction of her gaze and saw Gabriel Lambert, fighting his way to the front of the crowd, tears pouring down his face. He thought of the letter he had received that morning, delivered by the concierge of Alistair Quimby's hotel. It made no sense to him. Clearly that guy Quimby had some kind of grudge against him. Varian had simply screwed up the note and tossed it in the bin. Gabriel Lambert's file was closed. It's too late now, and there are hundreds of other people needing our help. Whatever the truth is, Lambert gave up his passage, and that's a selfless act. Good luck to them. I hope they find one another again. Varian smiled sadly at the love written clear on Gabriel's face as he waved, as he blew kisses to Marianne with both hands.
They had only just managed to get her papers in time. Varian remembered Gabriel pacing up and down in the office the day before.
"Please, you must help her."
"Listen, old chap—"
"You must!" Gabriel slammed his hands down on the desk.
"Look. I can see how much you care about this girl, but she's not..."
"Not what? Not a great artist? Not important enough?" Gabriel was shaking. He held out a golden cloth, an embroidery that spilled onto the table. "She made this."
"Lambert, it's hardly on a par with your work."
"She is young, and bright, and good. What price is a life, Varian? What makes one life more important than another?" He rubbed the heel of his hand between his eyes. "How can you say that my life is worth more because of a few paintings?"
Varian sat back in his chair and sighed. "You are willing to sign over your passage to America to her?"
"Yes, without hesitation."
"Listen. We've always had a rule here that we will only help people to escape that are known by people we trust. Can I trust you?"
"Yes."
"The ARC cannot officially help her to obtain papers."
"I'll do anything," Gabriel said. "I will pay whatever it takes to get her papers in order, for forgeries—I'll do them myself if necessary...." His voice trailed off as Varian looked at him sharply. "I don't... I don't know quite how you have managed to do what you have done here, but I am begging you, please help Annie."
Varian sighed. "All right. You're in luck. It would have been different a few months ago, but now that they are letting boats sail for Martinique, all she will need is a French exit visa."
"Thank you, thank you..." Gabriel clasped his hand. "I'll never forget this, Varian, never."
"There will be time for sentiment later. I'll get one of the men to bring Annie's visa to you tonight. Let's make damn sure she is on that boat, too."
"You'll never know how much this means." Gabriel hesitated. "Why do you do it? Why do you do all of this, put yourself at risk for us all?"
"Why?" Varian said, as though it had never occurred to him to ask the question. "You know, I keep a phrase of Emerson's with me always." He gazed out of the window, reciting from memory: "'There are men to whom a crisis, which intimidates and paralyzes the majority, comes as graceful and beloved as a bride.'" He turned and smiled at Gabriel. "I do this simply because it is the right thing to do. Charlie always said it's the duty of the strong to protect the weak. All I can tell you is I've met the most remarkable men and women of my life in the last few months. I've met people whose work I've loved my whole life. You know," he said, laughing, "all I knew about people outsmarting the Gestapo was what I'd seen in movies, but when the committee in New York said 'You're it' and sent me out here, I had to learn fast." He stood and guided Gabriel out of the office. "The thing is, I believe in freedom. I feel a deep love for these people and gratitude for the happiness their work has given me. I had to come and help them when they needed it." He took off his glasses and rubbed the bridge of his nose. "You remember Miriam? She always said it made her think of that verse in Ruth: 'Your people are my people.' That's what it comes down to."
"You are remarkable. If I ever have a son, I'll name him for you."
"I always hated my name," Varian said, laughing softly. "I wanted to be called Tommy." They shook hands. "Good luck, Gabriel. What will you do now?"
"I thought I'd head into the countryside. I hope I can be useful to the Resistance."
"Talk to my men. They can help you."
"What about you?"
"Me? I'm going to get Danny out of jail, and then I'll hold out here as long as I can. Besides, it is easy to become attached to a place, especially when it is a country as beautiful as France." He opened the door for me. "You know, we are all at war as surely as our brave boys on the front lines. One must bring them all back home, or at least one must try. That is all I can do, bring as many of you home as I can."
* * *
Varian stumbled as the crowd surged, fought to stay on his feet. A guard nearby brandished his pistol, cocked it up into the air.
"Arretez!" he warned.
If only we could round them all up, every single name on that list, and spirit them away on a boat like this, Varian thought. He waved as Jacqueline lifted Aube to see the boat leave the Vieux-Port. The sea lurched and swelled, slapping against the quay as the engines roared. A woman on the deck dropped red rose petals, spiraling, drifting down to the sea.
"Varian!" Mary Jayne yelled, pushing her way through the crowd. He reached out to her and put his arm around her, shielding her from the crowd. They gazed silently at each other for a moment, like equally matched creatures deciding whether to fight or make peace. "Well done," she said, and dug him in the ribs. "You did it."
"We did it," he corrected, "but only just." He shielded his eyes and looked up to the deck. "Are you okay? Where's Killer?"
"He's gone." She looked up at Varian. "Thank you."
"For what?"
"For not saying 'I told you so,' like everyone else has." She shielded her eyes. "Where are they?" He pointed up to the deck. "I'm so glad," she said, her voice catching. "Good-bye!" she called. "Good-bye!"
Varian cupped his ear, craning to hear what André was shouting as the ship pulled away, guided out of the harbor by the pilot boats. He shook his head, mimed that he couldn't hear. "I'll see you soon!" he cried, waving as the boat pulled away. "I'll see you in New York!"
"So," Mary Jayne said, glancing up at him. "What now? Friends?"
"Friends." Varian offered her his arm, and they walked in silence along the quay. "I think it's time we got you safely home, too, don't you?"
"Home? I don't even know where that is anymore." She looked toward the cathedral, the sun glinting on the Virgin high above the city. "Nothing will ever be as extraordinary as this year in Marseille, will it? We shared our finest hours, my friend, our finest hours."
FIFTY-SIX
FLYING POINT, LONG ISLAND
2000
SOPHIE
An old red truck pulls up in front of the beach house, and Marv clambers out. "Hey, Tommy," he says.
"Hey, Marv, how are you doing?" Tom lifts the last of the boxes into the back of the station wagon. "Who's this?" He squints his eyes in the sunlight.
"Pleased to meet you." Sophie steps forward and offers her his hand. "I think you're expecting me? Sophie Cass."
"This is Tom Lambert, Gabe's eldest," Marv says. "This here's the girl Gabe's been talking to himself about for the last few weeks."
"I know who you are," Tom says. "We tried to cancel a hundred times, but you wouldn't listen. When your mother rang, the only reason I agreed to see you was to tell you face-to-face. Gabe doesn't need this. Dad is... well, he's not been well for some time, but the last few days he's grown more confused. It's like he's lost in his own world. He really doesn't need you dragging up all this stuff about the war, and Vita, not now. It's all so long ago—what does any of it matter anymore?"
"I promise, I won't take up much—"
"Anyway," Tom says, stepping toward her, "it's too late. You've missed your appointment, and we have to get going."
"Look, I've walked for miles." Sophie folds her arms. Her skin is tanned, her hair windblown. "I wouldn't have been late if Harry hadn't dumped me on the wrong beach."
"I found her thumbing a lift on the road to town." Marv leans in to Tom. "Did I do the right thing bringing her here?" he whispers too loudly. "Gabe was talking to himself again, earlier on in the café—well, it was like he was talking to her." He points at Sophie. "I thought it might be important."
Tom throws a tarpaulin over the boxes. "Sure you did right," he says clearly, angling his head toward Marv's good ear. "What does it matter anymore?"
"Talk of the devil," Sophie says as Harry's pickup bounces along the dirt road and pulls to a stop beside Marv's.
"I'm sorry," Harry says, holding up his hands as he walks toward Sophie. "I felt bad and I went back to find you, but you'd gone already." He glances at his father. "We'd decided this morning we were going to send you on a wild-goose chase, and hoped you'd just give up if Gabe had gone by the time you found the house." His eyes are clear and blue as he looks at Sophie. "I was just trying to protect Gabe. He's not been that strong lately, especially since Grandma died."
"Annie died?" Sophie's eyes widen. "Oh God, when?"
"We lost her a few days ago—" Tom kicks at the ground, his voice breaks. "We haven't made it public knowledge yet. She was only diagnosed a few months ago, and she went downhill fast. Gabe... he's taken it pretty hard."
"I just can't imagine Gabe without her," Harry says. Sophie looks across at him, holds his gaze, tears pricking her eyes. "He's not well himself, but he was with her every moment she was awake, right up to the end."
"I'm so sorry." Sophie's throat is tight. "Why didn't someone tell me? I would never have—" She looks up at a sound from the house, her eyes glistening. Some of the great-grandchildren are playing in the garden, the flames of a last bonfire sparking into the sky. "Excuse me." Sophie walks toward the fire. Her hand shakes as she reaches for her phone.
"Honey?" Paige picks up on the first ring. "Are you okay? I saw you'd tried to call."
"I'm here, finally."
"Have you seen Gabe?"
"Not yet." Sophie closes her eyes. "She died, Mom, Annie died."
"Oh God, when?"
"A few days ago. I feel terrible for pushing and pushing to see Gabriel over the last few months. No wonder they were trying to keep me away. I don't know what to do—" She sensed something and glanced back at Harry, who was leaning against his truck, watching her. "I can't put him through this when he's just lost the love of his life."
"Do the right thing, Sophie."
"Mom, I'm so confused. And Jess told me..."
"What's he done now?" Sophie hears the edge in her mother's voice.
"He's just told me that Dad was illegitimate."
"Oh, that?"
"Mom, what do you mean, 'Oh, that?'" Sophie's voice rises. "Why didn't any of you tell me?"
"Darling, what does it matter? Of course, it was still a big deal in the fifties. You know what your grandmother was like, she always was independent, said she didn't want Sam to think she was trapping him into marriage. But Sam made her see sense in the end. They married after your dad was born. My guess is your grandmother was so stuck on the idea of doing it all herself that she wouldn't give Sam's name when she registered the birth. Simple as that. Maybe they never bothered changing the certificate later. You know what they were like, always so focused on the day. My bet is they just didn't look back. I promise you, your grandparents had one of the best relationships I've ever come across."
"Thank God. Vita, the story..." My story, Sophie suddenly realizes. She feels like she is watching a carefully constructed puzzle slot into place. "I thought... I thought it meant we weren't related. Dad always loved telling me the stories about her."
"Honey, you only have to look at you. You're the spitting image of her—Jack always said you reminded him of the paintings he'd seen of her."
"I know."
Paige clicks her tongue. "Typical, Jess couldn't even let you have that, could he? Just like that control freak to think he can destroy that link to your dad. Forgive me, darling, but I always said Jess is a narcissist. Love isn't about what you take from someone, it's about what you give."
"I was just... I was thinking about Dad." Sophie sits down by the bonfire, hugs her knees. "I wish—"
"Sophie, it was just his time."
"He died because of me." Her fingertips tremble, running across her right collarbone, searching for the indentation, the daily reminder of where the bullet nicked her shoulder. She had told Jess it was from a broken collarbone—described the pony, the fall, in vivid detail.
"He died protecting you," Paige says gently. "I know you never talk to me about your work lately, because you think it will upset me, but you don't have anything to prove. You don't have to pick up where he left off."
"If I hadn't insisted we go to the drugstore, he would still be alive. Think of everything he would have written."
"Sophie, you can't think like that. It was your dad's choice. He tried to talk the gunman out of holding up the drugstore, and he protected you when the guy started shooting."
Sophie closes her eyes. She hears the crackle of the bonfire, and her nostrils flare at the acrid smoke. She remembers it all. The sound of her father's voice, calm and sure, telling the guy to hand over the gun. She remembers his silhouette blocking the light. Then the screams as the shots rang out, the weight of her father's body, shielding her. How the bullet nicked her shoulder as he slumped down. "He died in my arms."
"I know, honey, I know." Paige falls silent. "You have to let go of this and move on, live your own life." She laughs softly. "Your dad was no angel, trust me, and he'd be the last person who would have wanted you to think that. He died as he lived, doing what was right, standing up to the bad guys. It's as simple as that. He was doing the right thing, the brave thing." She pauses. "I was just thinking about you out there today, about Gabe and Annie."
"What about them?"
"It's nuts—I've never admitted this to anyone before, but your dad had a huge crush on Annie. Every man who met her did—she was like the Long Island Bardot. You know that photo I gave you, of Annie and Gabe?"
Sophie searches in her bag. "It looks like she's in her thirties?"
"I hadn't even met your dad yet, when he took that. I found his journals after he died, and that was tucked in there."
"Did he and Annie—"
"God, no!" Paige laughs. "Not for want of trying on his part. It may have been the swinging sixties, but there was only one guy for Annie. Gabe didn't like it one little bit, though. Well, just look at his expression in the photograph."
Sophie smiles. "It looks like Annie's holding him back."
"Gave your dad his first black eye, just after that photo was taken." Sophie hears her mother sigh. "First of many."
"Did you know Gabriel well?"
"No, I never could understand why he wasn't friendlier—I mean, with the connection to Vita and all." Paige thinks for a moment. "Maybe like your grandparents, he just didn't want to look back. A lot of that generation just wanted to put the war and all the losses behind them. But I'm convinced that's why they ended up out here—Vita's mother was from East Hampton originally. Maybe after he lost Vita and his son during the war, it was a way of being close to them."
"His son?" Sophie hesitates. It is on the tip of her tongue to tell her mother about her doubts. "What about Annie?"
"She was older than me, obviously, but she was lovely, very down-to-earth. We'd see them both at parties and gallery openings occasionally. I lost touch with her when your dad convinced me to move to the city." Paige hesitates. "I don't know why I never looked them up again, when I moved back to Montauk. Maybe... maybe I was a little jealous of her, if I'm honest. It all seemed so natural and easy with them. The success, the troop of beautiful kids, that gorgeous house." Paige laughs sadly. "She was so kind to me, but I couldn't help envying her. After I lost your dad, I just couldn't face seeing all... well, all that happiness when I was trying to cope raising you by myself. Even you've been won over by Gabe and Annie's love story, haven't you?"
"I guess I have." The knowledge settles in her, strong and true. "I want that, you know?"
"I know," Paige says. "If I've learned anything, you have to fight for it. Sure, they made it look easy, but marriage is as much about sacrifice and compromise and forgiveness as moons in June." She laughs softly. "Maybe if each of you is willing to give more than you get, you stand a chance."
"Was it like that with Dad?"
"When I first met Jack, I thought we'd set the night on fire, we loved one another so much." She pauses. "Oh, God, it goes by too fast. What a fool I've been. I wish I'd seen Annie again."
"I'll tell Gabriel you say hi."
"You will be careful, won't you? I know he's what—ninety-something, now? He had a hell of a temper on him in the old days, though."
"I'll be careful, I promise."
"See you soon?"
"Sooner than you think. I've decided to take you up on the offer of the barn."
"Seriously? That's wonderful."
"I've even found a contractor." She glances back at Harry, who is watching her. "I don't need much—just water and electric."
"All mod cons? We can do better than that." Paige laughs. "It's the right decision, take your time and find your feet. You're always happy when you're out here, and Mutt will love it."
"I've got to go."
"Sure. And Sophie..."
"Yes?"
"It's just beginning, you know. Forget about all the what-ifs and what-might-have-beens." Sophie hears the love in her mother's voice. "Your story's just begun."
FIFTY-SEVEN
FLYING POINT, LONG ISLAND
2000
GABRIEL
I lie back on the sand and close my eyes, listening to the surf. I try to push that anxiety from me, the bittersweet relief of seeing Annie leave Marseille. I'm there, on the docks again, seeing the only sure light in my world sailing away.
"Don't go." I'm floundering, caught in nets I wove myself.
"Go? I'm not going anywhere."
I blink, shield my eyes from the overhead sun. The girl is above me now, her blond hair ablaze in the light.
"You?"
She comes closer so I can see her face.
"What is it, Gabriel? It's like you've seen a ghost."
Someone is crying out, a strangled whimper. It's me, I'm frozen like a dreamer caught in a nightmare, stifled and mute.
"Hush," she says, brushing the hair back from my brow. "Don't fight it now."
"You?" The breath is tight in me now, no respite. "Vita?"
"Sophie? Vita? What's the difference? Names are irrelevant." Her face hovers over mine, and I see, I see. Those lips, the pillow crease, her golden hair tumbling around me like flames.
"No, not you," I say, blinking away tears. "I don't want you here."
"Tell me the end of the story, Gabriel."
"I can't. I'm tired... It's too long ago."
"It's important," she insists. "You need to remember it all." She shakes me so my head rolls from side to side. "Did you do it, Gabriel? Did you kill Quimby?"
"I lied to Annie, one last time. I did it. I did it for her." My voice is a whisper. I never meant to kill him. I followed him down from our last meeting at the cathedral on the hill to the city, to the Vieux-Port. I just wanted to find out where he was staying, so I could get back as much of the money as I could for Annie and destroy the photos. I saw him go into some seedy hotel, and I waited until nightfall. As I waited, I got angry, thinking of everything he had done to me. I didn't trust him to leave me and Annie alone. Then I remembered he had a photo in his wallet, too. I decided I'd follow him and rough him up a bit, scare him away. If Varian and his guys could get tough with the gangsters and bullies, then so could I. He came out just after six. He must have realized he was being followed. I saw him go into the alley behind the café Au Brûleur de Loups. He was hiding in a doorway. For years, the memory of his face looming at me out of the darkness has haunted my nightmares. Quimby had a blade, and he jumped me, slashed at my face. I saw the surprise on his face when he saw it was me, and then he went for me. He had me backed into a dead end, waving that knife around. He wanted me dead, he said. I had no choice but to fight him. I knew he'd never let me go.
It's not like in the movies. The fight didn't last long. I was younger than him, and stronger, and I managed to get hold of his arm and knock the knife from his hand. Quimby backed away and put his glasses in his top pocket, then raised his fists. I told him to give up, but he said nothing would give him greater pleasure than to smash my face in. He was bluffing. I could see him peering at the floor, trying to see where the knife had gone, but he couldn't see a damn thing, and the alley was full of potholes and puddles of oily water. I hated him at that moment. I had no choice, I fought for my life, for Annie, and I thumped him with all my strength. He reeled away toward the wall, and I pushed past. I could hear him coming after me, the sound of his feet on the cobblestones, and then... then it just stopped. He fell, you see. Fell or tripped on something and hit his head on one of the sharp stone doorways jutting out into the alleyway. I went back to check, but he wasn't breathing.
"It wasn't your fault," the girl says.
"He wouldn't have been there if I hadn't followed him from the hotel."
"But you didn't kill him."
"I just wanted to scare him, that's all. He'd threatened to tell Varian. They wouldn't have helped Annie."
"Fry's team never knew who you are? Only Quimby and Annie knew the truth?"
"How could I tell them, even later? I was so ashamed." That is the greatest punishment of all, perhaps. All this, this beautiful life with its fragile happiness, has been built on a lie. Is it your own life, if you can't claim it as your own?
"Tell me about Varian."
"He... I lost track of him, toward the end."
"Did you contact him again?"
I shake my head. "I gave to the International Rescue Committee, as Fry's organization became, as much... as I could."
"Anonymously," the girl says.
"Of course."
"But you couldn't bring yourself to see him. Were you guilty because of what you had done?"
"Yes."
"Good. So you should be, lying to good people like that."
"Please don't..."
"Poor Varian. Everything that came after lived in the shadow of Air-Bel." She sighs. "And Mary Jayne, what of her? She wrote to Fry just before he died, and do you know what she said?"
I know. She said: We shared our finest hours, my friend. "But there have been other fine hours since then, simple hours full of work, and love, and family."
"Mary Jayne spent the last of her days in a villa in the south of France she christened 'Air-Bel.'" The girl reaches out her hand and points at the house, at my home. "Just like you, Gabriel." I think of Annie, sitting on her chair out on the deck looking out to sea, wrapped in blankets, and behind her, the old flaking sign we painted together: Air-Bel.
"Annie," I gasp.
"Tell me about Annie, Gabriel. Did she make it safely to America?"
"Annie... waited. Years. Years and years."
"You were trapped in France, weren't you?" Her voice soaks into me like sunlight. It comes from far away. "You went into hiding, fighting with the Resistance, didn't you, Gabriel, do you remember now?" I try to nod. "Just like Danny and the others." I feel her shaking me. "Stay with me, Gabriel. Do you think you made amends, for all you did and didn't do? For Vita, and your father, and Quimby?" I whimper. "Shhh," she murmurs, strokes my cheek. "Do you think all the years of working, all the years of loving your family and living quietly, atoned for that? How do you live a good life with such guilt?" The girl pauses. "What about the man who killed Annie's parents? Do you think he felt guilty?" I close my eyes as her voice, Vita's voice, whispers close to my ear. "All those nights Annie stood on the shore wondering what happened to her mother and father. Do you want to know what happened? They were captured, Gabriel, just like Annie always feared. Her father was shipped out to a concentration camp." I feel her breath on my cheek. "When you were all playing games in Marseille, no one had even dreamed of the horror to come, of the Holocaust, but it swept them away. Her mother was shot on the platform as they took old Bouchard away. She tried to run past the guards, to go with him."
"How do you... How do you know all this?"
"I'm everywhere and nowhere, Gabriel. That's what you will find."
"Please," I whisper. "Get Annie..."
"I'm almost done, Gabriel," she says. "After the war, you finally reached New York, didn't you? Do you remember arriving at Ellis Island, and Annie was waiting for you?"
"Annie." My breaths are shallow, useless.
"You were young, and penniless, but with Gabriel Lambert's name and your contacts at Air-Bel behind you, your work began to sell." I close my eyes as she strokes the lids with her fingertips. The images come thick and fast then. Annie sitting up in bed holding our first son, the love and amazement on her face as she looked at him. The old van we parked on the spot overlooking the beach, our first night talking by a bonfire here on the shore, our dreams for the future unfurling to the stars. We lay there watching the midnight-blue velvet sky lighten and the morning star shine for us. Then I see the timbers marking out the space that we called home. Annie in dungarees, painting the walls, her stomach swollen with our second child. One after another, the images come, fragments of a simple life. Our life. And I feel such happiness, such joy.
FIFTY-EIGHT
FLYING POINT, LONG ISLAND
2000
GABRIEL
Inside our sanctuary, our Air-Bel, my other son and daughter and their children are closing up the house for the season. Dust sheets billow like sails, falling silently across the furniture in the shadows for the last time.
"Marv? You okay?" Tom says, touching my old friend's arm.
"Oh, me, sure. Listen, like I said, Gabe was in the café just now."
Tom slams the tailgate closed. "Damn, is that where he's got to? He said he was going for a walk. Albie was supposed to be keeping an eye on him, but Dad gave him the slip."
"He seemed..." Marv chooses his words carefully. "Well, he seemed a little lost."
"Just so long as he wasn't upset. He has good days and bad days. We only just convinced him to throw out the Christmas tree."
"That was a good thing you did for your mother. I know how much he wanted her to have one last Christmas here, even if it was August." He looks up as my second son strides over, arms laden with bags. "Hey, Albie," he says. "How are you kids holding up?"
"Day by day." Albie leans on the car, dumps the bags into the back. "Dad was amazing, you know, he cared for her right up to the end, would hardly let anyone near her."
"It's good that she came home," Marv said. "They loved it here."
Tom looks back at the house, watching Sophie. She is talking to the kids now. She squats beside them, helps the littlest toast her marshmallows over the fire. The men wander over. "You know, we used to say we hoped they'd go together. It was impossible to imagine the one without the other." He shakes his head. "Dad's been pushing himself too hard. The last few weeks, all he's wanted to do is work when he wasn't with Mom. He spends hours in the studio, just talking to himself."
"Can I see him, now?" Sophie asks.
"Shame you weren't really in the café with him," Marv says. "It was like he was pouring out his whole life story to you." He shakes his head. "Even ordered you a stack of pancakes."
"That's Dad for you, always did have a good heart." Tom thinks for a moment. "He knew you were coming. His lawyers had told him about some old photographs, or something? It seemed to upset him."
"Oh God, I'm sorry—" Sophie says.
Tom sticks his hands in his old blue jeans. "Dad's still fine if he's working, but everyday things..." He shrugs. "We normally get to the mail before he does. As long as everything is routine, he's okay." He sorts through his bunch of keys.
"At least he's still working," Sophie says.
"Yeah, but no one can figure out what on. The whole studio is jammed with blue canvases. He spends days just painting a single canvas blue, then moves on to the next one, just the same as the last."
"That's it? Nothing but blue?"
Tom shrugs. "There's a tiny white dot, like an opal, on each one, but that's it."
Venus, Sophie thinks. The morning star, guiding him home.
"I hate seeing Gabe like this," Marv says. "He's always been amazing for his age." He shrugs. "Maybe it'll be good for him to be with you kids in the city for a few months. The winters are hard."
"I know." Sophie glances at Harry. "My mom lives in Montauk. I grew up out here, and I'm moving back." He holds her gaze and smiles. Impulsively, she pulls her files of research from her satchel. "In fact, my dad took this photo of Gabe and Annie, years ago, at some party."
"They were friends?" Harry looks at the picture Sophie hands him and passes it on to his father and Marv.
"At least, Mom and Annie were," Sophie says.
"Really?" Harry turns to her. "Your mom just mentioned that you were related to Gabe's old girlfriend during the war, Vita?"
"Yeah, Vita," Marv says. "Gabe's been yacking away to her too, the last few days. Vita this, Vita that..."
"Aren't you full of surprises?" Harry says to Sophie. His look quickens her heart, she feels her stomach free-fall.
"Unexpected?" she says.
"Look at that." Marv's face softens, gazing at the photo. "The image of Gabe, that's what you are, Harry. Can't be much older than you are now."
"Keep it," she says to Harry.
"You sure?"
Sophie nods. "I'm sorry," she says quietly to him. Do the right thing. Her mother's voice comes to her. She throws the file of notes and photographs onto the fire, watches the paper buckle and hiss in the flames, the faces of Gabriel, his father, and Vita disappear. Harry stands close beside her.
"Is that your story?"
"It was."
"I don't get it? Why...?"
"I can't do this to Gabe now. I never would have hounded him if I'd known about Annie." She pauses. "Besides, something my mom said really hit home. I've got to let the past go and live my own life. What good would it do now, pulling apart Gabriel's life for the sake of a story? Sometimes we need to believe in fairy tales."
"Thank you." Harry squeezes her hand.
"I'd better be on my way, Lil's waiting on me." Marv pulls down his cap. "See you kids in June at the start of the season?"
Tom hesitates a moment too long. "Sure."
"You are going to bring Gabe out here again next year?"
"Marv, the doctors... they just don't know how long he's got left."
"Damn." Marv blinks rapidly, his yellow-tinged eyes pooling with tears. "I... Oh, damn it. This place just ain't going to be the same without Gabe and Annie."
"Maybe you'll come and see him in town?" Tom checks his watch. "Listen, we've got to get going. Harry, why don't you and Miss Cass—"
"Dr. Cass." He glances at her.
"Sophie," she says. "I'm so sorry. You've lost Annie, and now Gabriel?"
"The doctors said he has a little time," Tom says. "We just don't know how much. Harry, why don't you drive over to the diner to fetch Gabe while I get the kids in the car and the shutters bolted down before the light fades? Thanks, Marv."
"Sure, Dad," Harry says, and gestures to Sophie to join him. They begin to walk to his truck, his hand resting on the small of her back.
"No, that's just it," Marv interrupts. "I tried to call you, but your phone's been cut off. He walked out of there about ten minutes ago, still talking away to... well, her." He points at Sophie. "Damn it, you know what I mean."
"Which way did he go?" Albie starts to run toward the coast path.
"He was heading toward the beach...."
"Mary Jayne!" Tom calls out to his sister. "Dad's missing."
* * *
They come running then, my children, Sophie among them. I honored Varian just like I said I would—the names of the good people who changed the course of my life live on in my children and in theirs. Their heels kick up the sand as they run to the crest of sea grass. My grandchildren run with them, down the steps, across the empty white sands.
"Dad!" Tom yells.
Then they see me, lying on the shore, a streak of palest blue.
FIFTY-NINE
FLYING POINT, LONG ISLAND
2000
GABRIEL
I am here, and there, Vita is right—everywhere and nowhere. With them, and not.
"Not bad," Vita says, looking around the walls of my studio. "A Chagall, Matisse..." She peers closer. "Duchamp? These should be in a museum."
"They will be, soon," I say, "when the house is cleared." Oh, I know the kids haven't been able to tell me what I've known for weeks. My heart is giving out. Today they were going to take me away, away from my home of over fifty years. They are good kids, but I don't want to go. I take a last look at my collection, at the art that has filled my soul for years. "None of us would be here without Varian, without all of them. Sometimes I think we left our hearts at Air-Bel...." And I think of him, walking down the drive that last time when the police came and arrested him and forced him out of France. The last time I shook his hand, he was holding the copy of Terre des Hommes that Danny had given him. That's what great art does—that's why these men and women counted. It shows us what makes life worth living.
"Is this great art?" she says, gesturing at the stacks of blue canvases. "What is all this?"
"You have to imagine it, when it is installed." I picture the canvases laid out as I have numbered them, in a circular white-walled space. The blues merge to infinity, from darkest indigo through Prussian, ultramarine, cobalt, to heavenly cerulean and back, and there she is, a tiny white dot of life, Venus, the morning star, always there night and day. It is our endless sky, mine and my love's, the sky we saw when we lay on the beach and looked to heaven and talked of the future. "It's for Annie, always," I say.
"Tell me, do you ever think of me, of Vita?"
I beckon her on and go to a locked wooden chest in the corner of the room beneath the window. I take the key down from its hook, and the smell of cedar fills the air as I lift the lid. I lift out a fold of linen and unwrap the gold embroidered cloth.
"I carried you with me," I said as I put the gold wreath of myrtle leaves on her head. "I'm sorry I couldn't save you, and our child. I'm sorry I couldn't save my father. Forgive me."
"There," she says after a time, and touches my forehead. I feel myself lift, like a ship weighing anchor. "We got there in the end. It's time to let go," she says gently, and I am on the beach again, and my children are running toward me. "I'll wait with you, while they come," she says. "You carried us all. You weren't the only young boy who took on the burden and fortune of his family." She strokes my forehead. The light is flickering, flashing behind my eyes. I see my father's face next to mine, in the photograph. The light flares. Our faces merge, become one. My eyes blink open, see endless blue. "When I look into your eyes, I see the ghosts of your father and mother, of all of us, staring back." I know she tells the truth.
"I can't... what if, what if Sophie tells everyone? What will it do to my children if they find out I've lived a lie?"
"It will be okay, Gabriel, you're almost there, now." She soothes my brow. "I know some nights the weight of carrying us, of all that guilt, has been too much, but a new day always begins." She looks out across the beach. The children are running toward us. "You know, sometimes it's been like a spectral army standing shoulder to shoulder with you, fists in the air, showing them that they didn't win, that we live on and we didn't give in." She leans down and kisses my brow, and the light fills my eyes, all fades to white. "You'll live on in them, too. Nothing matters more than the love we leave behind." The sea, the light, it is so beautiful. "Relax. She's coming for you. Not long now."
* * *
"Dad?" I hear Tom's voice calling me back. I hear Harry's voice above him, frantic, calling for an ambulance.
"I know CPR." Sophie kneels beside me on the sand as they gather around. I feel her lips against mine, breathing life into me, her hands against my heart. I am like an old engine, winding down. Just for a moment, it catches. "Gabriel," she says, turning my face to hers. "Gabriel, stay with me."
I try to raise my hand to touch her face. "Vita?"
"Sophie, my name is Sophie Cass. Jack and Paige's daughter." She smiles down at me. There are tears in her eyes. "Vita was my great-aunt."
"You look... like her." My voice is a whisper. Sophie leans closer, unable to hear. "Please, don't destroy"—I gasp for air—"my family."
"I won't. Vita loved you. Both of you. That's what matters, not some story."
Relief floods through me, and I close my eyes.
"She's proud... of you."
* * *
It's true, you know, what they say, that your life flashes before you. I feel Sophie's hands on me again, her urgent lips forcing air into my lungs as I slip back, but I am far away, free at last to run down the years as the people I love most in this world gather around me, and the sea rolls on regardless. I see my arm fall to my side, the shepherd tumble one last time from my hand.
Then, she is there. Annie, my Annie, is walking along the shoreline, and she is young, and beautiful, and she has the sun in her. She's come back, this woman I have loved my whole life, and she has come for me.
And it has been a good life, because of one good man. And I have lived a simple life and done good work. And I have loved, oh, I have loved. My heart is light with thankfulness.
I see her now, Annie, my Marianne, dancing and turning in the snow at Air-Bel, dancing to the music of life itself, and I see her, and I feel her touch. I see her holding our children, and I see them, and I feel them, I am back, I am back. I never knew, I never knew I was alive until I held them.
I see her now.
Annie walks to me with the fluid grace of the girl. She tilts her head as if to say, What are you doing there, Gabe? and she kneels beside me in the sand, takes me safe in her arms. Annie lies down beside me, and I'm not scared anymore. I've missed her, so much, but now she is here with me, and we'll never be apart again. She lies with me, and I am home. I am filled with so much joy I could fly right up there above the beach, the sea, our world, with her. I can't bear it, it is so beautiful.
It all falls away.
I loved, I am love, I am free.
AUTHOR'S NOTE
This novel weaves fiction with factual events at the American Relief Center (Centre Américain de Secours) in 1940–41. My admiration for Varian Fry and his team grew immeasurably the more I learned about them and the artists they saved. The "real" characters are fictional versions of just a few of the remarkable people involved:
Varian Fry (born 1907) was arrested and forced out of France on August 29, 1941. The ARC was raided and closed down on June 2, 1942, but the members of his team continued his work underground.
Fry received little thanks for his remarkable work during his lifetime. Now, the Consulate General of the United States in Marseille sits on place Varian Fry. After the war he divorced Eileen, but he married again and raised a family. In 1971, when Fry published the Flight portfolio of prints in aid of the International Rescue Committee (the organization that the ERC became), he struggled to convince artists to take part—though he was responsible for saving many of their lives.
He was honored with the International Rescue Committee's medal in 1963 and the Croix de Chevalier de la Légion d'Honneur by the French government in 1967. He died alone in his sleep later that year at only fifty-nine. A manuscript lay at his side. Varian Fry died surrounded by his incomplete notes for a new memoir, by his memories of Marseille.
In 1991, the U.S. Holocaust Memorial Council awarded him the Eisenhower Liberation Medal. In 1994, he was named "Righteous Among the Nations" by Yad Vashem—an honor bestowed on non-Jews who helped Jews during the Holocaust. It was an honor shared with Oskar Schindler and Raoul Wallenberg. He was the first American to be honored in this way. Warren Christopher (U.S. secretary of state at the time) said: "Even today, Varian Fry's tale of courage and compassion is too little known in the United States.... We owe Varian Fry our deepest gratitude, but we also owe him a promise—a promise never to forget the horrors that he struggled against so heroically, a promise to do whatever is necessary to ensure that such horrors never happen again."
SOME OF FRY'S COLLEAGUES
Danny Bénédite was released from prison and took over running the ARC after Varian was forced out of France. The onetime police official went on to become a leader of the underground Resistance. He died in 1990.
Hiram "Harry" Bingham IV (born 1903), Varian's "partner in crime," was abruptly removed from his diplomatic post as vice-consul in Marseille in 1941. He went on to serve in Portugal and Argentina, where he traced Nazi war criminals. When he was overlooked for promotion, he left the U.S. Foreign Service in 1945. Like Fry, Bingham has been honored with several awards for his remarkable humanitarian work. He died in 1988.
Dr. Miriam Davenport (born 1915, Boston) married her fiancé, Rudolf, and escaped with him to America. They divorced, and she remarried twice. She worked for Albert Einstein and became a prize-winning artist. She gained a Ph.D. in 1973 and died in 1999.
Charles Fernley Fawcett (born 1915) followed up his daring work at the ARC with spells in the RAF and the French Foreign Legion. After the war, he performed in more than a hundred films and was a veteran of several conflicts. He continued to help humanity as a freedom fighter and was involved in Afghanistan in the 1970s and the cause of refugees. This modest hero received many decorations, including the French Croix de Guerre and the Eisenhower medal, and died in London in 2008.
Bill Freier (Wilhelm Spira) survived Auschwitz and a series of concentration camps including Buchenwald and Theresienstadt. Mina gave birth to their son during his imprisonment but suffered a nervous breakdown after the war and died in 1953. Spira died in 2000.
Mary Jayne Gold (born 1909, Chicago) escaped to the United States in 1941. After the war, she divided her time between New York and the south of France and named her Saint-Tropez villa "Air-Bel." She never married or had children, but she maintained her contact with Killer. She died in France in 1997.
Raymond Couraud escaped France in April 1941. Killer rewarded Mary Jayne's faith in him and became a hero of the British Special Operations Executive and the Special Air Service.
Albert O. Hirschman (born 1915, Berlin; aka Albert Hermant, aka "Beamish") had a long and distinguished academic career and became one of the world's leading experts on economics. He taught at Columbia, Yale, and Harvard Universities and was Professor Emeritus at Princeton University. He died in 2012.
Justus "Gussie" Rosenberg (born 1923) attempted to escape via the Pyrenees but was caught and arrested. He escaped, joined the Resistance, and managed to get to the United States after the war, where he completed his studies. He is Professor Emeritus of Languages and Literature at Bard College and co-director of the Varian Fry Foundation.
SOME OF THE CLIENTS
André (1896–1966) and Jacqueline (1910–1993) Breton arrived safely in New York after being detained in Martinique. They later divorced, and both remarried. Their daughter, Aube Breton-Elléouët (born 1935), is a distinguished visual artist.
Marc Chagall (born 1887) was finally convinced to leave France after he was arrested by the police in Marseille. Fry secured his release. He died in 1985.
Peggy Guggenheim (born 1898) and Max Ernst (born 1891) finally met in Marseille. Max said, "When, where, and why shall I meet you?" Peggy said, "Tomorrow, four, Café de la Paix, and you know why." They escaped to the United States together and married. They later divorced, and Guggenheim established a museum of modern art in Venice, founded on the paintings she rescued from war-torn France. She died in 1979 and Ernst in 1976.
* * *
As Fry himself complained writing his memoir, Surrender on Demand, trying to write this story with its "hundreds of characters is worse than War and Peace." In the confines of fiction, it is necessary to simplify the true story—it was not possible to include all the people involved in the remarkable rescue operation in Marseille, but this in no way diminishes the contribution of Fry's unnamed colleagues. The Varian Fry Institute and Varian Fry Foundation both have excellent Web sites with information about all the people who helped Fry in Marseille and the full events of this time.
Many more of the world's greatest artists and intellectuals were saved than the original two hundred names on Fry's list. Some fifteen thousand people came to the ARC seeking help, many of them "ordinary" relief cases. The organization gave aid to more than 560 families and took food parcels to those detained in concentration camps. More than two thousand people were rescued.
There is a dedication at Harvard University that expresses Varian Fry's tenacious humanitarian spirit well: "To one who dared to defy authority and attempts at restraining the human impulse for good." The Emergency Rescue Committee of New York, for which he did such remarkable work, became the International Rescue Committee—it continues to operate in more than thirty countries, aiding refugees and victims of oppression whenever and wherever it is needed. Fry's legacy lives on.
ACKNOWLEDGMENTS
I would like to thank Professor Justus "Gussie" Rosenberg for his generous help writing this novel and Aube Breton-Elléouët for her kind permission to conjure a version of her five-year-old self. Thank you to Dr. Sarah Wilson, Jean-Jacques Lebel, Pierre Sauvage, Richard Kaplan, Paul B. Franklin, and Marisa Bourgoin of the Archives of American Art; Michelle Harvey of the Museum of Modern Art, New York; Constance Krebs of the Association Atelier André Breton; and Laurene Leon Boym for their help with my research.
Thank you to Professor Jon Stallworthy and Lorna Beckett of the Rupert Brooke Society, for their kind permission to quote from Rupert Brooke's "The Great Lover."
My thanks to the incomparable Sheila Crowley, Rebecca Ritchie, and the team at Curtis Brown and to my wonderful editor, Anne Brewer, and all at Thomas Dunne for their help with this story.
As ever, love and thanks to my husband and family for their support and encouragement. This much. Always.
Also by Kate Lord Brown
The Perfume Garden
ABOUT THE AUTHOR
Kate Lord Brown is the internationally bestselling author of The Perfume Garden. She grew up in a wild and beautiful part of Devon, England, and was first published while at school. Kate won the BBC International Radio Playwriting Competition, Middle East region, in 2014; was a finalist in ITV's The People's Author competition 2009; and has a master's in creative writing. The Perfume Garden was short-listed for the UK Romantic Novel of the Year 2014. She lives in the Middle East with her family. Visit her at www.katelordbrown.com, or you can sign up for email updates here.
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CONTENTS
Title Page
Copyright Notice
Dedication
Epigraph
From the Desk of Sophie Cass
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Chapter 27
Chapter 28
Chapter 29
Chapter 30
Chapter 31
Chapter 32
Chapter 33
Chapter 34
Chapter 35
Chapter 36
Chapter 37
Chapter 38
Chapter 39
Chapter 40
Chapter 41
Chapter 42
Chapter 43
Chapter 44
Chapter 45
Chapter 46
Chapter 47
Chapter 48
Chapter 49
Chapter 50
Chapter 51
Chapter 52
Chapter 53
Chapter 54
Chapter 55
Chapter 56
Chapter 57
Chapter 58
Chapter 59
Author's Note
Acknowledgments
Also by Kate Lord Brown
About the Author
Copyright
This is a work of fiction. All of the characters, organizations, and events portrayed in this novel are either products of the author's imagination or are used fictitiously.
THOMAS DUNNE BOOKS.
An imprint of St. Martin's Press.
THE HOUSE OF DREAMS. Copyright © 2016 by Kate Lord Brown. All rights reserved. For information, address St. Martin's Press, 175 Fifth Avenue, New York, N.Y. 10010.
www.thomasdunnebooks.com
www.stmartins.com
Grateful acknowledgment is made for permission to reproduce from the following:
"The Great Lover" by Rupert Brooke, 1914, is used with the kind permission of the Rupert Brooke Society.
Cover design by Lisa Marie Pompilio
Cover photographs: woman © Jayne Szekely/Arcangel; landscape © Ayal Ardon/Arcangel
The Library of Congress Cataloging-in-Publication Data is available upon request.
ISBN 978-1-250-08453-8 (hardcover)
ISBN 978-1-250-10982-8 (Canadian)
ISBN 978-1-250-08454-5 (e-book)
e-ISBN 9781250084545
Our e-books may be purchased in bulk for promotional, educational, or business use. Please contact the Macmillan Corporate and Premium Sales Department at 1-800-221-7945, extension 5442, or by e-mail at MacmillanSpecialMarkets@macmillan.com.
First Edition: May 2016
First Canadian Edition: May 2016
## Contents
1. Title Page
2. Copyright Notice
3. Dedication
4. Epigraph
5. From the Desk of Sophie Cass
6. Chapter 1
7. Chapter 2
8. Chapter 3
9. Chapter 4
10. Chapter 5
11. Chapter 6
12. Chapter 7
13. Chapter 8
14. Chapter 9
15. Chapter 10
16. Chapter 11
17. Chapter 12
18. Chapter 13
19. Chapter 14
20. Chapter 15
21. Chapter 16
22. Chapter 17
23. Chapter 18
24. Chapter 19
25. Chapter 20
26. Chapter 21
27. Chapter 22
28. Chapter 23
29. Chapter 24
30. Chapter 25
31. Chapter 26
32. Chapter 27
33. Chapter 28
34. Chapter 29
35. Chapter 30
36. Chapter 31
37. Chapter 32
38. Chapter 33
39. Chapter 34
40. Chapter 35
41. Chapter 36
42. Chapter 37
43. Chapter 38
44. Chapter 39
45. Chapter 40
46. Chapter 41
47. Chapter 42
48. Chapter 43
49. Chapter 44
50. Chapter 45
51. Chapter 46
52. Chapter 47
53. Chapter 48
54. Chapter 49
55. Chapter 50
56. Chapter 51
57. Chapter 52
58. Chapter 53
59. Chapter 54
60. Chapter 55
61. Chapter 56
62. Chapter 57
63. Chapter 58
64. Chapter 59
65. Author's Note
66. Acknowledgments
67. Also by Kate Lord Brown
68. About the Author
69. Newsletter Sign-up
70. Copyright
## Guide
1. Cover
2. Table of Contents
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See Inside This Architect-Owned Modernist Home in Arlington
Yours for $1.279 million.
Written by Michelle Thomas
A rare find in Arlington, this five-bedroom home was reworked by its owner, an architect, and it's full of features that design fans will appreciate. A short list of the best details: The concrete fireplace mantle, chalkboards in several rooms, wall-width bookshelves, and a blend of natural wood trims, ample sunlight, and industrial accents that lend the place a creative, organic vibe. The sleek little studio office off the master suite and awesome roofed deck round out the modernist aesthetic. 6425 26th St. N was relisted at $1.279 million this morning, a $110,000 drop from its original asking price. Take a look inside below, then go to Frankly Real Estate for the complete details.
More: Listing We LoveOpen House
Contemporary Masterpiece in Heart of Dupont Circle
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Home > Lifestyle
Chester Borrows: Easy living in a lucky country
11 Jan, 2019 5:00pm 4 minutes to read
Local Focus: Legendary lifeguard's career spans 40 years
12 Jan, 2019 5:30am 2 minutes to read
LIFESTYLE | Food & Drink
Gluten intolerant Kiwis can eat bread overseas but not in New Zealand
12 Jan, 2019 5:00am
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As thousands of perfectly healthy people ditch gluten in the hope it might help them to lose weight, it seems there are even more for whom gluten really could be a problem, yet who are still tucking in to bread and biscuits completely unaware of the effect this could be having on their health. Source: asapSCIENCE
By: Amy Wiggins
amy.wiggins@nzme.co.nz
Melanie Camoin grew up eating baguettes in France.
In traditional French fashion it was a staple food in her household but when she moved to New Zealand in 2011, all that changed.
The Waihi resident found herself feeling bloated, having a sore stomach, bowel issues, mood swings and becoming lethargic.
"It's just terrible. I feel so s*** and so tired. I had to sleep 12 hours a day and had no energy at all."
Three months later she cut out bread and wheat products like pizza, pasta and pastries and the symptoms stopped.
"I actually was poisoning myself."
But when she goes home or travels to other parts of Europe, or even the United States, she can dig in to all the gluten-filled food she can't usually eat and feels fine. She can even make her own food using T55 flour imported from France with no problem.
And she's not alone.
So why can many non-Coeliac gluten-sensitive Kiwis eat wheat-based foods overseas without experiencing the gut-wrenching pain they feel at home?
Principal scientist at Plant and Food Research Nigel Larsen is trying to answer just that question.
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"It is an issue which seems to be real but we don't know why," he said. "So far it's a mystery to us as to why we hear stories like that because that's one of the things that prompted us to start doing research on the issue.
"There's all sorts of things that could be different. It could be the wheat varieties, it could be the way we grow our wheat, it could be the way we process our wheat – who knows. It's just something we don't understand and we're trying to get to the bottom of."
When in Europe, Melanie Camoin can dig in to all the gluten-filled food she can't usually eat in New Zealand and feels fine. Photo / Alan Gibson
Plant and Food Research have teamed up with the Baking Industry Association of New Zealand to fund research into where the differences could be and why wheat seemed to have such a huge affect on many Kiwis.
Larsen had already looked into the way dough was mixed in New Zealand but did not find an answer there.
He had started studying proteins called amylase-trypsin inhibitors which were present in wheat. Their function in grains was to stop insects from eating them and there had been research suggesting they may cause inflammation. But, there was nothing to indicate the levels in New Zealand were any higher than overseas, he said.
Larsen was also looking into the proteins that aggravated Coeliac disease and how they could breed new wheat varieties with lower levels as well as which sourdoughs lowered the gluten levels of bread.
The yeast and bacteria in a sourdough starter worked together to digest the gluten proteins meaning the gut did not have to work as hard to get rid of them.
But not all sourdoughs are the same, Larsen found. While he was hesitant to divulge which types were better than others, he did reveal San Francisco sourdough was better for those intolerant to gluten.
Close-up of the expensive bread Waihi Beach woman Melanie Camoin is now forced to buy after becoming gluten intolerant since moving to New Zealand from France. Photo / Alan Gibson
Similar to sourdoughs, the rising process in bread could also play a part, but that would not explain the difference when eating pastas and pastries, Baking Industry Association of New Zealand president Kevin Gilbert said.
When bread rises, a fermentation process is taking place where enzymes begin to break down and convert proteins like gluten. The longer bread is fermented for, the more the proteins are broken down and the easier it becomes for the gut to process.
In the 60s, a new process of bread making was developed called chorelywood. It allowed bakers to go from flour to a loaf of bread bagged in about three hours. The traditional style of bread-making involved anywhere from three to 60 hours of fermenting alone, Gilbert said.
That was part of the reason many artisan breads, which were more common in Europe, were easier to stomach than loaves of sliced bread from the supermarket, he said.
While all wheat contained the same proteins, different grades of flour were used in Europe and had a different protein ratio whereas New Zealand flour usually only came in one grade.
Traditional Italian pasta was also usually made from durum flour rather than wheat flour, he said.
Professor of Nutrition at the Liggins Institute David Cameron-Smith agreed some types of flour had less gluten and when those low-gluten flours were used to make bread in the traditional style they had less impact on those sensitive to gluten.
David Cameron-Smith, Professor of Nutrition at the Liggins Institute. Photo / Supplied
He believed part of the problem was that we had become reliant on high gluten strains of wheat that allowed bread to rise and become soft in a very short period of time.
But, it was possible gluten was not the issue at all for many people.
Paediatric gastroenterologist at Otago University in Christchurch and Christchurch Hospital, Professor Andrew Day said it could be the sugars from carbohydrates which were the problem, and Larsen agreed.
Cereals like wheat all contained sugars, such as fructose, but some strains contained more than others, Day said.
Overseas studies had shown that many people who reacted badly to cereals were reacting to the sugars rather than the gluten, he said.
"If those sugars can not all be absorbed in the small bowel they get through to the large bowel and get fermented which contributes to gas production leading to bloating, diarrhoea, belly pain and so on," he explained.
He believed the most likely answer to the mystery was that strains of wheat with higher amounts of sugar triggered symptoms in people whose bodies did not absorb the sugar as well.
But, for now, those like Camoin will have to rely on a process of trial and error to work out what they can and can't eat.
What is gluten?
• A general name for the proteins found in wheat, rye, barley and triticale.
• Gluten helps foods maintain their shape, acting as a glue that holds food together.
• Gliadin is the protein in wheat which coeliacs react to.
• It is estimated about 1 in 70 New Zealanders have Coeliac disease but many more people believe they are intolerant to gluten (non-Coeliac gluten sensitivity).
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The remake looks amazing but does it charm like the animated original?
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{"url":"http:\/\/www.pearltrees.com\/u\/7493560-american-mathematical-homepage","text":"# American Mathematical Society\n\nRokhlin actions of finite groups on UHF-absorbing $\\mathrm{C}^*$-algebras This paper serves as a source of examples of Rokhlin actions or locally representable actions of finite groups on -algebras satisfying a certain UHF-absorption condition. We show that given any finite group and a separable, unital -algebra that absorbs tensorially, one can lift any group homomorphism to an honest Rokhlin action of on . belongs to a certain class of -algebras that is classifiable by a suitable invariant (e.g. -theory), then in fact every -action on the invariant lifts to a Rokhlin action of . of a Rokhlin action on a UHF-absorbing -algebra, an inductive limit decomposition is obtained in terms of and . is assumed to be abelian, then the dual action is locally representable in a very strong sense. -groups. -algebras to a question about certain finite group actions on\n\nHealthFitness At HealthFitness, we are always looking for top talent to join our team \u2013 leaders, strategic thinkers, team players and innovators who are dedicated to our mission of improving the health & well being of the people we serve. If you share this passion for making a difference in people\u2019s lives, we invite you to apply on-line today. HealthFitness is an equal opportunity\/affirmative action employer and we do not discriminate in hiring or employment on the basis of race, religion, color, sex, age, national origin, disability, veteran status, genetic information, or any other characteristic protected by law. HealthFitness provides reasonable accommodations to applicants and employees with disabilities and takes affirmative action to ensure that hiring and employment decisions are made without regard to protected class status. If you need a reasonable accommodation as part of the application process, please contact our Recruiting Department at 1-800-636-3304.\n\nPhysics Summer School Physics Summer School *This event is suitable for students who have completed AS Level Maths, or reached a similar standard in other qualification structures. This five day Summer School will take students on a tour through the exciting world of further physics study. The goals are threefold. Mathematics Books Online General Elementary Algebra & Trigonometry Abstract Algebra Mathematical Analysis & Calculus Differential Equations Linear Algebra\n\nWeb of Knowledge [v5.5] - IP Error WEB OF Science Your ideal single research destination to explore the citation universe across subjects and around the world. Web of Science provides you access to the most reliable, integrated, multidisciplinary research connected through linked content citation metrics from multiple sources within a single interface. And since Web of Science adheres to a strict evaluation process, you can be assured only the most influential, relevant, and credible information is included - allowing you to uncover your next big idea faster.\n\n10 Websites for College Students to Find Jobs and Internships Compiled by Forbes, the following is a summary of what are considered to be the best of the best when it comes to finding a job, networking and helping your career take flight. LinkedIn.com This website, not even ten years old, has been shown through research to be the number-one resource for hiring managers, HR personnel and prospective employers. Math Camps: Summer Camps and Tutors for Mathematics Kids! Math Camps create a vibrant community of kids sharing a common love of learning about and passion for mathematics. Mathematics Camps are designed for students with the mission of providing a challenging, intellectually stimulating environment where they can learn all about this stimulating subject. Only at a math camp can your mathematical-minded child explore his or her genius.\n\nGowers's Weblog A few analysis resources March 12, 2014 This will be my final post associated with the Analysis I course, for which the last lecture was yesterday. It\u2019s possible that I\u2019ll write further relevant posts in the nearish future, but it\u2019s also possible that I won\u2019t. This one is a short one to draw attention to other material that can be found on the web that may help you to learn the course material. It will be an incomplete list: further suggestions would be welcome in the comments below. The Dental Assistant Career Path Find the path that's right for you As a dental assistant, you have so many options to grow professionally and expand your career. The biggest challenge you\u2019ll have is figuring out where you want to grow. When weighing your options, it is important to keep in mind your state\u2019s requirements and available job titles. Once you have that covered, there are two main types of dental assistant career paths you can follow: broadening your role chairside, or shifting your focus in the dental field.\n\nFaculty of Science Helix Summer Science Institute runs for four weeks during the month of July at York University\u2019s Keele Campus. Interested applicants can find out how to apply here. Helix 2015 Schedule Public Service Loan Forgiveness The Public Service Loan Forgiveness (PSLF) Program forgives the remaining balance on your Direct Loans after you have made 120 qualifying monthly payments under a qualifying repayment plan while working full-time for a qualifying employer. What is qualifying employment?What is considered full-time employment?\n\nRelated:\u00a0\u00a0Professional Organizations","date":"2019-06-24 23:00:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.1810685098171234, \"perplexity\": 1955.8115834371426}, \"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-2019-26\/segments\/1560627999740.32\/warc\/CC-MAIN-20190624211359-20190624233359-00426.warc.gz\"}"}
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← rewrites are from h-e-l-l. i know it.
How is it the beginning of a new year? I feel like I've lived in a cave this past year with all of the writing I've done. Well over a million words in total–5 novels, 2 short stories, edits, rewrites, and blogging. I was always told I was talkative.
In 2013: Last year, I learned quite a few things about becoming an author. The querying process alone is still a work in progress, but I think I've improved and thanks to some helpful tips from other authors, I may have it down now. I learned that when I'm on a roll in my writing, I can't stop the story, even if it means continuing on with a second or third book without having the first one published. I learned that if you self-publish a book, it's frowned upon in most publishing circles–as in you have to sell 100,000 copies of your book at $2.99 and up or something like that to be considered a true author. I learned to properly format a draft, and how to write continuously on that draft.
I know there will be more learning as I go through the processes this year. I have three different novels in three different genres to pitch now. I'm currently learning how to rewrite an entire manuscript without losing my voice, but taking out what truly isn't necessary… which is the most difficult part of this process. Deletion of certain things means other things will have to change in the other two novels, and I have to remember to change them or else there will be this weird little conundrum that happens. In Back to the Future speak, it would create a paradox of epic proportions.
In 2014: I hope this year will bring an agent, and maybe a publish date for 2015 at the latest. I still intend on self-publishing that story I wrote years ago, if I can ever get done creating new ones… just to learn if I like self-publishing. I have three stories completely planned and intend on writing, although I would love to finish more than five stories this year. And I just hope that I can remember to keep it fun. I have managed not to get all upset or stressed with rejection letters, and I hope to keep that up as well. I just ask that someone is willing to tell me why I was rejected, because then, I can work on it.
And lastly, I'd like to thank all the authors, agents, and producers I've spoken with over Twitter and at SD Comic Con who have given me valuable information and some of the best pep talks I have ever had. Without their encouraging words, I'd feel more alone in this process. And what a weird, wonderful process it is. I'd also like to thank you all for reading this blog and commenting, also some great helpful advice and some personal thoughts that have made this a great journey this past year. Hope to have more of it this year.
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De 7. SS-Freiwilligen-Gebirgs-Division Prinz Eugen was een divisie van de Waffen-SS. De eenheid had haar oorsprong in maart 1942 en gaf zich op 11 mei 1945 over aan Joegoslavische partizanen.
Gedurende het grootste deel van haar bestaan heeft de divisie op de Balkan gediend. In april 1945 werd de divisie overgeplaatst naar Oostenrijk waar ze tijdens de rest van de oorlog verbleef.
Het embleem van de divisie was een odal rune, die oorspronkelijk 'geërfd eigendom' betekende en door de nazi's gebruikt werd in verband met hun "bloed en bodem"-theorieën.
Geschiedenis
Oprichting en formatie
De 7. SS-Freiwilligen-Gebirgs-Division Prinz Eugen werd in oorsprong gevormd in maart 1942, uit Volksduitsers uit de Kroatische en Servische gebieden van Joegoslavië. Ook uit Hongarije en Roemenië kwamen vrijwilligers om in de divisie te dienen. De eerste lichting van de divisie bestond vrijwel volledig uit vrijwilligers, maar dit veranderde naarmate de oorlog vorderde en mannen in dienst werden gedwongen.
De divisie werd genoemd naar Prins Eugène van Savoye, een commandant van de Heilig Roomse Keizer Leopold I. Hij vocht als commandant verschillende slagen uit in de Balkan, en wegens deze link verkreeg de divisie haar naam.
In juni 1942 bestond de divisie uit 21.500 soldaten en werd ze bewapend met buitgemaakte wapens en tanks: vooral Tsjechische machinegeweren en Franse lichte tanks. Na haar formatie werd de divisie naar Joegoslavië gestuurd om het daar in de bergachtige gebieden op te nemen tegen de partizanen.
Wapenfeiten in de periode 1942-1945
In oktober 1942 kwam Prinz Eugen voor het eerst in actie nabij de Servisch-Montenegrijnse grens, ten oosten van de Ibar. Na deze actie werd de divisie overgeplaatst naar het gebied rond Zagreb, waar ze deelnam aan Fall Weiss_I. Net zoals vele andere Duitse offensieven tegen de partizanen van Josip Tito liep het offensief op een sisser uit: de partizanen slaagden er telkens in om de hoofdaanval van de As-mogendheden te ontwijken. Aangezien de doelen niet waren gehaald, werd op 25 februari 1943 begonnen met Fall Weiss_II, maar ook dit offensief kon geen succes genoemd worden. Operatie Wit III stond gepland, maar werd uiteindelijk afgelast.
In mei nam de 7e SS-Gebirgs-Division deel aan Fall Schwarz, tegen de Servische partizanen in Montenegro en Herzegovina. Dit offensief verliep voorspoediger dan de voorgaande acties tegen de partizanen en de rebellen moesten zich terugtrekken naar Servië. Na de oorlog werd de divisie beschuldigd van verscheidene oorlogsmisdaden tijdens operatie Zwart.
Enkele maanden later, in augustus, werd Prinz Eugen overgeplaatst naar het Duitse XVe Gebirgs-Armeekorps en naar de Dalmatische kust gezonden. Na de wapenstilstand tussen Italië en de Geallieerden hielp de divisie daar de Italiaanse troepen te ontwapenen. Vervolgens kreeg Prinz Eugen bevel om verschillende eilanden, met name Hvar, Brač en Korcula, eveneens als het schiereiland Pelješac te beveiligen.
De divisie werd opnieuw overgeplaatst, nu naar het Ve SS Freiwilligen-Gebirgs-Korps, en nam deel aan operatie Kugelblitz en operatie Schneesturm, gericht tegen de partizanen van Tito. Opnieuw konden de partizanen ontkomen. De volgende maand werd de divisie uit de strijd gehaald en naar Split overgebracht, waar ze een extra opleiding kregen. Deze periode duurde twee maanden en in maart 1944 nam Prinz Eugen terug haar positie in om deel te nemen aan verdere acties tegen de partizanen, zoals operatie Meibaum in april.
In mei 1944 ondernam Prinz Eugen samen met SS-Fallschirmjäger-Batallion 500, het 1. Brandenburg Regiment en massale ondersteuning van de Luftwaffe een aanval op het tijdelijke hoofdkwartier van Tito nabij de stad Drvar. Het doel van de aanval was om Tito gevangen te nemen of te doden. De operatie mislukte en Tito wist te ontsnappen. Van mei tot juli nam de divisie deel aan haar laatste operaties tegen de partizanen. Vanaf augustus werd Prinz Eugen in de strijd geworpen tegen de oprukkende Sovjets. De divisie leed daarbij zware verliezen. Op 21 september verloor de divisie haar eerste commandant, SS-Obergruppenführer Artur Phleps.
In november 1944 nam de 7e SS Freiwilligen-Division deel aan een cruciale actie in Macedonië. Samen met de 13. Waffen-Gebirgs-Division der SS Handschar (kroatische Nr. 1) en de overblijfselen van de 23. Waffen-Grenadier-Division der SS Kama (kroatische Nr. 2) en 21. Waffen-Gebirgs-Division der SS Skanderbeg (albanische Nr. 1), wist Prinz Eugen een corridor tot stand te brengen tussen 350.000 Duitse soldaten die terugtrokken uit Griekenland en de Duitse linies in Macedonië. De divisie kwam zwaar gehavend uit de strijd, maar de operatie was een succes. SS-Kavallerie-Abteilung 7 werd wegens te weinig manschappen ontbonden en de overblijfselen van de Gebirgs-Division Skanderbeg werden opgenomen in SS Freiwilligen-Gebirgs-Regiment 17 van Prinz Eugen. Vanaf dan vocht de divisie in de achterhoede van de terugtrekkende Duitse troepen uit Joegoslavië.
In januari werd er nog hevig gevochten tussen de divisie en eenheden van de Sovjets en de partizanen van Tito nabij Otok en Vukovar. In april bereikte Prinz Eugen Kroatië. Daar kreeg de divisie bevel om stand te houden bij Karlovac tot 2 mei 1945. Na die datum trok de divisie zich verder terug, in de hoop de Geallieerde linies te bereiken en zich niet te hoeven overgeven aan de Sovjets of de partizanen. Op 10 mei werd Celje in Slovenië bereikt, maar het was te laat. Een dag later gaf de divisie zich over aan eenheden van de partizanen.
Bekende oorlogsmisdaden
Een bataljon van de divisie heeft deelgenomen aan wraakacties op de dorpen Dorfer Otok, Cornji, Ruda en Dolac Delnji in Dalmatië. Deze acties leverden uiteindelijk 834 doden op.
Sonderkommando Bothmann (ook bekend als Sonderkommando Lange), verantwoordelijk voor moordpartijen in vernietigingskamp Chełmno, werd in april 1943 overgeplaatst naar de divisie.
Commandanten
Samenstelling
SS Freiwilligen-Gebirgs-Regiment 13 "Arthur Phelps"
SS Freiwilligen-Gebirgs-Regiment 14 "Skanderbeg"
SS Freiwilligen-Gebirgs-Artillerie-Regiment 7
SS Panzer Abteilung 7
SS Panzer Compagnie
SS Gebirgs-Panzerjager-Abteilung 7
SS Kavallerie-Abteilung 7
SS Angreif-Kanonen-Abteilung 7
SS Flak-Abteilung 7
SS Flak Compagnie
SS Gebirgs-Intelligenz-Abteilung 7
SS Freiwilligen-Gebirgs-Aufklarüngs-Abteilung 7
SS Panzer-Aufklärungs Zug
SS Zweirad-Battalion
SS Zweirad-Aufklärungs-Abteilung 7
SS Gebirgs-Techniker Batallion
SS Gebirgs-Gewehr Batallion
SS Zufuhr Compagnie 7
SS Lagerhalter Batallion 7
SS Medizinischen Abteilung 7
SS Freiwilligen-Gebirgs-Veterinär Compagnie 7
SS Freiwilligen-Gebirgs-Aufklärungs Zug 7
SS Propaganda Zug
SS Feldwebel Zug 7
SS Feld Ersetzung Batallion 7
SS Reparatur-Abteilung 7
SS Geologische Batallion
Lijst met onderscheidingen
Bronnen
Vikingrune.com - symbolen van het Derde Rijk
Axishistory.com
Feldgrau.com
07
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,652
|
\section{Introduction}
Since the advent of neural networks and deep learning, major breakthroughs have
been made in many artificial intelligence tasks ranging from computer vision to natural language
processing.
However, a significant \emph{knowledge gap} still exist between machine and
human intelligence. In particular, humans often relate to and make
use of semantic knowledge outside of the task-specific data to make better decisions.
On the other hand, most machine learning algorithms including state-of-the-art deep
methods, only focus on the representation of the given data,
without leveraging any external knowledge that could benefit the given task.
Consider the surfing man example in
Figure~\ref{fig:surfing}(a). By only analyzing the pixels of the image (i.e.,
given data), it is difficult to conclude that the man is on a surfboard since most of
it is obsecured by the waves. However, given the
knowledge that a man cannot stand freely on water and surfing is a typical
sport at sea, it is straightforward to identify the surfboard in
the picture. Note that the knowledge crucial to recognizing the
surfboard is external to the raw data. Likewise, in the zoo example in
Figure~\ref{fig:surfing}(b), with the knowledge that
man-made structures containing polar bears are most likely zoos, the video can
be correctly classified as zoo even though it is not evident from the appearance
of the structure in the frames.
\begin{figure*}
\begin{subfigure}[t]{0.35\textwidth}
\subcaption{Image from Microsoft COCO\footnotemark}
\centering
\includegraphics[height=3.2cm]{figures/surf.jpg}
\end{subfigure}%
\begin{subfigure}[t]{0.64\textwidth}
\subcaption{Video frames from YouTube\footnotemark}
\centering
\includegraphics[height=3.2cm]{figures/zoo1.png}\hspace{2mm}
\includegraphics[height=3.2cm]{figures/zoo2.png}
\end{subfigure}
\caption{Bridging the knowledge gap between how humans and
machines learn in visual tasks: (a) Recognition of an obscured surfboard in an
image; (b)
Classification of zoo where the raw pixels in the video frames do not clearly
indicate a zoo.
\label{fig:surfing}}
\end{figure*}
In this paper, we study the problem of video classification.
In contrast to traditional ``knowledgeless'' models, we aim to design an
end-to-end ``knowledge-aware'' framework that can integrate external knowledge
into the learning process. The incorporation of knowledge is especially
critical to large-scale video classification benchmarks such as the recently
released YouTube-8M dataset \cite{abu2016youtube}, which presents two major
challenges \cite{wang2017truly}.
First, videos can be very diverse in nature, with vastly different topics
(e.g., sports, politics, entertainment, etc.) and genres (e.g., animation,
documentary, etc.). Second, the class distributions are highly imbalanced,
where majority of the classes have only very few instances.
Such diversity and imbalance makes the classes not easily separable based
only on features in the videos. Thus, external knowledge can play a
vital role in complementing the video features to attain higher classification
performance.
More formally, knowledge is often represented as a \emph{knowledge
graph} \cite{DBLP:journals/semweb/Paulheim17}, modeling each real-world concept as a
node, and each semantic relationship between two concepts as an edge. A toy
knowledge graph is illustrated in Figure~\ref{fig:toygraph}. In
particular, the relationships ``person--on top of--surfboard'' and
``surfboard--found in--sea'' are likely to reinforce the recognition of
surfboard in Figure~\ref{fig:surfing}(a); similarly the relationship ``polar bear--live
in--zoo'' could help with the classification of zoo in
Figure~\ref{fig:surfing}(b).
While knowledge graphs have already seen widespread use in fields such as Web search and social
networks \cite{DBLP:conf/kdd/0001GHHLMSSZ14}, it has not been integrated
into visual tasks including video classification in a flexible and end-to-end fashion---Most existing knowledge-aware approaches are either
specific to a particular task and model, or applying external knowledge as a decoupled after-thought, which we will elaborate in related work.
\begin{figure}
\centering
\includegraphics[scale=0.3]{figures/toygraph.pdf}
\caption{Toy knowledge graph.\label{fig:toygraph}}
\end{figure}
Towards knowledge-aware video classification, we make the following
contributions in this paper.
\begin{itemize}
\item We propose to incorporate external knowledge graphs into video
classification, bridging the gap in existing state-of-the-art
approaches.
\item We unify knowledge graphs and machine learning including deep neural
networks in a novel end-to-end learning framework, which is flexible to work with most existing learning models.
\item We conduct extensive experiments on the largest public video dataset
YouTube-8M, outperforming state-of-the-art methods by up to 2.9\% in mean average precision.
\end{itemize}
The remainder of this paper is organized as follows. First, we review related
work. Next, we present the proposed approach, followed by experimental evaluation. Finally, we conclude our
paper and lay out directions for future research.
\addtocounter{footnote}{-1}
\footnotetext{http://cocodataset.org/}
\stepcounter{footnote}\footnotetext{https://www.youtube.com/}
\begin{figure*}[tbp]
\centering
\includegraphics[width=0.99\textwidth]{figures/overallframework.pdf}
\caption{Overall framework of the proposed end-to-end
learning with knowledge graphs (example video from YouTube).\label{fig:overall}}
\end{figure*}
\section{Related Work}
Video understanding has been an active research area in computer vision.
Significant progress has been made especially since the
release of large-scale benchmarks such as Sports-1M \cite{karpathy2014large},
YFCC-100M \cite{thomee2015new} and YouTube-8M \cite{abu2016youtube}.
The problem of video classification is usually addressed at frame or video
levels. The deep bag-of-frames (DBoF) \cite{abu2016youtube} is a typical
frame-level approach, inspired by various classic bag-of-words representations
\cite{laptev2008learning,wang2009evaluation}.
It feeds frame-level features from randomly sampled
input frames into a fully connected layer, whose parameters
are shared across the input frames.
Beyond a bag of frames, a video is naturally a temporal sequence of
frames, which can be modeled using a recurrent neural network.
Typically Long Short-Term Memory (LSTM) cells can be
employed to capture long-term dependencies in the temporal dimension
\cite{yue2015beyond}.
Furthermore, as an alternative to LSTM, Gated Recurrent Unit (GRU)
often achieves comparable if not better performance
\cite{chung2014empirical,chen2017aggregating}.
At video level, a fixed-length feature vector is often extracted from the
frames through simple aggregation (which we call ``aggregation of frame features'' or AoFF). As such, standard classifiers including
logistic regression and support vector machines can be adopted. In particular, the mixture of experts (MoE)
\cite{jordan1994hierarchical} classifier has shown superior empirical
performance on video-level representations \cite{abu2016youtube}.
In this model, a binary classifier is trained for each class, which is composed
of a set of ``experts'' or hidden states, and a softmax function is used to
model the probability of selecting an expert.
Apart from frame or video-level features directly extracted from the videos,
there is also initial success in exploiting text features associated with the videos,
such as the accompanied title and keywords in YouTube \cite{wang2017truly}.
All of the above methods are knowledgeless in the sense that they do not exploit external knowledge.
The use of external knowledge is emerging in some computer vision
tasks, including image classification
\cite{DBLP:conf/eccv/DengDJFMBLNA14}, motivation prediction \cite{DBLP:conf/cvpr/VondrickOPT16}, question answering
\cite{DBLP:conf/cvpr/WuWSDH16}, relationship extraction
\cite{DBLP:conf/eccv/LuKBL16}, as well as object detection
\cite{DBLP:conf/ijcai/FangKLTC17}. However, most of these works are task or
model specific, and thus cannot be easily applied to different scenarios.
While the recent work on object detection \cite{DBLP:conf/ijcai/FangKLTC17} can
work with any detection models, their proposed approach is not
end-to-end. Rather, it consists of two stages:
in the first stage, object localizations and class probabilities are obtained
using any existing model; in the second stage, the class probabilities are
re-optimized based on a knowledge graph. In particular, the use of knowledge in
the second stage is independent from the first stage, which means
there is a lack of feedback mechanism for the knowledge to directly improve
the parametrization of the existing model.
Finally, knowledge graph is a popular choice to represent external knowledge,
for capturing both concepts and their pairwise relationships. The
use of knowledge graphs have already demonstrated various degrees of success
in machine learning applications including Web search and social media
\cite{DBLP:conf/kdd/0001GHHLMSSZ14}.
Quite a number of large-scale knowledge graphs are available commercially or in
open source, which are
generally constructed based on human curation
\cite{DBLP:journals/cacm/Lenat95}, crowdsourcing
\cite{liu2004conceptnet,DBLP:journals/ijcv/KrishnaZGJHKCKL17}, and distillation
from semi-structured
\cite{DBLP:conf/www/SuchanekKW07,DBLP:conf/semweb/AuerBKLCI07} or unstructured data \cite{DBLP:conf/aaai/CarlsonBKSHM10,DBLP:conf/wsdm/FangC11}.
The details of knowledge graph construction is beyond the scope of this work.
\section{Proposed Approach}
We describe our end-to-end knowledge-aware learning in this section,
starting with some preliminaries, followed
by our choice of knowledge representation, as well as the eventual
knowledge-aware classification.
\subsection{Preliminaries and notations}
Consider a set of pre-defined class labels $\ensuremath{\mathcal{L}}=\{1,2,\ldots,L\}$ and a set of
videos $d$.
We address the multi-label classification problem for videos, where each video has one or
more ground-truth labels which form a subset of \ensuremath{\mathcal{L}}. We assume a supervised
setting where some training videos with known ground-truth labels are available.
Given a test video with hidden ground truth, the task is to estimate a series
of probabilities $(p_1,p_2,\ldots,p_L)$ where $p_i$
represents the probability of label $i$ on the video. We can subsequently rank
the labels in descending probability and take the top few as the final output.
In this work, we further assume a knowledge graph. Many off-the-shelf knowledge graphs \cite{DBLP:journals/semweb/Paulheim17} exist for our purpose.
A knowledge graph is formally a graph $G=(V,E)$: $V$ is a set of vertices and $E$ is a set of edges between the vertices.
In the context of knowledge graph, each vertice represent a concept or class
label\footnote{We use the terms \emph{concept} and \emph{label}
interchangeably hereafter.}, and each edge represent a relationship between two
concepts.
A typical large-scale knowledge graph often contains millions or billions of concepts, and hundreds or thousands of different relationship types.
\subsection{Overall end-to-end framework}
The overall framework of our proposed end-to-end learning with knowledge graphs
is presented in Figure~\ref{fig:overall}. Given an input video, we can first
extract video and audio feature vectors from each frame.
Note that in YouTube-8M, the pre-extracted video and audio features per frame
consist of 1024 and 128 dimensions, respectively, as exemplified in the
diagram. The frame-by-frame feature vectors are then feed into either frame or
video-level models, to produce ultimate input into the classifier. As our main
novelty, in addition to accounting for features from the video instance, our
classifier further integrates a knowledge graph to narrow the knowledge gap
between traditional machine learning and human intelligence. As such, in our
running example, while the man-made structure is not clearly a zoo from the
frame pixels, we are still able to predict it with the help of a knowledge
graph, which reveals the strong semantic tie between polar bears and
zoos.
The proposed framework embodies two advantages. First, it enables the
incorporation of most existing video classification algorithms, including both
deep and shallow models. Thus, our framework can be highly flexible,
without being approach or task-specific. Second, the
unification with knowledge graphs happens within an end-to-end framework, which means external knowledge can directly influence the feature-based models in a feedback loop through mechanisms such as
backpropagation. In contrast, one recent approach for the related task of object
recognition \cite{DBLP:conf/ijcai/FangKLTC17} also draws input from
knowledge graphs. However, it is not end-to-end; it consists of two decoupled
stages where external knowledge is independent of the feature-based model. Due
to the lack of a feedback loop, their performance turns out to be unsatisfactory in video classification.
\subsection{Knowledge representation}
While external knowledge is commonly represented as graphs, knowledge graphs are
inherently still symbolic and relational. Thus, quantifiable semantics must be
further extracted to enable integration with machine learning models which
typically operate over numerical representations.
The notion of \emph{semantic consistency} has been used
\cite{DBLP:conf/ijcai/FangKLTC17} to quantify the strength of semantic ties
between class labels. Generally two labels with high semantic
consistency suggests that they are likely to show up in
the same video. For instance, polar bear and zoo are two semantically consistent
concepts, whereas polar bear and volcano have weak or no semantic consistency.
We can encode semantic consistency in an
$L \times L$ matrix $S$, such that $S_{ij}$ represents the semantic
consistency between labels $i$ and $j$, $\forall ij \in \ensuremath{\mathcal{L}}^2$. In particular,
$S_ij$ can be established based on the edges connecting the nodes representing
labels $i$ and $j$ on the knowledge graph.
Note that two nodes can be either directly connected by an edge (e.g., polar
bear--zoo), or indirectly through a path of edges (e.g., person--surfing--sea),
improving the generalization ability for concepts without any direct edge.
There can also exists multiple paths between two labels for
robustness.
Intuitively, between two nodes on the knowledge graph, when there are more paths
and these paths are shorter, their semantic consistency is stronger.
Random walk with restart
\cite{DBLP:conf/icdm/TongFP06} is a well-known method to realize the above
intuition. Starting from one node representing label $i$, we compute the
probability $R_{ij}$ of reaching another node representing label $j$ through
random walk.
The higher probability $R_{ij}$ implies that there are more and shorter paths
from $i$ to $j$ and thus the semantic consistency $S_{ij}$ is also higher. As
$R_{ij}\ne R_{ji}$ in general, but the semantic consistency matrix $S$ should
be symmetric in our context, we adopt the below definition follow the earlier
work \cite{DBLP:conf/ijcai/FangKLTC17}. We refer readers to existing work
\cite{DBLP:conf/icdm/TongFP06,DBLP:conf/icde/FangCL13,DBLP:journals/pvldb/ZhuFCY13}
on the computation of random walk probabilities $R_{ij}$.
\begin{align}
S_{ij} = S_{ji}=\sqrt{R_{ij}R_{ji}}
\end{align}
It is worth noting that semantic consistency can also be defined based on
the similarity of node embeddings, as enabled by recent representation learning
approaches on graphs \cite{grover2016node2vec,NIPS2013_5071}. However, our
proposed approach is orthogonal to the computation of semantic consistency,
which is beyond the scope of this paper.
Finally, for efficiency it is preferable to make the matrix $S$
sparser, by only focusing on the largest semantic consistency. To this end, we
consider the $K$-nearest neighbor (KNN) reduction for matrix $S$. A pair of
labels $i$ and $j$ are deemed KNN if $S_{ij}$ is one of the largest $K$ elements
in the $i$-th row or $j$-th row of $S$. Subsequently, we simply set
$S_{ij}=S_{ji}=0$ iff $i$ and $j$ are not KNN. The resulting matrix is much sparser, as it only encodes the strongest semantic consistency.
\subsection{Knowledge-aware classification}
Consider any classifier with a cost function $C$ and model parameters $\Theta$.
For a given video instance, we propose the following knowledge-aware cost function $K$,
where
$p_1,p_2,\ldots,p_L$ encode the label probabilities of the video and they are
functions of $\Theta$.
\begin{align}\label{eq:cost-pairwise}
K(\Theta)=C(\Theta)+\lambda \sqrt{\textstyle\sum_{i=1}^L
\sum_{j < i} S_{ij}(p_i-p_j)^2}
\end{align}
On the one hand, the original cost function $C$ captures the frame or video-level features,
whether they came from deep models or simple aggregation.
On the other hand, the new term here captures the semantics from the knowledge graph.
For a pair of labels $i$ and $j$, if $S_{ij}$ is large (i.e., the two labels have strong semantic consistency), minimizing
the cost function would force $p_i$ and $p_j$ to become similar. That is, it is likely that they either both appear in the video, or both not appear.
In contrast, if $S_{ij}$ is small (i.e., they are not semantically consistent), $p_i$ and $p_j$ become less constrained by the knowledge graph.
Note that the two cost terms, on the features and knowledge graph respectively, are balanced through a hyperparameter $\lambda \in (0,\infty)$.
While the above formulation is intuitive, it is not practical for implementation
with standard libraries such as
TensorFlow\footnote{https://www.tensorflow.org/}.
In particular, TensorFlow operations are organized into a dataflow graph, as
illustrated in Figure~\ref{fig:tensorflow}(a) for the pairwise computation in
Equation~\eqref{eq:cost-pairwise} with $L=4$ and batch size $M=1$ (i.e., for a
single video).
Evidently, the dataflow graph would contain $O(L^2M)$ nodes, which is
not scalable in terms of the time required to construct this
graph, as well as the memory overhead incurred by storing the computation of all the intermediate nodes.
\begin{figure}
\begin{subfigure}[t]{0.56\columnwidth}
\subcaption{Pairwise computations}
\centering
\includegraphics[scale=0.27]{figures/tensorgraph_pairwise.pdf}
\end{subfigure}%
\begin{subfigure}[t]{0.49\columnwidth}
\subcaption{Matrix computations}
\centering
\vspace{6mm}
\includegraphics[scale=0.27]{figures/tensorgraph_matrix.pdf}
\end{subfigure}
\caption{TensorFlow computation
graphs.\label{fig:tensorflow}}
\end{figure}
As such, we employ the Laplacian matrix transformation. It has been established
\cite{weiss2009spectral} that Equation~\eqref{eq:cost-pairwise} is equivalent to
the following:
\begin{align}
K(\Theta)=C(\Theta)+\lambda \sqrt{\Tr\left[\vect{p}(D-S)\vect{p}^T\right]}
\end{align}
where $D$ is a diagonal matrix such that $D_{ii} = \sum_{j=1}^L S_{ij}$ and
$\vect{p}=(p_1,p_2,\ldots,p_L)$ is a vector of label probabilities.
Note that $D-S$ is known as the Laplacian matrix. Using matrix computations, the dataflow graph is greatly
simplified as illustrated in Figure~\ref{fig:tensorflow}(b) with batch size
$M=1$. The total number of nodes simply become bounded by $O(M)$, improving the
scalability significantly.
Provided that the original cost
function $C$ and $\vect{p}$ are differentiable (which are generally true), our knowledge-aware cost
function is also differentiable, as follows. Thus, it can be optimized with the
gradient descent algorithm.
\begin{align}
\frac{\partial K(\Theta)}{\partial \Theta} & = \frac{\partial
C(\Theta)}{\partial \Theta} +
\lambda \frac{\partial \sqrt{\Tr\left[\vect{p}(D-S)\vect{p}^T\right]}}{\partial \vect{p}}\frac{\partial \vect{p}}{\partial \Theta} \nonumber\\
& = \frac{\partial C(\Theta)}{\partial
\Theta} + \lambda \frac{\vect{p}(D-S)}{\sqrt{\Tr\left[\vect{p}(D-S)\vect{p}^T\right]}}\frac{\partial
\vect{p}}{\partial \Theta}
\end{align}
\section{Empirical Evaluation}
In this section, we conduct empirical evaluations on the largest public video
classification benchmark to date, namely YouTube-8M.
We compare the performance of our approach against state-of-the-art video
classification models, and further investigate the impact of parameters on the
performance, and finally present some case studies to illustrate the reasons
that knowledge graphs can improve video classification.
\subsection{Experimental setup}
\subsubsection{Data}
We use the YouTube-8M benchmark\footnote{https://www.kaggle.com/c/youtube8m/data}, the largest public dataset
for multi-label video classification.
It contains over 7 million video instances and a diverse range of 4,716 classes
(entities), with an average of 3.4 labels per video. Pre-extracted and compressed
features at frame and video-levels are available, where the video and
audio features have 1024 and 128 dimensions, respectively.
We employ the off-the-shelf knowledge graph ConceptNet
5\footnote{http://conceptnet.io/}.
Following previous work \cite{DBLP:conf/ijcai/FangKLTC17}, we
only adopt its English subgraph, and remove self-loops and the so-called
``negative'' relationships (e.g., NotDesires, NotCapableOf,
Antonym and DistinctFrom). After these filtering steps,
we obtain a knowledge graph with 1.3 million concepts and 2.8 million
relationships. To further compute semantic consistency, we set the
random walk restarting probability to 0.15 as well.
To map the concepts in ConceptNet to class labels, we simply apply exact
string matching. As a result, 1,867 labels that have a path to at least one
other label are found in ConceptNet. To demonstrate the
advantage of using knowledge graphs, we only consider these 1,867 class labels,
which cover about 97\% the videos. Furthermore, these labels
account for almost 80\% of all label frequency. We emphasize that
obtaining better concept-class mapping for more coverage is not the focus of
this paper, and the current mapping already include the majority of the video
instances and label occurrences.
We use the given training set for training, and the given validation set for
testing since the ground truth of the original test set is not known.
\subsubsection{Evaluation metric}
For each test video, a ranked list of class labels is
produced, and we consider up to top 20 predictions per video for the following evaluation metrics.
\begin{itemize}
\item Mean average precision (MAP): the mean value of the areas under the precision-recall curve of each video.
\item Hit ratio (HIT): the percentage of test videos with the top one prediction belonging to the ground truth.
\item Global average precision (GAP): area under the precision-recall curve over a global list of predictions consisting of all the predictions of all videos.
\end{itemize}
\subsubsection{Knowledgeless models}
Our framework is flexible to integrate knowledge graphs with different
``knowledgeless'' models (i.e., models without using external knowledge), including frame-level deep models and video-level models.
Thus, we consider four different state-of-the-art baseline models, namely,
AoFF, DBoF, LSTM and GRU. For the first three models \cite{abu2016youtube}, we
use the implementation by Google\footnote{https://github.com/google/youtube-8m};
for GRU \cite{miech2017learnable}, we use the implementation by Miech et al.\footnote{https://github.com/antoine77340/Youtube-8M-WILLOW}
More details of these models have been discussed in Related Work.
To train the models, we adhere to the setup in the two studies, as follows.
\begin{itemize}
\item \textbf{AoFF}: learning rate 0.01, video-level model.
\item \textbf{DBoF}: learning rate 0.01, 30 frames per video.
\item \textbf{LSTM}: learning rate 1e-4, cell size 1024, all
frames.
\item \textbf{GRU}: learning rate 2e-4, cell size 1200, all
frames.
\end{itemize}
Note that MoE classifier is used in all models, with 2 experts and 5 epochs.
We further set a batch size of 1024 for AoFF and 128 for the other three models.
\subsubsection{Knowledge-aware models}
We name our proposed end-to-end approach \textbf{E2E}. Each of the knowledgeless models (AoFF, DBoF, LSTM and GRU) can be coupled with knowledge graphs in our E2E framework.
We use $K=5$ for the KNN reduction of the semantic consistency matrix,
and $\lambda=0.01$ for the trade-off between feature-based cost and knowledge-based cost, which are generally robust values with stable performance.
We will vary these hyperparameters to study their impact on the performance as well.
We also compare to a previous knowledge-aware method
\cite{DBLP:conf/ijcai/FangKLTC17}. This method is originally designed for object detection in images, which can be adapted for multi-label video classification as well. It involves two stages,
where the first stage uses an existing knowledgeless model, and the second
stage uses a knowledge graph two re-optimize the output from the first stage.
Thus, the two stages are independent and their approach is not end-to-end.
We name this method \textbf{2STG}. We use $K=5$ for KNN as well, and choose $\epsilon=0.9$ which is found to be the best setting.
\subsection{Comparison of performance}
\begin{table*}[tb]
\center
\setlength\tabcolsep{4.5pt}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{c|ccc|ccc|ccc|ccc}
\hline
& \multicolumn{3}{c|}{\bf AoFF} & \multicolumn{3}{c|}{\bf DBoF} & \multicolumn{3}{c|}{\bf LSTM} & \multicolumn{3}{c}{\bf GRU}\\
& MAP & HIT & GAP & MAP & HIT & GAP & MAP & HIT & GAP & MAP & HIT & GAP\\\hline
- & 0.370 & 0.846 & 0.810 & 0.287 & 0.834 & 0.791 & 0.279 & 0.838 & 0.800 & 0.337 & 0.856 & 0.823\\\hline
\bf 2STG & 0.364 & 0.841 & 0.804 & 0.286 & 0.830 & 0.787 & 0.275 & 0.838 & 0.797 &0.331& 0.855& 0.819\\\hline
\bf \multirow{2}{*}{E2E} & \bf 0.384 & \bf 0.849 & \bf 0.817 & \bf 0.301 & \bf 0.847 & \bf 0.794 & \bf 0.296 & \bf 0.854 & \bf 0.808 & \bf 0.340 & \bf 0.857 & \bf 0.824\\[-0.5mm]
\bf & \small (+1.4\%) & \small (+0.3\%) & \small (+0.7\%) & \small (+1.4\%) & \small (+1.3\%) & \small (+0.3\%) & \small (+1.7\%) & \small (+1.6\%) & \small (+0.8\%) & \small (+0.3\%) & \small (+0.1\%) & \small (+0.1\%)\\\hline
\end{tabular}
\caption{Performance comparison between E2E and 2STG across four state-of-the-art knowledgeless methods. The first row records the performance of the knowledgeless models;
the second row records the performance of 2STG that adopts the corresponding knowledgeless model in its first stage; the third row records the performance of E2E that couples with the corresponding knowledgeless model.
Bold entries represent the best value in each column. \label{table:perf}}
\end{table*}
\begin{table*}[tb]
\center
\setlength\tabcolsep{4.3pt}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{c|ccc|ccc|ccc|ccc}
\hline
& \multicolumn{3}{c|}{\bf AoFF} & \multicolumn{3}{c|}{\bf DBoF} & \multicolumn{3}{c|}{\bf LSTM} & \multicolumn{3}{c}{\bf GRU}\\
& MAP & HIT & GAP & MAP & HIT & GAP & MAP & HIT & GAP & MAP & HIT & GAP\\\hline
- & 0.292 & 0.828 & 0.788 & 0.206 & 0.788 & 0.726 & 0.211 & 0.807 & 0.757 & 0.253 & 0.819 & 0.759\\\hline
\bf \multirow{2}{*}{E2E} & 0.321 & 0.829 & 0.785 & 0.212 & 0.801 & 0.737 & 0.232 & 0.814 & 0.767 & 0.259 & 0.826 & 0.776\\[-0.5mm]
\bf & \small\bf (+2.9\%) & \small (+0.1\%) & \small (-0.3\%) & \small (+0.6\%) & \small\bf (+1.3\%) & \small\bf (+1.1\%) & \small\bf (+2.1\%) & \small (+0.7\%) & \small\bf (+1.0\%) & \small\bf (+0.6\%) & \small\bf (+0.7\%) & \small\bf (+0.7\%)\\\hline
\end{tabular}
\caption{Performance advantage of E2E across four state-of-the-art knowledgeless methods using only 10\% training data.
The percentage improvements of E2E are bolded if they are greater than or equal to the corresponding values in Table~\ref{table:perf}.
\label{table:perf10pct}}
\end{table*}
We first report the performance comparisons between the four knowledgeless
models and their respective knowledge-aware counterparts. Specifically, for
each knowledgeless model, we compare their results with those of both 2STG (previous
work) and E2E (our approach). The results are summarized in
Table~\ref{table:perf}. Note that we are only interested in comparing the values in each column, instead of comparing across different knowledgeless models.
Our approach E2E can achieve better performance every time, beating
the respective knowledgeless model by up to 1.7\% in MAP, 1.6\% in HIT and 0.8\%
in GAP. In contrast, 2STG performs poorly as it is not an end-to-end model. While 2STG can outperform knowledgeless models
for object detection on the Microsoft COCO \cite{DBLP:conf/eccv/LinMBHPRDZ14} and PASCAL VOC \cite{DBLP:journals/ijcv/EveringhamGWWZ10} datasets,
the two datasets involve only a restricted set of 80 and 20 classes, respectively. In contrast, our experiments deal with 1,867 classes over a much more complex and diverse range of topics, which could be the potential reason that the
decoupled two-stage method 2STG is unable to cope.
We further hypothesize that our knowledge-aware model has more advantage when the training set is smaller.
Intuitively, when there are fewer videos to learn from, the availability of external knowledge becomes ever more critical.
Using only 10\% training data, the performance of E2E with the four knowledgeless models is reported in Table~\ref{table:perf10pct}.
Not surprisingly, we observe slightly larger improvements than those using all training data, especially for the MAP and GAP metrics.
\subsection{Impact of parameters}
\begin{figure}[tbp]
\center
\begin{subfigure}[t]{0.56\columnwidth}
\center
\subcaption{Effect of $\lambda$}
\center
\hspace{-7mm}
\begin{tikzpicture}[scale=1][font=\small]
\begin{axis}[
minor x tick style = transparent,
ylabel=Performance,
xlabel=Values of $\lambda$,
enlarge x limits=0.1,
ymin=0.0,
ymax=1.0,
ytick={0,0.2,0.4,0.6,0.8,1.0},
xmode=log,
xmin=1e-5,
xmax=1000,
xtick={1e-5,1e-3,0.1,10,1000},
y tick label style={/pgf/number format/.cd,fixed,fixed zerofill,precision=1,/tikz/.cd},
width=1\textwidth,
height=5cm,
legend style={at={(0.03,0.35)},anchor=south west,nodes={inner sep=1.0pt},fill=none,draw=none,font=\scriptsize},
legend cell align=left,
legend columns=1,
xtick align=inside,
]
\addplot[red,mark=+] coordinates {
(1e-5, 0.279)
(1e-4, 0.318)
(1e-3, 0.329)
(0.01, 0.321)
(0.1, 0.322)
(1, 0.322)
(10, 0.320)
(100, 0.270)
(1000, 0.196)
};
\addplot[blue,mark=x] coordinates {
(1e-5, 0.824)
(1e-4, 0.830)
(1e-3, 0.831)
(0.01, 0.829)
(0.1, 0.828)
(1, 0.828)
(10, 0.814)
(100, 0.682)
(1000, 0.539)
};
\addplot[green!50!black,mark=o] coordinates {
(1e-5, 0.784)
(1e-4, 0.792)
(1e-3, 0.792)
(0.01, 0.785)
(0.1, 0.785)
(1, 0.785)
(10, 0.766)
(100, 0.598)
(1000, 0.414)
};
\legend{MAP,HIT,GAP}
\end{axis}
\end{tikzpicture}
\end{subfigure
\begin{subfigure}[t]{0.56\columnwidth}
\center
\subcaption{Effect of KNN}
\center
\hspace{-13mm}
\begin{tikzpicture}[scale=1][font=\small]
\begin{axis}[
ylabel=Performance,
xlabel=Values of $K$,
ymin=0.1,
ymax=0.9,
ytick={0.1,0.3,0.5,0.7,0.9},
symbolic x coords={1,2,5,10},
y tick label style={/pgf/number format/.cd,fixed,fixed zerofill,precision=1,/tikz/.cd},
width=1\textwidth,
height=5cm,
legend style={at={(0.03,0.35)},anchor=south west,nodes={inner sep=1.0pt},fill=none,draw=none,font=\scriptsize},
legend cell align=left,
legend columns=1,
xtick align=inside,
]
\addplot[red,mark=+] coordinates {
(1, 0.292)
(2, 0.311)
(5, 0.321)
(10, 0.328)
};
\addplot[blue,mark=x] coordinates {
(1, 0.827)
(2, 0.824)
(5, 0.829)
(10, 0.831)
};
\addplot[green!50!black,mark=o] coordinates {
(1, 0.788)
(2, 0.777)
(5, 0.785)
(10, 0.790)
};
\legend{MAP,HIT,GAP}
\end{axis}
\end{tikzpicture}
\end{subfigure}
\caption{Impact of parameters on the performance of E2E.\label{fig:paramimpact}}
\end{figure}
Next, we study the effect of parameters on the performance. There are two main
parameters for E2E: the trade-off $\lambda$ between the feature-based and
knowledge-based costs, and the choice of KNN for the semantic consistency
matrix. For brevity we only present their impact on AoFF,
as similar trends can be observed on other models.
In Figure~\ref{fig:paramimpact}(a), we vary $\lambda$ between 1e-5 and 1000,
while fixing KNN at $K=5$. Results show that the performance is generally stable
for a wide range of $\lambda$ between 1e-4 and 10. The performance only
deteriorates for very large values. Hence, it is robust to use $\lambda=0.01$ in
our experiments.
In Figure~\ref{fig:paramimpact}(b), we vary $K \in \{1,2,5,10\}$ for the choice
of KNN, while fixing $\lambda=0.01$. When we use larger $K$, there is a slight
increase in performance, especially in MAP, although the matrix $S$ becomes
denser and results in lower efficiency. Generally, using $K=5$ can
achieve a good balance between accuracy and efficiency.
\subsection{Result analysis}
\begin{table*}[tb]
\center
\renewcommand{\arraystretch}{1.3}
\resizebox{0.99\textwidth}{!}{%
\begin{tabular}{r|l|l|l|l}
\hline
& \bf Ground truth & \bf AoFF rank & \bf E2E rank & \bf Related concepts in the same video\\\hline
1 & fashion & 20 & 1 ($\uparrow$19) & hairstyle, bollywood, cosmetics \\
2 & origami & 20+ & 1 ($\uparrow$19+) & paper, toy\\
3 & amusement park & 20+ & 1 ($\uparrow$19+) & food, roller coaster, train\\
4 & disc jockey & 5 & 2 ($\uparrow$3) & nightclub, dance, album, guitar\\
5 & food, drink & 1, 5 & 1, 2 ($\uparrow$3) & recipe, cocktail, juice, cooking, bartender, bottle\\
6 & camera, photography & 4, 9 & 1 ($\uparrow$3), 4 ($\uparrow$5) & gadget, camera lens, smart phone\\
7 & hunting, deer & 4, 20+ & 1 ($\uparrow$3), 6 ($\uparrow$14+) & forest, tree, plant, animal, weapon\\
8 & vehicle, tool, drill & 2, 4, 20+ & 1 ($\uparrow$1), 2 ($\uparrow$2), 6 ($\uparrow$14+) & car, metalworking\\
9 & concert, lighting, festival & 2, 5, 16 & 1 ($\uparrow$1), 2 ($\uparrow$3), 8 ($\uparrow$8) & dance, album, Ibiza\\
10 & furniture, couch, bed, chair & 1, 3, 11, 20+ & 1, 2 ($\uparrow$1), 6 ($\uparrow$5), 4 ($\uparrow$16+) & living room, home improvement, house, television\\\hline
\end{tabular}}
\caption{Example videos that knowledge graphs can help with learning. Each row describes a video, where ``AoFF rank'' and ``E2E rank'' columns indicate the rank position of the ground truth label in the output of AoFF and E2E, respectively;
20+ means the ground truth is not found in the top 20; $\uparrow$ indicates the number of positions moved up in E2E output as compared to AoFF; related concepts
are listed if they have high semantic consistency with the ground truth and they are within top 20 of both AoFF and E2E.\label{table:casepositive}}
\end{table*}
\begin{table*}[tb]
\center
\renewcommand{\arraystretch}{1.3}
\resizebox{0.99\textwidth}{!}{%
\begin{tabular}{r|l|l|l|l}
\hline
& \bf Ground truth & \bf AoFF top & \bf E2E top & \bf Other concepts in the same video\\\hline
1 & telescope & \bf telescope & \it vehicle & \textbf{camera}, \textit{car}, \textit{boat}, \textit{bicycle}, \textit{motorcycle} \\
2 & transistor & \bf transistor & \it vehicle & \textbf{antenna}, \textit{train}, \textit{car} \\
3 & running, marathon & \textbf{running}, \textbf{hiking} & \textit{mountain}, \textit{nature} & \textbf{climbing}, \textbf{walking}, \textit{mountain pass}, \textit{trail}, \textit{lake} \\
4 & gardening, plant & \textbf{plant}, \textbf{gardening} & \textit{food}, \textit{news program} & \textbf{tree}, \textit{agriculture}, \textit{cooking}, \textit{television} \\
5 & banknote, money, dollar & banknote, dollar, money & paper, \textit{animation}, \textit{guitar} & \textit{manga}, \textit{art}, \textit{festival}, \textit{musician}\\\hline
\end{tabular}}
\caption{Example videos where knowledge graphs can hurt performance. Each row describes a video, where ``AoFF top'' and ``E2E top'' columns indicate the top prediction(s) of AoFF and E2E, respectively;
other concepts are listed if they have high semantic consistency with the top prediction(s) of either AoFF or E2E, and they are within top 20 of both AoFF and E2E; bold entries are
a group of concepts with strong mutual semantic consistency; likewise for italic entries.
\label{table:casenegative}}
\end{table*}
Finally, we conduct a more in-depth analysis of the results, using AoFF as the knowledgeless model.
At an aggregate level, we observe better predictions for 24.9\% of the videos after incorporting external knowledge with E2E,
whereas we witness worse predictions for 8.9\% of the videos. The remaining videos have no change in their predictions.
Given that the number of videos with better results are almost three times of the videos with worse results, our approach E2E does bring in net benefits,
consistent with the quantitative evaluation reported earlier.
We further zoom into some specific examples to understand the reasons behind the improvement.
In Table~\ref{table:casepositive}, we illustrated 10 videos where knowledge graphs can help with learning.
In these cases, E2E significantly improves the rank positions of the ground truth labels over AoFF, due to the evidence of the related concepts in the same video.
For instance, in example \#2, the concept of origami is semantically consistent with paper and toy, which helps E2E to identify origami correctly.
On the contrary, even though AoFF also recognizes the concepts paper and toy in the same video, it has no idea that those concepts are related to origami, especially when there are not enough training videos
involving origami, paper and toy.
\vspace{1mm}
Finally, we investigate some negative cases in Table~\ref{table:casenegative}, where knowledge graphs can hurt the performance.
In these cases, E2E often makes overgeneralizations based on the related concepts. In these videos, there exist an overwhelm of concepts that are semantically consistent to the incorrect top predictions
by E2E, whereas the concepts that are consistent with the ground truth are much fewer (e.g., \#1 and \#4) or even non-existent (e.g., \#5).
The root cause is that E2E treats each related concepts uniformly. However, in an ideal solution, we should only focus on the concepts related to the central theme of the video.
We leave the study of such ``focus'' concepts as potential future work.
\section{Conclusion}
\balance
In this paper, we studied the multi-label video classification problem. In
particular, we observed the knowledge gap between machine and human
intelligence. Towards bridging this gap, we proposed to utilize external knowledge graphs
for video classification, unifying machine learning including deep neural
networks with knowledge graphs in a novel end-to-end framework. Extensive
experiments on the largest public benchmark YouTube-8M showed the superior performance of our approach, outperforming
state-of-the-art knowledgeless models by up to 2.9\% in MAP among other metrics. Finally, we analyzed some case studies to understand the scenarios in which knowledge graphs can or cannot help.
As future work, we plan to extract features from knowledge graphs and directly
incorporate them into the deep neural networks. Moreover, it is also worth investigating that how we can identify focus concepts
that are related to the central theme of a video.
\newpage
\balance
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,436
|
package onion.tools.option
import scala.collection.mutable.{Map, Seq}
sealed trait ParseResult {
def status: Int
}
case class ParseSuccess(options: Map[String, CommandLineParam], arguments: Array[String]) extends ParseResult {
def status: Int = ParseResult.SUCCEED
}
case class ParseFailure(lackedOptions: Array[ValuedParam], invalidOptions: Array[ValuedParam]) extends ParseResult {
def status: Int = ParseResult.FAILURE
}
object ParseResult {
final val SUCCEED: Int = 0
final val FAILURE: Int = 1
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,044
|
This listing at 153 Poppy View Lane, Erie, CO 80516 is a residential-detached listing with 2 full baths, 1 three-quarter bath, 3 bedrooms and approximately 4,107 square feet on a lot of 6,098 sqft (or 0.14 acres). It was built in 2016. 153 Poppy View Lane is located in Erie and in ZIP Code 80516. It has been listed on our site since 2019-03-17.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,785
|
The smart thinking money app.
Try it now:
Follow Yolt
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When Banks are conducting maintenance or encounter a bug, this can have a knock-on effect on our connections. This page is your one-stop shop to keep up to date with all current bugs and connectivity issues.
This page is your one-stop shop to keep up to the date with all the current bugs and connectivity issues.
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As part of PSD2, banks have to adhere to a new requirement called Strong Customer Authentication (or SCA for short). As you may have noticed, HSBC has changed the way you login to your online banking account, asking that you use a Digital Secure Key.
Unfortunately, HSBC is not providing access for credit and savings accounts via API and, as mentioned previously, the current flow using a Digital Secure Key is not ideal. For this reason, we've made the decision to un-support HSBC credit and savings accounts until they can offer support via an API. We do hope to have them back in Yolt soon, click below to set up an alert so you're notified as soon as we've made progress.
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Yolt is a registered trade mark of ING Bank N.V. ('ING'). ING's registered office is in Amsterdam at Bijlmerplein 106, 1102 CT, The Netherlands, Trade Register number 33031431. ING is regulated and licensed by the De Nederlandsche Bank ('DNB') and the European Central Bank ('ECB') . ING is also regulated by the Autoriteit Financiële Markten ('AFM') and Autoriteit Consument & Markt ('ACM'). Information regarding supervision of ING can be obtained from DNB (www.dnb.nl), AFM (www.afm.nl) or ACM (www.acm.nl).
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,873
|
Q: react select similar function to getOptionLabel function in 1.3 version h guys ,
I am making a react app with react select 1.3 version , I need to add a custom function to drop down which includes 2 keys.
I noticed latest react select has a function for this
getOptionLabel
I want to find something similar to this function for react select version 1.3 .
could anyone able to help me on this ?
this is not supported in react select version 1.3 need a function similar to this
getOptionLabel={(option) => `${option.label}: ${option.rating}`}
A: You can relabel the options by mapping the original options array to a new one, like this:
const options = colourOptions.map(({ value, label, color }) => ({
value,
label: `${value}: ${label}, ${color}`
}));
Since the custom <select> is now using new objects, you need to make a change to handleChange so the original options are used in the state:
handleChange = (alteredOptions) => {
// map altered options to actual options using the value
const selectedOptions = alteredOptions.map((so) =>
colourOptions.find((co) => co.value === so.value)
);
this.setState({ selectedOptions });
};
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,168
|
{"url":"https:\/\/www.physicsforums.com\/threads\/why-did-mexico-free-itself-from-spain.60785\/","text":"Homework Help: Why did Mexico free itself from Spain?\n\n1. Jan 23, 2005\n\nConcealedDreamer\n\nHey, I\"m not sure if this was the right place to ask, but why did Mexico wanted to be free from rule from Spain? Any details?\n\n2. Jan 23, 2005\n\nquasar987\n\nBecause\n\n$$\\Gamma^l_{ki} = \\frac{1}{2} g^{lj} (\\partial_k g_{ij} + \\partial_i g_{jk} - \\partial_j g_{ki})$$\n\nHow could you miss that ?!?!\n\nNo this is probably not the best place to ask. Try General Discussion, or even Politics and World Affairs.\n\n3. Jan 23, 2005\n\nConcealedDreamer\n\nSorry, I'll post this there then.","date":"2018-11-13 21:06:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.21147601306438446, \"perplexity\": 2401.8104248874106}, \"config\": {\"markdown_headings\": false, \"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-2018-47\/segments\/1542039741491.47\/warc\/CC-MAIN-20181113194622-20181113215811-00051.warc.gz\"}"}
| null | null |
\section{Introduction}
\label{intro}
The unprecedent convergence of observational results that we are currently
witnessing has narrowed down the region of the cosmological parameter space
which is compatible with all the different and independent probes of cosmology:
Supernovae \cite{Riess98} \cite{Perlmutter99}, CMB \cite{deb00, Spergel07} and Large
Scale Structures \cite{Tegmark06, Guzzo08, Mar08}. Robustly growing evidence suggests that {\it
i)} ordinary matter is a minority ($\sim 1/6$) of all the matter content of
the universe, {\it ii)} matter -- mostly dark, non-baryonic
matter -- is a minority ($\sim 1/4$) of all the cosmological mass--energy
density, mostly contributed by an obscure form of energy referred to as `dark
energy', {\it iii)} the 3D spatial geometry of the universe is flat and {\it
iv)} the expansion of the cosmic metric has been accelerating for the last $\sim 7$
Gyrs of our universe lifetime.
Even though the big picture is in place, the two dominant
contributions to the stress-energy tensor -- i.e. dark energy and dark matter
-- still have a hypothetical nature and they have not been discovered
yet.
While there is widespread evidence for the existence of the non-baryonic dark matter
component producing the potential wells of large-scale structures \cite{Clowe06},
as yet no persuasive theoretical explanation has been able to elucidate the
physical nature of the dark energy component \cite{pebrat}.
As a matter of fact, unveiling the nature of dark energy and its role in cosmology and
gravitation is a difficult and subtle challenge.
In such a context, it should not be overlooked that the large roaming from model to model,
and the abundance of weakly predictive theories, might eventually limit the possibility
of discriminating between different competitors proposed so far for explaining the observed
dynamics of the accelerating universe.
In the absence of a compelling theoretical explanation for the dark energy
component, and in a minimal, zero-order approach, we explore here the
possibility of preserving the standard metric interpretation of the accelerated
expansion against possible alternative physical scenarios.
To this end, we first evaluate and then discuss the observable consequences of
local, non-gravitational mechanisms which could in principle accelerate matter
in our Hubble patch of the universe. We assume here that the universe is described by
general relativity, that it is dominated by components which satisfy the usual
energy conditions (according to which the universe can only decelerate) and that
the onset of recent accelerated expansion is the result of the presence of a
hypothetical non-gravitational force field.
Such an alternative explanation is rather conservative, since it assumes
neither a cosmological constant (or negative-pressure fluid) nor a modification
of general relativity.
Accordingly, we first work out a self-consistent, non-geometric model for the
cosmic acceleration that is able to reproduce the current observations of
standard candles (i.e. SNIa) and then we discuss a falsifiability procedure
aimed at testing its observational predictions.
The motivation behind this work is to put strong limits on a hypothetical (or
non usually considered) physics that is possibly missing in our picture of the
universe, and, in turn, to strengthen the evidence supporting the standard
paradigm with which we are currently explaining its past history, its present
stage and its future fate.
\section{Accelerated cosmological expansion with a non-cosmological,
large-scale, radial force field}
Our goal is to work out testable predictions that allow us to reject
non-metric acceleration models. To this end, we construct here a general,
phenomenological model in which the role of dark energy is mimicked by an
alternative mechanism of non-gravitational origin.
We consider here a background universe with a metric expansion as predicted by
general relativity. We assume that in such a universe a hypothetical
large--scale, non-gravitational force field influences the overall dynamics of
large scale structures in a patch of the universe with typical dimensions of
the local Hubble volume. In this scenario every object which at time $t$ sits
on the shell of a sphere of proper radius $r(t)$ centered on the observer feels
a peculiar acceleration field $\gamma_p(t)$ that is radially directed and time
dependent.\\
We further speculate that the only cosmological component contributing to the
stress-energy tensor is dark matter. In other words, we assume that there is no
dark energy at all in the universe and that what we interpret as apparent
isotropic acceleration of the metric is indeed the combination of the decelerated cosmological expansion predicted by general relativity plus the Doppler effect sourced by matter which is
accelerating outward under the effect of the radial force field we have added
to this cosmological scenario.
In such a case, by appropriately tuning the non-cosmological contribution to
the observed redshift, one can reconstruct a functional form for the
luminosity distance $d_L$ (see eq. \ref{eq.dlgeneral} below) which reproduces
the standard one derived within a model with cosmological constant (or dark
energy): i.e.
\begin{equation}
d_{L}(z_{obs}, {\bf p}_{obs})=d_{L}(z,{\bf p}, {\bf \gamma_p}) \; .
\label{dleq}
\end{equation}
In other words, the standard $\Lambda$CDM cosmological parameter set (represented by the
vector ${\bf p_{obs}}$) observationally inferred by simply plugging in the observed
redshift $z_{obs}$ into the luminosity distance formula are biased with respect to the
{\it true cosmological} values (represented by the vector ${\bf p}$) that would be
naturally inferred by recognizing the presence of a physical mechanism responsible for the
acceleration of matter {\it in situ}.
Having outlined the phenomenological model we are pursuing, let us now
elucidate its finer details, i.e. the dependence of the luminosity distance
(Eq. 1) on the peculiar acceleration term $\gamma_p$.
\\*
The luminosity distance of an object standing at the observed redshift
$z_{obs}$ is given, in a pure matter scenario, by
\begin{equation}
d_L = (1+z_{obs}) {c \over H_0 \sqrt{|\Omega_k|}} S_k
\bigg[ \sqrt{|\Omega_k|} \int_0^{z_{obs}} \frac{dz} {E(z)} \bigg] \; ,
\label{eq.dlgeneral}
\end{equation}
where $\Omega_k= 1- \Omega_m$, $S_k(x) = \sin(x)$ (if $k=1$), $S_k(x)= x$ (if
$k=0$), $S_k(x)= \sinh(x)$ (if $k=-1$), and $E(z)=[\Omega_m (1+z)^3 + \Omega_k
(1+z)^2]^{1/2}$.\\
While in the standard model the measured redshift $z_{obs}$ has a pure
cosmological interpretation, in the accelerated model it additionally includes
the contribution of the peculiar velocity $v_p$ induced by the radial non
gravitational field and it can be written as
\begin{equation}
z_{obs}=z+\frac{v_p}{c}(1+z).
\label{eq.zo}
\end{equation}
Clearly, there are an infinite variety of cosmological models that, when
combined with an equally arbitrary variety of radial acceleration fields
$\gamma_p$, could in principle reproduce the observational features of the
standard cosmological model. So, for the sake of simplicity, and in order to
capture the essential physics of the problem, we assume in the following that
the universe is flat, that its total density is contributed only by matter
$\Omega_0=\Omega_m=1$, and its expansion rate is
$H(z)=H_0(1+z)^{\frac{3}{2}}$
\\*
Under these hypotheses, the luminosity distance can be written as
\begin{equation}
d_{L}(z, v_p)=\frac{2c}{H_0} \Big\{ \Big(1+z)\Big(1+\frac{v_p}{c}\Big)
\Big\} \Big\{ 1-\Big[(1+z)\Big(1+\frac{v_p}{c}\Big)\Big]^{-\frac{1}{2}}\Big\}
\; ,\label{eq.dl_om1}
\end{equation}
\noindent nonetheless, we stress that these working assumptions do not influence the generality
of the conclusions presented in $\S 3$: analogous considerations hold, in fact,
for a low density universe, open universe (with, e.g., $\Omega_m \sim 0.2$).
We can now compute how a peculiar velocity $v_p$ in this scenario
depends on the peculiar acceleration field: $v_p=v_p[\gamma_p(t)]$ at
a given cosmological redshift $z$.
\\*
Using the expression of the proper distance to a given coordinate,
$r(t)=a(t)x(t)$, we obtain the expression of the acceleration $\gamma(t)$ by
means of which a perturbed metric expands radially:
\begin{equation}
\gamma=\frac{\ddot{a}}{a}r+\frac{\dot{a}}{a}v_p+\frac{dv_p}{dt} \; ,
\label{eq.alpha}
\end{equation}
where $v_p=a\dot{x}$ and the peculiar acceleration term is given by
\begin{equation}
\frac{dv_p}{dt}+Hv_p=\gamma_p(t) \; .
\label{eq.dwdt}
\end{equation}
After transforming this equation from the time to the redshift domain, we find
the general solution
\begin{equation}
v_p(z)=(1+z)\Big(K - H_0^{-1} \int \frac{\gamma_p(z)}{(1+z)^{7/2}}dz\Big) \; .
\label{eq.wz}
\end{equation}
We model the peculiar acceleration term in this equation by means of a general, non-divergent
polynomial function
\begin{equation}
\gamma_p(z)=\gamma_0+\sum_{i=1}^n{\gamma_i(1+z)^{-i}}
\label{eq.gamma}
\end{equation}
where the $\gamma_i$ are free parameters to be adjusted with the constraint given
by Eq. 1. Note that in the absence of a peculiar acceleration acting on cosmic
matter, we recover the standard result that any primordial peculiar velocity is
damped by the background expansion (i.e., $v_p \propto a^{-1}$).
\subsection{Fitting the data}
We show here that by appropriately tuning the parameters $\gamma_i$ we can
match the luminosity distance $d_L$ of standard candles like SNIa as inferred
within the standard model of cosmology.
\\*
To this end, we first set the integration constant $K=0$ by imposing that, in
the limit $z\rightarrow \infty$, the peculiar velocity $v_p=0$. We also set
$\gamma_0=0$ in order to avoid any primordial acceleration field. Then, we fit
Eq. \ref{eq.dlgeneral} to reproduce the currently available supernovae data. We
find that even a flat, matter--dominated ($\Omega_m=1$) cosmological model, when
supplied with an appropriate local acceleration field, would be able to
reproduce the luminosity distance derived by fitting the observed data assuming
(`{\it wrongly'}) a cosmological origin of the source redshift (see
Fig.\ref{fig2}). \\
We obtain an acceptable fit ($\chi_{\nu}^2 \sim 1$) by expanding up to order $n=4$ in Eq. 8.
The best fitting coefficients in the redshift
range $0<z<1.7$ (for a background cosmological model with $\Omega_m=1$) are:
$\gamma_1=2.1(\pm 0.5)\cdot
10^{-8}$m/s$^2$, $\gamma_2=8.7(\pm 0.6)\cdot 10^{-8}$m/s$^2$, $\gamma_3=1.1(\pm 0.5)\cdot
10^{-7}$m/s$^2$, $\gamma_4=-4.3(\pm 0.3)\cdot 10^{-8}$m/s$^2$.
\begin{center}
\begin{figure}[h]
\includegraphics[width=75mm,angle=0]{fig1.eps}
\includegraphics[width=68mm,angle=0]{fig2.eps}
\caption{The best fitting peculiar acceleration $\gamma_p$ {\it (left)} and
peculiar velocity $v_p$ {\it (right)} needed to reproduce current SNIa data (in
the redshift range $0<z<1.7$) are shown as a function of redshift in a flat,
decelerating $\Omega_m=1$ background cosmology.} \label{fig 1}
\end{figure}
\end{center}
\begin{center}
\begin{figure}[h]
\includegraphics[width=75mm,angle=0]{fig3.eps}
\includegraphics[width=75mm,angle=0]{fig4.eps}
\caption{{\em Left :} the SNIa distance modulus is shown as a function of redshift for
different cosmic expansion models: the best fitting $\Lambda$CDM model
obtained by using distant supernovae \cite{Riess07} (solid), a decelerating
Einstein deSitter model (dotted) and a model in which a large scale peculiar
acceleration field acts on top of the EdS world model (short-dashed curves).
The shaded area shows the state-of-the-art dispersion in the observed SNIa
samples from the SNLS collaboration \cite{Astier06} and from the HST gold sample
\cite{Riess07}. {\em Right :} We show the convergence, at high redshift, of the peculiar
acceleration model to the underlying background cosmological model (an Einstein de Sitter
model in this case).} \label{fig2}
\end{figure}
\end{center}
We find that the large scale peculiar acceleration required to fit the SNIa
data (see Fig.\ref{fig2}) is negligible with respect to other local
acceleration fields. Given the smallness of its amplitude, it modifies almost
undetectably the inertial nature of the reference systems to which it is
applied at high redshift.
Note that, in general, this is the order of magnitude of any large-scale, peculiar acceleration field one needs to introduce in order to reinterpret SNIa data over the redshift range $0 < z < 1.7$.
For $z \to \infty$, the distance modulus predicted within this paradigm converges to the standard
behavior expected in the underlying, decelerated, cosmological model because the peculiar perturbation vanishes at early epochs.
We stress here that, by definition, the acceleration induced by such an
hypothetical non-gravitational force should act differentially on the baryonic
and non-baryonic matter components. Therefore, any possible non-gravitational
accelerating mechanisms are physically viable only if it does not destroy
(e.g., by segregating it into parts) the astronomical object on which they act
(for instance, by segregating the dark matter halo from the luminous baryonic
component hosted within it). In other words, the acceleration time scale
$\tau_{acc}$ in a cosmological frame has to be much larger than the typical
dynamical time scale $\tau_{dyn}$ of a system trapped inside a dark matter halo.\\
By inserting the time-averaged value of Eq. 8
($\langle \gamma_p \rangle \approx 5.7 \cdot 10^{-10} m s^{-2}$, if
$\Omega_m=1$) into Eq. 5 and solving by using $v_p=a\dot{x}$, we obtain hence
\begin{equation}
\tau_{acc} \equiv \bigg[2.9 {R H_0^{2/3} \over \langle \gamma \rangle }
\bigg]^{3/4} \approx (G\rho)^{-1/2} \; ,
\end{equation}
where $R$ is the comoving radius of the dark matter halo assumed in virial
equilibrium, and $\rho$ is its mean density. This simple physical argument puts
stringent constraints on the viability of such a class of models. Unless one is
willing to consider non standard model of gravity sourced only by baryons
[e.g., MOND (Milgrom 1983) or TeVeS theories (Bekenstein 2004)], or as yet
unexplored interactions mediated by the dark matter particles, one can use the
previous argument to doubt the physical reliability of such
non-gravitational acceleration mechanism.
Moreover, models of non-cosmological acceleration that are able to reproduce the
metric acceleration without invoking the effect of dark energy are not
positively defined over all the epoch of cosmological interest. Specifically,
to explain the acceleration of distant standard candles without violating local
constraints, one would need to invoke an acceleration of opposite sign at low
and high redshift (see the left panel of Fig. 1). Additionally, these models yield
also quite high peculiar velocities, which can even be of the order of a tenth
of the speed of light at a characteristic epoch of $z \sim 1-2$ (see the right
panel of Fig. 1). In principle, the strong sensitivity of the value of the
large-scale peculiar velocity in such scenarios can also be used to put
constraints on the viability of the non-gravitational acceleration models.
On one hand, one must mention the ability of these models to handle the
age-problem in FRW cosmologies with no dark energy. A recurring argument for a
dark energy component has come from considerations that the observed stellar
ages are too long to be consistent with the observed Hubble parameter
$H_0\sim 70$ km s$^{-1}$ Mpc$^{-1}$ for an universe which has always been decelerating. Since
our results have been presented, for the sake of illustration, for a flat CDM
model, they are consistent with the stellar estimates of the age of the
universe only if low values of $H_0\sim 50$ km s$^{-1}$ Mpc$^{-1}$ are viable \cite{Blanc03}.
Although there are authors claiming -- given the still large systematic
uncertainties affecting $H_0$ measurements -- that at present the age argument
cannot be used to definitively exclude the Einstein de Sitter model and,
therefore, arguing for a cosmological constant \cite{Row02} \cite{Sark08}, we
note here that the non gravitational acceleration models can naturally
reconcile decelerated background cosmological models with the age of the
oldest stars in the universe.
As a matter of fact the observed Hubble constant evaluated by comparing the observed
redshift ($z_{obs}$) with redshift-independent distance indicators (d) would
overestimate the true universal value $H_0$ obtained by using, as prescribed by
theory, the cosmological redshift (z).
The SNIa Hubble diagram is constructed in such a way to be independent of the
values of $H_0$. Anyway, one can directly estimate the amplitude of the $H_0$ bias.
In the accelerated model the expansion rate in the local
universe is overestimated by a factor $\delta H_0=v_p/d$.
Therefore, at $z\sim 10^{-2}$,
the isotropic peculiar expansion needed to recover
a Hubble parameter of $\sim 50$ km s$^{-1}$ Mpc$^{-1}$ is of order of $v_p/c=10^{-2}$.
We stress here the analogy between this '{\it reconciliation mechanism}' and
the one invoked within the `Hubble bubble' scenarios. In fact, in such models
\cite{Zehavi} we are supposed to live inside an underdense region (Hubble
bubble) that is expanding faster than the average. Measurements of the Hubble
constant within the underdense region would therefore overestimate the
universal value by $\delta \rho/\rho=-3\delta H/H$ thus offering a different
way to bypass the age problem within a decelerating background metric.
We will see in the next session, however, that a crucial observational test
can be devised to reject such non-gravitational acceleration mechanisms.
\section{Testable predictions}
We now discuss the observable effects that a hypothetical
non-gravitational force field, say a `{\it dark force field'}, should have on
the global expansion of the universe, as traced by a set of distant standard
candles (e.g., SNIa). We show that the physical imprints of such a
{\it `dark force'} can be unambiguously contrasted with those of a {\it
`dark energy'} component. In particular, it is possible to discriminate
between them directly in the Hubble diagram rather than, as usually done, in
the equation of state parameter space.
Indeed, if such a non-gravitational, radial force field exists (see, e.g., the
preliminary suggestion that a large-scale magnetic tension might mimic the
effects of dark energy \cite{Contopoulos07}, or the equivalent proposal of a
large local void \cite{Calsteb}), we can show that it accelerates cosmological
objects (test particles) at the same radial distance in a different way. In
fact, since only the gravitational force has the property to depend on the
gravitational mass -- which via the equivalence principle is equal to the
inertial mass -- it follows that, in the proposed `{\it dark force}' scenario
SNe are not inertial systems in free-fall along geodesics of the space-time.
The acceleration of SNe will thus have a unique signature: it will depend on
the mass of the hosting system. Since in this {\it dark force} scenario the
luminosity distance $d_{L}$ of a cosmic object depends on its acceleration
(see eq. \ref{eq.dl_om1} and \ref{eq.wz} ), the different inertial masses of
the test particles will cause predictable deviations from the mean, best
fitting curve shown in the left panel of Fig. 2.
One can parameterize the mass dependence of the peculiar acceleration experienced
by a given system $\gamma_p^*$ in terms of the best fitting mean acceleration
$\bar{\gamma}_p$ using a simple scaling law
\begin{center}
\begin{equation}
\gamma_p^*=\Bigg(\frac{\bar{M}}{M^*}\Bigg)^{\alpha} \bar{\gamma}_p
\label{acmo}
\end{equation}
\end{center}
\noindent where
the exponent $\alpha$ characterizes the specific physical mechanism responsible for
the acceleration of supernovae. The limiting case $\alpha=0$ describes the action
of the gravitational field showing that, in this case,
the acceleration of test particles is mass-independent.
In contrast, the acceleration generated by non-gravitational force fields is generically
described by setting $\alpha \neq 0$. In particular, the case $\alpha=1$ illustrates the
large class of force fields in which the acceleration is inversely proportional
to the mass of the test particle (for example models in which the strength of the
force is independent of the mass of the object experiencing the field).
\begin{figure}[h]
\begin{center}
\includegraphics[width=140mm,angle=0]{fig5.eps}
\caption{The SNIa distance modulus is shown as a function of redshift in the
`{\it dark force}' scenario (black dashed curve). The lower (upper) solid
curves bracketing the extremal envelopes of the dispersion in the Hubble
diagram (shaded green area) are obtained by assuming $\alpha=1$ in Eq.
\ref{acmo} and by letting the minimum and maximum mass of the test particles
(SNe host systems) vary over five orders of magnitude i.e., from normal isolated
galaxies to galaxy clusters.}
\label{fig3}
\end{center}
\end{figure}
For the sake of simplicity let us discuss first the case $\alpha=1$. A direct
consequence of this assumption is that, at fixed redshift, SNe hosted in isolated
normal galaxies (with typical mass of $\sim 10^{10} h^{-1}M_{\odot}$), when subject to
the force field we are challenging here, will experience an acceleration of
order $\sim 10^{5}$ stronger than SNe hosted in rich galaxy clusters (with
typical mass of $\sim 10^{15} h^{-1}M_{\odot}$). (We assume here that the
masses of the host systems remain constant with time). This effect
systematically shifts the distance modulus of SNe in low (high) mass hosts
towards higher (lower) values with respect to the reference, best fitting curve
characterizing supernovae hosted in some fiducial system of mass $\bar{M}$ (see
Fig. \ref{fig3}).
Similarly, and without loss of generality, we find that for all the
values $|\alpha| \gtrsim 0.1$, (i.e. even in the extreme cases of an
acceleration field which has a weak dependence on the host mass), the family of
distance moduli associated to SNe hosted in systems whose mass differs by up to
five orders of magnitude still spans the whole dispersion region characterizing
current data. The systematic deviation from the best fitting relation of the
distance moduli of supernovae hosted in a large variety of systems, going from
small to large masses, is therefore practically insensitive to the exponent
$\alpha$ parameterizing the mass dependence of the acceleration felt by
supernovae. This general result follows from the fact that a) the distance
modulus calculated by including the peculiar acceleration term cannot be
smaller than the one computed in the associated background cosmological model,
and b) to each value of the exponent $\alpha$ one can always associate an
appropriately tuned value of the fiducial mass in the range
$10^{10}<\bar{M}/M_{\odot}<10^{15}$ in such a way that a change of five orders of
magnitude in the mass of the host gives distance moduli which are always
confined between the upper and lower envelopes of the dispersion in the SNIa
Hubble diagram.
Therefore, we conclude that in a {\it dark force} scenario, part of the
scatter in the SNIa Hubble diagram has a physical origin: for a fixed redshift
$z$ and $\alpha \gtrsim 0.1$ ($\alpha \lesssim -0.1$), SNIa with larger values
of $m-M$ in Fig. \ref{fig3} should be hosted in small (big) systems, while SNIa
with smaller values of $m-M$ should be hosted in rich clusters (isolated normal
galaxies).
To summarize, a null-test of the metric nature of the accelerated expansion can
be easily performed by means of an environmental analysis of the cosmic
structures in which SNIa are found. Such a study could be optimally performed
by future large-area sky surveys of distant SNe such as SNAP \cite{alde06}. The
strong sensitivity to the host system mass of the test strategy that we suggest here
will allow to exclude any hypothetical non-metric large-scale interaction which
could be in principle responsible for the observed kinematics of the universe.
Such a result would hence shed a much clearer light on the nature of the
assumed dark energy component dominating the late stages of the cosmic
evolution.
\vskip 1.truecm
\noindent {\bf Acknowldgements}. We thank the referee for useful comments.
We acknowledge stimulating and useful
discussions with D. Fouchez, P. Taxil and J.M. Virey. S.C. thanks
the Centre de Physique Th\'eorique de Marseille for hospitality.
\section*{References}
|
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| 2,232
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| null | null |
\section{Introduction}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{overview.pdf}
\caption{\textbf{Overview.} An input of 5 consecutive frames is passed through an autoencoder (our proposed VQU-Net) trained to predict the next frame. A saliency map is formed from per-pixel prediction error with the real next frame. The anomaly detection score is formed by summing over the whole frame. The real next frame and saliency map are then passed to the explainability module to explain the detected anomalies.}
\label{fig:overview}
\end{figure}
Detecting anomalies has an important application in many types of video monitoring settings. For example, it is crucial for self-driving cars to hypothesise about normal future video content and detect deviations from the norm \eg pedestrians normally walk road-side but a fall into the road would be anomalous. Anomaly detection also shows a huge potential in video surveillance, particularly for crime prevention or for health and safety \eg detecting an object left unattended in a train station, a drunk person provoking by-standers, falls or abnormal behaviour in care homes.
In both of these applications explanation of anomalies is essential, because the required response, ranging from subtle change of direction to coming to an abrupt stop in the case of the self-driving car, depends on the nature of the anomaly. Moreover, anomaly detection systems used in public settings (\eg anomaly detection in CCTV videos of train stations) have to be interpretable in order to prevent any bias (\eg against minority groups).
This is challenging, because (a) neural networks at the forefront of this problem are not interpretable, (b) detecting and understanding the appearance of anomalous objects is often impeded, because the objects can be blurry, occluded and in uncommon locations (\eg flying up in the air), (c) interpreting the anomaly requires high level understanding of the video context, \eg is anomalous motion in the scene due to people pushing each other in a train station, or someone walking in the wrong direction? and (d) systems used in practical settings cannot miss important anomalous events, \eg someone carrying a gun.
In this work we focus on anomaly detection datasets in which all videos come from the same scene. In an effort to address all four of these challenges, we make the following contributions to the field of interpretable anomaly detection: (1) we propose a general method for explaining anomalies based on per-pixel prediction of future frames in video. Thus we remove the need to classify action or objects for the anomaly detection stage, (2) we use temporal information for explaining anomalies with the use of an action recognition module, (3) develop a Vector Quantized Autoencoder (VQA) to ensure our models cannot reconstruct anomalous objects or actions, resulting in a justifiable behaviour for it's good performance and (4) achieve attractive qualitative and quantitative results against state-of-the-art, improving by 61\% mean Average Precision (mAP) and 74\% mAP on the task of anomaly explanation on the public X-MAN labels~\cite{szymanowicz_2021_xman} for UCSD Ped2~\cite{mahadevan_2010} and Avenue~\cite{lu_2013} anomaly detection datasets, respectively. Project page: \url{http://jjcvision.com/projects/vqunet\_anomally\_detection.html}
\section{Recent works}
\subsection{Anomaly detection}
The most successful anomaly detection approaches are based on deep learning methods and can be split into two groups. The first group learns and uses feature representations directly to detect anomalies. These methods use out of distribution detection algorithms~\cite{hendrycks2018deep,hsu2020generalized,ren2019likelihood} applied to the task of anomaly detection based on learned feature representations~\cite{mousavi_2015,sabokrou_2015,zhao_2016}. More recently and successfully, one-vs-all cluster classification~\cite{ionescu_2019_cvpr,ionescu_2019_wacv} is used based on learned feature representations.
The second group of approaches to anomaly detection with deep learning is to reconstruct or predict future `normal' video frames from sparse feature representations~\cite{hasan_2016,luo_2017,wang_2018}, sometimes augmented with memory modules~\cite{park_2020}, and/or optical-flow images~\cite{liu_2018,GANs,Cross_channel_GANs}. With a one-stage approach (not requiring object detection), these methods are more robust. Anomalies are detected based on the assumption these models will find it difficult to generate abnormal frames, \eg if there are no bikes observed in the training data, the trained models are expected to fail to generalise when attempting to reconstruct a bike in an anomalous frame. In such situation where the predicted frame differs significantly from the observed one, an anomaly is declared. The downside to these approaches is that they do not provide high level explanation to the anomalous events.
Our method is most similar to the memory-augmented autoencoder~\cite{park_2020}, where the features at the bottleneck are appended to the closest entries from a learnt codebook containing a small number of codes. This guides the reconstructions to be similar to `normal' events, therefore making bigger errors in reconstructing anomalous frames. We observe in the method by Park \etal~\cite{park_2020} that (1) the size of the bottleneck of the method is twice as big as the input, hence the model could potentially copy the input when reconstructing and (2) the codebook could simply be ignored by the decoder as it has access to raw bottleneck features. Performance therefore relies on careful tuning so that it generalises enough to reconstruct only normal frames, which is difficult and time consuming to achieve. We address these shortcomings by using a vector quantizer~\cite{oord2017vq} at the bottleneck of our autoencoder, therefore discretising the feature maps. We show fixing the possible set of high-level feature maps to a discrete set of embeddings provides a better guarantee that anomalous events will not be reconstructed. This discrete version replaces the original raw feature map so that the decoder has no access to any low-level information from the input, reducing generalisation performance when reconstructing anomalous frames. To the best of our knowledge, we are the first to propose the use of a vector quantizer~\cite{oord2017vq} module for anomaly detection.
\subsection{Anomaly explanation}
In this work we consider anomaly explanation as the process of labeling anomalous events with high-level human interpretable labels, \eg `running', similarly to Szymanowicz \etal~\cite{szymanowicz_2021_xman} and Hinami \etal. This is different and one step further to visual explanation of anomalies~\cite{Liu_2020_CVPR} where the system only has to highlight anomalous regions in an image. The method of Szymanowicz \etal~\cite{szymanowicz_2021_xman} explains the decisions of anomaly detectors based on feature representations and gives high-level explanations of anomalous events. However, the method of Szymanowicz\etal~\cite{szymanowicz_2021_xman} and MT-FRCN~\cite{hinami_2017} are brittle under object detection failure. We argue that this is a significant shortcoming, because in practical settings anomalous objects are likely to (1) be in motion, hence they might be blurry, (2) be in uncommon locations (3) have unusual appearance and (4) be from out of domain classes to the object detector. Hence two-stage anomaly detection designs where object detection as a pre-processing stage are unlikely to be successful in practical anomaly detection. In contrast, we chose to explain the decisions of more robust anomaly detectors based on future frame prediction. Our method provides high-level explanations of the vector that was used in these kind of methods to declare a frame as anomalous, \ie the per-pixel prediction error. As opposed to previous works that describe anomalous events~\cite{hinami_2017,szymanowicz_2021_xman}, we additionally use temporal context to understand actions, leading to an improvement in anomaly explanation performance.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{vqunet.pdf}
\caption{\textbf{VQU-Net.} Architecture of our prediction network. The resolutions of input and output are the same. $n$ is the number of frames at the input to the network -- they are concatenated along the channel dimension. The output of the encoder, $\boldsymbol{z}_{e}(x)$, is quantized, resulting in a quantized feature map $\boldsymbol{z}_{q}(x)$, which is then used as the input to the decoder.
\label{fig:unet}
\end{figure*}
\section{Method}
The method consists of two stages, first an encoder/decoder architecture is used to produce saliency maps for detecting anomalies. The second stage is an explainability module which interprets the saliency maps and provides spatial location and high-level human interpretable labels for the anomalous event, see Figure~\ref{fig:overview}.
\subsection{Econder/Decoder architecture}
The network architecture is based on U-Net~\cite{ronneberger2015unet}, which has been successfully applied to the task of reconstruction and future frame prediction~\cite{park_2020, liu_2018}. Our contribution is the proposal of a learnable codebook using a vector quantization module~\cite{oord2017vq} at the output of the encoder (see Fig.~\ref{fig:unet}), forming a Vector Quantized U-Net (VQU-Net).
Following~\cite{park_2020} we remove the last batch normalization layer and the last ReLU activation layer, because ReLU cuts off negative values, possibly restricting the diverse feature representation. We also pad the input to convolutions to keep image size unchanged between downsampling or upsampling layers.
The input $x$ consists of $n$ consecutive frames $I_{t}$ at time indices $t=t_{0}$ to $t=t_{0}+n-1$ inclusive, concatenated along the channel dimension. The output of the encoder (last feature map before the first deconvolution) is denoted $\boldsymbol{z}_{e}(x)$ -- a set of $H \times W$ $D$-dimensional vectors $z_{e}(x)$ (see Fig.~\ref{fig:unet}). The output of the decoder $\hat{I}$ is trained to predict the frame at time index $t_{0}+n$, $I_{t_{0}+n}$. The reconstruction task is equivalent to simply setting $n=0$. When training the network for frame reconstruction, skip connections, are removed so that the network cannot learn to simply copy the input.
The learnable codebook is placed between the output of the encoder and the input to the decoder. The codebook is a set of $K$, $D$-dimensional embedding vectors $e_{i} \in \mathcal{R}^{D}, i=1,2,\ldots,K$. For an input feature vector $z_{e}(x)$, the quantizer retrieves and outputs $z_{q}(x)$, the closest entry $e_{k}$ in the codebook, measured by Euclidean distance.
\begin{equation}\label{eq:codebook_retrieval}
z_{q}(x) = e_{k}, \quad \text{where} \quad k = \text{argmin}_{j} \lVert z_{e}(x) - e_{j} \rVert_{2}
\end{equation}
The operation described in Eq.~\ref{eq:codebook_retrieval} is repeated for all vectors $z_{e}(x)$ in the feature map $\boldsymbol{z}_{e}(x)$, outputting a quantized feature map $\boldsymbol{z}_{q}$.
Following~\cite{oord2017vq}, $\boldsymbol{z}_{q}(x)$ is passed to the decoder during the forward pass. The $\text{argmin}$ operator is non-differentiable, but the gradient with respect to the encoder parameters is approximated by copying the gradient during the backward pass from the decoder to the encoder.
\subsection{Training losses}
The total loss function consists of the prediction loss $\mathcal{L}_{pred}$, embedding loss $\mathcal{L}_{embed}$, the so-called commitment loss~\cite{oord2017vq} $\mathcal{L}_{commit}$ and feature separatedness loss~\cite{park_2020} $\mathcal{L}_{sep}$.
\begin{equation*}
\mathcal{L} = \mathcal{L}_{pred} + \lambda_{e}\mathcal{L}_{embed} + \lambda_{c}\mathcal{L}_{commit} + \lambda_{s}\mathcal{L}_{sep}
\end{equation*}
Prediction loss $\mathcal{L}_{pred}$ is the L2 norm of the error between the prediction and the target frame.
\begin{equation*}
\mathcal{L}_{pred} = \lVert \hat{I} - I_{t_{0}+n} \rVert_{2}^{2}
\end{equation*}
Embedding loss trains the retrieved embeddings $\boldsymbol{z}_{q}$ to be close to the input features $\boldsymbol{z}_{e}$ by minimising the L2 norm of the error between them, assuming the input features are held constant. This is implemented with the stop-gradient operator~\cite{oord2017vq} (denoted as $\text{sg}$), which is an identity on the forward pass and has zero derivatives on the backward pass, effectively setting the argument as a constant in the backward pass.
\begin{equation*}
\mathcal{L}_{embed} = \lVert \text{sg}\left[ \boldsymbol{z}_{e}(x) \right] - \boldsymbol{z}_{q}(x) \rVert_{2}^{2}
\end{equation*}
The commitment loss ensures that the encoder outputs values close to the ones present in the codebook, therefore forcing the encoder to `commit' to the discrete representation defined by the codebook. Scaling factor $\beta$ is set to $0.25$ as in~\cite{oord2017vq}.
\begin{equation*}
\mathcal{L}_{commit} = \beta \lVert \boldsymbol{z}_{e}(x) - \text{sg}\left[ \boldsymbol{z}_{q}(x) \right] \rVert_{2}^{2}
\end{equation*}
Finally, separatedness loss~\cite{park_2020} is used to help learn a diverse feature representation and improve the discriminative power of the codebook. The encoded features $\boldsymbol{z}_{e}$ are the anchor, the closest entries in the codebook $\boldsymbol{z}_{q}$ are a positive sample and the second closest entries $\boldsymbol{z}_{n}$ are a negative sample. The loss then helps push the negative samples away from the queries, while the other losses push the positive samples close to the queries. This results in codebook features being placed far from each other improving diversity.
\begin{equation*}
\mathcal{L}_{sep} = \gamma \left[ \lVert \text{sg}\left[ \boldsymbol{z}_{e}\right] - \boldsymbol{z}_{q} \rVert_{2}^{2} - \lVert \text{sg}\left[ \boldsymbol{z}_{e} \right] - \boldsymbol{z}_{n} \rVert_{2}^{2} + \alpha \right]_{+}
\end{equation*}
\subsection{Saliency maps and anomaly detection}
Saliency maps are produced by calculating the per-pixel error between the predicted frame and the ground truth. These can be visualised as heatmaps (Figure~\ref{fig:overview}) with zero error as dark blue and becoming light green as it gets larger. From these saliency maps an anomaly score is formed based on it's L2 norm. Different to Szymanowicz\etal~\cite{szymanowicz_2021_xman}, this anomaly score is now computed globally for the whole frame rather than based on the detected objects, this results in improved performance while not being reliant on object detection accuracy.
\subsection{Explainability module}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{explanation_method.pdf}
\caption{\textbf{VQU-Net explanations.} Top: 5 consecutive frames are input to VQU-Net trained for the prediction task. The predicted frame is compared to ground truth to obtain a map of per-pixel prediction error. Bottom: ground truth frame is passed to object detection and action recognition modules. Right: per-box prediction error is computed and thresholded to identify the anomalous regions and their corresponding objects and actions, i.e. anomaly explanations.}
\label{fig:vq_explanation_method}
\end{figure*}
Explaining the decision behind the anomaly requires the system to specify which actions and objects are responsible for the error in prediction. Given a predicted frame and ground truth we first produce a heatmap of the per-pixel squared error. Next, we run an object detector (Faster-RCNN~\cite{ren_frcnn_2015}) and action recognition module (SlowFast~\cite{Feichtenhofer_2019_ICCV}) on the ground truth frames (see Fig.~\ref{fig:vq_explanation_method}). The "heat" is summed within the each bounding box of detected actions and objects to obtain per-box anomaly score. The classes of objects and actions corresponding to the boxes with highest anomaly scores serve as anomaly explanations. Note that this frame prediction framework will fail to explain an anomaly if the object detector fails however the system can still return a successful anomaly detection. This is in contrast to methods which rely on object detectors as a pre-processing step where the anomaly would be missed.
\section{Experiments}
Our approach is evaluated for both detecting anomalies and explaining them. For anomaly detection we compare to state of the art on existing public datasets described in Sec.~\ref{sec:detection_datasets} and for anomaly explanation we use the X-MAN~\cite{szymanowicz_2021_xman} dataset and the metric described in Sec.~\ref{sec:explanation_dataset}.
\subsection{Datasets \label{sec:detection_datasets}}
For the task of \textit{anaomaly detection} two public datasets are used: UCSD Ped2~\cite{mahadevan_2010} and Avenue~\cite{lu_2013}. For \textit{anomaly explanation} evaluation is conducted on the public X-MAN Dataset~\cite{szymanowicz_2021_xman}.
\paragraph{UCSD~\cite{mahadevan_2010}.} A standard benchmark for anomaly detection. The training data contains only normal events, while testing data contains some abnormal events. 19600 frames captured using two different cameras: UCSD Ped1 and UCSD Ped2 which contains 16 training and 12 testing videos. Normal events include pedestrians walking, while abnormal events include trucks, cyclists and skateboarders. Following~\cite{hinami_2017} we evaluate on Ped2 only as Ped1 is very low resolution.
\paragraph{Avenue~\cite{lu_2013}.} This dataset contains contains 16 training and 21 testing videos. All captured from the same scene, a total of 30,652 (15,328 training, 15,324 testing) frames. This is a challenging dataset because it includes a variety of events such as ``running", ``throwing bag", ``pushing bike" and ``wrong direction". We train from the videos in Avenue that contain normal events. This dataset focuses on dynamic events \eg walking in an uncommon area in the scene and regards abnormal static events as normal \eg standing in the same uncommon area.
\paragraph{X-MAN~\cite{szymanowicz_2021_xman}.} A recent dataset for evaluating anomaly explanation methods. Consists of 22,722 manually labelled frames in ShanghaiTech (17,362), Avenue (3,712) and UCSD Ped2 (1,648). Each frame contains between 1 and 5 explanation labels, each label being a different reason why the frame is anomalous (many frames contain multiple anomalous events, \eg one person running and one riding a bike). In total, there are 40,618 labels across all frames. The majority of anomalies (22,640) are due to actions, followed by anomalous objects (14,828). The remaining anomalies are due to an anomalous location. There are 42 anomalous actions and 13 anomalous objects. We use X-MAN labels for anomalies in UCSD Ped2 and Avenue.
\subsection{Evaluation metrics \label{sec:explanation_dataset}}
Two separate evaluation metrics are used, one for anomaly detection and the other for explanation.
\paragraph{Anomaly detection metric.} All test video frames from all datasets are marked as either containing or not containing an anomaly. Measuring the true and false positive rates against this ground truth, we use the standard metric of evaluating abnormal event detection: the area under the ROC curve (AUC).
\paragraph{Anomaly explanation metric.} The metric, first fully explained by~Szymanowicz\etal~\cite{szymanowicz_2021_xman}, is the mean average precision (mAP) in predicting the labels of anomalous events. The mean is taken across the different explanation classes in order to weight rare explanation classes equally to common ones.
\subsection{Implementation details}
A codebook with $K=256$ entries is used with a separation margin of $\alpha=1.0$ and separation loss weighting $\gamma=0.01$. The network is trained with a learning rate of $2\times10^{-5}$ for the reconstruction task and $2\times 10^{-4}$ for the prediction task. We train the network for 60 epochs. For object explanations, we use Faster R-CNN~\cite{ren_frcnn_2015} models for object detection, implemented in the Detectron2~\cite{wu2019detectron2} framework and pre-trained on MS-COCO~\cite{lin2014microsoft} dataset. For action explanations we use the implementation from the authors of SlowFast~\cite{Feichtenhofer_2019_ICCV} pre-trained on AVA Kinetics~\cite{li2020avakinetics} dataset.
\section{Results}
\subsection{Qualitative analysis}
\paragraph{Reconstruction.} We first compare the reconstructions of anomalous frames from the baseline U-Net model and from the model with the codebook (VQU-Net). Fig.~\ref{fig:vq_qualitative_results_rec}. shows the attempted reconstructions. While the baseline model does produce slightly blurry reconstructions, the shapes are still clearly visible - the model is only able detect anomalies due to limited generalisation, which cannot be guaranteed in a practical setting. In contrast, our model with the codebook is clearly strictly limited to certain shapes and appearances: in Fig.~\ref{fig:vq_qualitative_results_rec}. our model (1) fails to reconstruct the truck completely and replaces it with person-like blobs, (2) completely removes the bikes from the input frames and replaces cyclists with pedestrians and (3) is seen to produce a much worse reconstruction of a bending person. Removing these anomalous objects when attempting the reconstruction is advantageous for the task of anomaly detection and shows that our method is purposefully limited by the codebook, and not a simple side effect of limited generalisation.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{vq_reconstruction_qualitative.pdf}
\caption{\textbf{Reconstruction examples.} Example reconstructions of the input frames (left column) with baseline U-Net (middle column) and VQU-Net (right column). VQU-Net is constrained by the codebook and is not able to reconstruct anomalous objects, as intended: the truck is missing (top row), bicycles are removed and cyclists are replaced with pedestrians (middle two rows) and person picking up a bag is largely missing (bottom row).}
\label{fig:vq_qualitative_results_rec}
\end{figure}
\paragraph{Anomaly detection.}
Saliency maps are formed by the per-pixel reconstruction error.
Examples produced by our prediction network are shown in Fig.~\ref{fig:vq_saliency_pred}. Examples of correctly detected anomalous frames include anomalous objects, actions and objects in anomalous locations. The saliency maps closely align with the regions in the image where the anomalies occur.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{vq_saliency_prediction.pdf}
\caption{\textbf{Prediction saliency maps.} Example saliency maps (per-pixel reconstruction squared error) from the output of VQU-Net trained for prediction task. The prediction saliency maps align closely with a variety of anomalous objects and actions. Top right shows a frame with anomalous objects: a bicycle and a car. Other examples show frames with anomalous actions: top right -- running, middle left -- catching a bag, middle right -- throwing a bag. Bottom right shows an example of a frame with an object in an anomalous location: pieces of paper flying in the air.}
\label{fig:vq_saliency_pred}
\end{figure}
\paragraph{Anomaly explanation.} Combining the prediction error saliency maps with object detection and action recognition modules allows for explanations of anomalies detected by our method. Examples of explained anomalous objects and actions are shown in Fig.~\ref{fig:vq_explanations}. The examples show that our method is capable of explaining anomalies due to unexpected actions, \eg riding a skateboard, practicing martial arts or bending / bowing at the waist. Our method can also explain anomalies due to anomalous objects, \eg trucks or bicycles.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{explanations.pdf}
\caption{\textbf{Anomaly explanations.} Examples of correctly explained anomalies. Left column shows the input frame with bounding boxes of detected objects. Middle column shows the saliency maps produced by passing the input frame through the prediction network. Right column shows the bounding box that was found anomalous and the corresponding action or object category.}
\label{fig:vq_explanations}
\end{figure}
\subsection{Anomaly detection state-of-the-art (SOTA) comparison.}
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Method & UCSD & Avenue \\
& Ped2 & \\
\hline\hline
Kim \etal~\cite{kim_2009} & 59.0 & - \\
Mahadevan \etal~\cite{mahadevan_2010} & 82.9 & - \\
Lu \etal~\cite{lu_2013} & - & 80.6 \\
Hasan \etal~\cite{hasan_2016} & 90.0 & 70.2 \\
Luo \etal~\cite{luo_2017} & 92.2 & 81.7 \\
Liu \etal~\cite{liu_2018} & 95.4 & 85.1 \\
Park \etal~\cite{park_2020} & 97.0 & 88.5 \\
\hline\hline
Hinami \etal~\cite{hinami_2017}& 90.8 & - \\
\hline
Szymanowicz \etal~\cite{szymanowicz_2021_xman} & 84.4 & 75.3 \\
\hline
VQU-Net (Ours) & 89.2 & 88.3 \\
\hline
\end{tabular}
\end{center}
\caption{Abnormal event detection accuracy in AUC (\%). We compare the results from VQU-Net trained for the prediction task against SOTA methods. Only Hinami \etal~\cite{hinami_2017} and Szymanowicz \etal~\cite{szymanowicz_2021_xman} propose methods with explanations of anomalies, hence these are the main methods we compare against. \label{tab:results}}
\end{table}
Table~\ref{tab:results} summarises the AUC on all datasets.
Our VQU-Net method performs better than the method of Szymanowicz \etal on the task of anomaly detection. This illustrates the advantage of single-stage approaches for anomaly detection. VQU-Net performs on par with other SOTA methods, while additionally providing explanations of detected anomalies.
We believe that in a practical setting providing anomaly explanations at the cost of slightly lower anomaly detection performance is advantageous because it allows human operators of monitor systems to decide on appropriate responses. It also allows for grading of the alert level raised by the anomaly, i.e. anomaly due to a gun is more alarming than an anomaly due to a person jumping.
\paragraph{Ablation study.} We investigate the effect of including temporal information (i.e. if the network attempts to predict a future frame, or simply reconstruct a current frame) and the effect of using the codebook. As seen in Tab.~\ref{tab:ablation}., including temporal information improves performance on the anomaly detection task by more than $15$ percentage points on UCSD Ped2 and around $7$ percentage points on Avenue. Hence, it can be concluded that temporal information (i.e. motion) is crucial for identifying anomalies. Analysis of Tab.~\ref{tab:ablation} also reveals that using the learnable codebook in the reconstruction model improves performance on Avenue by almost $3$ percentage points. Including the codebook in the prediction model improves performance on both datasets by around $1$ percentage point, suggesting that restricting the network to a discrete feature map is advantageous for detecting anomalies. The codebook has a smaller effect on the prediction network due to the presence of skip connections -- not all feature maps are quantized in the network. It is hypothesised that quantizing the feature maps at all levels (i.e. quantizing the skip connections too) would improve performance even further, but this could be slow, because quantization requires retrieving the nearest neighbour in the codebook.
\begin{table}
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Temporal & Codebook & UCSD Ped2 & Avenue
\\
\hline\hline
\xmark & \xmark & 71.1 & 78.3 \\
\hline
\xmark & \cmark & 66.7 & 81.5 \\
\hline
\cmark & \xmark & 88.1 & 87.9 \\
\hline
\cmark & \cmark & 89.2 & 88.3 \\
\hline
\end{tabular}
\end{center}
\caption{\textbf{Ablation study.} Abnormal event detection accuracy in AUC (\%). We compare the effect of the temporal information and the codebook on the performance of VQU-Net method.\label{tab:ablation}}
\end{table}
\subsection{Anomaly explanations.}
\paragraph{Comparison against the method from Szymanowicz \etal~\cite{szymanowicz_2021_xman}.}
\begin{table}
\begin{center}
\begin{tabular}{|l|l|l|l|l}
\hline
Method & UCSD & Avenue \\
& Ped2 & \\
\hline\hline
Szymanowicz \etal~\cite{szymanowicz_2021_xman} &
41.6 & 6.82 \\
\hline
VQU-Net + Explainability module &
\textbf{67.2} &
\textbf{11.9} \\
\hline
\end{tabular}
\end{center}
\caption{Abnormal event explanation mean Average Precision (mAP)
evaluated on the full X-MAN dataset. \label{tab:explanation_results_full}}
\end{table}
Tab.~\ref{tab:explanation_results_full} shows the mAP achieved by our system on the anomaly explanation task. VQU-Net is seen to outperform the method from Szymanowicz \etal~\cite{szymanowicz_2021_xman} on both datasets. There are 2 main reasons for this. Firstly, SlowFast~\cite{Feichtenhofer_2019_ICCV} (used for explanations in the VQU-Net method) is a temporal method, while DRG~\cite{gao_2020_drg} (used for explanations in the method from Szymanowicz \etal~\cite{szymanowicz_2021_xman}) operates on a single frame, hence it is expected that SlowFast will recognise actions / interactions better than DRG, hence resulting in a higher explanation mAP. Secondly, the X-MAN dataset contains classes that follow mostly the COCO and AVA datasets, but the method from Szymanowicz\etal~\cite{szymanowicz_2021_xman} is trained on the V-COCO dataset, hence there is a mismatch of class labels in the case of the method from Szymanowicz\etal~\cite{szymanowicz_2021_xman}, which results in lowering the mAP score. Finally, in Tab.~\ref{tab:results} we have shown that VQU-Net outperformed method from Szymanowicz \etal~\cite{szymanowicz_2021_xman} method on the task of anomaly detection, which is the first stage in anomaly explanation, hence improving the quality of explanations.
\paragraph{Upper bound analysis.} Our explainability module is not able to explain anomalies due to anomalous location, hence in this section we evaluate explanations on the subset of X-MAN dataset excluding the location classes. Tab.~\ref{tab:explanation_results_vq} illustrates that the codebook has little effect on the explanation capacity of the network -- it can be hypothesised that the prediction error is equally informative in both methods for anomaly explanation. Last row in Tab.~\ref{tab:explanation_results_vq} was obtained by using the ground truth anomaly segmentation maps as prediction error, therefore evaluating only the capacity of the evaluation module. Comparison of the scores obtained with VQU-Net reveals that the major limitation on the explaantions is from the limited performance of action recognition and object detection modules. Results in Tab.~\ref{tab:explanation_results_full} reveal that explanations using ground truth saliency maps result in a similar score to explanations using VQU-Net saliency maps, suggesting the saliency maps from VQU-Net identify anomalies well.
The explanation scores are limited, because action recognition is a difficult task and even state-of-the-art methods like SlowFast~\cite{Feichtenhofer_2019_ICCV} achieve less than 5\% AP on action recognition on classes such as throwing or pushing. The task of action recognition is even more difficult in anomaly explanation, where the combinations of objects and interactions are unlikely to have been seen in training.
\begin{table}
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Method & UCSD & Avenue \\
& Ped2 & \\
\hline\hline
U-Net, prediction &
67.0 &
13.6 \\
\hline
VQU-Net, prediction &
67.2 &
13.4 \\
\hline
Ground truth anomaly & 69.7 & 15.1 \\
\hline
\end{tabular}
\end{center}
\caption{Abnormal event explanation mean Average Precision (mAP)
evaluated on the subset of X-MAN dataset, excluding location classes. The last row is obtained by evaluating using ground truth anomaly saliency maps. \label{tab:explanation_results_vq}}
\end{table}
\section{Conclusions}
We proposed a new system for detecting and explaining anomalous events in video. A novel architecture for learning discrete representations of video (VQU-Net) was shown to produce high quality saliency maps. We showed how this architecture aids the downstream task of explaining anomalous events while also decoupling anomaly detection performance from the accuracy of object / action classifiers. To the best of our knowledge this is the first time a neural vector qunatizer has been used for the task of anomaly explanation. Qualitative analysis showed how it restricts the network and only allows for generation of normal frames to produce high quality saliency maps. These saliency maps were shown to be both useful for anomaly detection and also for accurate event explanation. On a subset of the X-MAN dataset (containing single scene videos from Avenue~\cite{lu_2013} and UCSD Ped2~\cite{mahadevan_2010}), this lead to a 60\% mAP improvement over the state-of-the-art anomaly explanation methods. To further extend our method to handle datasets with videos from multiple scenes, for example ShanghaiTech~\cite{liu_2018}, we would like to investigate training an ensemble of models, one per scene. In a practical setting this could entail training one model for each camera. Finally, our upper bound analysis on event explanation suggested that to improve scores significantly further, focus should be aimed towards improving understanding of actions in video.
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\section{Introduction}
Planet formation and evolution are heavily dependent on the circumstellar environment. The circumstellar material can dictate formation, composition, orbital parameters, and migration of planets. While much has been learned in recent years about circumstellar disks, especially with ALMA \citep{AndrewsDiskSubstructuresHigh2018}, the evolution from protoplanetary disk to debris disk is not well understood. This time period is crucial for the final stages of the growth of terrestrial planets and the early evolution of their atmospheres \citep[e.g.][]{KenyonTerrestrialPlanetFormation2006, OlsonNebularatmospheremagma2019}. Additionally, ALMA typically images only the outer regions of disks; it is also of interest to understand the inner few AU of a system where many exoplanets are found and where the systems' habitable zones are.
Protoplanetary disks consist of gas, dust, and eventually planetesimals. All three components play a crucial role in the formation and evolution of planets. Measurements of micron-sized dust are relatively easy, as dust can produce detectable amounts of infrared (IR) emission, even through the debris disk stage. However, the primordial gas, which is mostly molecular hydrogen, is assumed to be 99\% of the protoplanetary disk mass \citep[see review by][]{WilliamsProtoplanetaryDisksTheir2011} and controls much of the disk dynamics, such as altering orbits of planetesimals and planets \citep[e.g.,]{WeidenschillingAerodynamicssolidbodies1977, GoldreichEccentricityEvolutionPlanets2003,YoudinStreamingInstabilitiesProtoplanetary2005, BaruteauPlanetDiskInteractionsEarly2014} and potentially producing rings and spirals \citep{LyraFormationsharpeccentric2013, GonzalezSelfinduceddusttraps2017}. Lower gas fractions and optically thin gas are expected in debris disks \citep{WyattEvolutionDebrisDisks2008}, although the precise gas fraction is poorly constrained \citep{MatthewsObservationsModelingTheory2014} and possibly varies significantly between disks. But even small amounts of optically thin gas can still have a large effect on disk dynamics \citep[e.g.][]{TakeuchiDustMigrationMorphology2001, LyraFormationsharpeccentric2013}. Thus, to understand the evolution of the circumstellar environment, we must understand how the hydrogen evolves.
However, H$_2$ is notoriously hard to detect. Its only allowed electric dipole transitions are in the ultraviolet (UV). In most circumstellar environments, those transitions require excited H$_2$, which can occur in circumstellar disks with warm gas \citep{NomuraMolecularhydrogenemission2005, AdamkovicsFUVIrradiatedDisk2016}. Warm gas is common in protoplanetary disks, but is less likely to be found in debris disks because of the generally large distance of the gas from the central star. Chromospheric and transition region lines, such as Ly$\alpha$, pump the H$_2$ molecule from an excited level in the ground electronic state to the first (Lyman) or second (Werner) electronic levels. Because of extremely high oscillator strengths, the excited molecule immediately decays back to the ground electronic level in a fluorescent cascade, emitting photons. The set of emission lines produced by transitions from a single excited electronic state to the multiple allowed ground electronic states is called a progression. Within a given progression, H$_2$ line fluxes are proportional to their branching ratios \citep{WoodAnalysisH2Emission2002, HerczegFarUltravioletSpectraTW2004}. Because of this, far UV spectra are a powerful way to characterize the warm H$_2$ gas. Emission from these lines is a probe of gas temperature. But as all these transitions are in the UV, they require data from space-based observatories, thus limiting the number of observations currently available. There are magnetic quadrapole transitions in the IR that have been detected in protoplanetary disks \citep[e.g][]{WeintraubDetectionQuiescentMolecular2000, BaryDetectionsRovibrationalH22003}; unfortunately, they are weak and require much larger amounts of warm H$_2$ than debris disks typically have in order to detect them \citep[e.g.][]{BitnerTEXESSurveyH22008, Carmonasearchmidinfraredmolecular2008}.
To try to get around these issues, other molecules, most notably IR and millimeter transitions of HD and more commonly CO, have been used to trace the H$_2$ \citep[e.g.][]{TrapmanFarinfraredHDemission2017}. However, neither is a perfect tracer, and both rely on an assumed ratio to H$_2$. For example, disk mass estimates have often used the ISM CO/H$_2$ of $\sim$10$^{-4}$, consistent with the value found by \citet{FranceCOH2Abundance2014} based on CO and H$_2$ observations in the UV, but other recent studies have shown that CO appears depleted in protoplanetary disks \citep{FavreSignificantlyLowCO2013, SchwarzUnlockingCODepletion2019, McClureCarbondepletionobserved2019}. Furthermore, the difference in chemistry and masses between the molecular species mean that neither HD or especially CO trace H$_2$ perfectly \citep{MolyarovaGasMassTracers2017, AikawaMultiplePathsDeuterium2018}.
Molecular hydrogen emission has been detected in every protoplanetary and transition disk that have far UV spectral observations \citep[e.g.][]{ValentiIUEAtlasPre2000, ArdilaObservationsTauriStars2002, HerczegOriginsFluorescentH22006, InglebyFarUltravioletH2Emission2009, FranceHubbleSpaceTelescope2012, YangFarultravioletAtlasLowresolution2012, France1600EmissionBump2017}. Debris disks are not defined by their gas content --- they are instead defined by secondary dust produced from planetesimal collisions, which observationally gets translated into a fractional luminosity, $f=L_{disk}/L_*$ less than 10$^{-2}$ --- but all evidence indicates that compared with protoplanetary disks, they have a smaller gas-to-dust ratio and less gas in total \citep[e.g.][]{ChenDustGasPictoris2007}. Other gas species, like CO, have been detected in debris disks \citep[e.g.][]{RobergeDebrisDisksNearby2008, MoorMolecularGasYoung2011, DentMolecularGasClumps2014, HiguchiDetectionSubmillimeterwaveEmission2017}, but the only previous potential detection of H$_2$ in what is clearly a debris disk is from AU Mic \citep{FranceLowMassH2Component2007}. This is not unexpected. While comets in our own Solar System produce CO, they do not produce H$_2$ \citep{MummaChemicalCompositionComets2011}. Thus, it is likely that secondary H$_2$ is not produced in the same manner as secondary CO. In several cases, there are arguments for the reclassification of systems based on the discovery of H$_2$, such as RECX 11 \citep{InglebyEvolutionXrayFarultraviolet2011}, HD 98800 B (TWA 4 B) \citep{YangFarultravioletAtlasLowresolution2012, RibasLonglivedProtoplanetaryDisks2018}, and potentially DoAr 21 \citep{BaryDetectionsRovibrationalH22003, JensenNoTransitionDisk2009}. But exactly when and on what timescale the H$_2$ dissipates is not known.
Since even small amounts of H$_2$ gas can have a significant impact on planetary systems at ages $\sim$10 Myr, we have begun a program to examine UV spectra of young stars that show no evidence of near-infrared (NIR) excess. One specific way that gas can impact a system is by limiting the IR flux from dust produced by planetesimal collisions. \citet{KenyonRockyPlanetFormation2016} show that there is a discrepancy between the incidence rate of dust expected to be produced by terrestrial planet formation (2 to 3\% of young systems) and the incidence rate of close-in terrestrial planets (20\% of mature systems). Gas, however, could sweep away that dust via gas drag, making it harder to detect. Thus, it is critical to understand the evolution of H$_2$ in the terrestrial planet forming regions.
\section{Target and Observations}
TWA 7 is an M dwarf that is part of the $\sim$7-10 Myr TW Hya Association \citep{WebbDiscoverySevenTauri1999}. Recent spectral classifications assign an M2 or M3 spectral type \citep{ManaraXshooterspectroscopyyoung2013, HerczegOpticalSpectroscopicStudy2014}; we adopt M2.5. The star is surrounded by a debris disk that was first detected due to its IR excess at 24 and 70 $\mu$m by \citet{LowExploringTerrestrialPlanet2005} with the Spitzer Space Telescope. However, the lack of near IR excess \citep{WeinbergerSearchWarmCircumstellar2004} and typical accretion signatures \citep{JayawardhanaAccretionDisksYoung2006} strongly imply that it is a ``cool'' debris disk, making it one of the few known M stars with a debris disk \citep{TheissenWarmDustCool2014}. The dust in the disk has since been detected in the FIR at 450 and 850 $\mu$m using the James Clerk Maxwell Telescope \citep{MatthewsMassTemperatureTWA2007} and at 70, 160, 250, and 350 $\mu$m using the Herschel Space Observatory \citep{CiezaHerschelDIGITSurvey2013}. No [O I] was detected by Herschel at 63 $\mu$m \citep{Riviere-MarichalarGasdustTW2013}, but CO has been recently detected using ALMA in the J=3-2 transition \citep{MatraUbiquityStellarLuminosity2019}. The disk has been imaged in the IR with SPHERE showing spiral arms near 25 AU \citep{OlofssonResolvingfaintstructures2018}. \citet{YangFarultravioletAtlasLowresolution2012} and \citet{FranceHubbleSpaceTelescope2012} both failed to detect H$_2$ around TWA 7 in UV spectra. \citet{YangFarultravioletAtlasLowresolution2012} used a less sensitive prism spectrum; \citet{FranceHubbleSpaceTelescope2012} looked at 12 H$_2$ features separately, as opposed to detecting the combined H$_2$ emission from many features as we do in this paper. (See Section \ref{cc_sec}.)
We can put some constraints on the expected H$_2$ based on dust and CO measurements. For TWA 7, the total dust mass in the disk, M$_d$, is 2$\times$10$^{-2}$ M$_\oplus$ \citep{BayoSubmillimetrenoncontaminateddetection2019}, while the mass of CO in the disk, M$_{CO}$, is 0.8-80$\times$10$^{-6}$ M$_\oplus$ \citep{MatraUbiquityStellarLuminosity2019}. Based on these estimates, if TWA 7 has an ISM value for the CO/H$_2$ ratio of $\sim$10$^{-4}$ then we can expect M$_{H_2}$ to be on the order of M$_d$. If it has a lower CO/H$_2$ of $\sim$10$^{-6}$, as TW Hya has \citep{FavreSignificantlyLowCO2013}, we can expect M$_{H_2}$ to be 100$\times$ larger than M$_d$, consistent with the ISM gas-to-dust ratio \citep{SpitzerPhysicalprocessesinterstellar1978}. Models that have explored gas-to-dust ratios between 0.01 and 100 indicate that gas can significantly influence the disk dynamics \citep{YoudinStreamingInstabilitiesProtoplanetary2005, LyraFormationsharpeccentric2013, GonzalezSelfinduceddusttraps2017}, so in either case, H$_2$ could play an important if not dominant role in TWA 7's disk dynamics. Given the presence of this distant reservoir of gas, we explore here the possibility that an (as yet unseen) reservoir of gas is also present at smaller disk radii, in the terrestrial planet region of the disk
For this work, we use archival HST-Cosmic Origins Spectrograph (COS) observations of TWA 7 from May 2011 (PID 11616, PI: G. Herczeg). The data were acquired with the far UV medium resolution modes of COS: G130M and G160M. These spectra have a spatial resolution of 1'' and a wavelength uncertainty $\sim$15 km/s \citep{cosmic2020}. The observations are at a range of central wavelengths that allow us to get a contiguous spectrum that spans from 1133 to 1795 \AA\ (Figure \ref{spectrum}). In addition to TWA 7, we also analyze spectra of classical T Tauri stars (CTTS) and main sequence M dwarf stars for comparison purposes (Table \ref{compstars}) taken between December 2009 and August 2015. The CTTS were chosen from the stars analyzed by \citet{FranceHubbleSpaceTelescope2012} that had extinction values measured by both \citet{HerczegOpticalSpectroscopicStudy2014} and \citet{FurlanSpitzerInfraredSpectrograph2011}. The main sequence M dwarfs were from \citet{KruczekH2FluorescenceDwarf2017}, chosen because they had H$_2$ detected from the stellar photosphere and COS spectra that covered a comparable wavelength range. One of the six M dwarfs --- GJ 581 --- has a cold, faint debris disk \citep{LestradeDEBRISdiskplanet2012}, but it is much older (2-8 Gyr) and less active \citep{SchoferCARMENESsearchexoplanets2019} than TWA 7 or the CTTS. Its disk is also significantly less luminous than that of TWA 7 \citep{ChoquetFirstimagesdebris2016}. The remaining five M dwarfs have no detected disks. All spectra were observed with COS in a similar manner. Spectra were reduced by the CALCOS pipeline. Multiple observations were then co-added into one spectrum as described by \citet{DanforthEmpiricallyEstimatedFarUV2016}. The TWA 7 spectrum we analyzed is plotted in Figure \ref{spectrum}.
\begin{figure}
\centering
\includegraphics[width=6.4in]{spectrum_labled.png}
\caption{The spectrum of TWA 7 used for analysis with the most prominent stellar features labeled. Note that the Ly$\alpha$ profile is largely geocoronal airglow emission and was thus not used. \label{spectrum}}
\end{figure}
We also used archival HST-STIS spectra of TW Hya, reduced with the STIS pipeline. For each observation, we combined the orders to create a single spectrum. We then co-added the observations in a similar manner to the way we co-added the observations from COS.
\begin{deluxetable}{llrrrl}
\tabletypesize{\small}
\tablewidth{0pt}
\tablecaption{Stellar Properties}
\tablehead{
\colhead{Object} & \colhead{PID/PI} & \colhead{Distance} & \colhead{RV} & \colhead{A$_V^a$} & \colhead{A$_V^b$}
\vspace{-5pt}
\\
\colhead{} & \colhead{} & \colhead{(pc)} & \colhead{(km s$^{-1}$)} & \colhead{(mag)} & \colhead{(mag)}
}
\startdata
\textbf{TWA 7} & \textbf{11616/Herczeg} & \textbf{34.0} & \textbf{11.4} & \nodata & \textbf{0.00}$^c$ \\
\cutinhead{Classical T Tauri Stars}
AA Tau & 11616/Herczeg & 136.7 & 17.0 & 1.9 & 0.40 \\
BP Tau & 12036/Green & 128.6 & 15.2 & 1.0 & 0.45 \\
DE Tau & 11616/Herczeg & 126.9 & 15.4 & 0.9 & 0.35 \\
DM Tau & 11616/Herczeg & 144.5 & 18.6 & 0.0 & 0.10 \\
DR Tau & 11616/Herczeg & 194.6 & 21.1 & 1.4 & 0.45 \\
GM Aur & 11616/Herczeg & 159.0 & 15.2 & 0.6 & 0.30 \\
HN Tau & 11616/Herczeg & 136.1 & 4.6 & 1.0 & 1.15 \\
LkCa 15 & 11616/Herczeg & 158.2 & 17.7 & 1.0 & 0.30 \\
SU Aur & 11616/Herczeg & 157.7 & 14.3 & 0.9 & 0.65 \\
UX Tau & 11616/Herczeg & 139.4 & 15.5 & 0.5 & 0.00$^c$ \\
\cutinhead{Main Sequence M Stars with H$_2$}
GJ 176 & 13650/France & 9.5 & 26.2 & \nodata & \nodata \\
GJ 832 & 12464/France & 5.0 & 13.2 & \nodata & \nodata \\
GJ 667 C & 13650/France & 7.2 & 6.4 & \nodata & \nodata \\
GJ 436 & 13650/France & 9.8 & 9.6 & \nodata & \nodata \\
GJ 581 & 13650/France & 6.3 & -9.4 & \nodata & \nodata \\
GJ 876 & 12464/France & 4.7 & -1.6 & \nodata & \nodata \\
\cutinhead{STIS Spectra}
TW Hya & 11608/Calvet & 60.1 & 13.4 & \nodata & 0.00
\enddata
\tablecomments{Properties of stars analyzed in this paper. \\
$^a$ A$_V$ from \citet{FurlanSpitzerInfraredSpectrograph2011}\\
$^b$ A$_V$ from \citet{HerczegOpticalSpectroscopicStudy2014} with uncertainties of 0.15 mag.\\
$^c$ The measured value was negative. Since this is unphysical, we adapted an extinction of 0.0 mag. \\
Distances are from \citep{Bailer-JonesEstimatingDistanceParallaxes2018} based on Gaia DR2 \citet{CollaborationGaiaDataRelease2018}. RVs of the T Tauri stars are from \citet{NguyenCloseCompanionsYoung2012}, from Gaia DR2 for the M stars, and from \citet{TorresSearchassociationscontaining2006} for TW Hya and TWA 7.
Based on extinction measurements from stars in the Local Bubble \citep{LeroyPolarimetricInvestigationInterstellar1993}, we assume these Main Sequence M stars have no extinction.
\label{compstars}}
\end{deluxetable}
\section{Analysis and Results}
\subsection{Methods for Cross-Correlation}\label{cc_sec}
In protoplanetary disks and nearby M dwarfs, the strength of the H$_2$ lines make them clearly detectable above the noise; however, this is not the case for systems with smaller amounts of H$_2$ flux. Instead, we take advantage of the many weak H$_2$ lines in the system and use a cross-correlation function (CCF), a technique that has been used previously with IR data to study gas in protoplanetary disks \citep{HartmannFurtherevidencedisk1987}. The CCF allows us to combine the signal from multiple lines into one signal by calculating how well the spectrum correlates with that of a model template \citep{Tonrysurveygalaxyredshifts1979}. Our full template was created using the procedure from \citet{McJunkinEmpiricallyEstimatedFarUV2016} for a temperature of 2500 K and a column density of log$N$(H$_2$)=19 cm$^2$. As temperature and density have little effect on the relative strengths of these lines in protoplanetary disks \citep{FranceHubbleSpaceTelescope2012}, we used the same template for all the stars. Ly$\alpha$ also has a significant impact on H$_2$ line strengths, but its profile is contaminated from self-absorption, ISM absorption, and geocoronal airglow, and thus cannot be used for all of our targets. As a result, we do not consider the shape of the Ly$\alpha$ profile when defining our template. We use a single FWHM of 0.047 \AA\ for the lines in the template, chosen solely because it is the width that maximizes the CCF for TWA 7. Templates for individual progressions were created by picking the lines from the full template based on \citet{AbgrallTableLymanBand1993} (Figure \ref{template}). Although there are many H$_2$ lines, focusing only on the strongest lines gives the clearest signal. We chose the minimum H$_2$ line strength in the template that maximized the CCF detection for TWA 7 for each progression. The minimum line strength is dependent on how strong the lines in that progression are: progressions with weaker fluxes require smaller minimum line strengths. While we analyzed 12 different progressions (Table \ref{pros}), chosen because all 12 were detected by \citet{FranceHubbleSpaceTelescope2012} in protoplanetary disks, our analysis focused on the progressions that typically produce the most H$_2$ flux in CTTS --- [1,4], [1,7], [0,1], and [0,2]. Each of these progressions is excited by Ly$\alpha$ and can decay to multiple lower states, resulting in a set of H$_2$ emission lines throughout the UV. The total summed flux in an individual progression is a function of having enough Ly$\alpha$ photons to pump the H$_2$ molecule to the excited state (Figure \ref{cartoon}), the filling factor of H$_2$ around the Ly$\alpha$, the column density in the excited rovibrational level of the X electronic state, and the oscillator strength of the the pump transition \citep{HerczegOriginsFluorescentH22006}.
\begin{deluxetable}{lcrcclc}
\tabletypesize{\small}
\tablewidth{0pt}
\tablecaption{Progressions Analyzed}
\tablehead{
\colhead{[\hbox{$v^\prime$},\hbox{$J^\prime$}]} & \colhead{$\lambda_{pump}$} & \colhead{velocity} & \colhead{TW Hya H$_2$ Flux} & \colhead{oscillator strength} & \colhead{[\hbox{$v^{\prime\prime}$},\hbox{$J^{\prime\prime}$}]} & \colhead{E$^{\prime\prime}$}
\vspace{-5pt}
\\
\colhead{} & \colhead{(\AA)} & \colhead{(km s$^{-1}$)} & \colhead{(10$^{-15}$ erg cm$^{-2}$ s$^{-1}$)} & \colhead{($\times$ 10$^{-3}$)} & \colhead{} & \colhead{(eV)}
}
\startdata
$[3,13]$ & 1213.356 & -571 &\ \ 4.7 & 20.6 & [1,14] & 1.79 \\
$[4,13]$ & 1213.677 & -491 &\ \ 2.4 & \quad 9.33 & [2,12] & 1.93 \\
$[3,16]$ & 1214.465 & -297 & 14.9 & 23.6 & [1,15] & 1.94 \\
$[4,4]$ & 1214.781 & -219 &\ \ 8.9 & \quad 9.90 & [3,5] & 1.65 \\
$[1,7]$ & 1215.726 & 14 & 16.2 & 34.8 & [2,6] & 1.27 \\
$[1,4]$ & 1216.070 & 99 & 36.0 & 28.9 & [2,5] & 1.20 \\
$[3,0]$ & 1217.038 & 338 & \ \ 3.5 & \quad 1.28 &[3,1] & 1.50 \\
$[0,1]$ & 1217.205 & 379 & 37.9 & 44.0 & [2,0] & 1.00 \\
$[0,2]$ & 1217.643 & 487 & 33.4 & 28.9 & [2,1] & 1.02 \\
$[2,12]$ & 1217.904 & 551 & 18.4 & 19.2 & [1,13] & 1.64\\
$[2,15]$ & 1218.521 & 704 & \ \ 3.1 & 18.0 & [1,14] & 1.79 \\
$[0,3]$ & 1219.089 & 844 & \ \ 2.1 & 25.5 &[2,2] & 1.04 \\
\enddata
\tablecomments{Velocity is from Ly$\alpha$ center. TW Hya H$_2$ Flux as measured by \citet{HerczegOriginsFluorescentH22006}. Oscillator strengths of the pumping transitions calculated by \citet{HerczegOriginsFluorescentH22006} based on \citet{AbgrallTableLymanBand1993}. \\
$[$v$^{\prime\prime}$,J$^{\prime\prime}$] and E$^{\prime\prime}$ are the lower level in the electronic ground state for the pumping transition and the corresponding energy for that state.\\
Each of these progressions is pumped by Ly$\alpha$ flux and can decay to multiple lower states, resulting in a set of H$_2$ emission lines throughout the UV.
\label{pros}}
\end{deluxetable}
\begin{figure}
\centering
\includegraphics[width=6.4in]{progressiontemplates.png}
\caption{Templates of the most prominent progressions used for cross-correlation. A cutoff for a minimum line strength of 5$\times$10$^-{15}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ is also shown. \label{template}}
\end{figure}
Since we want to be sure that we are only cross-correlating continuum and H$_2$ emission (plus the associated noise) and not emission from hot gas lines from the chromosphere or transition region, we masked out FUV lines commonly seen in lower mass stars from \citet{HerczegFarUltravioletSpectrumTW2002}, \citet{BrandtDor94Hubble2001}, and \citet{AyresFarUltravioletUpsDowns2015}. As these lines have different widths in different stars depending on numerous properties, we erred on the side of masking the wavelength regions covered by the broadest of these features to minimize the chance of a false positive from a line that was not H$_2$.
\begin{figure}
\centering
\includegraphics[width=6.4in]{ly_alpha_cartoon.png}
\caption{Schematic Ly$\alpha$ profiles. The solid purple line corresponds to a broader profile, like seen with CTTS, while the dashed pink line is indicative of Main Sequence stars with a more narrow profile. The pumping wavelengths of some prominent are marked. The strongest progressions have not only substantial Ly$\alpha$ flux but also larger oscillator strengths and smaller lower state energies (Table \ref{pros}). \label{cartoon}}
\end{figure}
Cross-correlating the entire spectrum with the entire masked template returns a tentative detection. However, this involves cross-correlating a significant amount of noise which can weaken the detection. Therefore we created segments of spectrum for each H$_2$ feature of $\sim$1 \AA\ ($\sim$200 km/s) wide centered around the wavelengths of expected H$_2$ lines, which is wide enough to get the entire line profile without adding too much continuum flux or noise. (We cannot be certain as to whether the photons detected outside of lines are from the star itself, as M dwarfs have very little stellar continuum flux in these regions, so we will refer to the region outside of lines as ``continuum/noise.'') To calculate the final CCF, we explored two procedures to verify any findings. With both methods, if the flux for every line is emitted at a similar relative velocity (within $\sim$10 km s$^{-1}$), the CCF's signal will grow stronger. In the first method, we created one long spectrum by putting all the individual segments end-to-end. We do the same for the corresponding template segments. We then cross-correlated this pieced together spectrum with the same regions from the template. In the second method, we cross-correlated each segment of spectrum individually with its corresponding template segment and added the cross-correlation functions. Because of this, we chose to use a CCF that has not been normalized for length, which is usually the last step of calculating the CCF. Unnormalized CCFs work equally well for both of our methods; normalized CCFs of different lengths cannot be added linearly, because longer CCFs should be weighted more
\subsection{\texorpdfstring{H$_2$ Detection and Verification}{H2 Detection and Verification}}
\begin{figure}
\centering
\includegraphics[width=6.4in]{ccfs_3_wlines.png}
\caption{Cross-correlation functions of TWA 7 for the most prominent H$_2$ progressions. \label{ccf}}
\end{figure}
We detect peaks near the stellar radial velocity in the CCFs of the spectra of TWA 7 (Figure \ref{ccf}) using a minimum H$_2$ template line strength cutoff of 5$\times$10$^{-15}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$. The peaks are detected with both methods --- segmented spectrum and added CCF --- for calculating the CCF. Although the peak is strongest when all of the most prominent progressions are included, we also see significant ($>$3$\sigma$) detections when some individual progressions are analyzed. While the peak for [0,1]+[0,2] is slightly off-center from the systemic velocity of 11.4 km/s, we attribute this to the uncertainties in the wavelength calibration that can lead to shifts in the resulting velocities by up to 15 km/s, as described in \citet{LinskyUltravioletSpectroscopyRapidly2012}
The strength of the cross-correlation function is dependent on the S/N in the H$_2$ lines. Since we are trying to measure the height and significance of the CCF peak, we need to understand the noise properties of the observed spectrum. This is made more difficult, because there are so few FUV photons that reach us. For example, \citet{LoydMUSCLESTreasurySurvey2016} looked for FUV continuum in our M dwarf sample, obtaining a significant detection for only 3 of 6 targets. As a result, noise in our spectrum cannot be approximated to be Gaussian as it can when there are hundreds or thousands of counts. Typical continuum/noise regions in our TWA 7 spectrum have flux distributions that look approximately like that seen in Figure \ref{kde} where we show a histogram of flux levels found in the continuum/noise of TWA 7.
There are several potential issues in modeling this noise. The first is that there are undoubtedly unidentified lines that we do not mask, as possibly seen in the increase around 0.3$\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$ . However, since other unidentified lines could possibly overlap with the H$_2$ lines, we choose not to remove this peak from the distribution. Another issue is that because of the low flux level, when the detector background gets subtracted, we end up measuring ``negative'' flux in some wavelength bins. To deal with this, we estimate the noise in two separate ways: with a scaled Poisson distribution and using the actual distribution fit with a kernel density estimator (KDE) \citep{RosenblattRemarksNonparametricEstimates1956, ParzenEstimationProbabilityDensity1962}, as shown in Figure \ref{kde}. The scaled Poisson was determined by calculating the skew of the distribution of continuum/noise, $\gamma_1$. The mean of the Poisson distribution $\lambda$ is then $\gamma_1^{-2}$. We then convert from counts to flux using a constant scaling factor determined by the mean of the distribution. The KDE was calculated with a Gaussian kernel using a bandwidth (equivalent to the sigma parameter) of 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$. We randomly draw our noise from these distributions. These two noise models cover the range of possibilities of the underlying true noise, so a robust detection will only be evident if it occurs using both noise models.
\begin{figure}
\centering
\includegraphics[width=3.4in]{hist_kde.png}
\caption{Typical continuum/noise distributions, which we model in two ways: by creating a KDE of the distribution and by scaling a discrete Poisson distribution to the data. Both were normalized so that the total area is equal to 1. \label{kde}}
\end{figure}
To determine the significance of the detection, we used our noise models to create spectra containing only noise. We then cross-correlated these noise spectra with the template in the same ways we did for the TWA 7 spectrum. We then record the CCF maximum within 15 km s$^{-1}$ of the systemic velocity. We chose this range, because this was the range we used to look for a detection, as COS has a velocity precision of 15 km s$^{-1}$. This procedure was repeated multiple times (see Table \ref{pros_sig}) for each type of noise to produce the distributions shown in Figure \ref{ccf_max_noise}. We then compared the CCF maximums to TWA 7's CCF maximum within 15 km s$^{-1}$ of the systemic velocity. The fraction of times the noise's CCF maximum was equal to or larger than TWA 7's CCF maximum is taken to be the probability of a false positive. The significance ($\sigma$) values we report are the the equivalent probabilities for a Gaussian distribution.
\begin{figure}
\centering
\includegraphics[width=6.4in]{just_noise.png}
\caption{Distribution of CCF heights for simulations of noise cross-correlated with the [0,1] and [0,2] progressions of the H$_2$ template. Based on 1,200,000 simulations for each set, we detect H$_2$ at a level of 4.5$\sigma$ for Poisson noise with the added CCFs, 4.6$\sigma$ for CDF sampled noise with the added CCFs, 3.9$\sigma$ for Poisson noise with the segmented spectrum CCF, and 4.0$\sigma$ for CDF sampled noise with the segmented spectrum CCF. \label{ccf_max_noise}}
\end{figure}
We did an initial trial of 32,000 simulations with each method to see if we could detect each progression individually. We obtain significant detections for [0,1], [0,2], [1,4], and [1,7], with significant being defined as $>$3$\sigma$ detections for all four methods; we also get a a marginal detection ($>$3$\sigma$ for some but not all methods) for [0,3] (Table \ref{pros_sig}). We then investigated the detected progressions further. Using a line strength cutoff of 5$\times$10$^{-15}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$, as shown in Figure \ref{template}, we detect H$_2$ at a significance $>$5$\sigma$ for all of our noise models and CCF types for the combination of the [1,4], [1,7], [0,1], and [0,2] progressions based on 3,500,000 simulations of each. For just the progressions on the wing --- [0,1] combined with [0,2] --- we did 1,200,000 simulations for each measurement. We detect H$_2$ at a level of 4.5$\sigma$ for Poisson noise with the added CCFs, 4.6$\sigma$ for KDE noise with the added CCFs, 3.9$\sigma$ for Poisson noise with the segmented spectrum CCF, and 4.0$\sigma$ for KDE sampled noise with the segmented spectrum CCF. The segmented spectrum CCF produces similar distributions for both types of noise, as shown on the right of Figure \ref{ccf_max_noise}, because it is more robust to slight differences in noise models.
\begin{deluxetable}{lrrrrrrc}
\tabletypesize{\small}
\tablewidth{0pt}
\tablecaption{Detection Significance of H$_2$ in Progressions}
\tablehead{
\colhead{} & \multicolumn{2}{c}{\underline{Added CCF}} & \multicolumn{2}{c}{\underline{Segmented Spectrum CCF}} & \colhead{} & \colhead{Minimum line strength} & \colhead{Lines}
\vspace{-5pt}
\\
\colhead{Progression} & \colhead{KDE} & \colhead{Poisson} & \colhead{ \hspace{7pt} KDE} & \colhead{ \hspace{17pt} Poisson} & \colhead{Simulations} & \colhead{erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$}
&\colhead{ Included}}
\startdata
$[3,13]$ & 2.2 & 2.2 & \hspace{2pt} 1.6 & 1.5 \quad \hspace{2pt} & 32000 & 0.1$\times$10$^{-15}$ & 9 \\
$[4,13]$ & 0.7 & 0.7 & \hspace{2pt} 0.8 & 0.8 \quad \hspace{2pt} & 32000 & 1.7$\times$10$^{-15}$ & 2 \\
$[3,16]$ & 2.4 & 2.4 & \hspace{2pt} 1.8 & 1.8 \quad \hspace{2pt} & 32000 & 1.2$\times$10$^{-15}$ & 9 \\
$[4,4]$ & 1.2 & 1.2 & \hspace{2pt} 2.0 & 2.0 \quad \hspace{2pt} & 32000 & 1.3$\times$10$^{-15}$ & 6 \\
$[1,7]$ & $>$4.0 & $>$4.0 & \hspace{2pt} $>$4.0 & $>$4.0 \quad \hspace{2pt} & 32000 & 7.5$\times$10$^{-15}$ & 2 \\
$[1,4]$ & $>$4.0 & $>$4.0 & \hspace{2pt} $>$4.0 & $>$4.0 \quad \hspace{2pt} & 32000 & 3.0$\times$10$^{-15}$ & 12 \\
$[3,0]$ & 0.7 & 0.7 & \hspace{2pt} 0.0 & 0.0 \quad \hspace{2pt} & 32000 & 3.5$\times$10$^{-15}$ & 2 \\
$[0,1]$ & 3.5 & 3.4 & \hspace{2pt} 3.2 & 3.1 \quad \hspace{2pt} & 32000 & 9.0$\times$10$^{-15}$ & 2\\
$[0,2]$ & $>$4.0 & $>$4.0 & \hspace{2pt} 3.3 & 3.3 \quad \hspace{2pt} & 32000 & 5.0$\times$10$^{-15}$ &2 \\
$[2,12]$ & 2.2 & 2.3 & \hspace{2pt} 2.6 & 2.6 \quad \hspace{2pt} & 32000 & 2.0$\times$10$^{-15}$ &2 \\
$[2,15]$ & 2.8 & 2.8 & \hspace{2pt} 2.7 & 2.7 \quad \hspace{2pt} & 32000 & 3.0$\times$10$^{-15}$ & 4 \\
$[0,3]$ & 3.3 & 3.1 & \hspace{2pt} 2.9 & 2.8 \quad \hspace{2pt} & 32000 & 3.0$\times$10$^{-15}$ & 5 \\
\hline
$[0,1]+[0,2]$ & 4.6 & 4.5 & \hspace{2pt} 4.0 & 3.9 \quad \hspace{2pt} & 1200000 & 5.0$\times$10$^{-15}$ & 6 \\
$[1,4]+[1,7]+[0,1]+[0,2]$ & $>$5.0 & $>$5.0 & \hspace{2pt} $>$5.0 & $>$5.0 \quad \hspace{2pt} & 3500000 & 5.0$\times$10$^{-15}$ & 20 \\
\enddata
\tablecomments{Significance of detection in each individual progression for our four different methods, as well as for the combination of [0,1] and [0,2] and the combination of [1,4], [1,7], [0,1], and [0,2].
\label{pros_sig}}
\end{deluxetable}
There are two unclassified background objects in the sky near TWA 7. However, the two sources are not expected to be in the COS-PSA aperture, given the COS-PSA aperture size of 2.5" in diameter and the distance of the background sources to TWA 7, at 2.5" \citep{Neuhauserpossibilitygroundbaseddirect2000} and 6" \citep{BayoSubmillimetrenoncontaminateddetection2019}, given we confirmed that TWA 7 was centered on the aperture. Furthermore, as we required that the CCF peak at the radial velocity of the object fall within the error of the spectrograph, we feel these background sources are an extremely unlikely source of the H$_2$.
\subsection{\texorpdfstring{Determining the Origin of the H$_2$}{Determining the Origin of the H2}}\label{origin}
Simply detecting H$_2$ does not indicate that the H$_2$ is circumstellar in origin, because some M stars are known to show H$_2$ emission pumped by Ly$\alpha$ \citep{KruczekH2FluorescenceDwarf2017}. Active M stars, like TWA 7 \citep{YangMagneticPropertiesYoung2008}, have strong chromospheric Ly$\alpha$ emission, which can pump H$_2$ in star spots or in their lower chromospheres. Since TWA 7's debris disk is nearly face on at an inclination of 13$^\circ$ \citep{OlofssonResolvingfaintstructures2018}, and the resolution of COS is 15 km/s, we cannot use velocity information to differentiate between circumstellar and stellar H$_2$. Instead, we looked at the flux ratios between different progressions. The [1,4] and [1,7] progression are both pumped by emission from the center of the Ly$\alpha$ line profile (Figure \ref{cartoon}). Other progressions are pumped from the wings of the profile, so strong emission in these lines is only possible with a broader Ly$\alpha$ line indicative of active accretion. The two most prominent examples are [0,1] and [0,2] which are pumped at velocities 379 and 487 km/s from line center. These progressions should only be bright if the Ly$\alpha$ profile is especially wide, as shown in the purple profile in Figure \ref{cartoon}, but they will be much fainter if the star's Ly$\alpha$ profile is more narrow, similar to the pink curve. Stars that are accreting have much broader Ly$\alpha$ profiles than active main sequence stars \citep{SchindhelmLyaDominanceClassical2012, YoungbloodMUSCLESTreasurySurvey2016} and are therefore expected to produce more emission in the [0,1] and [0,2] progressions relative to the [1,4] and [1,7] progressions in comparison to non-accretors.
Using the segmented spectrum, we took the ratio of the CCF maximum of all the non-central progressions
to the ratio of the CCF maximum of [1,4]+[1,7] for all the stars with previously detected H$_2$ in our sample. Dividing by the height of [1,4]+[1,7] for each system acts as a normalization factor to deal with spectra with different S/N or different line widths due to rotational broadening. Since this ratio can be affected by extinction, with lines at shorter wavelengths appearing fainter than they are intrinsically, we have to de-redden the spectrum first, which we do based on the extinction laws from \citet{CardelliRelationshipInfraredOptical1989}. We examine three different sets of ratios: ratios with spectra uncorrected for extinction, ratios with spectra corrected by the extinction values found by \citet{FurlanSpitzerInfraredSpectrograph2011}, and ratios with spectra corrected by extinction values from \citet{HerczegOpticalSpectroscopicStudy2014}. We assume the main sequence M stars and TWA 7 have no extinction based on extinction measurements from stars in the Local Bubble \citep{LeroyPolarimetricInvestigationInterstellar1993}.
To estimate the 1$\sigma$ limits (the gray areas in Figures \ref{ratios} and \ref{ratios_app}), we used a similar procedure as we used to calculate the significance of detections for TWA 7. We sampled the noise from each spectrum, calculating a KDE like we did for TWA 7, and created spectra of pure noise to cross-correlate with the template. We then took the maximum of each CCF within 15 km/s of the RV of that star. The gray regions represent the inner 68\% of ratios calculated based on those maxima. These 1$\sigma$ regions are biased towards positive numbers, because we chose the maximum CCF value, which even in a normally distributed, random noise sample will bias to positive values.
\begin{figure}
\centering
\includegraphics[width=6.4in]{ratios_2.png}
\caption{The CCF height ratios of [0,1] and [0,2] to that of [1,4]+[1,7] for our selection of T Tauri stars (open symbols), M stars (filled symbols), and TWA 7. The gray areas are the 1$\sigma$ regions for a null result if that progression had no flux and just noise. They are greater than zero because we select the maximum of the CCF within 15 km s$^{-1}$, which would typically be greater than zero even for random noise. TWA 7's ratios matches better with the T Tauri stars, suggesting that its H$_2$ is also circumstellar. The ratios for the other progressions are shown in the appendix in Figure \ref{ratios_app}. \label{ratios}}
\end{figure}
The [0,1] and [0,2] ratios differentiate the samples most clearly regardless of extinction correction. Based on their data, TWA 7's H$_2$ appears to be more similar to that from the CTTS (Figure \ref{ratios}). However, other progressions are not as clear. To analyze all of the ratios, we created a Support Vector Machine classifier \citep{Platt99probabilisticoutputs} with a 4th order polynomial kernel using the data from the M dwarfs and CTTS --- excluding TWA 7 --- to determine where the H$_2$ was coming from. We then applied this classification scheme to the TWA 7 data. Based on the observed ratios, this test categorizes TWA 7's H$_2$ as similar to CTTS' 99.2\% of the time for the set uncorrected for extinction, 98.2\% of the time for the set corrected using the extinction values from \citet{HerczegOpticalSpectroscopicStudy2014}, and 98.3\% of the time for the set corrected using the extinction values from \citet{FurlanSpitzerInfraredSpectrograph2011}. This implies that the TWA 7's H$_2$ is being pumped not only from the core, but also from the wings of the Ly$\alpha$ profile as with CTTS. We expand upon this in Section \ref{sec_disc}.
\subsection{\texorpdfstring{Estimating the Amount of Circumstellar H$_2$}{Estimating the Amount of Circumstellar H2}}\label{mass_est}
To estimate the amount of warm H$_2$ in TWA 7, we compared its H$_2$ emission with that of a transition disk system, TW Hya. We chose TW Hya because in comparison with TWA 7, it has a similar age \citep{WebbDiscoverySevenTauri1999}, inclination \citep{PontoppidanSpectroastrometricImagingMolecular2008}, and a relatively similar spectral type \citep{HerczegOpticalSpectroscopicStudy2014}. While we do not think that the line profile of TW Hya would be identical to that of TWA 7, as TW Hya is accreting enough to be measured by conventional methods, it is the best match from the data available. Our goal was to find a constant scale factor that is the ratio between the H$_2$ line strengths in TWA 7 and those in TW Hya. We used a least squares fit to calculate this scale factor with the only other free parameter being the radial velocity difference. We coadded the 19 brightest H$_2$ features from the most prominent progressions --- [1,4], [1,7], [0,1], and [0,2] --- and compared the line flux from the coadded profile to the coadded profile from TW Hya. We also measured the uncertainty of this ratio by measuring the noise in the spectrum in comparison to the flux. We found a ratio between the coadded profiles of (6.9$\pm$0.7)$\times$10$^{-4}$. as shown in Figure \ref{coadd}. Adjusting for differences in distance, TWA 7 has (2.2$\pm$0.2)$\times$10$^{-4}$ of TW Hya's H$_2$ line strength, and, as a result of its similar inclination and line widths, its H$_2$ luminosity is assumed to be less than TW Hya's by the same factor. \citet{FranceHubbleSpaceTelescope2012} measure TW Hya's H$_2$ luminosity as (16.2$\pm$2.0)$\times$10$^{29}$ erg s$^{-1}$. This gives us an H$_2$ luminosity of (3.6$\pm$0.6)$\times$10$^{26}$ erg s$^{-1}$ for TWA 7. By comparing the flux values measured by \citet{KruczekH2FluorescenceDwarf2017} in star spots to our value for TWA 7' s flux, even if there is a contribution from star spots to this value, we expect that the circumstellar gas is more than 50\% of the total H$_2$ luminosity.
\begin{figure}
\centering
\includegraphics[width=3.8in]{coadded_flux.png}
\caption{The co-added H$_2$ line profile of TWA 7 compared to the co-added line profile of TW Hya, scaled by the best fit ratio of (6.9$\pm$0.7)$\times$10$^{-4}$.\label{coadd}}
\end{figure}
From this scaling factor, we can also put a lower limit on the mass of warm H$_2$ assuming the gas is all circumstellar. The flux observed in a specific H$_2$ line, $F_{obs}$, is a function of the Einstein A value for that emitting transition, $A_{Bl}$, the distance to TWA 7, $d$, the frequency of the emitting transition, $\nu_{Bl}$, and the number of H$_2$ molecules that have been pumped to the required electronic excited state, $N_B$:
\begin{equation}\label{eq_fluor}
F_{obs} = N_B\frac{A_{Bl}h\nu_{Bl}}{4\pi d^2}
\end{equation}
where $l$ is a lower energy state than $B$.
$N_B$ depends on the number of H$_2$ molecules in the lower state of the pumping transition, $N_X$ and the rate at which those get excited. This rate is dependent on the oscillator strength $f$, the Ly$\alpha$ flux of TWA 7, and the optical depth of the warm H$_2$. Since we do not know the Ly$\alpha$ flux, the H$_2$ filling factor, or the optical depth, we instead choose to estimate an upper limit for the excitation rate, which turns into a lower limit for $N_X$ as described by Equation \ref{eq_pump}:
\begin{equation}\label{eq_pump}
N_X \frac{4\pi J_\nu}{h\nu_{p}}\frac{\pi e^2}{m_ec}f\leq N_B\sum A_{Bl}
\end{equation}
where $A_{Bl}$ are all the relevant Einstein A values for the upper state, J$_\nu$ is the Ly$\alpha$ flux at the pumping wavelength, and $\nu_p$ is the frequency of the pumping transition. We do not consider dissociation, as the probability of dissociation for [1,4], [1,7], [0,1], and [0,2] is predicted to be negligible \citep{AbgrallTotaltransitionprobability2000}. This $N_X$ is dependent on the total number of warm H$_2$ molecules, $N_{H_2}$, and the temperature, which gets factored into the Boltzmann Equation, $q(T)$:
\begin{equation}\label{eq_part}
q(T)N_{H_2} =N_X.
\end{equation}
We calculate a $q(T)$ based on the assumption that the H$_2$ is being thermally excited \citep{AdamkovicsFUVIrradiatedDisk2016}.
While Ly$\alpha$ flux varies in time, and the HST FUV observation of TWA 7's Ly$\alpha$ flux is contaminated by ISM absorption and geocoronal emission, we estimate that TWA 7's Ly$\alpha$ is less than 0.03 of TW Hya's based on comparison of the spectra at velocities $>$400 km/s \citep{HerczegFarUltravioletSpectraTW2004}. We estimate the flux observed for a given transition using our scaling factor found above and the flux observed for TW Hya by \citet{HerczegFarUltravioletSpectrumTW2002}. All pumping transition properties are described in Table \ref{pros}, while the Einstein A values are from \citep{AbgrallTableLymanBand1993}. We calculate a separate N$_{H_2}$ for each line flux measured for TW Hya, which then converts into a line flux for TWA 7; we then average these values together to get our final result. For a gas temperature of 1500 K, we get a rough estimate for the minimum amount of warm H$_2$ of $\sim$9.9$\times$10$^{-11}$ M$_\oplus$. If spread out in a ring with a radius of 0.3 AU --- a radius at which H$_2$ is commonly seen \citep{FranceHubbleSpaceTelescope2012} --- this corresponds to a minimum column density of $\sim$2.8$\times$10$^{15}$ cm$^{-2}$. This is consistent with the upper limit on H$_2$ column density reported by \citet{InglebyFarUltravioletH2Emission2009} of 3.0$\times$10$^{17}$ cm$^{-2}$ using a less sensitive prism spectrum of TWA 7. Based on the spread of line fluxes, we adopt a range of 10$^{15}$ to 3.0$\times$10$^{17}$ cm$^{-2}$ for the vertical column density of H$_2$ in TWA 7.
\section{Discussion}\label{sec_disc}
The H$_2$ progressions ratios from TWA 7 (Figure \ref{ratios}) more closely resemble that from CTTS than that from M stars. However, these ratios do not guarantee that the H$_2$ is circumstellar. TWA 7 is much closer in age to the CTTS and is thus likely to have higher chromospheric activity than an average M star. Chromospheric activity produces Ly$\alpha$ emission, which can then excite the H$_2$ in star spots on mid M type stars. We suggest this is not the primary source of H$_2$ emission in TWA 7 as chromospheric activity affects the core of the Ly$\alpha$ profile significantly more than the wings \citep{LemaireHydrogenLyaLyv2015}.So while there is some Ly$\alpha$ emission in the wings from all of these stars, the amount of flux induced solely from chromospheric activity is likely not enough to excite the outer H$_2$ progressions. \citet{YoungbloodFUMESIILya2021} looked at how Ly$\alpha$ varied with stellar parameters, showing how increased Ly$\alpha$ is correlated with higher chromospheric activity and lower gravity, both of which are correlated with youth. However, the profiles from \citet{YoungbloodFUMESIILya2021} show that the ratio of the flux between the peak and the wings can remain constant with varying chromospheric activity and gravity, even if the overall flux changes. Thus the most likely explanation is that TWA 7 is still weakly accreting circumstellar gas from an inner disk.
Accretion rates for weakly accreting stars are notoriously hard to measure accurately. There are cases, like MY Lup, that have FUV accretion signatures but lack optical ones \citep{AlcalaHSTspectrareveal2019}. Previously, TWA 7 was considered a standard, non-accreting WTTS. It shows no accretion signatures in the optical. The hot FUV lines seen in TWA 7's HST-COS spectrum, such as C IV or N V, have profiles that do not look like those of CTTS \citep{ArdilaHotGasLines2013}. It also lacks the NUV flux and Ca II] $\lambda$2325 emission of known accreting stars \citep{InglebyNearultravioletExcessSlowly2011,InglebyAccretionRatesTauri2013}. However, most of the accreting gas is expected to be hydrogen in the ground state \citep{MuzerolleDetectionDiskAccretion2000}, so Ly$\alpha$ should be more sensitive to small accretion rates than any other line. There is a similar system in TWA, TWA 4 B, a K5 star with circumstellar H$_2$ FUV emission discovered by \citet{YangFarultravioletAtlasLowresolution2012} despite not showing obvious accretion signatures. Given TWA is close in age to when the typical prototplanetary disk is predicted to evolve into a debris disk, these systems could represent a short-lived phase of disk evolution with residual gas that does not accrete at the high levels detectable in optical spectra. FUV spectra of more stars in TWA would allow us to further investigate the gas evolution at this crucial age.
Assuming the H$_2$ we observe is indeed circumstellar, the next question concerns its origin. One possibility is that the H$_2$ originates from the inward migration, sublimation, and the subsequent photodissociation of H$_2$O ices in comet-like nuclei. The H$_2$O photodissociates into H, OH, and O, and the newly available H atoms can then reform into H$_2$. If true, there should also be some oxygen gas species in the inner disk. \citet{Riviere-MarichalarGasdustTW2013} give an upper limit on the oxygen mass of 2.3$\times$10$^{-5}$ M$_\oplus$ from Herschel data. This upper limit is more than the oxygen that would accompany the H$_2$ we detect if the H$_2$ originates from dissociated H$_2$O and assuming the warm H$_2$ is confined to the inner few AU of the disk, making this a potentially viable source of the observed H$_2$. Future observations could better constrain the oxygen mass in the inner disk and allow us to determine whether H$_2$O ice evaporation is a possible origin of the circumstellar H$_2$ around TWA 7. Additionally, detection of dust from these comets could lend support to this theory \citep{PearceGastrappinghot2020}.
Another possibility is that the H$_2$ we see is residual protoplanetary disk gas. Regardless of its origin, an H$_2$ formation pathway is needed to balance ongoing UV photodissociation of H$_2$. Molecular hydrogen forms most efficiently on grain surfaces, as in the ISM, but it can also form via gas phase reactions (e.g., through H + H$^-$ $\rightarrow$ H$_2$ + e$^-$) when there is less dust surface area available \citep{BrudererSurvivalmoleculargas2013}.
To explore the possibility of grain surface formation of H$_2$ in the case of TWA 7, we can estimate the upper limit on the surface area of warm grains by looking at its spectral energy distribution (SED). Although the W3 band from WISE \citep{WrightWidefieldInfraredSurvey2010} shows no excess IR emission from dust \citep{OlofssonResolvingfaintstructures2018, BayoSubmillimetrenoncontaminateddetection2019}, we can put a limit on the amount of warm dust by assuming the dust can generate the equivalent of the 1$\sigma$ uncertainty for the W3 flux. Under that assumption, we compute the SED using the model described by \citet{Isellashapeinnerrim2005} to put an upper limit of $\le$5.1$\times$10$^{-8}$ M$_\oplus$ on the amount of warm ($\sim$1000 K) silicate particles between 1 $\mu$m and 1 mm. We chose a lower particle size limit of 1 $\mu$m, because in more evolved systems, particles smaller than that near the star can get blown away by stellar winds. Based on this estimate, grains with radii between 1 $\mu$m and 1 mm could make up a significant surface area, up to a surface area of 10$^{23}$ cm$^2$. If the grains are spread out evenly over the inner 0.3 au, the mass column density is 2$\times$10$^{-5}$ g cm$^{-2}.$
With the above constraint on the possible dust content of the inner disk of TWA 7, we can use the results of \citet{BrudererSurvivalmoleculargas2013} to estimate the H$_2$ reservoir that can be sustained in the inner disk. In modeling the inner regions of transition disks, \citet{BrudererSurvivalmoleculargas2013} considered two physical models: a dusty inner disk and a very dust-poor inner disk with dust column densities of $\Sigma_d$=3$\times$10$^{-4}$ g cm$^{-2}$ and 3$\times$10$^{-9}$ g cm$^{-2}$ respectively at 0.3 au. The upper limit on the dust surface density of TWA 7 we find above is an order of magnitude below the surface density of the dusty inner disk model but many orders of magnitude above the surface density of the dust-poor disk model. Thus the dust-poor disk model provides a relatively conservative estimate of the H$_2$ density allowed for TWA 7.
The other relevant parameter in the \citet{BrudererSurvivalmoleculargas2013} model is the gas surface density. Figure 6 of \citet{BrudererSurvivalmoleculargas2013} shows the results for a case in which the inner disk and is very dust poor and has a gas column density of 0.3 g cm$^{-2}$ at 0.3\,au. The H$_2$ fraction in the disk atmosphere is $\sim$3$\times$10$^{-6}$ relative to hydrogen or an H$_2$ column density of N$_{\rm H_2}$=3$\times$10$^{18}$ g cm$^{-2}$. \citet{BrudererSurvivalmoleculargas2013} do not show the temperature of the H$_2$, although much of it is likely to be warm, as the disk is dust poor, and dust is a coolant for the gas through gas-grain collisions. If 0.1\% of the total H$_2$ column is warm ($\sim$1500 K), this scenario predicts a warm H$_2$ mass similar to that inferred for TWA 7.
Note that this result is obtained despite using a model with a dust density several orders of magnitude below our dust upper limit. Thus, it seems plausible that even a dust-poor inner disk can sustain a warm H$_2$ column density in the range we estimate for TWA 7 in Section \ref{mass_est}. While the models from \citet{BrudererSurvivalmoleculargas2013} were not tuned specifically to TWA 7's parameters --- the model assumes a hotter 10 L$_\odot$ star and a given polycyclic aromatic hydrocarbons (PAH) abundance --- these two factors should impact the H$_2$ production in opposite ways: the higher UV flux of the more massive star enhances photodestruction of H$_2$, while the PAH abundance enhances H$_2$ production. We therefore believe it is plausible that the H$_2$ we detect is sustained via some combination of gas phase reactions in the circumstellar environment of TWA 7. Future observations between 3 and 12 microns with telescopes like JWST could detect PAHs in the disk and lend further support to this possibility \citep{SeokPolycyclicAromaticHydrocarbons2017}.
Although we do not have the requisite measurements to conclusively determine why there is is warm H$_2$ in the circumstellar environment of TWA 7,
regardless of its origin, warm gas in a region without detectable warm dust is not unique to this star. Primordial warm H$_2$ is detected inside the inner edge of the dust disk in transitional disk systems \citep{FranceHubbleSpaceTelescope2012, ArulananthamUVtoNIRStudyMolecular2018}. Warm CO has also been detected in these regions \citep{PontoppidanSpectroastrometricImagingMolecular2008, SalykCORovibrationalEmission2011}. Clearly, warm gas can outlast detectable amounts of warm dust. Thus, the physics resulting in warm gas in the cavities of transitional disks could also be the cause of the H$_2$ we detect in TWA 7.
\section{Conclusions}
We have detected molecular hydrogen from four progressions ([1,4], [1,7], [0,1], and [0,2]) in TWA 7, a known debris disk system. The ratios between CCF peaks of the detected H$_2$ progressions (Figure \ref{ratios}) resemble those from CTTS. This suggests that the H$_2$ in TWA 7 is circumstellar, as it is for CTTS. This is highly unexpected, because H$_2$ is not typically detected in debris disk systems. This star joins a small group of systems that have H$_2$ but are not accreting by typical diagnostic standards. Assuming the H$_2$ is circumstellar, we have estimated a column density of 10$^{15}$ to 3.0$\times$10$^{17}$ cm$^{-2}$. While we cannot determine the origin of the gas conclusively, it is likely to be generated from residual protoplanetary disk gas.
\acknowledgements
Based on observations with the NASA/ESA Hubble Space Telescope obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. Support for program number (GO-15310) was provided through a grant from the STScI under NASA contract NAS5-26555. GJH is supported by by general grant 11773002 awarded by the National Science Foundation of China. L.F. would like to thank Andrea Isella for help with the S.E.D. modeling. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published by \citep{WengerSIMBADastronomicaldatabase2000}. This research has made use of the SIMBAD database,operated at CDS, Strasbourg, France. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement.
\facility{HST (COS, STIS)}
\software{SpecTres \citep{CarnallSpectResFastSpectral2017}, NumPy \citep{oliphant2006guide, van2011numpy}, Scikit-learn \citep{scikit-learn}, Pandas \citep{reback2020pandas}, Scipy \citep{VirtanenSciPyFundamentalAlgorithms2019}, Matplotlib \citep{Hunter:2007} }
\restartappendixnumbering
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,341
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On the west coast of Argentina is Buenos Aires, the country's capital which overlooks the Rio de la Plata estuary. It's home to two of South America's most renowned football teams, Boca Junior and River Plate, between whom there is a long-standing rivalry. Visit their respective stadia and museums to understand the passion and history of these South American titans. An arena of a different stripe sits between these two sporting icons: the Teatro Colón, one of the greatest opera houses in the world. Join us here and you'll see parts other visitors do not, including the workshops that produce the fabulous costumes and the backstage area. A short walk north will take you to the affluent Recoleta neighbourhood – where you'll get to stop at the impressive Recoleta Cemetery, last resting place of Eva Peron. In chic, upmarket Palermo Viejo you'll be welcomed into the home of a family who will show you how country bread is made.
View our Argentina tours featuring Buenos Aires
In Bariloche lies Nahuel Huapi National Park, a setting of outstanding natural beauty. Ride the Campanario Hill chairlift to the summit to admire spectacular views of the forests, Mount López and vast bodies of water. For a less hair-raising but no less fascinating experience, visit the Chocolate Museum to learn about the history and production of chocolate. We'll also give you the chance to visit a family ranch on the Patagonian Steppe, the largest desert in Argentina, where the owners have lived for generations. Saddle up and explore the countryside, where you might spot iguanas and grey foxes, before a delicious meal back at the ranch and a sheep-shearing demonstration.
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Weather in Bariloche
In Argentina's deep south lies El Calafate, an appealing town by Lago Argentino. Named after an indigenous shrub that yields tasty berries, it's said that once bitten you're sure to return to the region. Its picturesque location is reason enough to visit; it is also the perfect place to explore Parque Nacional Los Glaciares. 65 million years in the making, the park is a vista of mountains, lakes and woodland, a geological nirvana with the ice fields of the Perito Moreno Glacier at its centre. Time your visit well and you might witness the 'birth' of an iceberg – where smaller portions of ice break away from the main glacier with a tumultuous cracking sound.
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Weather in El Calafate
We've pulled together and answered some of our guests' most popular questions. Here you'll find everything you need to know – from who you'll be travelling with to whether tips are included.
"Trafalgar continues to be the best"
Susan , Jul 2014
Recommended Argentina tours
Impressions of South America
Explore Buenos Aires, see Christ the Redeemer and ascend Sugarloaf Mountain on this At Leisure trip.
South America Landscapes
Visit South America's most influential and intoxicating countries, including Brazil and Chile.
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BLOG TAGS Meet our Professors Why Study at CKGSB? Professor Viard's Column Globalization Market Trends Digitalization RCEP Leadership Finance Coronavirus Analysis and Data Economics Strategy ESG & Social Innovation
Achieving the SDGs in a Post-pandemic World: The Role of Social Innovation
At the turn of the last decade, the momentum around the Sustainable Development Goals (SDGs), the ambitious set of UN objectives for a more just and sustainable future, seemed to be fizzling out, as their scope and complexity made implementation especially hard and costly. When the COVID-19 pandemic hit and political and financial capital got diverted to fronting the health emergency, some feared that the SDGs would not survive. As we learn to coexist with the virus, however, it is possible to see a silver lining: rather than patching up the existing system with all its flaws and shortcomings, the global reset triggered by the pandemic provides the opportunity to "build back better" and realize the future envisioned by the SDGs.
Social innovation plays a critical role to get us there. Defined as the "conceptual, process, product or organizational change which improves the welfare and well-being of individuals and community," social innovation shares the SDG's ultimate objective of a better world, and offers a powerful instrument to achieve them.
Take SDG 1, for example, to eradicate poverty in all its forms by 2030. China, the country where I currently serve as the UN Resident Coordinator, has been extremely successful in its fight against poverty, having lifted some 750 million people from it in less than two generations. Challenges, however, remain, especially in bridging a lingering rural/urban divide. However, through rural revitalization, the government is now anchoring social innovation in rural areas by investing heavily in infrastructure, internet penetration and education in those regions. The private sector is also seeing the potential of social innovation.
Taobao Villages, equipped with technology and infrastructure to engage in e-commerce, have seen their income shooting up since they were first established. Social innovations that are well executed have the potential to yield positive spillover effects. In Lancang Lahu Autonomous County, in China's Yunnan Province, Zhu Youyong teaches local farmers how to plant potato varieties during the harsh winter season. Undeterred by the initial skepticism, Zhu Yuoyong took to the Pinduoduo ecommerce platform to promote winter potato planting, also looping in thousands of students from the local agricultural university to popularize the initiative. These farmers are now able to sell their produce through ecommerce to different provinces and have significantly improved their incomes and livelihoods.
The UN Country Team in China is also using social innovation to help advance rural revitalization in this country. Through public-private jointly financed business plans and Public-Private-Producer Partnership model, we support inclusive value chain and rural business development, benefiting the most vulnerable groups in rural areas.
What makes social innovation so powerful to help us achieve the SDGs is that it allows us to address the interlinkages that exist across the goals. One of the fundamental characteristics of the SDGs is the realization that development is complex. Operating in silos, in the way we have traditionally done to individually tackle each of the economic, environmental, or social dimensions of development challenges, is insufficient, and often counterproductive. Social innovation allows us to address multiple challenges simultaneously, thus hitting more than one goal.
Ending hunger as per SDG 2, for example, requires addressing the root causes such as poverty (SDG 1), which often comes from lack of access to health services (SDG3) or education (SDG 4) or discrimination (SDGs 5, 10 and 16) or unemployment (SDG 8) or from the effects of climate change (SDG 13). As a result, tackling hunger in a rural community requires social innovation and a combination of interventions, including climate-smart agricultural practices, access to agricultural inputs and advisory services, weather-indexed insurance, post-harvest management practices, and vertical linkages to markets.
Social innovation is not only done by government. All resources, public and private, national, and international, need to be brought to bear. The private sector has an especially important role to play, not just for the funds it can contribute but, perhaps more importantly, for its frame of mind, technical capability, and execution ability. Wawira Wanjira is this year's UN Person of the Year in Kenya, where I last served before my current role. Through her organization, Food 4 Education, she provides low-cost nutritious meals to thousands of schoolchildren in the country. Her innovative model allows parents to load digital money to electronic wristbands worn by their children, who then use them at school to pay for their meals with a tap-to-eat device.
The potential for achieving the SDGs through social innovation is not limited to rural areas, with much able to be done in cities. Chinese online retailer JD.com has developed an innovative solution to address SDG 12 on Responsible Consumption and Production through a Corporate Social Responsibility initiative. Working with residential communities in Beijing, they teach residents waste sorting through an application that rewards users for correctly disposing of their urban waste. In addition to SDG 12, this smart use of technology (SDG 9) also educates residents (SDG 4), makes communities sustainable (SDG 11), protects the environment (SDGs 14 and 15), and strengthens compliance with government regulations (SDG 16).
As we redouble our efforts to achieve the SDGs in the face of the pandemic, social innovation provides a formidable tool to build back better and realize a more just and sustainable world by 2030. What remains urgent, and is our debt to posterity, is the concerted efforts to make the next nine years count.
By Siddharth Chatterjee, United Nations Resident Coordinator in China
Mr. Siddharth Chatterjee took office as the United Nations Resident Coordinator in China on 16 January 2021 and presented his letter of credentials to the President of the People's Republic of China, Xi Jinping, on 14 April 2021. The Resident Coordinator (RC), is the highest ranking representative of the UN Development System. The RC is the designated representative of – and reports to – the UN Secretary-General. Mr. Chatterjee has more than 25 years of experience in international cooperation, sustainable development, humanitarian coordination and peace and security, which he has acquired at the United Nations and externally.
What can China Learn from the European Union Emissions Trading System (EU ETS)?
Case Study: Ji'an Project
What I Learnt from the Pandemic about Education
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\section{Introduction\label{sec:intro}}
The origin of $CP$ violation has been an outstanding issue
since its discovery
in $K^0_L \rightarrow \pi^+\pi^-$ decays 35 years ago \cite{KL_CP}.
In 1972, before charm was discovered,
Kobayashi and Maskawa \cite{CKM} proposed that
quark mixing with 3 (or more) generations was the cause.
In this case, the CKM matrix relating
the mass and weak eigenstates of quarks
possesses, in general, a complex physical phase
that violates $CP$.
Unfortunately, the $K^0$ has been the only place to study $CP$ violation.
Despite precision $K^0$-studies,
a complete picture of $CP$ violation
is still lacking.
$CP$ tests have encompassed $B$ mesons,
but the violations in
inclusive studies \cite{B_CP}
are too small (\mbox{$\sim\!\!10^{-3}$}) to as yet detect.
In the '80's it was realized
that the {\it interference} due to mixing of $B^0_d$
decays to the same $CP$ state could show large violations \cite{CarterSanda}.
$B^0_d/\overline{B}{^0_d} \!\rightarrow\! J/\psi K^0_s$ is
the ``golden'' mode for large effects, with
little theoretical uncertainty
relating it to the CKM matrix.
A $B^0_d$ may decay directly to $J/\psi K^0_s$,
or it may oscillate into a $\overline{B}{^0_d}$
and then decay to $J/\psi K^0_s$.
The two paths have
a phase difference, and
the interference results in an asymmetry:
\begin{equation}
{\cal A}_{CP}(t) \equiv \frac{\overline{B}{^0_d}(t)-B^0_d(t) }
{\overline{B}{^0_d}(t)+B^0_d(t) } =
\sin(2\beta) \sin (\Delta m_d t),
\label{eq:cp_asym}
\end{equation}
where $B^0_d(t)$ [$\overline{B}{^0_d}(t)$] is the number of
$J/\psi K^0_s$ decays
at proper time $t$ from mesons
produced as $B^0_d$
[$\overline{B}{^0_d}$].
${\cal A}_{CP}$
varies as $\sin (\Delta m_d t)$ because
it is shifted by a $\frac{1}{4}$-cycle
relative to the $\cos (\Delta m_d t)$ mixing oscillation
by the mixed/unmixed decay
interference.
The am\-plitude is $\sin(2\beta)$, with
$\beta =$ $ \arg(-V_{cd}^{\,}V_{cb}^*/V_{td}^{\,}V_{tb}^*)$
for CKM elements $V_{qq'}$.
$\beta$ is also an angle from the so-called ``unitarity triangle''
of the CKM matrix.
\section{\boldmath
The $B^0/\overline{B}{^0} \rightarrow J/\psi K^0_s$ sample
\label{sec:sample}}
\vspace*{-12.0cm}
{ \flushright \large
$\,$\\
FERMILAB-Conf-99/228-E \\
}
\vspace*{11.05cm}
We exploit the large $B$ cross section at the
Tevatron and obtain a sample of $J/\psi K^0_s$ decays
to measure $\sin(2\beta)$. We start
from the Run I $J/\psi \!\rightarrow\! \mu^+\mu^-$ sample
($p_T(\mu)$ above $\sim\!\!1.5\,$GeV/$c$)
of $\sim\!\!\frac{1}{2}$ million events.
The $K^0_s \rightarrow \pi^+\pi^-$ reconstruction
tries all oppositely charged
track combinations (assumed to be pions).
The $p_T(K^0_s)$ must be above 0.7 GeV/$c$,
its decay vertex displaced
from the $J/\psi$'s by $>\!5\sigma$,
and $p_T(J/\psi K^0_s) > 4.5$ GeV/$c$.
After imposing the $J/\psi$ and $K^0_s$ masses,
the fitted $J/\psi K^0_s$
mass $M_{FIT}$ and error $\sigma_{FIT}$
are used to construct $M_N \equiv (M_{FIT} - M_0)/\sigma_{FIT}$, where
$M_0$ is the world average $B^0_d$ mass.
The $M_N$ distribution is shown in Fig.~\ref{fig:mass}.
A likelihood fit
yields $395 \pm 31$ $B^0_d/\overline{B}{^0_d}$'s.
\begin{figure}[t]
{\centering
\epsfxsize=16pc
\epsfbox{eps_k0sall_1.eps}
\caption{
Normalized mass distribution $M_N$ (see text)
of the $J/\psi K^0_s$ candidates. A Gaussian signal
plus linear background fit to the data is shown by the curve.
}
\label{fig:mass} }
\end{figure}
The $\bar{p}p$ collisions spread beyond CDF's
Si-$\mu$vertex tracking detector (SVX), so only about half
($202 \!\pm\! 18$ vs$.$ $193 \!\pm\! 26$)
of the $J/\psi$'s have both muons
in the SVX.
The precision life\-time information from the SVX
allows one to make a time-dependent fit
to Eq.\ref{eq:cp_asym}. However,
the $CP$ asymmetry remains even when integrated in time;
so although life\-time in\-for\-mation is basically lost in ``non-SVX'' data,
they are still useful. The statistical power
for meas\-ur\-ing $\sin(2\beta)$ is only reduced by $\sim\!{1}/{3}$
for this sub\-sample.
The ``SVX'' subsample was the basis for our previous $\sin(2\beta)$
measurement~\cite{CDFcp}.
\section{Flavor tagging\label{sec:tagging}}
Observing the asymmetry ${\cal A}_{CP}(t)$
is predicated upon determining
the $b$ ``flavor''---whether the $B$ meson is composed
of a $b$ or a
$\bar{b}$ quark---at the time
of production.
If the initial flavor is correctly tagged
with probability $P$,
then the observed asymmetry
\endcolumn
\noindent
is attenuated by the ``dilution''
${\cal D} = 2P -1$, {\it i.e.}
${\cal A}_{CP}^{Obs} = {\cal D} \sin(2\beta)\sin(\Delta m_q t)$.
A method
with tagging efficiency $\epsilon$ yields an error
on $\sin(2\beta)$ which scales as $1/\sqrt{\epsilon {\cal D}^2 N}$ for
$N$ background-free mesons. Thus,
$\epsilon {\cal D}^2$ measures the effective tagging power.
An analysis can be im\-prov\-ed by using
several taggers, the combined effect is approximately the sum
of the respective $\epsilon {\cal D}^2$'s.
The tagging needs for this analysis are similar to those
employed in $B^0$-$\overline{B}{^0}$ oscillation
measurements of $\Delta m$. CDF has performed six
$\Delta m_d$ analyses that demonstrate three types of tagging methods.
The CDF average
$\Delta m_d$ is $0.495 \!\pm\! 0.026 \!\pm\! 0.025 \,{\rm ps}^{-1}$
\cite{CDFMixSum},
which is of similar precision to other experiments and
agrees well with a world average \cite{Artuso}.
We call the first method ``same-side tagging'' (SST),
as it relies on the charge of a particle ``near'' the $B^0$ \cite{Gronau}.
The idea is simple.
A $\bar{b}$ quark forming a $B^0_d$
combines with
a $d$ in the hadronization,
leaving a $\bar{d}$.
To make a charged pion, the $\bar{d}$
combines with
a $u$ making a $\pi^+$.
Conversely, a $\overline{B}{^0_d}$ will be as\-so\-ci\-ated with a $\pi^-$.
Correlated pions also arise from
$B^{**+} \!\rightarrow B^{(*)0} \pi^+$
decays.\footnote{A CDF
$B^{**}$ analysis of $\ell D^{(*)}$ data found
the fraction of $B_{u,d}$ mesons arising as
$B^{**}$ states to be $0.28 \pm 0.06 \pm 0.03$~\protect\cite{Dejan}.
} Both sources have the same correlation,
and are not distinguished
here.
The SST tag is the candidate track with the smallest momentum
transverse to the $B$+Track momentum.
A valid track candidate
must be
within $\Delta R \!=\! \sqrt{(\Delta\eta)^2\!+\!(\Delta\phi)^2} \!\le\! 0.7$
of the $B$,
have $p_T \!>\! 400$ MeV/$c$,
reconstructed in the SVX,
and have its impact parameter within $3\sigma$ of the primary
vertex.
SST was studied in a $\Delta m_d$-analysis \cite{SSTPRD},
and used in our earlier measurement
of $\sin(2\beta)$. The SST dilution for
the $J/\psi K^0_s$ sample was found
to be $16.6 \pm 2.2$\% \cite{CDFcp} for events reconstructed in the SVX.
This method has been extended to events out\-side
the SVX coverage (the impact parameter cut is removed);
and we find ${\cal D}^{SST}_{nonSVX} = 17.4 \pm 3.6$\%.
Two ``opposite-side'' taggers, where the
other $b$-hadron signals
the flavor of the $B^0$, are also used.
The lepton charge from $b \!\rightarrow\! \ell^-$ decay
of the other $b$-hadron tags the $B^0$ flavor,
{\it i.e.} $\ell^-$ ($\ell^+$) implies $B^0$ ($\overline{B}{^0}$).
Lepton ($e$ and $\mu$) identification criteria
are applied to all charged tracks\footnote{
Lepton identification limits the tracks to $|\eta| \!<\! 1.0$. Also,
identified conversion electrons are explicitly removed.}
with $p_T$ thresholds of 1.0 (2.0) GeV/c
for electrons (muons).
The dilution is measured using a $B^+ \rightarrow J/\psi K^+$
sample ($\sim\!1000$ events),
and we find ${\cal D}^{lep} = 62.5 \pm 14.6$\%.
The other opposite-side method is ``jet-charge.''
The tag is a charge average
of an opposite-side jet. The jet is formed by a mass-clustering
algorithm which starts with ``seed'' tracks of $p_T \!>\! 1.75$ GeV/$c$,
and combines other tracks with $p_T \!>\! 0.4$ GeV/$c$, up to a cluster
mass approximating the $B$ mass. The $B^0$ decay products are explicitly
excluded from the jet, as are tracks within $\Delta R \!<\! 0.7$
of the $B^0$.
If mul\-ti\-ple jet clusters are present, the one most like\-ly to be
a $b$-jet is chosen based on track impact par\-a\-met\-ers
and cluster $p_T$.
The jet-charge for a cluster is:
\begin{equation}
Q_{jet} =
\frac{{\Sigma}{_{i}}\, q_i
p_{ Ti} (2-T_i)}
{{\Sigma}{_{i}}\,
{p_{Ti} } (2-T_i) },
\label{eq:jet-Q}
\end{equation}
where $q_i$ and $p_{Ti}$ are the charge and $p_T$ of the $i$-th
track in the jet with $p_T \!>\! 0.75$ GeV/$c$,
and $T_i$ is the probability that the track
is from the primary vertex.
A $B^0$ ($\overline{B}{^0}$) is implied if $Q_{jet} \!<\! -0.2$ ($>\! 0.2$),
otherwise it is untagged.
The dilution is measured from the $B^+ \!\rightarrow\! J/\psi K^+$
sample to be $23.5 \pm 6.9$\%.
By coincidence, each tagger has an $\epsilon {\cal D}^2$
of $\sim\!2$\%. The total $\epsilon {\cal D}^2$ is $6.3\!\pm\!1.7$\%,
so our sample of 400 events corresponds to $\sim\!25$
perfectly tagged $J/\psi K^0_s$ decays
plus background.
\section{\boldmath Extracting $\sin(2\beta)$ \label{sec:sin2beta}}
The three taggers are applied to the sample.
A lepton tends to dominate the jet-charge if a lepton tag is in the jet.
Lepton tagging has low efficiency
but high dilution, so the correlation between lepton and
jet-charge tags is avoided by dropping the jet-charge tag
if there is a lepton tag. This means each $B^0$ is tagged
at most by two methods. If the tag result for an event by method-$i$
is $s_i$ ($s=+1, \; -1, \; 0$ for $B^0$, $\overline{B}{^0}$,
untagged), then the effective dilution for two
tags is ${\cal D}_{ij} =
|s_i{\cal D}_{i}+ s_j{\cal D}_{i}|/(1+s_i s_j {\cal D}_{i}{\cal D}_{j})$.
An unbinned likelihood fit is performed using
the flavor tags (and the effective ${\cal D}_{ij}$'s),
$M_N$, and lifetime information from the data;
and it computes the likelihood probability that an event is signal
or background (either prompt or long-lived).
The treatment of the SVX and non-SVX data in the likelihood
is different, but both are part of the same fit.
The $B^0$ lifetime and $\Delta m_d$ values are fixed to
world averages ($1.54 \pm 0.04$ ps
and $0.464 \pm 0.018 \,{\rm ps}^{-1}$~\cite{NewPDG}).
The fit also incorporates
allowances for (small) systematic detector biases.
The fit yields $\sin(2\beta) \!=\! 0.79^{+0.41}_{-0.44}$
({\it{stat. $+$ syst.}})
for the combined taggers~\cite{CDFNewCP}.
The fit is shown in Fig.~\ref{fig:sin2beta} along with
a dilution weighted average of the sideband-subtracted data.
The result corresponds to
$0 \!<\! \sin(2\beta)$ for a 93\%
unified frequentist \cite{Feldman} con\-fidence interval.
Although the exclusion of zero has only slightly increased
from our previous result~\cite{CDFcp},
the uncertainty on $\sin(2\beta)$ is cut in half.
We applied our taggers and fitting
machinery to a sample of $\sim\!450$ $B^0_d \rightarrow J/\psi K^{*0}$ decays
as a cross check.
We find $\Delta m_d \!=\! 0.40 \pm 0.18 \;{\rm ps}^{-1}$, in
accor\-dance with the precision of
the $\sin(2\beta)$ analysis.
\begin{figure}[t]
{\centering
\epsfxsize=18.3pc
\epsfbox{newsin2beta_edit.eps}
\caption{
The $CP$ asymmetry of the data with the fit
result. The SVX data is shown in proper-time bins on the left, and
a single bin for non-SVX data on the right.
}
\label{fig:sin2beta} }
\end{figure}
\section{Summary and prospects \label{sec:summ}}
We have directly measured $\sin(2\beta)$, and our result provides
evidence for large $CP$ asymmetries in $B^0$ mesons
as expected from indirect determinations,
{\it e.g.} $0.52 \!<\!\sin(2\beta)\!<\! 0.94$ at 95\% CL~\cite{Ali}.
A critical test, however, requires much greater precision.
CDF will attain this in Run II. Starting
in 2000, a 2-year run
should deliver $20\times$ the lum\-inos\-ity ($\sim\!\!2\,$fb$^{-1}$),
and be exploited by a greatly enhanced detector~\cite{CDFup}.
We project $\sim\!10^4$ $J/\psi K^0_s$'s
from dimuons, for a $\sin(2\beta)$ error of about $\pm 0.08$.
Triggering on $J/\psi \!\rightarrow\! e^+e^-$
may boost the sample by $\sim\!50$\%.
CDF is also working on
a Time-of-Flight system which will aid flavor tagging.
We expect to achieve sensitivities
in the range projected for
the dedicated $B$ factories.
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AS HUMANS AND MONSTERS CLASH AND ROYAL BLOOD FLOWS, A NEW AGE WILL DAWN...
Prince Alexander of Macedon has battled both men and monsters, but his final war will determine his fate...and the future of all mankind. While Macedon's enemies close in from all corners of the earth, Alexander must fulfill one last prophecy that dictates only he—and he alone—can ensure humanity's survival against the age of the deadly Spirit Eaters.
As the threads of fate draw Alexander closer to his destiny, an exiled queen will meet a runaway princess, a young sorceress will set the final path of her heart and generals will choose their final battles. Before the light of victory can shine, enemies must become allies, Death must be tamed and hearts must break.
Who will rise and who will die? All is revealed in the epic finale to New York Times bestselling author Eleanor Herman's rich and fantastical Blood of Gods and Royals series.
Books by Eleanor Herman available from Harlequin TEEN
Blood of Gods and Royals series
Full-length novels in reading order:
Legacy of Kings
Empire of Dust
Reign of Serpents
Dawn of Legends
Ebook novellas:
Voice of Gods*
Queen of Ashes**
*Can be read at any time without spoilers
**Recommended to read after Empire of Dust
ELEANOR HERMAN
Dawn of Legends
To Starr Thompson and Pam Gilbert, for their frustration and fury when each of the books in this series ended!
Contents
ACT ONE
CHAPTER ONE
CHAPTER TWO
CHAPTER THREE
CHAPTER FOUR
CHAPTER FIVE
CHAPTER SIX
CHAPTER SEVEN
CHAPTER EIGHT
CHAPTER NINE
ACT TWO
CHAPTER TEN
CHAPTER ELEVEN
CHAPTER TWELVE
CHAPTER THIRTEEN
CHAPTER FOURTEEN
CHAPTER FIFTEEN
CHAPTER SIXTEEN
ACT THREE
CHAPTER SEVENTEEN
CHAPTER EIGHTEEN
CHAPTER NINETEEN
CHAPTER TWENTY
CHAPTER TWENTY-ONE
ACT FOUR
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
ACT FIVE
CHAPTER THIRTY-THREE
SIX MONTHS LATER
CHAPTER THIRTY-FOUR
CHAPTER THIRTY-FIVE
ONE YEAR LATER
CHAPTER THIRTY-SIX
TWO YEARS LATER
CHAPTER THIRTY-SEVEN
ACKNOWLEDGMENTS
AUTHOR'S NOTE
EXCERPT FROM _REIGN OF SERPENTS_ BY ELEANOR HERMAN
ACT ONE
CATACLYSM
Great things are won by great dangers.
—Herodotus
CHAPTER ONE
JACOB
Jacob runs.
A hot pain radiates through his skull, blurring his vision—a nameless, wordless pain, the pain of a loss so great he cannot even think it.
Once, not so long ago, Jacob ran across the fields of Erissa in pursuit of a girl who raced gazelles. In pursuit of Kat. The vast sky loomed above him as large as his dreams.
Now his boots pound the labyrinth-like corridors of the Byzantine palace, three hundred miles from Macedon's countryside. Now _he's_ the one being hunted.
As he races down the wide marble corridor on the second floor of the palace, strains of discordant music from the open windows wrap their eerie tendrils around him. He inhales the salt smell of the sea. Catches a glimpse of crowds on the docks below celebrating the royal wedding, and fishing boats bobbing in the crowded harbor just steps away from the palace.
He crashes through a small door at the end of the corridor into a winding servants' staircase, almost too narrow for his broad shoulders, and takes the steps down three at a time. He hears a loud, rhythmic pounding, but he's not sure if it's a wedding drummer or the pumping of his own heart. He explodes onto a wide marble hallway, hoping it will lead him to an exit—but it doesn't. Just more doors, more chances to make the wrong choice.
_The queen._
_Bright red blood. Silver-blond hair turning dark and matted._
He didn't mean to do it.
He didn't mean to kill her.
One of the many doors flings open directly in Jacob's path, nearly causing him to collide with it. A man and woman lurch out of the chamber behind, the man's mouth smeared with the woman's lip stain. The woman wears the sheer dress of the dancing girls he'd seen earlier; the man reeks of sandalwood cologne and wealth.
Something wild and angry lurches inside him—the carelessness of these people. The obliviousness. He should be grateful for it—for their distraction—but all he feels is a dizzying disgust. Shoving past them, Jacob finally spots an open door leading outside. He can make out the trill of a flute, the low hum of men's voices, and high lilting notes of ladies' laughter. There are smells, too: spiced wine, wood smoke, and the pungent tang of urine.
He may still have time to escape. It will be dark soon. If he can slip into the anonymous crowd of drunken revelers, he could find safety. He runs for the door, this time racing through the wedding guests, passing a man vomiting in the corridor and a boy playing the flute, and then—he's out.
A chilly wind off the nearby harbor ruffles his stolen red cape and snakes across his face with cold fingers. He turns around, disoriented; he has careened into a square central courtyard—four sides of turrets, balconies, and columns in green and pink and white marble. He curses. He's _not_ out. He's wandered deeper into the palace complex.
The flowers are gone now, the ornamental bushes bare of leaves. But on the cobblestone paths, wedding guests huddle for warmth around brightly burning braziers in the last rays of the afternoon sun, pulling cloaks around them. A plump gray-haired manservant picks up fallen wine cups and places them on a tray. An old man with a cane walks slowly through the crowd, leaning heavily on a younger man's arm.
"Wine, my lord?" asks a high-pitched male voice in the soft accent of Byzantium.
Jacob turns to see that a servant boy has come up behind him, a teenager, his face a mass of red pimples. He's holding a tray of full wine cups.
"No." Jacob clears his throat. "No, thank you."
"But it's to their majesties' health," the boy replies earnestly.
Jacob hardly hears him. He needs to get down to the harbor and on the first ship out of here. But he can't risk making a scene. He reaches for the offered cup. Before his fingers even graze the cool metal of the stem, the boy lurches back. The goblet topples off the tray and clangs loudly on the cobblestones, spilling dark red wine clotted with black lees.
It takes Jacob a moment to realize what has startled the boy so badly. Then he notices his own hands—white-knuckled, caked in something dark and red. Blood.
Before the boy can recover, Jacob whips around and loses himself in the crowd.
How did he not notice before? How long has he run through the palace, his crime blazing on his hands for all to see? He is too overwhelmed to care. In the middle of the courtyard sits a marble fountain in the shape of a griffin, wings tucked against a lion's body, water foaming out of its open eagle's beak into a painted basin below.
Jacob stumbles toward it through the crowd and plunges his arms up to the elbows into the water. He can hardly feel the harsh cold of it against his slashed flesh as he scrubs, scratching and rubbing until he's sure the skin itself will peel away. Until the churning water has become cloudy and brown with his blood. He watches it swirl, overcome.
Jacob meant only to kidnap Queen Olympias—not kill her.
The plan was to hold her hostage. He had been planning to use her to lure out her lover, Riel the Snake. The last living god.
But when he went to her chamber, she recognized him. And then, smiling, told him she had savagely murdered his entire family.
She'd gone after his first love, Katerina, and when his family hadn't known her whereabouts, the queen had set their home on fire, had personally watched Jacob's mother and father and little brothers scream in agony as the soldiers butchered them and the flames devoured them. And from staring into her hard green eyes, Jacob had known she wasn't lying.
Now, as the water swirls before him, his dizziness returns; he replays in his head the moment he lunged for the queen and she swiveled on him, wielding her own two-pronged hairpin like the fangs of a snake, lashing out, and scratching open his arm. But she was too slow, too unprotected. He caught her slender wrist and snapped it like a twig. Then, without even thinking, he slammed her head hard against her cosmetics table. She slumped to the floor. He stared, stunned, at her limp, lifeless form, her blood draining into the cracks between mosaic tiles on the floor...before fleeing her private chambers.
Bent over the fountain and blinking to try to clear his vision, Jacob takes stock of the fact that he has lost the god, that the queen is dead, that his family is gone forever. If Jacob is caught, he will be executed for murder. But that's not even the worst of it. The plan he spent months pursuing—capturing Riel to satiate the Spirit Eaters with the flesh of the Last God—has been compromised by his own passion, his own rage.
Riel walks free.
And as for the Spirit Eaters—their evil and their hunger will only spread.
Once, Jacob knows, Riel had been good. He had, in fact, been humanity's savior in the battle between the gods and the Spirit Eaters, who fed on magic. Who fed on _gods_. But in that ancient war, Riel lost much of his power. He became a fallen god, trapped in this world while the others fled, desperate to return to the realm of divine beings. He grew hateful, and for centuries he devoted himself to murdering his own kin. He had become a terror, a force of evil.
And yet, the Spirit Eaters were the far greater danger. Frightening, monstrous, they had been biding their time. That was all thanks to the Aesarian Lords, who had dedicated themselves to persecuting people with magic, capturing them...and feeding them to the Spirit Eaters.
For hundreds of years, that had been enough to keep the monsters satiated, contained among their caves in the Eastern Mountains. But now magic is running out. The Spirit Eaters are growing restless again.
And hungry.
They have begun to consume and destroy entire villages in the central regions of Persia, leaving only collapsed homes and bones sucked dry of marrow. Soon, they will come for the great cities. Then the empires, one by one, will fall, until one day the monsters will devour the world itself.
The long twin scratches on his left forearm from Olympias's hairpin throb painfully now, as if to remind him of the present moment. Blood still oozes from the jagged tracks in his skin. _A man never cries over a scratch_ , his little brother Calas used to say, proudly sticking out his chest.
A fresh wave of nausea sickens him, and he leans over the basin, retching into the fountain. Cal is dead.
Cal is ash.
Because of the queen. She deserved to die.
"Too much worship of Dionysus?" a wedding guest calls, and a group of men nearby guffaw loudly.
Rage pounds Jacob's head. His heart, he's sure, is going to explode right out of his chest. He was the oldest son, the strongest. He should have been there to protect Cal, to protect all of them. He wants to vomit again but there is no way to purge the real sickness—the grief. The guilt. He's gripping the rim of the fountain basin so hard he is shocked it doesn't crumble. He looks at the tormented face in the soiled water, rippling in the wind, causing his reflection to warp and scatter, unfamiliar.
Except it's not wind that makes the water shake and foam. Earth magic, the hot shame of it, is shooting up through his body, roaring through his veins, bursting into his knuckles and fingertips. Jacob just realizes what's happening before his mind shuts down, as the sickening fury—the raging pain of loss—goes still within him, and the power of soil and rock, of wave and wind, of the oneness of all earthly things courses through him, a scorching heat like the inside of the smith god Hephaestus's mountain forges. It burns, corroding his insides, singeing away all his anger. It pours from his hands into the stone of the fountain, crashing into the water and down, down into the earth below. It is horrible in its intensity, beautiful in its raw power. And then—
—it ends.
Jacob feels empty, liberated. Gasping for breath, he stumbles back from the fountain.
Slowly, he becomes aware of a silence in the courtyard around him. The performers' flutes have stopped, their final notes lingering sourly in the air. The chatter and laughter of the wedding guests have ceased.
The fountain trembles.
Suddenly, the pink-and-green-painted basin cracks open, releasing its water in a torrential gush. The griffin on the pillar lurches forward and dives toward the earth, falling in a deafening clatter on the broken basin, as water shoots out of the center like a geyser, five times the height of a man.
The ground, he realizes, is trembling. The world shifts suddenly and Jacob falls on one knee just as a jagged crack appears in the palace facade in front of him. The upper part of the wall collapses to the ground with a crash, revealing a dainty bedroom.
Someone screams. A plume of water gushes out of the ground at the other end of the courtyard, and three, four, five more erupt, pumping water higher than the palace itself.
All around him, water gathers on the ground, rising, rising to his ankles.
To his knees.
Somewhere in the distance, horses whinny in fear. All around him, people are screaming, tripping over one another to try to escape, some of them falling into the water and getting trampled. Braziers tip over, the burning logs and hot iron hissing when they hit the water, all the warm, flickering light becoming plumes of black smoke. The entire courtyard is plunged into the chaos of choking darkness and deafening noise.
Jacob pushes through the many-limbed dark, water filling his boots and sloshing into his clothes, weighing down the ends of his cloak. Somehow, he manages to make his way back into the building, which shudders and groans. Because of him. All of this, because of him.
He sent his rage into the earth.
The earth has answered.
He enters the rapidly filling main hall as people push their way toward the front door. He tumbles down the marble steps into the front courtyard and barely has time to register why everyone has frozen in place.
Then he sees it. Roaring up the Bosporus Strait from the Sea of Marmara toward the harbor at their feet is a towering wave, higher even than the palace on the hill, like the mouth of the coming night opening its massive jaws to an ever greater and more depthless dark. The setting sun, low on the horizon, colors the foaming white crests on its apex bloodred. And Jacob knows that evil—a magic far greater than most could ever fathom—has found its way to him at last.
CHAPTER TWO
HEPHAESTION
They have killed a god.
But all Hephaestion can think, as Alexander brushes his thick light hair from his face, is that they have saved the god's son. The prince lives—wounded but rapidly recovering as he lies on the bed in this private chamber, high above the royal wedding festivities in Byzantium. And Katerina lives, too.
The prophecy has been fulfilled in ways Heph never imagined.
For months he held the secret safe from both of them: that Kat was destined to kill Alexander. It came true, but not in the way anyone would have thought. She killed him in order to save him. The spirit of their father, the god Riel, has fled Alexander's body, and if the prophecies are true, he is gone forever.
Now he watches the way Kat tends to her twin brother's wounds and knows that he was wrong to think she could ever _really_ hurt him. Heph doesn't know how to feel—relieved, overwhelmed, and something else—a pang of jealousy. The two people he loves most in this world have a bond that he will never fully understand...and will never share. He will always stand outside that circle of love and light, a stranger to it.
But something else is bothering him, too.
It had also been prophesied that Riel could be killed only by an Earth Blood.
Heph once again picks up the ivory hairpin shaped like a snake, sticky with blood, that Kat used to stab Alexander, killing him—but only to release Riel's possession over him. There had _already_ been blood on the deadly sharp hairpin.
Not just any blood. Earth Blood. Heph wonders whose it could be and where he or she is now. The needle on the Atlantean Mechanism in his hands swings wildly as if to echo his swirling thoughts.
Propped against large tasseled pillows on the ornate four-poster bed, Alex hisses in pain as Kat washes the scratches on his neck with wine. "Sit still," she commands, kneeling beside him. "I must clean it."
Pushing the ministrations of his sister aside, Alex slides off the high bed. He holds out a hand to steady himself, clearly dizzy.
"I'm fine," he says, as Kat tries to tug him back into bed. "Heph is right. We must discover who this Earth Blood possessor is."
As if the words themselves have conjured the response, the ground below them begins to tremble. Heph looks at Alex and Kat, whose expressions show the same surprise and alarm that must be in his. The floor continues to move beneath his feet. An oil lamp falls off a table with a crash as the ceramic shatters.
"An earthquake," Alex says. "We must..."
Kat, who has moved to the window, interrupts him. "Not just an earthquake," she says, looking out over the harbor below and into the strait beyond. "Oh gods," she whispers.
Heph is beside her before she can say what she has seen. It is like something from a haunting dream.
In the failing light, they see that the harbor is _empty_... It has no water.
All the little boats tilt at odd angles on the sand, which is littered with gasping silver fish. People who were celebrating the royal wedding on the docks moments earlier stand stock-still, gaping at the watery horizon. Coming relentlessly toward them from across the sea is a wave that, though still in the distance, must be taller than the Byzantine hills themselves.
"Our Earth Blood," Kat repeats, her voice soft with awe.
He should be horrified, but for a moment, Heph is filled with a wonder as wide as the sea. He learned about Earth Blood and their powers during his time on Meninx, the Island of the Lotus-Eaters. They are those who have the power to heal fatal wounds, melt metal, and make the earth tremble with their rage...and this particular Earth Blood must be _furious_.
Alex has joined them at the window. Despite the terror of what they're seeing, the prince seems to have the same exact thought Heph is having. "We need him," he says. He doesn't have to explain. An Earth Blood with this much power is someone they must have on their side.
Especially if they are going to take on the Spirit Eaters.
After all, there are no gods left on earth to help them.
Kat reaches for Heph's hand, and he can feel her own trembling. And then, seamlessly, she plies the Atlantean Mechanism from his grip. "Stay up here with Alex. I'll go..."
Heph grabs her wrist. "Absolutely not." He's not going to risk losing her, not when they've come this far. But just then, screams echo from the harbor below. Turning to the window, Heph sees crowds of people fighting their way toward the palace gates, others racing into the maze of narrow streets alongside the palace, madly dashing up the hill.
During his moment of distraction, Kat slips from his fingers and pushes out of the room.
He tears down the marble corridor after her, knowing that Alex is moving more slowly than usual behind him. "Kat!" She has always been so fast— _too_ fast.
Two turns, three, down a set of stairs, and Heph is nearly bowled over by wedding guests scrambling _up_ the stairs past him. He doesn't understand at first why he feels damp until he realizes the guests are all drenched, their garments flinging stray water onto him as they pass. They are fleeing an internal flood of some kind. He fights his way down to the landing, Alex not far behind him now, trying to find Kat in the mayhem.
The courtyard in the center of the palace is no longer a courtyard; he can see through the landing window that it is more like a massive pool, with water as high as Heph's waist.
Once again, Heph can't understand what he is seeing. He saw the coming wave—but it hasn't reached them yet. The water is coming up from cracks in the earth, he realizes, fountains of it, like underground springs exploding skyward.
_Kat._ Where is Kat in all this shoving, writhing, frenzied mass of humanity?
"I don't see her!" Alex yells, as men and women push past them in their desperate efforts to run upstairs.
On Heph plunges. Now he's level with this panicked mob, and he realizes his mistake: he can't see anything once he is _in_ it.
"My wife is pregnant!" shouts a bearded man carrying a woman who looks as if she's fainted. "Help me get her upstairs!"
"Mother! Where are you?" yells a teenage girl.
Two men in front of Heph move away, and then he sees her, standing perfectly still as people jostle around her, like a mermaid emerging from the depths, her hair, wet now, streaming down her back. She is reaching out an arm to someone sloshing toward her—a man with broad shoulders and the red cloak of a Macedonian soldier. Jacob.
Heph remembers him. And with sudden clarity, he understands. He doesn't have to see the mechanism to comprehend where it has led Katerina—and why.
_Jacob is the Earth Blood._
Cold water continues to flood from the courtyard, rising higher, making it impossible to run. They must swim. Those who cannot will surely drown. And though he can't see it, just outside the palace, the wave must be roaring ever closer toward them.
Something inside him twists sharply as he pushes his way toward Kat and Jacob, though he can't quite yet acknowledge what this could mean. Slogging through the rising water, forcing his way in front of panicked people, at last he thrusts himself between Jacob and Kat. He grabs the Atlantean Mechanism from Kat and looks at the needle. It has moved so that it still points at Jacob.
"Can't you stop it?" he yells at Jacob, gesturing to the watery chaos around them. "You did this, didn't you? So make it stop."
"I cannot stop an arrow I have unleashed on the field of battle." Jacob looks pale and terrified.
Heph's military instincts take over. "Down here," he shouts, pulling them toward the banquet hall, as he secures the mechanism in the pouch on his belt. "There has to be another stairway to the second floor."
They struggle through the freezing water and up the four steps to enter the empty two-story room where hundreds of guests recently celebrated King Philip's wedding. The water inside is shallower, only ankle-deep. Tables and benches are upended on the floor. Coals glowing orange in the central fire pit hiss and spit as water trickles onto them.
"Over there!" Heph clambers to the small door behind the two thrones set up on the dais. It must lead to a staircase.
He hears it first.
A low growl quickly rising to a roar.
And then he senses it. Though he has no magic in his bones like Kat and Alex with Snake Blood—or Jacob with Earth Blood—he feels the presence of something almost alive, and very, very powerful.
He steps toward a wide-open window facing the harbor and there he sees it: the wave, dark but littered with the sparkle of white foam that makes it look, for a second, like a giant segment of fallen sky.
For a moment, it's as though the wave is alive, holding its breath.
And then...it exhales.
Heph reaches for Kat, but his hand brushes air—Jacob has already pulled her to the door behind the thrones on the far side of the hall.
And then finally, the water fist punches into the palace, shaking it to its core, exploding into the banquet hall through the two rows of windows and down from the wide smoke hole in the roof.
A cold blast from the lower windows hits Heph in the gut, sending him hurtling into the wall behind him, and another pours in through the higher row, pounding him into the floor. A bench knocks him hard in the chest, taking the breath out of him.
He climbs to his feet, sputtering and shivering, but before he can inhale, another wave pounds the palace, followed by another, until it seems as though the sea itself is intent on swallowing all of Byzantium.
The freezing water is everywhere now, lifting Heph off his feet and pummeling him into something hard. He can no longer see the floor, can hardly tell which way is up. He sees Kat struggling, gasping for air, and calling to him for help. He fights the roiling water to swim toward her, but more water rushes in and suddenly he is below the surface, buffeted by strong currents. He opens his eyes underwater, hoping to get his bearings, and sees the huge iron chandelier... _below_ him. Has it snapped off and started falling to the floor, or has Heph floated above it? Or does he no longer know what is up and what is down?
Somehow, he rises above the surface, coughing, and sees that the harborside wall has completely collapsed, allowing in the surging, sucking sea. Nearby, Kat's arms flail madly, while the rest of her is underwater. A new wave pushes him toward her and he grabs her robe, trying desperately to pull her up, to keep her from getting pulled out to sea. But she is caught on something.
He takes a deep breath, plunges below the surface, and opens his eyes. He sees swirling greenness. Floating furniture. Kat's struggling body. And then...
Cold.
A cold so numbing that Heph wonders for a second if he is not in water at all but in ice. And below him, from the depths, comes a wide shadow. At first, he thinks it is a shark. He can't tell its exact shape—the water is too dark, the salt stings his eyes—but he has the strangest impression that the water itself has sprouted massive black wings, twin shadows, darker than the rest of the dark. No, it is four wings. Or...six?
Kat, instead of struggling, goes limp. Tiny bubbles escape her mouth as she loses consciousness and lets her last breath float out into the water.
Heph takes his dagger out of its sheath on his belt and starts slashing at the shadows that seem to be holding her down. An unearthly wail reverberates through the dark underwater world.
Heph's chest feels as though it's on fire, and his lungs scream. He needs air.
With one hand, he reaches for Kat, wrapping his fingers around her arm, and with the other, he holds his knife, slashing into the darkness again and again as the water roars, the sound low and warped.
Before he can drag her fully to the surface, something lunges into his gut, separating him from Kat. Her body floats upward and away, arms and legs spread, hair and robes swirling out. He fights his way to the surface at last, gulping in air that burns his throat and chest. Breathing is almost as painful as not breathing. He looks around. A gilded throne floats past him. No Kat. No Jacob, either. He panics at the thought that perhaps they have been sucked out to sea through the gaping hole in the wall.
Though his lungs still ache, he takes a deep breath and dives back down. But still, he doesn't see her in the murky depths. He swims this way and that, his numb, prickling limbs slow to obey, his head heavy and his thoughts scattered. They must be out there, in the churning ocean that Byzantium has become. Something hits him in the head, and pain explodes through him. Crying out, he inhales water. His lungs scream.
Raging waters push him down. He has no breath left.
An image of a girl with bright green eyes flashes in front of him. She's smiling...but not at him.
Wet darkness closes in on Heph as he stops fighting...
* * *
...and then...
A firm hand pulls him upward, yanking him into air. Someone throws him onto something solid and begins to hit him hard on the back.
Even as Heph tells himself to breathe, his waterlogged lungs don't respond. His entire chest is racked with pain and totally paralyzed. The hard blows keep raining down mercilessly on his back, and he knows whose hand it is by the strength of it, by the love of it.
Alexander.
The prince, only recently convalescing, seems suddenly stronger than he has ever been. He violently pounds Heph's back. "Come on, Heph! You can do it, dammit! Spit it out!"
Heph vomits an enormous amount of salt water and collapses as Alexander continues to beat his back. He throws up more water and, a moment later, more. When the vomiting stops, he gulps in air, his aching lungs making strange moaning sounds like a dying animal. His back is bruised. His abdomen throbs with pain. He's vaguely aware of moving on water, of being pulled away by a strong current.
Dizzy and shivering uncontrollably from the cold, he squints open his eyes. Alex has dragged him onto a giant slab of wood—a tabletop. They are no longer within the palace. Everywhere is darkness and water and floating objects. Heph has the unnerving sense that this is what death looks like—endless and undifferentiated darkness, no ground, no sky. Then he has the horrifying fear that he _is_ alive, but that this is what has become of the entire world. That the sea has consumed everything they ever knew.
He sleeps fitfully. When he wakes, he sees a full moon slipping from behind a cloud, and a horse swimming beside them, its eyes wide with fear. Sometime later, a canoe passes by, carrying a black goat calmly chewing its cud. Shivering, Heph hears a baby's cry, turns, and sees a wooden cradle floating by.
He wonders if they have drowned and are floating to the Underworld on the black and unforgiving River Styx, a place of unending nightmares.
* * *
The night is long.
The cold is bone-chilling.
He clings to Alexander for warmth, the two of them balanced on their raft in the middle of the nothingness, convinced that this is the end, that they have fought and won battles only to die this way, this young, stripped of everything, unmoored from the world.
But somehow, unbelievably, the dawn returns.
And with it, they are washed ashore, borne onto solid land once more.
CHAPTER THREE
TIMAEUS
Disgust rises in Timaeus, sour and strong, as he takes in the marble masterpiece in front of him, the pride of the Dardanian palace.
It's a statue of Aphrodite, bathing naked, each curve lustful and full. Or at least, it would have been a statue of Aphrodite, had not her head been hammered off and replaced with a carving of the Dardanian king's head.
The _late_ king.
Whoever made this bust was talented—almost too talented. He'd perfectly captured King Amyntas's ridiculous grin, too wide for his narrow face. And even though Amyntas has been dead since last night, his painted pale gray pupils stare at Timaeus, unnerving him, seeming more alive than the overbright stare of the mad king himself.
Aphrodite's head is not the only one that has been replaced with one of King Amyntas. All the statues in the garden and in the Dardanian palace itself have suffered the same mutilation: Apollo, Zeus, Hera—everyone. Evidence in stone of the king's insanity.
"One of our first orders of business will be to replace all the statues," says High Lord Gideon in his deep, sonorous voice.
Tim looks away from the grinning head and toward the leader of his regiment of Aesarian Lords.
"Perhaps we should keep one to remind the Dardanians of life under their unfit ruler," Tim suggests, as they continue walking the garden paths. "Just in case they ever complain about Aesarian rule."
The brisk sea wind whips his black leather cloak around him. It feels so good to be in his uniform again, to feel the weight of his horned helmet on his head.
He is finishing up his tour of the Dardanian palace with the High Lord and has shown him, from the vantage point of the highest tower, the rich kingdom he has conquered for the Aesarian Lords. A tidy city of white limestone houses with red roofs wedged between rugged gray mountains and a sparkling blue fishing harbor. And beyond those mountains are towns and farms and mines.
"The people will no doubt rejoice," Gideon says.
Tim's chest swells. Everyone has always underestimated him just because he's approximately the size of a twelve-year-old girl. People have routinely overlooked him, bullied him, and called him names. Growing up, he developed his fighting skills to beat off village tormentors. Then, last summer, he competed in the royal Blood Tournament—held in Macedon's capital, Pella—for a chance to become a soldier, and was chosen because of his acrobatic skill to join the Lords, along with the tournament's victor, Jacob. But Jacob, his best friend, betrayed him, convincing High Lord Gideon to send him away from their powerful regiment to this sleepy little kingdom on the far western side of the Greek mainland. Tim easily infiltrated the Dardanian court—who would suspect such a silly, tiny man of being an Aesarian spy?—and became King Amyntas's court jester. The suspicious king gradually trusted Tim—or Papari, as he was known, a name that can mean _balls_. The name he chose because, while he didn't have height, he did have balls.
And it was exactly that trust that Papari had craved. Because it was just what he needed in order to enact the ultimate betrayal.
Tim flushes with an inner thrill when he remembers the events of last night. How he had brought towels and hot water, promising the mad king a clean shave under the full moon. How he lifted his sharpened razor...and sliced it deeply across the king's throat.
"However," Gideon adds, "it may pose a problem for us that the culprit has disappeared."
His dark eyes hold Tim's, and it's clear from his gaze that while _he_ knows the truth—that Timaeus is the one who killed the king—the Dardanians can never know it. They would not take kindly to foreign interference. Which is why they have pinned the crime on the missing Queen Cynane.
Besides, if Cynane returns, there is a chance the people would see fit to have her become their ruler. Tim can't allow that.
Without a king or the possibility of an heir from the traitorous queen, the Dardanians will be grateful for the Aesarian Lords' protection.
Tim turns away from Gideon and looks out over the windswept garden, the twig-like bushes, empty flower beds, and dried leaves dancing in a breeze around the paths. A winter garden is not truly dead, not even at rest. Life, he knows, lies tightly coiled just beneath the soil, preparing itself to spring forth again when the time is right.
Boots crunch on gravel, and Tim turns to see Otus, a barrel-chested Dardanian steward, carrying a lead pitcher. Trailing behind him, connected by a metal chain and handcuffs, is his mute slave boy, Aesop, holding a tray with two cups.
Timaeus had convinced Amyntas to chain these two together, night and day, as the beginning of what he hopes will be a rewarding experiment. Aesop darts a frightened look at Gideon, but his glance lingers on Tim.
Timaeus smiles at him and the child smiles back. Aesop has always been nearby, always watching. He reminds Tim of himself. Tim is overlooked because of his size, Aesop because of his muteness. They are both disregarded, underestimated, invisible. Which is why Tim gave Aesop the honor of being chosen first to test his theory.
Gideon takes a cup and hands it to Tim, and then takes one for himself. Otus pours hot spiced wine, and it steams slightly in the cold air as Tim inhales the aroma of thyme and rosemary.
"I salute you, Lord Timaeus," Gideon says, raising his cup. "I predict a great future for you among the Lords."
_Greater than you can imagine._ Tim clinks his cup with Gideon's and drains it rapidly. He hasn't told the High Lord about Smoke Blood yet, how he discovered the kind of Blood Magic mankind has forgotten.
He hasn't even begun to experiment with the magic fizzing and burning inside him. But once he knows more—knows how to control it—he will tell the Lords how they can stop the Spirit Eaters' destruction of the world. He will quickly rise high in the hierarchy of Aesarians. Surely, he will one day become Supreme Lord himself, ruler of all Aesarian regiments from the headquarters in Nekrana in Persia. Not to mention, the savior of all mankind.
All of it hinges on the mute slave boy.
* * *
Tim leads the way back into the palace, down a long marble corridor, and through a doorway into the odeon, an open-air structure of eight tiers of marble benches rising in circles from a small round floor and topped by a sturdy, brightly painted roof. Because of its acoustic perfection—a whisper can be heard on the highest row—the odeon is the location of all council debates.
Gideon and Tim stride down the stairs and onto the floor. Other Lords are already there—Turshu, the little bow-legged Scythian; Ambiorix, the huge blond Gaul; Gaius, the sleek Roman; and others. Dardanian nobles wrapped in furred cloaks crowd the benches, and their babble of conversation abruptly ceases as Gideon and Timaeus enter.
"Is it true?" asks the lean, bearded General Georgios, the late king's minister of war. "Is King Amyntas dead?"
Gideon nods. "It is—and by the queen's hand." He holds up his hand to quiet the burst of outcry. "Lord Timaeus, will you explain?"
Tim steps forward, relishing the height the tall horned helmet gives him, enjoying the awe in the eyes of those noblemen who scorned him only yesterday. He hears a few in the crowd murmur his false name, Papari. Well. He is Papari no more.
"It is as our High Lord says," he announces, taking in the room of gaping Dardanians. "Several months ago, the Lords heard of a plot by King Philip of Macedon to put his daughter on the throne of Dardania, not merely as queen consort, but as ruling monarch. The Lords placed me here, in disguise, to monitor the situation. And last night, just as my brother Lords arrived at the palace, I found the king's body in his bedchamber—his throat slit open."
"And where is she now?" a noble asks. "Where is Queen Cynane?"
Tim holds out his hands, waiting for complete silence before he says, "She's vanished—by magical means, one can only imagine. The woman was a known witch."
A murmur of alarm rumbles around the benches.
"We will offer a generous reward for her capture," Gideon says. "The Lords are experts in controlling chaos—and this woman has brought chaos to your doorstep. She must be stopped, and she will be."
Tim smiles inwardly. The wayward Macedonian princess with a tangle of black hair and an eye for the kill brought much more to the land than chaos. In search of Smoke Blood herself, she unknowingly revealed to Tim much that he hadn't known from the magic archives. As a sooty twist of smoke and ashes curls in his chest and wraps around his soul, he can no longer contain his joy and grins for all to see.
"It is true she hated him," says Kyros, the baggy-eyed minister of the treasury. "She wanted to dispose of him and had plans to rule in his stead—she told many of us so. But magic! Are we in danger of witchcraft?"
Gideon shakes his head. "Not with the Lords here. We have tools to detect unnatural abilities and... _methods_ we can use to contain witches like Cynane."
"But what will we do now?" asks Simon, minister of religion, his huge barrel-shaped body rising from the bench. "Amyntas wasn't much of a king, that is true, but at least we _had_ a king."
"And Cynane has not yet had time to bear an heir," Kyros adds. "Amyntas had no brothers and sisters, and no living cousins other than Eumalia, who ten years ago married the king of Crete. She has two sons. Should we ask her to send us the second one as king?"
"We don't want a child to rule us," General Georgios says angrily, waving his maimed hand as if swatting at an annoying fly. "And certainly not a Cretan child."
As the nobles begin to argue among themselves, Gideon steps forward. "As you know, the Aesarian Lords have a high code of honor. We do not pillage, burn, or rape. We do not murder innocent civilians. We do not steal. If you allow us, we will protect your kingdom for as long as your country needs."
Tim feels a rush of joy. This is it. His moment of victory. He sees admiration and jealousy in the eyes of the other Lords, much larger, stronger Lords, that he—a tiny fool, an acrobat, a comedian—can deliver a rich kingdom into the hands of the Aesarian brotherhood without a drop of blood...other than that of the mad king, of course.
"But there _is_ no need," says a stranger in a commanding voice. A tall, dark-haired young man in military garb emerges from the shadowy doorway at the top of the benches. The crowd murmurs as the man makes his way down to the floor to face Gideon. "I am General Pyrolithos, late of Paeonia, but I was born in this palace, son of Nearchas, brother of Amyntas's father. By right and tradition, _I_ am the king of this land."
Tim is hardly aware of the crowd's reaction because a rage he has never known kindles in his heart and burns outward, consuming him in a flash of red light. He wants to kill this man, this stranger who has shown up at the last moment to foil his lifetime of working and training to become _somebody_. If he were Lord Ambiorix, the brawny, brutish blond Gaul, he would cut off this imposter's head before anyone knew what was happening...but Tim's strength lies in his brain, not his muscle.
Think. He must _think_. Gritting his teeth, Tim steps outside his fury. Who is this man?
And why is there something familiar about him?
CHAPTER FOUR
PYROLITHOS
On the circular white marble center of the odeon, Pyrolithos looks up at the shocked faces of the Dardanian nobles on the benches above.
"Liar!" says Cilix, the hawk-nosed owner of the mountain silver mines.
"Treason!" cries the brawny minister of the games, Branchios, standing and unsheathing his sword. The rising buzz of conversation reminds Pyrolithos of visiting the palace beehives in that other, vanished life of his.
He turns slowly, confidently, recognizing all the personalities of the Dardanian court. But naturally, they don't recognize him. He recognizes, too, the Aesarian Lords standing next to him, eyeing him keenly, and hopes they can't hear the pounding of his heart or smell his fear of them. The sight of High Lord Gideon's proud, dark face, and the sound of his deep, rich voice make Pyrolithos want to run out of the odeon as fast as he can.
For all his personal flaws, however, cowardice was never one of them. "Hear me!" he says, walking around the edge of the floor, raising his hands to quiet the outbursts of surprise. He looks down at those seated in the first row, then up at those in higher rows. "Hear me."
The odeon falls silent. His heart calms. He can do this.
"You know me—you all do. Do you remember ten years ago, a hunt in the Mantean woods?" He fingers the hilt of his hunting knife. "It was in celebration of the rites of Artemis during the fall equinox."
"I remember that day well," Kyros says, standing abruptly. "It was the hunt where Nearchas and Damia lost their ten-year-old son. His name was—"
"Pyrolithos," Pyrolithos cuts in.
"Yes," Kyros breathes. "We offered prayers and sacrifices in all the temples, but to no avail. No trace of him was ever found. I think the grief killed Nearchas and Damia."
Pyrolithos lowers his head in respect, forcing himself to remember the death of a parent, hoping the truth of his mourning will be etched on his face.
Slowly, heads nod up and down. A man with a long gray beard says in a trembling voice, "With no body, we burned Pyrolithos's effigy on the pyre to help him on his way to the underworld."
"For that, Damianos, I thank you," Pyrolithos says, enjoying the man's shock that he knows his name. "But as you can see, I've climbed back out of Hades and am here to reclaim my throne."
Pyrolithos lets the silence dangle, watching the nobles as they weigh his words. Out of the corner of his eye, he sees the Aesarian Lords standing rigidly, shifting their weight.
"Take off your helmet!" commands a man with long black hair. "We want to see you!"
Pyrolithos removes his helmet and pivots in a slow circle so all can study him. The Dardanians elbow one another and point. He starts to smooth his hair but drops his hand to his sword belt.
"But he looks just like Nearchas!" says Kyros. "And just like Nearchas's sister, Audata, who married King Philip of Macedon, and her daughter, Queen Cynane. The family resemblance is amazing."
"There are many tall, dark men in Greece, you fool," General Georgios retorts, folding his muscular arms over his chest. "That means nothing."
"If you are who you say you are, where did you go?" asks a red-faced, white-bearded nobleman Pyrolithos recognizes as Phalces. This man, at least, seems ready to accept him. Phalces stands, pulling his thick gray cloak more tightly around him. "Why didn't you return?"
Pyrolithos stares at the floor. "Why, indeed," he says, almost in a whisper. "My parents had taken me to Delphi late that summer, do you remember?" He sees heads—white and gray and black, some bearded, others clean-shaven—nodding.
"After giving my parents prophecies, the oracle asked to see me alone, and my parents withdrew. The gods, the Pythia said, had a message for me, a warning. My cousin, Amyntas, heir to the throne and still a child himself, was jealous of my popularity at court and was plotting to kill me. She said I must do all in my power to thwart his plan."
He looks into the shrewd dark eyes of the Aesarian leader, High Lord Gideon, a frank, man-to-man look that he hopes conveys respect and sincerity.
"I asked the oracle how I was to avoid being murdered," Pyrolithos continues, sliding his gaze to the audience, "and she said that I must tell no one, not even my parents, and that I must flee Dardania. I was not to return until the gods sent me a sign that Amyntas's death was near. And so, I disappeared at the hunt and made my way to live among the Paeonians, telling no one my true identity. Then last month, when I sacrificed to Zeus in the main temple of Stobi, the high priest removed the organs for examination and found the goat had no liver."
"No liver!" The words repeated around and around the tiers of marble benches sound like a circling wind. Sacrificing an animal with no liver is the worst omen imaginable.
"This is preposterous!" thunders Memnos, the wealthy sheep trader, in his annoying high-pitched voice. "Are we really supposed to believe such nonsense? Who will take the time to travel all the way to Stobi and confirm this man's story? Clearly, he is an imposter!"
Pyrolithos casts what he hopes is a cool gaze around at the men in the odeon. But his courage begins to falter. What if he says the wrong thing? His throat seems to close. Angry at his weakness, he forces himself to continue.
"Perhaps," he says, slowly unsheathing his hunting dagger, "some of you will remember this." He walks up three steps and hands it hilt-first to Georgios.
Georgios fingers the hilt slowly. The ivory overlay is the winged lion of Dardania. "You could have had this made." He grunts, thrusting the dagger back.
Pyrolithos says, "Look again, Georgios. The wood beneath is seared with a thunderbolt of Zeus, imbued with his divine lightning."
The general looks, then pushes the dagger back at Pyrolithos.
Pyrolithos grins. Now he holds the dagger up for all to see, "Surely some of you here remember this heirloom? It has been in my family for generations."
"I remember it," says Tisias, a mild-mannered man who served as Amyntas's private secretary. "Upon coming to the throne, Amyntas had me search for it, but it has been missing these past several years."
Relief sweeps over Pyrolithos. They don't remember.
"Thank you, Tisias. Now, who else remembers this dagger?" he asks. "You, Phalces?" The old man nods. "You, Thrasullys? Vettias? Olus? Agamedes?" he asks, pointing at each one. "If I am an imposter, a liar, how do I know your names?"
"Someone could have given you the dagger and told you about us," Georgios says, but his voice is weak.
Pyrolithos cocks an eyebrow. "I remember the day you returned from fighting Labatean cattle raiders, Georgios, who, before you slew them, took two of your fingers and most of your ear," he says. "Despite the pain, you never cried out or even groaned. My father praised your bravery."
Georgios falls silent and stares at his knees.
Simon heaves his bulk off his bench, pushes past several men with difficulty, and clambers down the steps. "Welcome back, Pyrolithos, son of Nearchas," he says gruffly before embracing him. "Praise be to all the gods that kept you safe to rule over us!"
"Praise be to all the gods!" several men cry. "Praise be!"
Pyrolithos stifles a small sigh of relief. Because, though his people might be ready to accept him, he can still feel the tension of the Aesarians standing behind him. The small one, Papari-turned-Lord Timaeus, looks at him with unadulterated loathing. This act is not yet done. Pyrolithos will need to win the Lords over, too.
"Silence! Silence!" he cries. As the room settles down, he continues, "The Aesarian Lords are right about our need for their guidance and protection," he says. "Amyntas took no joy in training our armies. Let the Lords as our allies reorganize our defenses, train our men, and develop our strategy, together with our own brave General Georgios. We can all work together. And our first order of business must be to find the murderous Queen Cynane."
Something changes in High Lord Gideon's face. He steps forward, smiling slightly. "People of Dardania," he says in the sonorous voice that causes Pyrolithos to shudder. "It is clear you are beloved by the gods. You have found a new king from the old royal line. You are fortunate, indeed."
* * *
Lamps flicker in the spacious bedroom used by Dardanian kings. Warm tapestries line the walls, and the cabinets and chairs are of the finest ebony inlaid with ivory and tortoiseshell. But beneath the thick wool rug, Pyrolithos found bloodstains on the olive-wood floor. Evidently, no amount of scrubbing could remove them. And he is unwilling even to touch the wide four-poster bed where that cackling threat slept. The past few nights, he has been sleeping on a pallet on the floor.
Tomorrow is the day he has waited for all his life. The day he will be crowned and take up his rightful role as king of Dardania. Since earliest childhood, he wanted to be a king and a general of armies. Until a few days ago, it seemed impossible, a ridiculous dream, a childish fantasy. _No. Never. Not allowed. Take off that armor. Stop your military training. You won't be allowed to rule._
They were wrong, all of them, wrong. He has achieved all his wildest dreams. He wants to let all his naysayers know what he has done, to rub their noses in his victory the way you housebreak a puppy by rubbing its nose in an ill-placed turd.
Frustratingly, ironically, he cannot tell a soul. Yet.
If he can ever tell anyone depends on tonight's private errand. For he could never trust men, who lie and plot and kill for their own advantage. He will only put his faith—his life—in the loyal hands of women.
Wearing a wool tunic, supple leather boots, and a thick cloak, he walks the windy streets of Dardania until he finds what he's looking for—an ornate door in an otherwise windowless wall. Aphrodite's Grotto, the most elegant brothel in town.
Two guards, broad and muscular—more like slabs of beef than men—stand in torchlight that flickers on the myriad knives in their belts. They look him up and down as he approaches, eyeing approvingly the cut of his cloak, and step aside.
Inside, he's met by a gracious dark-haired woman in shimmering robes, who smiles warmly.
"Welcome. I am Xanthe. First time here, sir?" she asks in a sultry voice, tilting her head and gazing at him through thick lashes. "I never forget a face. Are you looking for something in particular?"
"Alecta," he says gruffly, heart pounding. "I've heard...of Alecta."
The woman bows her head. "One of our most popular girls. You won't be disappointed. Won't you follow me?"
She sashays down a wide corridor and shows him into a small but elegantly outfitted chamber with two couches, a table, and a bed. A bronze brazier keeps the little room warm, and aromatic smoke rises from a perfume burner on the wall.
A few minutes later, Alecta sidles in, her red hair gleaming golden in the lamplight, her flawless skin rich and smooth as cream, her teal robe just sheer enough to tantalize rather than reveal. She has brought with her the delicate scent of violets, which caresses him like a gentle hand.
She doesn't recognize him, but he recognizes her. Her lithe beauty makes his breath catch in his throat. Smiling, the entertainer pours two cups of wine, hands him one, and takes the other. He stares at the red-and-white paintings of figures on the black glazed cup, clearly meant to arouse: naked lovers embracing. He sips it: a rich, expensive Chian.
Alecta has settled herself on a couch like a sinuous cat, long legs folded neatly beneath her. "It's not every day a handsome stranger asks for me," she says in a low, musical voice. "How can I serve you, my lord?"
Desire surges through his body, this strong, masculine body he has many plans for. Man's desire, he thinks, is so different from woman's desire. It is immediate and all-consuming, rather than the slow burn of a woman's secret yearnings. No wonder, he thinks cynically, prostitution has been thriving longer than perhaps any other business in the history of mankind.
But tonight is truly business. Lust must wait.
"I am here at Queen Cynane's request," he begins in a low voice.
Alecta's sensuous demeanor drops, and she sits up straight.
"But the queen's vanished," Alecta says, her voice much less musical and more practical. "There is a large reward for bringing her in alive."
He nods. "There is, but she did not kill Amyntas. The Aesarian Lords killed him and are blaming it on her. She is arranging the next step of her plan, and in the meantime, she has sent me here on her behalf."
"For what purpose?" Alecta asks, eyes narrowing.
Pyrolithos takes a sip of wine. "I am her ally and kinsman—and I am a general. She wants me to train the women of Aphrodite's Grotto to become her personal army. She doesn't trust men. Doesn't much like them, either." He smirks.
Alecta's stormy dark eyes take on a wary look. "Then why does she trust you?" she asks.
Pyrolithos feels his cocky confidence extinguished like a lit lamp wick in a strong wind. He wants to come up with a witty answer but cannot.
"Perhaps you are a spy sent by the Aesarian Lords, looking for an excuse to close down this establishment?" she continues, relentlessly.
He has always had a quick response. It must be Alecta. She does this to him.
"I think," she says, rising, "you had better leave."
This will not be easy, then. Yet even as he curses himself for his slow-wittedness, he can't help admiring Alecta's caution.
"The queen will send you a sign," he says, collecting himself and standing. "Gifts in preparation for your training."
"Goodbye, General. The guards at the door will expect full payment for my time."
Pyrolithos rises to his full height. He had not planned on revealing this, not yet, but perhaps he should. "Let me be clear. Tonight, I am a general. Tomorrow, I will be your king."
Alecta sucks in her breath. "My apologies. Shouldn't you be getting ready for your coronation tomorrow, your majesty?" she asks.
He's about to reply when he sees the mocking gleam in her eyes.
He bows his head. "Come to the main palace entrance tomorrow at midday. I will have your name on the list of guests for the banquet."
And before she can say a word, he leaves, making sure to pay the guard Alecta's due.
That night, when his dreams should be filled with the coronation, he dreams instead of long red hair and flashing dark eyes. And by the time he wakes the next morning, bathes, dresses, and appears for the ancient coronation rituals, he thinks he will explode with impatience.
He has plans, so many plans with Alecta and the other women, and aches to get started, but first he must endure long prayers on the high palace tower as dawn warms and brightens the land. Then he must take a ritual bath in front of the royal council.
"I wash my hands to clean them of all bad actions," he intones, plunging his hands into the steaming bowl of fragrant water.
"I wash my mouth to cleanse it of all falsehood," he says, drinking from a cup of warm spiced wine and water, and spitting it out.
"I wash my face to eradicate all shameful behavior," he affirms, splashing cold spring water on his face.
"I wash my feet from having walked in all false paths," he announces, stepping into a large basin of hot water and dried rose petals.
Servants robe him in a blazing white tunic and place a crown of laurel leaves on his head. Then they form a solemn procession through the streets to the shining Temple of Zeus, where a white bull is slaughtered, and the chief priest, after examining its liver, predicts a long and successful reign. The priest replaces Pyrolithos's laurel crown with a golden one, and when the new king steps outside between the tall columns, the jubilation begins.
In the amphitheater back at the palace, he watches the king's champion—he picked Lord Timaeus, who seemed honored at the request—defeat the masked demon of Chaos in a mock battle. Then, finally, comes the feast.
Seated on his throne in the banquet hall, he searches the hundreds of excited faces but doesn't see the one face he wants— _needs_ —to see above all others.
Then he notices, in the back of the hall, a veiled matron standing as tall as a willow. Her eyes, dark as night in the pallor of her face, latch onto his with the ferocity of a predator pouncing on its prey. He feels their bite and sting. And then she pulls her veil over her head and walks back out the door, the very sway of her hips a challenge.
If he were not within view of his new nobles, Pyrolithos would jump off his throne and chase after her. As it is, he wilts with relief. She came. She saw him.
It's clear she still doesn't trust him, but that can be dealt with later. Maybe she will agree to work with him when she sees the gifts he sent the brothel: an oxcart full of swords, helmets, breastplates, shields, spears, bows, and arrows, all concealed under a canvas cloth. _A gift from the Queen_ , was the message, _and her servant, the King. When do we commence our training?_
Pyrolithos endures the rest of the celebrations, waiting until most of the Dardanians drink themselves into stupors, and then he leaves.
After slipping into his room, he looks in disgust at the dead king's possessions. But he is king now, he can live where he chooses. In the morning, he will instruct his servants to move his things to Queen Cynane's old room.
A soft knock on the door interrupts his thoughts. The mute slave boy, Aesop, comes in, chained to his slave master, a bald, burly man with eyes that wander in two different directions. The boy's eyes dart nervously to the spot on the floor covered by the new rug.
"Yes?" Pyrolithos says, annoyed at the disturbance.
"A message for you, my lord," says Otus. He elbows Aesop hard, and the boy, trembling, holds out a tiny scroll tied with a scarlet ribbon. Heart pounding, Pyrolithos unrolls it.
_We give thanks for the generous gifts. We'll be sure to make good use of them._
Pyrolithos smiles a slow smile. Clearly, Alecta made her words coy enough that no one would guess at her true meaning if the message were intercepted.
He locks the door behind Aesop and waits. Waits until the hour of the night when even the whores have gone to bed, when the palace cooks have not yet arisen to begin the day. The hour of utter silence.
Quickly, quietly, he bolts the door and closes the shutters. Then he unsheathes his hunting dagger, the precious family heirloom that convinced the nobles that he was the lost boy, Pyrolithos. But Pyrolithos never had this dagger when he disappeared in the woods. It left Dardania soon after, when the boy's aunt, Audata, went to Macedon to marry King Philip, as part of her dowry.
Standing before the polished bronze table mirror, he lights a lamp. Then, carefully, slowly, he pricks himself in the fleshy part of his upper arm, just enough for a drop of blood to flow. Pain sparks, helping him to push the Smoke Blood through his veins as he concentrates. Not enough to last more than a few minutes. He feels the power of fire and glowing cinders spreading through him, reshaping his face, his shoulders, his chest, his hands. Just for a moment. Just to remind himself of who he really is.
In the mirror, Cynane smiles.
CHAPTER FIVE
OLYMPIAS
Urgent voices. Worried faces. Careful, prodding fingers. Tiny sips of warm spiced wine.
A world of gray and haze.
A cool cloth wipes Olympias's face, and she smells the clean scent of mint. _Mint water calms the ill._
Her blankets are plucked off. She feels a cold rush of air and then heat radiating from a warming pan as the blankets are tucked back around her. Her eyelids are so heavy it takes enormous concentration to open them. But when she does, she sees the slight form of a girl bustling about a tiny room that dips and sways slightly. A small, high window lets in a shaft of light and a stream of frigid air.
She is on a ship.
It seems impossible, as she vaguely remembers being in a palace on the Bosporus Strait. A pang moves through her, a feeling of loss more jarring and heavy than the weight of her illness.
She feels as though she has been brought back from the dead—again.
Perhaps, in fact, she has.
She opens her mouth to speak, but her tongue and lips resist the formation of words.
"Your Majesty," the girl beside her breathes. Her pretty, pointed face seems dimly familiar. She places one hand behind Olympias's neck to support her head and tips a cup of sweet, rich wine into her mouth. It slides down her parched throat, warming her.
Olympias tries hard to focus her thoughts, even though the effort hurts her head. Memories and images blur and shift. She was sitting in her bedroom in the Byzantine palace, getting ready for the eighth wedding of the king, her husband, to a brazen blue-eyed girl: the eighteen-year-old Cleopatra, niece of Attalus. The wedding where, according to the words of the gods themselves, Philip was destined to die.
What of them now? Olympias had tried to poison Philip, had given Cleo a pot of poisoned lip stain, knowing the king couldn't resist the young temptress he'd chosen to make his newest wife—and knowing, too, that he wouldn't survive his toxic reaction to the rare Persian mulberry.
"Tell me where..." she begins, her voice like the scrape of one pottery shard on another.
"Don't tire yourself, my queen," the girl says, adjusting the slats in the shutters on the small window. "King Xander will take care of you once we arrive. I am sure of it. He will make certain you are safe from those who might mean you harm."
"Safe..." So, they are bound to Epirus, home to her brother, Xander, the stern, distant man she never really cared for, despite having named her son Alexander in his honor. Xander never protected her before—not when they were children and she was beaten and neglected by their cruel and coldhearted parents. Not when they were fair-headed teens, and she was sought after by a god.
_Riel._
A cold dread moves through her. What has happened to the god who once loved her? He was at the wedding, waiting for her to do the deed. For the prophecy said King Philip would die at the wedding of Cleopatra by the hand of a lover. Riel and Olympias decided she would be that lover. He was waiting for her in the banquet hall, but she never made it downstairs...
She shudders hard, the chill suddenly unbearable, and desperately tries to tug at the blankets but realizes something is wrong with her right arm. She raises it and squints. Her wrist is in a splint, tightly bandaged. She puts her other hand to her head. That, too, is bandaged. She can't remember how this happened. "Who hurt me?" she croaks.
The girl shakes her head. "No one hurt you, my lady," she says. "You, like so many others, were injured in the great waves that Lord Poseidon sent to batter Byzantium. Then you developed a high fever."
The girl's words make no sense, and Olympias wonders if she heard her correctly. Questions just beyond her reach prick at her like small knives.
Then she remembers the truth: that peasant boy turned Aesarian Lord Jacob did this to her.
There was no one to save her from his rage when he learned that she had killed his family. She managed to slice his arm open with her hairpin, but he snapped her wrist and slammed her head hard onto her cosmetics table. She remembers waking in a jumble of blood-soaked broken glass and, terrified Jacob would return, crawling like a wounded animal into the servants' passage.
If there's one thing Olympias has learned on this earth, it is how to survive. Sometimes she thinks it is all she was built for.
"Why...Epirus?" Olympias manages to gasp out.
"Your lord brother worries that a fatal accident might befall you now that King Philip has taken a new bride," the girl says, pulling up a stool and sitting beside the bed. She holds a long strip of laundered bandage and starts to roll it. "They both survived the devastation unharmed, and King Xander didn't want Cleo adding anything _unusual_ to your medications. He'd rather his own healers attend you in your childhood home."
Olympias shuts her eyes. Philip is alive, then.
And her brother—he doesn't care about her well-being. Only his fragile alliance with Macedon. He must be planning to use her in winning some advantage from Philip.
But how could Philip have lived? The prophecy of the oracle of Delphi—that Philip would die at the wedding of Cleopatra—was wrong...
_The oracle..._
"I know you," Olympias says, her voice sudden and harsh in her throat. She pushes herself up. Pythia. The oracle of Delphi. _That's_ where the queen last saw this girl. During her and Riel's visit to Delphi, seeking answers.
Immediately, Olympias's moment of recognition morphs into the acrid taste of suspicion. Delphi is far from Byzantium. This can be no coincidence. "What are you doing here?" she snaps. "Who are you _really_?"
The girl drops the bandage in her lap, suddenly looking achingly young and vulnerable. For a second, the queen's suspicion makes room for something else. Something like pity. But it is gone before she can feel it. Olympias's life has no room for pity. No room for fragile, breakable things.
"My real name is Patra," she says.
Olympias sits up straighter, digesting this stunning piece of news. "Is that short for Cleopatra?" she asks, her voice harsh with rising excitement. For if so, here is another Cleopatra. Here is another chance.
"Yes," the girl replies, her voice low. "Too grand a name for a slave. Because oracles may be the voice of gods, but in reality, we are nothing more than slaves." She swallows and straightens her slender shoulders. "I escaped Delphi during the great earthquake, and I will never go back."
Olympias relaxes back on the pillows, fear draining like water as a wave pulls back out to sea. The girl's name opens a world of possibilities for the queen. And besides that, truth rings in her voice. A truth she recognizes because it was once her own. When she was a girl, her father ignored her. Her stepmother beat her. All she could dream of was running away.
She shuts her eyes, unwilling to remember any more. Unwilling to care.
"Patra," she says, raising her shoulders in a delicate shiver, "would you put more wood in the brazier and bring it over here? I am so terribly cold." The pull of the beyond has not yet fully released her. At any moment, she could still let go, could slip away. The burning, raging fever has broken. But is the coolness she now feels the chilling presage of death?
Patra pushes the portable brazier closer to her. She gathers sticks of wood from the tray below and places them in the elaborately carved bronze bowl on top. They catch quickly, and a sweet-scented smoke rises like curling white fingers. Patra coughs and stumbles. Olympias's eyes flash open. The smoke.
Smoke to an oracle is like sparks to tinder. Someone told her that once. Someone she loved dearly.
Helen. Who raised Katerina as her own daughter. The only friend Olympias ever had.
_No._ No more memories.
Olympias blinks again and watches as the smoke swirls around Patra's face. Her eyes are drifting away from the queen... They start to roll back in her head and muttered sounds move between her lips.
The few words Olympias can make out send a jolt of terror through her.
Patra shakes her head rapidly, flinging her hair from side to side. Wild tendrils are strewed over her face, which is as blank as new parchment. Her hazel eyes are flat and soulless as river pebbles.
"All gone. All dead," Patra murmurs.
"Who?" Olympias's voice is a raw scratch against her throat.
"The last of us. The last of us. The last of us. No more gods walk this earth. No more ever shall."
_Riel._ That chill of certainty moves through her again, the dull thud of a loss she cannot fathom, cannot allow herself to feel.
"Gone...gone...gone..."
And then Patra collapses, breathing heavily as the Voices of the gods leave her. She moans on the floor and curls into herself, but Olympias barely notices. The queen falls back on the pillows, light-headed, too numb to feel pain.
All that she possesses is a yawning emptiness.
She is an abyss.
An empty space.
Nothing.
Life before she met the green-eyed god had been unbearable. He gave her love, ambition, secrets, and plans.
Life with him was also hard. But even then, all those years when he had been trapped in snake form, she had had him close by her, knowing she would release him with a powerful spell as soon as the changing of the age allowed.
And a life without him now would be impossible.
A wave of raw pain rips through her, a wave as huge and destructive as any that hit the Byzantine palace. Tears prick her eyes—she cannot remember the last time she wept. It takes everything in her, every last fortress against disaster, every force of will that has kept her alive this long, to hold it in.
She is alone. Alone. Every single day, for as long as she lives, she will be totally, utterly alone.
But she has this. _She has this._ This will to live. This will to _win_. And, most important, the will to _get revenge_.
She opens her eyes and sees Patra distorted through the murk of her tears.
"My queen... The god is dead," the girl rasps, pushing herself up from the floor, her wild hair covering her face, though the fear in her voice is clear as a light breaking open in the sky. "But the monsters." She gasps. "The monsters are coming."
CHAPTER SIX
ZOFIA
_There's only one way to change your fate_ , Kohinoor had said.
In the predawn chill, Zofia pulls her overlarge blue cloak tighter around her shoulders and hurries down one of the winding streets toward the center of Athens. Unspeakable dangers may lurk in the shadows all around her. It is never safe for a woman to travel alone—Zo, of anyone, should know that. But she is no mere woman—she is an escaped princess, promised to the great Prince Alexander himself.
And more than a princess, she's a pawn...in what could become the largest war the world has ever seen. The Assassins Guild is hunting for her, even now. She was not actually meant to wed Alexander but to be murdered by the Assassins, who would have made it look like the prince had done it, her death the planned catalyst to ignite war between Persia and Macedon.
And what none of them know—not Alexander, whom she has yet to meet, nor the Assassins, nor the Great King of Persia—is that she is with child.
But her secret has become more difficult to hide. Her hands automatically move to her belly, now taut and round, where she feels a stirring. Her child will live. Zofia will never stop running until she finds a way for that to be true. Until she understands what freedom is.
Not so very long ago, Zofia had been a romantic. She'd fallen in love first with Cosmas, the father of her baby, with whom she made love once in the Sardisian palace basement. Then she had fallen for Ochus, son of the head of the Assassins himself, Darius, chief advisor to the Great King of Persia. Ochus had promised to protect her, no matter what came. No matter if it meant disavowing his father and everything he'd been raised to believe.
That last sweet night with Ochus at the Pellan palace, where she had been waiting for her fiancé, Prince Alexander, to return from Byzantium, she told Ochus the name of her first lover and the father of her unborn child. When the first rays of dawn struck the gilded statues on the roof, Zo found Cosmas dead, murdered along with soldiers both Persian and Macedonian. Ochus had vanished. He had either fled, been captured, or died. It didn't really matter which fate he had met, because in each scenario, she was completely alone. Utterly without protection against the Assassins, who would kill her the moment Prince Alexander returned.
And so, she had run away.
She hadn't planned to come here. But once she realized her life was in danger in the Pellan palace, she had thrown gold coins, all her expensive dowry jewels, and a change of plain clothing into a pack, slipped out of the palace complex, and made her way to Lakeport. There she boarded the first ship out, which happened to be going to Athens.
For a brief time—and truly, for the first time in her life—Zo had a taste of freedom, though it was a perilous, precious kind of freedom, for she knew at any moment it could be snatched away again. Still, she hoped—hoped Athens would provide the safety she hadn't found elsewhere.
And, at first, it seemed to. She spent days in awe of the wonders Athens has to offer. She watched as bearded philosophers hotly disputed the nature of the human soul and sculptors chiseled gods from blocks of stone. She gaped at tanned, brawny wrestlers in loincloths grappling with each other as men wagered and shouted. She listened as a statesman called Demosthenes enraged crowds of citizens against the evil King Philip of Macedon, who threatened the freedom of all of Greece, the orator's powerful voice and persuasive rhetoric sweeping so forcefully into her heart that, as if in a trance, she found herself raising her fist and chanting against the northern tyrant who almost became her father-in-law.
But then, the shadows returned. Or maybe they had never left her. Lately she has had the feeling of being hunted, like a stag in the forest as riders slowly close in on her. Everyone she sees—the plump baker's wife, the cheerful wine merchant—could be an Assassin lying in wait. And what about the ones she doesn't see?
There's movement in the shadows of a lamp maker's shop, and Zo grabs the knife in her belt. But it's just a lean black cat, rising on legs stiff from the cold. It stares at her with glowing yellow eyes that make the hairs on the back of her neck rise, then turns to sniff the smell of frying fish. _It's just a cat_ , she thinks, chiding herself for the heart-churning panic that shot into her so fast she could hardly stop it. That's been happening more and more. There have been darkly hooded figures in the eaves of doorways, watching her. If the Assassins have followed her here, she must move on.
But where can she find safety? Bound together with the urgent need to do something is the incapacitating fear of doing the wrong thing. What do the Fates have in store? Didn't all of this begin because of her desire to change her own?
A hooded figure staggers out of a low dwelling and pauses, staring at her. In the iron-colored early light, she can't see the figure's face, and the panic returns full-fledged, a stampede of hooves inside her chest. She blinks. The figure sways and the hood slips back, revealing a haggard painted face. A whore, Zo realizes, leaving the room of her latest customer. The woman slinks around the corner and into the purpled shadows of an alley. Slowly, Zo begins to breathe again. The city is full of women who sell their bodies—women like this who do so for a few obols, as well as hetaerae, the city's most elegant, educated, and expensive courtesans. She has seen them in the marketplace, wafting clouds of sweet perfume in their wake, their golden anklets tinkling like tiny bells.
Well-heeled prostitutes operate in every major city, Zo knows. But she has heard that in Dardania common prostitutes are training with military weapons, perhaps armed by the warrior-queen Cynane herself, who murdered her insane husband and disappeared. There have been rumors that women from across Greece have been leaving abusive husbands and flocking to a brothel called Aphrodite's Grotto. Men who try to take back their wives are beaten black-and-blue by the prostitutes.
These women, Zo knows, are searching for the one thing women are always denied: freedom.
The sun is about to rise, the sky brightening above her. Zo takes another breath and presses onward, praying that today she will get answers. She is running out of time.
The baby will not wait.
Neither will the Assassins.
Last summer, when she was held captive by rogue slave traders, Zo received a prediction from Kohinoor, an old soothsayer locked in the cage next to hers. _Your blood is fated to mix with that of Prince Alexander_ , the violet-eyed crone had mumbled. _The only way to undo the threads of Fate that have been woven for you is to find the Spirit Eaters._
Zo was repulsed at the thought of marrying Alexander, a cripple, it was said, with mismatched eyes. And so, she traveled deep into the Eastern Mountains to find a Spirit Eater and change her fate. She did briefly find one—or it found her and raked her back with its claws as she raced away on a flying horse—but she doesn't believe the encounter was one that could have changed her destiny, especially after Great King Artaxerxes packed her off to Macedon as soon as he saw her.
Every day in Athens she has visited at least one temple or altar, imploring the gods for guidance, seeking a divine omen to tell her how to change her fate, where to go, and how to protect the precious life growing inside her. In short, how to alter her fate. But none has given her any sign at all. Even the goddess of the biggest temple, the Parthenon, great Athena herself, was silent. The enormous statue, many times the height of a man, with robes of beaten gold and skin of gleaming ivory, stared down at her with cold crystal eyes. Zo left feeling smaller and more alone than ever before.
But yesterday Zo learned of another temple, this one outside the city, ancient and mostly forgotten, called the Temple of the Moirai or the Three Fates, and hope rose in her chest. Perhaps her best path now is to find the Greek goddesses who spin the threads of Fate and negotiate with them herself.
She cuts through the agora—the main marketplace—where shopkeepers throw open shutters with loud bangs and drag out tables of merchandise. Men set up booths and unload wagons of cheese and wine, olives and fish, glinting weapons, fine leather goods, and brightly colored carpets. In the center, wood has been stacked for a huge bonfire. Tonight is the winter solstice, and drunken people will dance around the flames, praying to the sun god to turn his face back toward them and bring longer, warmer days.
Smoke rises from shops and houses, mixed with the mouthwatering aroma of fresh bread, which makes her stomach grumble. She should have brought some food with her, but it was too early to go into the kitchen of Berenike, the kindly landlady who has been renting her a room.
The freezing wind is stronger in this wide plaza, beating her with invisible fists, and she pulls her dark blue cloak more tightly about her. It was Ochus's; he had left it on a chair in her bedroom the evening before her escape. She should have shrunk from touching anything of his, but in those last frantic moments before she fled the palace, she realized her own cloaks were embroidered with gold and sparkled with gemstones, far too ornate for her to blend into the crowd and escape.
She marches through Kerameikos, the famous neighborhood of pottery shops, with pots and cups in every size and shape, beautifully painted with scenes of gods, heroes, and myths. Then she is out the Acharnian Gate, fighting her way past wagons of produce headed to market. Something glimmers in the corner of her eye. She turns and sees the first rays of the sun strike the temples crowning the Acropolis, setting them aflame in shades of gold and orange. The enormous flat-topped hill, jutting straight out of the center of Athens, has become a familiar sight to her, yet she is constantly thrilled by the majestic beauty of its many-columned buildings and towering statues.
Athens was named in honor of the goddess Athena, Zo learned, because she gave the city the incomparable gift of the olive tree, used for food, oil, and wood. Now, as Zo trudges up the Acharnian Road, she passes through olive orchards as far as the eye can see, the thousands of leafless, silver-barked trees standing like burly armored soldiers on the field of battle.
She walks quickly to warm up, deftly avoiding steaming animal droppings and stepping out of the way of horses and carts. Then, a mile or so down the road, she sees the temple off to one side. At least she feels like she can breathe again—there are no mysterious hooded figures in the countryside. No one is following her. At least, not as far as she can see.
Then again, the deadliest enemies are the kind you don't see coming.
The Temple of the Moirai is nothing like the tall marbled masterpieces on the Acropolis. It looks centuries older and is made of wood, the black and red paint on its squat columns faded and flaking. And it is small, the size of a comfortable room. Gnarled trees bend over it protectively, and the wind streaming through their leafless branches seems to whisper ancient lullabies.
An old man wearing the white vestments of a priest sweeps dust off the porch with a broom of bent twigs. He stops and looks up. As soon as his eyes catch on her swelling belly, his bushy gray eyebrows lift knowingly, as though this is not the first time a pregnant woman in need has visited their humble temple. Then he gives her a toothless grin.
Leaning on the broom, he lisps, "You've come to the right place. We help many girls like you here."
Zo hesitates before nodding. Then she steps carefully through the open door. Inside, the pungent smells of mildew and old wood smoke nearly make her choke. It's dark here; a few shafts of light slanting through high openings reveal cobwebs hanging from every beam and fat spiders spinning. At the far end is a wooden statue of three old women at a loom. Zo knows she has seen the image before, but she can't remember where.
"Come here, my child," a voice croons from the shadows. Zo follows the command to a screen in the corner and sees a figure seated behind it, though it's too dark to make out the old woman's face. Chills race up her spine and she considers backing out, but she has come this far, and something takes hold of her body, compels her to obey. She sits on a low stool. It is much colder in here than in the sunlight outside.
"The three goddesses have told us you would come." The priestess's voice sounds like fall leaves scuttling in a whirlwind. Zo shivers again, but this time, not from the cold. She could swear she _knows_ that voice.
But that is impossible.
Still, the crone continues in that eerily familiar voice. "What is it you seek?"
_To change my fate_ , Zo thinks. _To save my child._ But she hesitates. "I—I..." She shuts her mouth as a cold breeze winds its way up the back of her neck. She is suddenly gravely aware of how the temple doors closed shut behind her with a low, muffled bang, sealing her in. How hard would it be for her to open them? "I think I've come to the wrong place. I'm sorry. Thank you for your time."
She starts to get up but hears the priestess laugh softly. Again, that stirring familiarity—but not the good and trusting kind.
Once, Zo was a trusting girl. She trusted that Cosmas wanted to marry her. She trusted that love would conquer fate; that if she were righteous enough, people would listen to her, that they would do the right thing.
That men existed not to hurt each other—not to wage wars but to build cities, not to take advantage of women but to adore them.
That even wise old soothsayers like Kohinoor meant only to protect and never to harm.
She doesn't believe those things anymore.
"Less than four months, I think, before your child is born," the priestess says.
Zo swallows hard. Even through the screen, in the dimness of the temple, her situation is no longer a secret.
"I must go," Zo says again.
"And where is it you think you'll go next? How far can you run from them?"
Zo freezes. "What do you mean, from _them_? What do you know?"
The priestess is silent for a moment, and in that silence, a creeping awareness begins to take hold of Zo. She _does_ know this priestess, somehow. It's more than just her voice that's familiar. It's the weight of her gaze; even though she can't see her face, she can feel how the woman is watching her, studying her. She hasn't felt this exposed, this uncomfortable, since she realized Kohinoor was healing the injuries from her fall off the Pegasus in her cave in the Eastern Mountains, not out of kindness, but to try to steal her child.
"Live with us here, girl. We will take care of you both," the priestess says gently. But her offer feels stifling, like a wet cleaning rag being stuffed down Zo's throat, choking her from the inside. She can't breathe. She can't breathe!
She bolts upright, the child giving a violent kick in protest, and the little stool falls backward with a crash. She pushes away the screen, its legs screeching against the floor like the harsh cry of hunting birds. There sits the priestess—her gnarled hands up in front of her, as though in surrender...as though she has been caught.
Zo squints.
The woman doesn't just remind her of Kohinoor.
She _is_ Kohinoor.
Kohinoor: the woman whom Zo last saw _hundreds of miles from here_. First on the slavers' trails outside of Sardis, then much farther east, by the caves of the Spirit Eaters, and now all the way here, across the sea, in this tiny temple outside Athens. Has she managed, somehow, to follow Zo all this way? That cannot be. Travel even for royals can be difficult and dangerous, but it must be impossible for those as weak and ancient as Kohinoor, so poor she lived in a cave and scrabbled for food in the hills.
Only some work of dark witchcraft could have transported her here.
It _can't_ be her.
And yet it is.
As Zo gapes in shock, Kohinoor smiles gently, her blind, violet eyes unblinking and terrifying. "Princess, you cannot escape fate," she says calmly. "On the longest day of the year, you will suffer your longest night."
_The longest day of the year._ Summer solstice. That is just six months away. Nauseous with fear, Zo stumbles toward the door and pushes her way out into the morning. She runs, even though her body aches, her belly taut and sore and resisting. The cold, slimy, creeping feeling still follows her; her breath comes fast and short.
Eventually she is forced to stop and throw up at the side of the road. Acid burns her throat. Why didn't she buy a roll in the marketplace for breakfast? She feels light-headed and wonders if she imagined the whole thing. It isn't possible she saw Kohinoor just now. She begins to doubt herself. And if it _was_ Kohinoor, and the woman did manage, somehow, to track Zo here, then is _anywhere_ safe?
She wipes her mouth with the back of her hand, takes a shuddering breath, and moves on.
It's not until she reaches the Acharnian Gate that she stops, leans against the thick walls, and catches her breath, sinking to the ground. She closes her eyes and inhales deeply to stop the hammering of her heart and the spinning of her head. She must calm herself and _think_.
Near her, wheels creak. Horses whinny. A donkey utters a cry of stubborn protest over a man's loud commands.
She opens her eyes on a chilly Athens morning with life going on as normal. But normalcy has fled from Zo's life. It's not just the Assassins who are after her, seeking her blood. Kohinoor is following her, too, for reasons she cannot fully understand.
All she knows, with a sudden certainty, is this: Kohinoor wants the child.
Zo clenches her eyes shut, remembering that day, three months ago now, when she wandered into a cave adjoining Kohinoor's living quarters in the Eastern Mountains and found ancient paintings brushed onto the rough walls. One of the paintings—crude, hand drawn—depicted three old women on a loom, she now remembers. That's where she saw that image before.
Another one was a dark-haired girl falling from a Pegasus—just as Zo had fallen after the Spirit Eater attacked her. She also saw the capital of Persia, Persepolis, burning, people falling from cracked and flaming towers. And a winged child, its tiny arms holding a wax tablet, rising from the flames. The pictures seemed to indicate that a child was destined to save the world from destruction.
Zo had had the dizzying suspicion that it was _her_ child.
Then the worst memory: the object Kohinoor had been diligently constructing, all while drugging Zo to keep her complacent. It had not been a table, as she said, but a crib. And next to it, Zo had found an adult-sized cage.
She didn't understand Kohinoor's plan then, and she doesn't now, but the woman had shown an unnerving amount of strength and cunning despite her feeble appearance, blindness, and age.
Zo tries to concentrate. She needs to get out of here.
If she sold some of the royal bridal jewels sewn into her robe, she would have enough money to go any number of places—Egypt, Rome, Crete, Jerusalem, Bactria... All the places Zo has ever learned about flash through her mind. Should she lose herself in the black northern forests of the Gauls? In the eternal grasslands of the Scythians? Or in the hot jungles of Africa, with their empires of wild beasts and mineral riches? Where in all the world can she hide, when not even the temples are safe?
That last night in Pella, Ochus asked her where she would go if she could go anywhere, do anything she wanted. She said Babylon. She wanted to study at the famous Royal Library, learn important things to teach her child, things she never cared about when she was a gossipy princess who ignored tutors she found deathly boring. But Babylon, in the heart of the Persian Empire, is not terribly far from Persepolis, home of Darius and his Assassins. It would be like wandering into a lion's den.
She needs a sign. She needs the Pegasus, whom she named Vata for the Persian goddess of the winds. If Vata landed here, on the Acharnian Road, and Zo hopped on, the magical creature would soar high above Athens and wing her way to exactly where Zo needed to be. _Pegasus sees the strings of Fate. Pegasus is never lost. Pegasus knows the way._ Zo remembers with a pang how she used to tell her little sister stories about the legendary winged horses, stories that made Roxana's dark eyes light up with excitement. But Roxana is dead, and the last time Zo saw Vata was in Persepolis, almost two thousand miles from here. No one is coming to tell her what to do, to show her the way.
She opens her eyes just as four tall Ethiopian slaves wearing leopard-skin cloaks and wide golden collars stride by, carrying an ornate litter, and set it down not far from where Zo sits crumpled on the ground. A white hand flashing an emerald ring pushes open the curtains, and a low, sensual voice calls to the commander at the gate, a barrel-chested man who grins and strides over. He pokes his head inside the curtains and laughs, agreeing to an assignation that evening. As he returns to his post, the woman pokes her head out and waves farewell, her gleaming chestnut hair piled under a golden tiara. She is clearly a hetaera.
This one is a different class entirely than the woman Zo saw staggering through the streets earlier this morning, before the sun had fully risen. Zo marvels at the whims of Fate—how fickle and unfair it is that some can be beaten down and bruised, while others are carried high above the heads of men, in control of their bodies and their destinies.
And it's then that clarity hits her. _This_ is the sign she's been looking for.
Suddenly, Zo knows _exactly_ where she must go.
CHAPTER SEVEN
DARIUS
The message clutched tightly in his hand, Darius, leader of the Assassins Guild, hurries through the torch-lit corridors of the many-domed palace of Persepolis. The Great King of Persia must hear the news from Darius himself.
This matter, like some of the rarest and deadliest blooms in Darius's poison garden, is delicate, to say the least.
Around him, the noise of dancing revelers swirls through windows facing the Courtyard of Palms. A raging bonfire in the outdoor enclosure celebrates the longest night of the year, the winter solstice, a time for music and joy and light in the nadir of darkness.
Darius knew better than to trust his son, Ochus, who had failed in his first mission to kill Princess Zofia. That was why, nearly three months ago, when Artaxerxes sent the girl to Macedon as Prince Alexander's bride, Darius inserted additional Assassins into the princess's escort. Several soldiers, of course, but also two attendants no one would ever have suspected—a eunuch and a sweet young handmaiden—and a few Greek-born Assassins who enlisted as Macedonian palace guards.
If Ochus didn't kill her this time, these others were supposed to do the job. Darius had envisioned himself telling Artaxerxes of the girl's murder at the hands of Alexander, an affront to Persia that would force the Great King to break the alliance and declare war on Macedon.
At last.
But that is _not_ the news Darius just received.
He takes the curling steps three at a time up into the tower, slipping into the viewing-deck room unseen. Artaxerxes has his back to him, his tall conical crown glittering in the dancing torchlight. The eunuch Bagoas, the Great King's other trusted advisor, stands beside him, looking out over the bonfires below.
"Sire," Darius says, his well-practiced voice sliding out as smoothly seductive as always.
"Grandson," Artaxerxes says coolly, turning. His onyx eyes glint, just slightly.
Darius freezes, his mind racing with calculations.
_He knows._
It is possible Bagoas has told the king already.
Perhaps Bagoas heard from his own spies and persuaded him that Darius is a traitor. Darius flicks his gaze subtly to the eunuch for some sign, but the dark eyes under heavy makeup are inscrutable.
Artaxerxes turns back to the railing, the chill wind whipping his long purple robes and white hair. With one word from Darius, his Assassins could kill the Great King's entire family and place Darius on the throne.
There's movement behind him. Darius turns to see four of the king's bodyguards with spears outstretched emerge on the tower.
Without turning, the king begins to speak. "Do you know why, Darius, I like this vantage point?"
Darius doesn't reply immediately. There is something odd in the king's tone that begs caution. He glances at the sprawling complex below—the gardens and courtyards alight with bonfires and celebrations, the barracks and racetracks, the stables for a thousand fine horses.
"To view your power, sire," he replies smoothly.
"Ah," Artaxerxes says, "a good reply, but not the right one." He finally turns again, and Darius drops into a deep bow. While his eyes are down, he hears a familiar sound: the scrape of a sword unsheathing. Instantly, Darius prostrates himself on the ground, hating the nearness of the dirty paving stones to his well-oiled beard, worried about the dust grinding into his flawless, finespun black robe. But such thoughts are foolishness. The other boot lands on his neck, followed by cold steel. Being the Great King's grandson will not help him, he knows. This king has even strangled his infant brothers in their cribs.
"I come up here," Artaxerxes continues, "to remember how thin the line is between king and peasant, power and death. Someone has betrayed me, Darius. Someone close in my ranks has been purposefully thwarting my goals for peace."
The sword presses more deeply into Darius's neck.
"Is that someone you?"
Darius inhales sharply, waiting for the fierce bite of the sword into flesh. "No, my king," he says. His voice is thin, yet steady. Deceit is one of his greatest skills.
"Think carefully," the king instructs. "The girl bride I sent to Alexander of Macedon was accompanied by _your_ most trusted escort."
"Yes, Most High," Darius gasps out, the king's boot increasing its pressure on his windpipe.
"And yet...the girl is gone. Your son is gone. And many in my military escort have been mysteriously slaughtered. Can you explain this?"
"Perhaps my son has rescued the princess from some Macedonian plot," he says calmly, as if he were sitting across from the king over hot spiced wine and not prostrating himself with his face pressed painfully into the stone floor.
"Oddly, King Philip and Prince Alexander hadn't arrived yet in Pella when Ochus and the princess disappeared. Perhaps _your son_ stole her away to prevent an alliance with Macedon," the king retorts.
"I don't know what happened, sire," Darius assures him, hoping the king will hear the truth ringing in his voice. "But I do know this." He inhales for the speech he has prepared, but the crushing of his windpipe under the king's boot turns his voice into an unpleasant gasp. "No matter whom I sent to escort the princess, the results would have been the same."
"And why is that?"
"Because the gods ordained it so," Darius says with the last of his breath.
The pressure of the foot lightens on his neck. "Explain," the king commands.
Darius inhales, conscious of the fluttering of his heart, subtle, like a frantic moth. Nothing an observer would notice, of course. "My king, the gods have been against this alliance since the beginning," he says. "First Princess Zofia was kidnapped by slave traders before she could travel to Macedon. Then we sent three other princesses as brides, and they were murdered on the Royal Road. Miraculously, Princess Zofia flew into Persepolis on a Pegasus and, ignoring all the messages the gods tried to give us, we sent her to Macedon, where she has disappeared."
"Are those dire messages from the gods or from men?" the Great King asks.
"The flooding of Byzantium, my king. Is _that_ a message from a man?" Darius replies, his voice rising along with his frustration. "Did a _man_ shake the earth and send waves taller than any building to batter King Philip's wedding?"
The king's boot is no longer at his neck, nor is the tip of any blade. Still, Darius doesn't stir. "Even the Greek gods themselves are attacking Macedon," he continues, more hurriedly than he had planned. "Poseidon, Earth-Shaker, ruined King Philip's wedding and destroyed his newly conquered city. Their gods have made it clear they will not lift a finger to protect Macedon. Now is the time for Persia to attack."
Darius hears the king's heavy jeweled robes flap in the brisk wind and knows he is considering his answers. Then Bagoas speaks in his oddly high, silken voice.
"Lord Darius is right, Most High. Verethragna, god of war and victory, has told our priests the same. We must no longer ignore the signs and omens of the gods. If we do, they will surely punish our stubbornness with earthquakes and giant waves."
Darius waits for what seems like an eternity. Finally, he hears Artaxerxes sheathe his sword.
"Rise."
He pushes himself up on all fours, like a dog, then stands awkwardly and brushes the dust off his cheek and his fine black robe.
The Great King smiles at him. "I choose to believe you...for now. We will hold a council of war."
The king's words wash over him as Artaxerxes ducks to pass through the doorway, followed by the guards. As their footsteps fade down the stairs, Darius's heart continues its shameful battering. His eyes rise to meet those of Bagoas, which now sparkle with sly humor.
A cold understanding passes between them that requires no words. Artaxerxes, though sharp in mind and healthy in body, is in his eighties. His numerous heirs are either at each other's throats or drunkards or cowards or all three.
Who will be the next king of the Persian Empire? Who will be left standing?
Darius, himself, of course.
And when he is, he will surely reward his allies.
But in the meantime, there's a war to be fought.
CHAPTER EIGHT
ALEXANDER
Despite the winter chill, sweat drips into Alexander's eyes as he sidesteps Heph in the training arena outside the Pellan palace. Heph's ax comes slicing out of nowhere, cutting through the cold dawn air, and Alex pivots just in time, hoping his weak left leg won't buckle. He raises his own ax, his muscles singing with the effort. The two handles meet with a bone-shaking _thud_. Alex feints to the right, but Heph doesn't commit. He's springy, light on the balls of his feet, watching Alex coolly, refusing to fall for any of his tricks. Suddenly Heph attacks, beating Alex back with calm determination.
Ever since Heph and Kat woke him from his possession, Alex has felt...altered. Bereft. He no longer sees deep-buried secrets in the eyes of men, no longer feels that heady transport from the current moment into one of the past. And he can't help but guess at the truth: that when Riel died, leaving Alexander's body, he took all of Alexander's Snake Blood Magic with him.
Alex's magic is gone.
And twinning that loss is the disappearance of Kat. They never found her after the waters receded in Byzantium. Instead, Alex received a note from a beggar boy several days after their return to Pella, which said simply, _She is too weak to travel, but trust that she is safe with me. She is healing_.
The note was from Jacob. The Earth Blood. Kat's first love. And it indicated he would pick up messages at a bakery near the Temple of Ares, behind the Byzantine palace.
Heph was worried, but Alex argued that if Jacob wanted to kill her or turn her over to the Aesarian Lords for her Blood Magic, why would he have written the prince to let him know she was safe?
Heph still wants to go back to Byzantium, stake out the bakery, find Kat, and bring her home. He has brought it up nearly every day since the note arrived, and Alex is sure he's still furious at him for insisting that they stay here in Pella, where they are needed most.
But for now, for this brief instant in the training arena, with sand covering his legs and shifting beneath his feet, all of the prince's focus narrows into the blade, into Heph's form as they circle each other. There is nothing but this moment. The dark, undulating memories of the Byzantine flood, which haunt his dreams, and the strange time of possession where he lost himself to the control of his father-god, Riel, have fallen away. The uncertainty about Kat. The loss of his Snake Blood Magic. The great emptiness yawning open inside him is, for these brief moments, disguised, and he feels alive.
Before he ever understood his magic, before he ever met his twin sister, before he woke up back in his own body at the disaster that King Philip's eighth wedding had become—before any of that, there was Heph. His oldest friend. His sparring mate. When Alex saw him floating, facedown, in the black swirling waters of the palace, a raw fear the like of which he had never before known sliced through him. And when the cold, corpse-like figure started to retch and cough, even as the table they floated on was sucked out into the foaming tide, Alex cried tears of joy and gratitude to the gods.
Now Heph's face is like the theater mask of a warrior—cold-eyed, tight-jawed, and absolutely imperturbable. His best friend has truly changed, Alex realizes, with a brief but not unpleasant flutter of surprise. Before, when they trained, Heph would show off, making the dazzling move, crowing with pride before he truly won, and often getting himself in trouble because of it. Now, he's pure concentration, all graceful skill and unyielding strength. He no longer bothers to comb that wild dark hair, and its disarray suits him, makes him seem older and more rugged. Familiar and yet different.
Heph's ax blade strikes Alex's as the clang of iron on iron rings out. Alex feels the strength of the blow shoot up his arm and into his shoulder, giving him a thrill of fear. Heph seizes the advantage, pummeling Alex's ax until it flies out of his hand and lands at an angle in the sand.
"Well done," Alex says, stripped of his weapon, breathless. He claps Heph on the back, his hand meeting the bulge of Heph's shoulder muscles.
Heph doesn't reply, and despite the physical release of sparring, Alex can still feel the tension in Heph's body—the fury, the frustration. And more than that. Heph is heartbroken over Kat.
"Are you thinking of her?"
Alex's question seems to startle Heph, and he nods. At least he hasn't lost the connection with his best friend—that feeling of two souls who are perfectly in tune with each other even without magic.
"If something was gravely wrong with Kat, Jacob would let us know," Alex says for what feels like the hundredth time, but his voice trails off. Despite his lost powers, the prince's bond with Kat is immutable, eternal. She could not leave the earth without him sensing it, without him knowing. "And in the meantime, we have to stay focused."
"Focused. So you've said." Heph sits down on a bench, opens a pot of beeswax, and inserts two fingers. "It's _almost_ like you're testing me," he says and starts spreading the pale yellow lump on the handle of his ax to lubricate the wood—without a good coating of wax, winter would bring out all its splinters.
Alex sits down beside him, resting his arms on his knees. Watching Heph carefully massage the ax handle, he marvels again at how much his friend has changed. Before, he would have defied Alex, would have gone against orders to search the entire city of Byzantium for her, but now he's finally focused on _duty_. And his duty— _their_ duty—is to protect Macedon and the world from its newest horror:
The Spirit Eaters.
Because the creature that attacked the wedding in Byzantium was a Spirit Eater, he's certain.
No, this isn't a test of Heph's loyalty. This is real. Heph said he'd seen something unsettling during the flood—a black spread of wings beneath the water, a shadow that seemed to morph like a cloud of claw and fang in the waves, reaching for Kat. And then there were the bodies...at least three corpses in the ruins of the palace itself that seemed to have been almost entirely devoured, and many others found in town. Based on everything he learned while possessed by Riel, combined with everything Heph learned from Ada during his time on the island, they've spent the last few weeks piecing together some semblance of a theory: the Spirit Eaters are, clearly, spreading their reach far beyond the Eastern Mountains, possibly north into Scythia and east toward Bactria. Some have found their way as far west as Byzantium.
Alex rubs his thumb over his ax handle, picks off a splinter, and flicks it away. "The prophecies warned that the world was poised between ushering in an Age of Men or an Age of Monsters. Riel would have allowed the monsters to rule the earth, and I would rule over the Age of Men. Riel is dead. That can only mean I will vanquish the Spirit Eaters." He feels like a broken marionette at this point, but adds, "What we need is a solid plan."
"What we need," Heph argues, forcing the cork back into the pot of beeswax, "is an _army_. The Spirit Eaters live hundreds of miles deep in Persian territory, and the last I've heard, the Great King's troops are poised to pounce the moment you get off the ship. It's a shame your father's skull is as thick as one of these," he says, handling the blade carefully. "Nothing gets through."
It's true—Philip has so far refused to believe anything they have told him with regard to the greatest threat to their nation. It seems the king prefers to focus on wars he knows he can win—wars with men, not monsters.
Dark smoke curls up from the blacksmith's next to the barracks, and the metallic pounding of hammer on iron rings out in the early-morning light. Even now, Brygos, the palace blacksmith, is making or repairing weapons for Philip's army, the army that has built Macedon from a hardscrabble land of feuding cattle raiders into a power strong enough to challenge Athens, or even Persia.
Alex shakes his head. "His pride won't let him accept the truth." That his own wedding was ill-fated, overtaken by a destructive force beyond the workings of nature. In fact, Philip becomes furious at the mere suggestion that people believe the disaster was a bad omen for Macedonian rule over Byzantium. His repeated defense is that the earthquakes and waves were caused by Lord Poseidon drinking too much in celebration of the wedding. If the gods were angry at him, he points out, why did he and Cleo emerge unscathed from their upstairs palace rooms while the waters killed and injured hundreds of others?
Heph smiles. "Maybe the goal shouldn't be to get him to believe us, in that case."
Before Alex can question his line of thinking, Heph looks at something behind him and closes his mouth.
Alex turns in time to see Straton, his father's elderly manservant, approaching, his dazzling white hair like snow in a sunbeam.
"My lord," says the man with a bow. "Your father would see you."
"This early?" Alex darts a glance at Heph. Ever since his wedding with the luscious young Cleo, the king has taken to rising quite late.
Straton nods, his coal-black eyes twinkling. "He says it's to do, my lord, with the disappearance of your bride."
_Zofia._ The woman promised to Alexander—the one who showed up in Pella while Alex was away, but disappeared before his return, almost as though she could sense from afar his disinterest in ever meeting her. Perhaps she had as great a reluctance to meet him.
Well, if it gets Philip to talk about Persia with him, it will be a fruitful morning. "We are on our way," Alex says, and Straton darts off quickly for a man of his age, dust and sand rising at his heels.
* * *
A chilly breeze swirls through the slatted shutters in the king's office, waving the bloodstained battle flags hanging from the walls. In between the banners are spears, shields, and swords, trophies of Philip's many victorious battles. Alex has always loved the king's office better than any other room in the palace. It is a man's room, a soldier's room.
Heph stands respectfully by the door as Alex walks up to the king's desk, careful to hide his limp. Philip drums his large fingers on the highly polished ebony surface, which smells of lemon and beeswax. Then he looks up toward the door and makes a dismissive gesture. "You may leave, Coxcomb," he says, using the nickname he gave Heph years ago for being vain about his appearance. "I want a private discussion with my son."
Without hesitation, Heph slams his right fist on his chest and walks out in long, smooth strides.
As Philip rubs his bushy red-brown beard, pondering, Alex studies the man he had always believed to be his father. He had, in the way of boys with their fathers, loved him, feared him, wanted to be just like him, wanted to be nothing like him, and wanted to be much better than him.
Now he is no longer a boy—and he knows the truth.
Still, some of Alex's earliest memories are of the king carrying him proudly in his arms at palace feasts, showing his beautiful son off to visiting ambassadors. Alex will never forget his excitement on his sixth birthday, when Philip gave him a hunting pup, the pick of the litter. When Alex was thirteen, the king took him into a skirmish—his first battle, really—against mountain cattle raiders, staying at his side the whole time to prevent him from being wounded.
There were also not-so-good memories. The time a ten-year-old Alex drank the king's best Chian wine, stumbled into the throne room when Philip was giving an audience to Persian emissaries, and promptly threw up in his father's lap. Philip took off his belt, pulled up his son's tunic, and whipped his bare bottom in front of everyone. Alex's humiliation hurt far more than the physical pain.
What would the king do now if he knew the truth?
"You trust him?" Philip asks, after the door has clicked shut behind Heph.
"With my life," Alex replies firmly.
The king grunts again, picks up an oenochoe, and sloshes wine into his cup. "You trusted that traitor Kadmus, too. By his own account, he was spying for Persia while serving as a general in my army."
"Kadmus only did it because Persia threatened to kill his family members living in the empire. He confessed everything to me, begging me to execute him as his just punishment." He clears his throat, remembering how Kadmus had confessed far more than regret or loyalty. "But I convinced him to make amends to Macedon by continuing to work as Persia's spy and reporting back to us. While you were in Byzantium, he was invaluable. But what does this have to do with my fiancée and her disappearing act?"
King Philip is silent. He plucks an ivory toothpick from his desk and digs in between his big yellow teeth. "The princess of Sardis was a pawn in a much larger game, it seems. My intelligence has it she was meant to be killed upon arrival, her murder blamed on us as a means to incite war."
This is a twist. "And presumably Artaxerxes is behind this plot?" Maybe convincing his father to take on Persia will not require as much work as Alex thought. And once in Persia, he will find a way to seek out the Spirit Eaters.
The king grunts and shrugs. "Perhaps Artaxerxes is playing a double game with us," Philip says, his words muffled by the toothpick, "or, he truly wants an alliance, but other forces within his great nation do not."
"So, the princess has escaped her fate," Alex says, thinking how glad he was to find he had escaped his, too. As with most royal marriages, neither the bride nor the groom in this one was eager for the wedding. When he first arrived in Pella from Byzantium and heard of her disappearance, he assumed she had run away in the chaotic aftermath of the guards' murders, grasping desperately for the freedom no princesses ever had. But now, knowing she is prey, the thought of the young foreign girl, alone, afraid, and hunted, touches something deep inside him. "Perhaps it is not too late to send investigators to detect her whereabouts," he adds. "She is in a highly vulnerable state, if what you say is true—and she could be an asset to us either way."
The king scoffs. "I'm not worried about the princess."
Philip's heartlessness irritates Alex. Kings must be brutal at times, he knows, but it would cost Philip nothing to try to find the forsaken girl. "If they find her, her own men will kill her," he protests.
"Her life doesn't matter to me. Listen, Alexander. There are many things for people to fear. But do you know one of the most feared? Us. Macedon. The Greek nations tremble in terror of us. They fear us, and they covet what we have. We cannot rush east, giving our neighbors ample opportunity to invade an undefended Macedon. We conquer the rest of the Greeks first— _then_ we take on Persia."
Alex is about to argue back that to focus on the Greek nations could be a fatal distraction from the real threat to Macedon—Persia and the Spirit Eaters—but stops himself. He's right, Alex suddenly realizes. Of course, he is. How could he have missed it? A heavy weight presses on his chest when he realizes that, for the first time in a long time, he feels like the youngest cadet. There is no immediate gratification in war—only a long and steady journey of building up one's forces, gathering strength, and perfecting strategy. You don't get what you want overnight. Alex should know that by now.
"I will march with you, Father," he says, his heart speeding up and mind racing even faster. Once in Persia, he will find a way to make Philip see the truth of the Spirit Eaters, the magnitude of the danger.
As soon as they are actually equipped to handle it.
Philip tilts his head and studies Alex. "And your regency?"
"You can select a regency council to govern while we go to war," Alex says, thinking swiftly. "I need not be a part of it."
Philip strokes his beard thoughtfully as his single dark eye bores into his son. "Based on your recent victories against the Aesarian Lords," he says slowly, "I believe that you would be of more service to your country on the front lines. Alexander, I would be pleased to have you join me."
Alex is shocked at the thrill that moves through him. After all this time, his father's approval still matters. More than that: this is the greatest compliment Philip has ever— _could_ ever—give him. Alex has inured himself to the king's gruff insults and harsh criticisms, but now Philip is willingly calling him a great warrior. It's a strange feeling.
"Thank you, sir," he manages to say with an air of confidence. "When do we depart?"
"We'll use the winter to prepare the army," he says. "And leave for Athens around the time of the spring equinox."
"Athens?"
"Yes," the king replies. "Only once Athens is crushed beneath our heel will we be able to defend ourselves against Persia. Demosthenes continues to stir the Athenians against me, and he will grow more dangerous if left unchecked."
The Athenian statesman, Demosthenes, Alex knows, is the greatest orator in the known world, whose powerful speeches could almost rouse the dead to enthusiastic applause. Unfortunately, he has devoted much of his talent into whipping the Athenians into an alliance against Macedon.
Alex nods and turns to leave.
"Wait."
Alex turns back. Philip steeples his large fingers, frowning, and says, "There are two other things I wish to discuss with you. They are...delicate matters."
Alex's heart starts to flutter, but he says nothing, nodding respectfully.
"The first one is this. It seems Cleo is, well, in a _condition_."
"Condition?" This is not the direction Alex was anticipating.
The king waves a dismissive hand. "Late in her bleeding. With child, they think. It is hardly confirmed, but I know she will go running that mouth of hers, and by dinner, the whole court will have heard. I wanted to be the one to tell you."
Alex feels cold as ice, unable to move, unable to breathe. Once, on a winter hunting trip with Heph, a snowstorm hit. Crossing a frozen stream, Alex tumbled in and got soaked, from his boots to his fur hat, quickly becoming too numb to feel anything besides panic. He is like that now, but worse. Philip knows the truth, and he will disinherit him for this child if it is a boy.
Still frozen with dread, Alex nods. "Excellent news, Father. Let us hope Cleo is indeed with child and carries a boy."
The king seems satisfied with the response. "It is a good thing for a king to have more than one son," he says. "And poor Arridheus, even if he turns up, could never be king."
A pang grips Alex's heart. Last summer, his twelve-year-old half brother, Philip's son from a servant girl, was kidnapped by unknown enemies and never found. Arri has the mind of a much younger child, and often Alex pictures him afraid, tongue-tied, abused, without the ability to escape or find help.
The king's gaze wanders over unrolled maps on his desk. "Second..."
So, there's more.
"This Katerina. Ever since the girl appeared at the Battle of Pellan Fields last summer, rumors have abounded. I hadn't planned on even dignifying them by discussing them, but still they continue." He fixes his eye on Alexander. "Who is she?"
Alex exhales, choosing his words carefully. "She is my sister, Father. My twin." He doesn't say, _Your daughter_.
Philip snorts. "How can you be so sure? As you know, it is treason for an imposter to claim royalty. It could be punished with death."
"Which is why someone who does so must have very good reason." Alex inhales deeply and tries to calm the hammering of his heart. "Look, Father," he continues, "when you were in Byzantium, I spent a great deal of time with Katerina. It was as if I found the missing part of my soul. We are so much alike. And she wants nothing from us. She even risked her life to go on a mission to Egypt while I was regent."
Philip shakes his head. "I just can't fathom why Olympias would have thrown this child away at birth and never told me. It's hard to believe that even she—a selfish witch with the morals of a Persian cutpurse—would do such a thing."
"You know how superstitious Mother is," Alex replies, ignoring the slur on Olympias—because of its truth, among other reasons. "Leonidas showed her a prophecy of ill omen."
Philip nods. "By all the gods, I most certainly do. I believe you, Alexander. But I don't want this ridiculous prophecy to get out."
"It's hard to stop rumors," Alexander says, thinking of the many that surround him. That he was born of a god. That he could speak with his horse. That he could practically read men's souls. But, of course, these are all true. _Were_ true.
"Then we must attempt to quell just one—that your mother had Katerina by another man. It makes me look the fool, her the whore, and you, quite possibly, illegitimate."
He tilts his head, a smile playing on his lips. Alex feels color flood his face.
Philip _knows_.
The king pushes back his chair with a scrape and, to Alex's surprise, puts a muscular arm around his shoulder and whispers, "And we can't have that now, can we?"
Is Philip saying he knows but doesn't care? That he will keep Alexander as his heir? And if so, is it pride that will prevent him from disowning his popular warrior son? Or something else?
The king chuckles and goes to the window overlooking the palace's entrance courtyard. "We will say this girl is my daughter by another woman," he says. "Bastards only burnish a man's reputation for virility, after all. And it's not too late for her to prove useful to me—she is the perfect age for marriage."
Alex's heart sinks. "Yes, Father."
"Do you know where she is?" he asks, the eye boring into him.
"No, Father," Alex replies truthfully. "I last saw her in Byzantium, the day of your wedding, and heard that she survived the flood."
"Send a search party for her," Philip commands, turning toward him. "Let me know when you have found her. I will send an honor guard to fetch her here. Perhaps we can marry her into one of the top families in Athens. Or... No, come to think of it, I have a better idea."
The king does not continue, and Alex has to bite his tongue to stop himself from asking what plan he has in mind for Kat. Alex feels as though he's used up his allocation of luck for the day.
"May I leave?" Alex asks.
The king flicks his fingers in a gesture of dismissal.
Alex bows, then hurries from the office. He's resolved to send agents to learn Kat's whereabouts as soon as possible. But not to bring her home. To warn her to stay far away from Macedon if she values her freedom.
CHAPTER NINE
AESOP
It is the depth of night. Aesop lies still in the slave quarters so as not to wake his master, Otus. Otus is chained to him, wrist to wrist, and currently snoring like a pig in Aesop's ear.
With his unchained arm, the boy reaches down and feels for the knife hidden beneath his tunic, his movements subtle and quiet as the night.
Aesop's tongue was cut out when he was five. Now he is eleven, which means he has lived more than half his life in silence, and all his life a slave.
But no more.
* * *
There are certain advantages to being mute. People say and do things right in front of you as if you were their dog—things they would never do in front of someone who could speak. They mistake a lack of response for a lack of understanding.
But Aesop understands far more than anyone knows.
He knows about Queen Cynane and her invisible Smoke Man. About the Blood of True Betrayal and the magic powers it brings. Hiding in the courtyard, he saw Queen Cynane kill the Smoke Man, and the blood of his heart burst over her.
He saw the smoke and fire enter her body, giving her the magic.
And he believes that Papari, now Lord Timaeus, killed the mad king to get the same magic. He and Otus were walking by when he saw Papari enter the king's rooms with hot water and a sharp razor shortly before he raised the alarm that the king was murdered.
Though he blamed it on Queen Cynane, Papari must have killed the mad king, who'd cut out Aesop's tongue.
Is the enemy of your enemy your friend?
Cold wind howls down the smoke hole of the kitchens, where the slaves sleep. Several mutter and groan, clutching thin blankets. If Otus makes a loud sound before Aesop is finished, he will wake the others and Aesop will lose his chance. He must be swift, then.
Sitting up, he is careful not to move his left hand, which is chained to Otus's right one. Slowly, he pulls the vial Lord Timaeus handed him in secret earlier from beneath his belt and removes the cork with his teeth. Then he drinks it and almost retches. It is as bitter as gall. He feels as if he has inside him all the foul dead things of the world. It sears like flames down his throat and spreads across his chest. Once the urge to vomit has passed, Aesop braces himself. Though all the torches are out, and the shutters closed, he can see well enough to do the deed. A sliver of moonlight floats through the smoke hole above the hearth, and some coals still glow orange.
He positions the knife right above Otus's heart and then, using the full force of his weight, he thrusts.
The blade pierces through skin and breastbone and muscle into soft, palpating flesh.
Otus's eyes open wide and he gasps and gurgles, but not louder than someone murmuring in his sleep. The eyes stay open and gleam white as hard-boiled eggs as air rattles out of Otus's throat. Blood blooms on his tunic. Dark, red, and glistening.
The blood of betrayal. Blood spilled by a slave who has betrayed his master.
Fiery pain explodes inside Aesop, as if someone has thrown him onto a bonfire. Flames and smoke, ashes and soot coat his throat, choke his lungs, eat his internal organs, burn his skin. He cups both hands over his mouth to avoid crying out as he writhes on his sleeping mat.
And then, just as swiftly, the pain flickers out. Aesop, too exhausted to stir, curls up on his side and gasps for air. He faces Otus's corpse, ghastly white in the moonlight, the blood on his tunic almost black.
Suddenly, Aesop feels as if a strong wind carrying vibrant sparks is hurling itself through his veins. Power rises inside him, vanquishing weakness and fatigue.
Slowly, he sits up and sets to work on his slave master's hand, sawing through bone. Finally, he picks up the bloody hand and sets it on Otus's stomach.
It is over.
He is free.
Just as Lord Timaeus promised.
ACT TWO
ENEMIES & ALLIES
You will never do anything in this world without courage.
It is the greatest quality of mind next to honor.
—Aristotle
CHAPTER TEN
KATERINA
_Shadows, wide and black winged and mas_ _sive_ _as clouds, engulf her._
_The water pours over and around her, drowning out the pain. Her screams are swallowed in the waves._
* * *
_Something wants her. Something has its claws in her, tearing through skin, through self, through soul. She feels torn and limp._
_The world goes black._
* * *
_Waters full of horror surround her, suck her down into a whirlpool of sadness. Trapped in the dark, airless bottom, she feels all the death and loss that anyone on earth has ever suffered. She is no longer Kat. She is Despair._
* * *
_Through the blackness comes a flickering lamp. A cackling laugh. A wrinkled face that becomes three faces._
_An old woman. Or three old women, each rippling into the others._
_Each unspooling a ball of yarn through the darkness._
_Only it isn't yarn—it shines as silver and glows as the sharp knitting needles that glint and clack, glint and clack. With each tiny move of a needle, Kat feels a stabbing, feels pulled, twisted, entrapped in an invisible web._
* * *
_Now she is in Ada's abandoned palace in Caria—covered floor to ceiling in an intricate snare of spiderwebs. Her magic tugs at her from within as if bursting to escape. Her soul has scattered into the corners of the darkness, tiny spiders racing off in every direction, until she is nothing._
_She is nothing._
_She is nothing._
* * *
She wakes screaming.
* * *
Jacob appears in the darkness, real, though wavering, as if he has emerged from the sound of her voice. How long had she been screaming before he came? She reaches out weakly to touch his face. She must feel the warmth of him. She must assure herself that this is not another dream.
By some miracle, it is him. And he is here. They are together again. It must be a dream. He had gone forever, become an Aesarian Lord, Macedon's enemy.
His family burned. Because of her.
He hates her. Now he is her enemy, too.
No. He saved her life. Didn't he?
On the battlefield, he kissed her. She nearly died then, too, but something changed.
His lips had been warm and gentle against her own. A pulse, a wave of heat moving through her, connecting them... Bringing her back to life. She remembers.
Kat tosses in her sweat-laden sheets.
Now his strong hand is on her skin, brushing stray hairs, damp with sweat, from her face.
_Yes._ She takes a deep, calming breath at his touch. Yes, he is real. He will help pull her from the pain of despair, of death and loss.
He stays by her side until she falls back to sleep.
* * *
In her weeks-long delirium, images and conversations, real and imagined, swirl around her:
Heph on the island of Meninx, confiding his deepest secrets.
Riel offering her the entire world if she would only betray Alexander. His hands around her neck, squeezing the life out of her.
The needle on the mechanism pointing right at...Jacob. The boy she had grown up with. The boy she had hunted with. The boy whom she had milked the goats and swept the kiln with.
He'd been there all along. An Earth Blood.
* * *
It has been a month since the flood, and still Kat's body is weak, her mind stuck and swimming in the darkness of her thoughts and memories.
The only thing anchoring her is this knowledge: Jacob is here. He forgives her. He is taking care of her. Through his touch, he has been healing the deep lacerations all over her body—her shoulder, her hip, both legs. She doesn't remember his healing her, exactly, only spasms of pain that are there and then gone. A warmth like a radiating light moving through her limbs at his touch.
Even as her physical body heals, the inner pain continues, a pain much worse than the cruel claws that raked her skin in the freezing maelstrom of the Byzantine flood. It's as if her spirit itself has been mauled and torn, pulled and sucked almost completely out of her. There is a sadness in her now, so strong and deep that she thinks it must be welded to her soul and she will never lose it.
She is grateful to Jacob, and yet...when she wakes to find him gone, out to get supplies or whatever he has been doing beyond the closed door of her little room, she wishes he'd just left her to die. Death would be better than this ache, this yawning, yearning emptiness at the heart of her.
She is afraid she will never be herself again.
She has been touched by something evil.
* * *
She wakes again, her mind clearer than it has been in a long time. Her hand goes automatically to her neck, searching for the silver lotus blossom pendant that had belonged to the woman who raised her, Helen. She feels its cool, smooth surface and smiles. She didn't lose it in the flood, then. She props herself up and glances around the little attic room. Today her forehead is cool.
Jacob is standing by the window, looking out on a cold gray day.
All she can do, lying weakly on the little bed, is drink him in. His height, the width of his shoulders, the strength of his legs. The broad, friendly face that seems handsomer now, chiseled by loss and longing.
This is—and has always been—her Jacob.
She has loved him since childhood, of course, for his kindness and sturdiness and laughter. But she overlooked him, too. She didn't really want to marry her childhood friend from the farm. She wanted someone completely new and exciting. Someone like Heph.
But now...
"Jacob," she says softly, barely louder than a whisper.
And when he turns, a smile illuminates his face so brightly that she thinks she can feel its heat from across the room.
He comes to her bed and sits, placing a hand on her forehead. She shivers slightly at his touch.
"The fever has broken," he says. He looks so happy that something inside her shatters all over again. There has been so much time lost. There's so much they don't know about each other.
She misses Heph—and Alexander. Her brother, her _twin_.
But at this moment, there is a comfort in being here—wherever they are—with Jacob. Just Jacob. The rightness of it being him. The power he exudes, partly the result of his Earth Blood blooming, partly because he has grown from a boy to a man. A stranger, handsome and hard and distant—but still her Jacob, the one person she knows the heart of better than anyone else in the world.
* * *
Over the next days and nights, they reacquaint themselves with each other. She learns that, in the aftermath of the flood, after they washed up on a nearby beach, Jacob hastily found a dry room for them on the top floor of this narrow, abandoned town house within walking distance of the Byzantium harbor.
He manages, somehow, to find that other attic room where she and Heph hastily left their belongings before they set out for the royal wedding, where the waves parted them. And he brings back her cherished sword and her bags, one of them containing some two dozen curious blue-glazed figurines shaped like little mummies—ushabtis. Kat saw the brawny warriors, all sharing the soul of the great Egyptian general Wazba, transform into the statuettes, and scooped them into her bag as Princess Laila's palace caught fire.
Gradually, Kat explains everything to Jacob—her trip down the Nile with Heph and the mechanism she and Heph created with the help of Aristotle and Ada of Caria. She tells Jacob how she discovered her true parentage—how Prince Alexander is her twin brother. And—though she pauses briefly, wondering if _Lord_ Jacob should know—she tells him about her Snake Blood abilities. That Riel, the Last God, was, in fact, her true father, and with Jacob's blood on the hairpin, she killed him, freeing Alexander from his evil possession.
In turn, Jacob explains how his blood came to be on the hairpin, that after the queen told him she had murdered his family, he grabbed her wrist and she sliced his arm with it. Then he killed her.
"I'm not so sure she is dead," Kat says now, watching the oil lamp next to her bed sputter, threatening to plunge them into darkness. "When I arrived in the room, there was no sign of her body." The details are coming back to her more clearly now: the hairpin, the streaks of blood on the mosaic floor.
"Then," Jacob says grimly, tipping a bit more oil into the lamp, "either she dragged herself out of the room or someone else disposed of the body."
He sets the oil jug down with a hard click, squeezes his eyes closed, and hangs his head so low that his chin almost touches his chest.
When she found out last summer, Kat, too, grieved deeply for the family who had raised her. And she will always mourn. But Jacob's grief is fresher, sharper, like a dark living thing gnawing his insides.
He stares into the glowing lamp flame. "Dragged herself out of the room," he repeats, shaking his head. "She's one person I wish I _had_ killed. Instead, I killed hundreds of innocent people."
He stands and starts pacing the little room. "My rage unleashed hell's deep. I am the cause of so much death and destruction, not just here, but in Delphi, too. Me." She can see how his entire body seems weighted with guilt.
"Jacob, it's not your fault—"
"It _is_ my fault," he says, approaching her. "Thousands here in Byzantium were injured. I couldn't control it. But I can try to make up for it. Because it's my fault."
_No._
It isn't his fault that he was born with this magic in his blood and that he was never taught how to use it. Just like it isn't her fault that her mother was the cruel Queen Olympias and her father the evil Last God, Riel.
Before he can resume his restless pacing, she takes his hand—so warm and strong—and curls her fingers around it.
"Jacob," she says. Because there is one thing she remembers more clearly than anything else—the reason she and Heph were searching for an Earth Blood in the first place. "Your powers _can_ be used for good. We must destroy the Spirit Eaters, or they will devour the world. You can help us defeat them."
Jacob looks uncertain, but he allows her to pull him down onto the edge of her bed again. "Perhaps," he says, though he seems doubtful. "But in the meantime, I have been doing all I can to make amends."
He tells her about the past few weeks of agony as he tended to her, waiting for her to get better. How he left every day while she slept to ask those he met in the streets if they knew of injured people who required a skilled physician. Quietly, he visited them, healing their wounds. Slowly, he has been undoing part of the damage he caused, one survivor at a time.
"I want to help you," Kat says, sitting up in bed.
"Not yet, Kat. You are still so weak."
* * *
But soon, with his help, she grows stronger, begins to stay awake longer. Even sitting at the tiny table to eat is exhausting at first, but she forces herself every day to do more than the day before. Slowly, she begins to walk again, practicing in the small room, one hand against the wall. And just when she thinks she will die of frustrated boredom, Jacob deems her well enough to leave the house.
Kat bites her lip, fighting back a cry of pain as she pushes her way through the bedroom door and down a narrow hall. It has been six weeks, she realizes, since she's been outside, and her body protests loudly as she walks down the three flights of stairs. She sees waterlogged plaster and smells pungent mold.
"Are you all right, Kat?" Jacob asks, turning back to check on her.
"Yes," she says, forcing a smile. "At least, I will be soon."
He forces open the swollen front door to the street. As Kat carefully emerges into an icy wind, she gasps at the devastation all around her. Doors and shutters have been torn off the lower floors of the mud-caked buildings. A fishing boat protrudes from a second-floor window across the street, and a building on the corner has collapsed entirely. Part of her wants to return to their cozy nest on the top floor, the small attic room easily warmed by a brazier.
But she's not sure she has the energy.
"It was much worse right after," Jacob says, gesturing. "The streets were filled with boats, furniture, dead horses. I even saw a dolphin. Many people left the city, afraid of more floods, but they have started to trickle back now, ready to clear out the rubble and rebuild."
Kat sees a bonfire at the far end of the street. A sickly sweet, acrid smell fills her nostrils and she rubs her nose. "What is that?" she asks, pointing.
"A pyre." Jacob's mouth twists. "Whenever they clear out rubble, they find bodies..."
Kat shudders. Though the air is cold and the wind off the harbor brisk, she is assailed by the smells of mildew and rot and the lingering scent of burning human flesh. _Pain here._ So much loss and death and grief she is almost overwhelmed. She wants to sink to the ground and cry. Instead, she squares her shoulders and forces the despair back down.
Looking around, she sees that most of the properties are abandoned, but there are also signs of returning life. The city reverberates with the sounds of hammering and sawing, and the creaking of carts loaded with rubbish on the way out of town, or wooden planks and tools on the way in. A few bakeries are open, and a tavern, and Jacob points out a shop selling warm cloaks where he bought Kat's thick russet one, probably looted from an abandoned home, he said.
She nods, unable to reply as the exercise has made it difficult to breathe. Holding on to Jacob's arm, she feels slightly dizzy. He stops abruptly.
"This is too much for you your first time out," Jacob says. "Let's go back. There's time."
"No," Kat says forcefully. "There _isn't_ time. We've lost so much already, it's unthinkable. I need to rebuild my strength. We have to be ready." She doesn't have to finish her sentence for him to understand what she means: they must be ready to fight.
When the monsters come.
He hesitates, then nods. Slowly, they make their way to the harbor. The gorgeous palace of pink, green, and white marble, of turrets and balconies, was heavily damaged in the flood. The banquet hall and one wing have completely vanished, and workmen crawl over scaffolding surrounding the remainder. All the long piers that jutted far into the sea are gone. Ships have anchored farther out, and Kat spies several rowboats, their white oars slicing through the choppy gray sea like the powerful wings of swans, ferrying people and goods back and forth.
Next to the palace complex, an entire block of houses and shops is gone. Men are still dragging away waterlogged wood and plaster and hoisting it into carts. But across the street, some houses still stand.
They visit several families, each one worse off than the one before. Each time, Jacob does his best to channel his magic to help the wounded. Even watching, Kat is awed by the raw power he possesses, the gentleness with which he wields it, clearly held back by something she recognizes: fear.
There is nothing worse than fearing what you, yourself, are capable of. Kat knows that fear all too well. The fear of a darkness bubbling in your veins, capable both of greatness and of terrible destruction.
As they leave the home of a young mother with a broken leg that healed badly, Kat knows she cannot enter another sickroom. It is as if she has absorbed all the pain and worry in each house they visited and none of the joy when Jacob left the injured person healed. Now, suddenly, the sheer weight of it hits her like a massive wave. She has been lost in her own sickness long enough, and she feels dizzy, overwhelmed, pushed to the limit. She can no longer bear the stuffy sickroom smell—overflowing chamber pots, sweat, and fear—along with the mold and mildew the waters left in their wake. The cold harbor breeze is what she needs, and the winter sun on her face. Perhaps the coldness and brightness will help clear away the dark thing growing inside her.
"I'll let you do the next one by yourself," she says. "I need some fresh air."
A look of concern crosses his face. "Are you all right?" he asks.
She smiles. "I'm fine."
Jacob squeezes her hand. "Once I'm done here, I will see if there are any messages at the baker's."
He enters a neighboring house, and Kat walks to the water. She takes off her boots and enjoys the feel of sand between her toes, even if that sand is cold. Then she sits and watches a rowboat ply its way toward shore from a merchant vessel some way out.
Despite the nightmares, the dark, writhing memories of near drowning, and the unthinkable _thing_ that came for her in the water—a Spirit Eater, she now knows—there is still something about the sea that draws her, calms her. It is as though a voice amid the waves is singing to her. For a moment, she could swear it was her mother, Helen, who died when she was only six. For a moment, she is back in time, a child in Helen's arms. They are safe, and she has not yet learned what or who she is.
The wind embraces her.
The voices ebb and rise within and around her, braiding together and then dissipating. They are the voices of infinite creatures in the sea, she realizes.
It is an amazing comfort how, after all this time, her magic comes back to her, easy and flowing, like a near-forgotten lullaby. She would cry with the relief of it, the blessed release, except that she is too tired, too lulled by its heavy calm.
Soon, the sounds of hammering, sawing, and men's voices fade entirely. Her breathing slows, and her mind drifts into the water, into darting fish and scuttling crabs and shellfish that drag themselves over sand with one muscular foot. She sinks into floating, pulsating creatures too tiny to see, her mind drifting, seeking that distant song. She has done this often with water—with the Aegean and the Nile and the North African sea off Meninx. Once, in Meninx, she thought she saw an entire civilization below the waves, a many-towered city with handsome people walking through the watery depths, their long hair and robes floating behind them. Atlantis.
But never before has she felt such a fresh, living wound, such lingering trauma. The creatures here, below this churning silver, were shaken first in a violent underwater storm, then pulled out, far out to sea, thousands of them left gasping for breath on harbor sand. Then they were cast into the air, somersaulting in the roaring power of the waves, and slammed into the city. Those that survived felt the horror and pain of the dying. Now, even six weeks later, having no sense of time, they tremble with dread. This unquiet sea is filled with fear.
Something else floats toward her, a bloom of tiny creatures almost lost in the tug and pull of the waves. The feel of them is foreign, alien, completely different from any creature she has ever inhabited. Curious, Kat slips inside the collective consciousness of the hundreds of watery forms. They are small; each could fit easily in the palm of a young child's hand. They are mostly transparent, with bright red centers and dozens of long, slender tentacles.
Jellyfish, but not any jellyfish. They are both young and old. Sinking deeper into their being, Kat feels herself age and renew, age and renew, again and again. The jellyfish have, by their nature, a kind of immortality, though something natural, crafted by the earth herself without magic. The parts of the jellyfish refresh themselves with new vitality, unlike the physical bodies of all other living things, which age, sicken, and die. Suddenly, Kat realizes the creatures are hundreds, possibly thousands, of years old. She has never before traveled into something so ancient, and the weight of their age nearly crushes her soul.
Floating on warm summer waters, she sees hundreds of black-sailed ships, bristling with men, sunlight glinting off their armor. But the ships and armor are nothing like anything she has ever seen in her travels. They aren't Macedonian, Persian, Egyptian, or Carthaginian.
Is this a recent memory of a different place, far away? Or is it from nearby, but long ago? The sun and moon rise and sink in the sky. Torches and campfires flicker on a beach. Men fight on a wide plain. Chariots clash together as horses and soldiers shriek in pain. Soldiers on high walls fire arrows. A city burns and collapses in upon itself, becoming its own tomb. Then, for what seems like an eternity, all is silence as the jellyfish glide past vacant beaches.
She feels a tap on her back and looks around. It's unsettling at first, to be in air, not water, and to see a human face peering at her. She blinks in confusion. It's Jacob.
"Found you," he says, his voice cracking as he seats himself heavily beside her on the cool sand. He looks drained, as if he has used every ounce of his Earth Blood Magic to heal the injured today. His powers are not inexhaustible, she knows, and it will take him some time to regain his strength. There will be no healing of Byzantium's injured tomorrow.
He reaches into his pack, pulls out three fluffy rolls, and hands her one, along with the cloth the baker wrapped it in. It is still warm. She puts a hand on his cheek, and he kisses it, his face brightening. "I've received another message from Timaeus," he says, pulling a small scroll from the pouch on his belt. "It seems that Spirit Eaters have spread as far west as Dardania. Farmers have been plucked out of fields, fishermen off boats. Archers now stand guard in fields and on boats, ready to shoot the creatures."
Kat shudders and holds the bundle of bread closer to her for warmth, though she is no longer hungry.
"My Aesarian regiment is leaving Dardania," he says. "For Troy. Timaeus has urged me to meet them there."
_Troy._ Kat has an icy feeling on the back of her neck. The ancient ships. The battles on the plain. The burning city. What she saw in the memory of the immortal jellyfish was the fall of Troy. It must be.
And it cannot be a coincidence that she has just had a vision of it, and now some new evil breeds in the ruins of Troy. She can feel the darkness in her bones. That is why the Lords have gone there.
"Why Troy?" she asks. "What's in Troy?"
Jacob shrugs. "I don't know. Perhaps it's just a meeting place where our regiment will join up with other Aesarian armies. He says there are orders from Nekrana for all regiments to combat the Spirit Eaters in any way they can."
"Do you trust Tim?" she asks quietly.
Jacob is silent for a moment. "I'm not entirely sure, actually. I suppose that's why I had him sent to Dardania. We were both always ambitious to rise above our humble origins, but...he changed. Became a bit too determined, somehow. And he—he knows. What I am. What I have." He looks at his hands, and Kat knows he is talking about his Earth Blood.
She looks out to sea again, wishing she could locate the jellyfish to learn more, but they have slipped beyond her reach.
Even still, she recalls that strange feeling of being drawn toward the water, toward distant shores. She cannot understand why, but there's something in this—something she must follow and uncover.
Besides, Kat knows she cannot return to Macedon. At least, not yet. The queen, if she is still alive, wants Kat dead. And Alexander sent her a message that King Philip decided to acknowledge her as his daughter by a servant woman and plans to use her in a marriage alliance with a rival nation. If she valued her freedom as much as Alex thought she did, she would keep clear of Macedon until he told her it was safe.
If not Macedon, where should she go?
She sinks back into her memory of the jellyfish, into the darkness of the ruins of Troy—now she's _sure_ that is what it was. A strange and beautiful thought comes to her: perhaps she and Jacob, with their powerful Blood Magic, are the ones fated to bring light to that darkness. And perhaps, in so doing, she will bring light to the darkest part of herself.
"I think we should go there," she says carefully. "But in secret." Regardless of Jacob's feelings, she knows her own: she will never trust any Aesarian Lord but him.
She wonders if Jacob will argue with her, but he says, "Absolutely. I don't want them getting anywhere near you with their Hemlock Torch." That's right—the tool they developed to detect the presence of Blood Magic. A far more rudimentary form of the mechanism she and Heph built on Meninx. "You and I are both in danger, as far as the Lords are concerned," he goes on, "but they know me as one of their own, and I can handle myself around them. If we go, stealth must be the order of the day until I can find a way to speak to Timaeus alone."
His gaze slides to four men heaving a rowboat onto the sand, splashing through knee-deep water, then moves toward the numerous ships anchored near the harbor. "We will need to find a ship sailing for Mytilene on the isle of Lesbos," he says. "From there it's just across the channel to Troy."
For the second time that day, Kat feels like she could cry—whether from relief or joy or the terrible weight of not knowing what may come, she can't say. But the tears, like the waves, rise only to recede as she reaches out to Jacob—her oldest friend, her greatest love, her only reason, right now, for being alive—threading her hand into his, aware that she is only just getting her strength back and a great journey awaits them yet.
* * *
"I've been thinking," Jacob says late that night, unrolling his sleeping mat in the narrow space next to Kat's bed. They have packed everything else they can and will ship out tomorrow. "It is said that Earth Blood runs in families, though I don't think anyone in my family ever had it."
Staring at the ceiling in the flicker of dim lamplight, Kat mulls this over. "When Helen needed to hide me from the queen, Ada of Caria told her to take me to Erissa. Perhaps she knew of an Earth Blood family there, Jacob. Your family. Maybe they learned to suppress it, knowing that if they didn't, the Aesarian Lords would take them away."
Though she can't see him from her position in the bed, she can hear his breathing, quick and short, as if nervous—or harboring an emotion he is trying, with difficulty, to hold back. "It's true that my father loved working with earth. Channeled all his skill into making household things—pots, bowls, vases, even the occasional little sculpture for me or my brother."
A wave of guilt and sadness engulfs Kat.
"It's my fault," she whispers into the near darkness.
"No," he starts.
"If I hadn't followed you to Pella, the queen would never have found out where I was—"
"Kat, no, you can't blame yourself—"
"—and your family would be safe—" Her voice breaks.
"It's not your fault." Suddenly he is no longer lying on the floor but sitting on the side of the bed, reaching out to her. She sits up and lets him fold her into his arms. "I'm the one who never should have left Erissa," he says. "I'm the one who was bursting with ambition. Father's kiln and the farm were never good enough for me. Now I would give anything to have them back..." His deep voice cracks.
She chokes back the heavy knot in her throat, touching his arm, his face, wanting to make it better, and knowing she can't. "To lose everything, everyone..." she whispers.
"Not my _whole_ family," he says, lifting his head. "I haven't lost _everyone_ , Kat." He looks into her eyes, and heat races through her body.
He puts a finger beneath her chin and lifts her face to his. She stares into his warm brown eyes and feels desire prickle and tease her. She parts her lips, and his mouth meets hers, as though he has been waiting to kiss her ever since that time on the battlefield, when he saved her life...or even before that. Since their first kiss, drenched in the pond outside of Erissa, urgency still coursing through her limbs from the hunt. It's as though every touch to heal her since has been leading to this moment. Her body feels like it is on fire, melting and shivering. His hand traces her back, pulling her to him as his tongue gently meets her lower lip, sliding into her mouth, making her want to pull him even closer. His body is hard against hers, guiding her back down onto the bed, and she clings to him even as she feels herself dissolving into movement and heat, into touch and breath and gasps.
The intensity of it burns through her until her whole body is aflame. But it is not just physical desire that consumes her—it is deeply rooted love, and the need to comfort him and to match his loyalty with her own, her magic with his, to bring everything within her together with everything in him.
And it is _right_. So very right. There are no doubts.
There is no holding back.
Outside their little room, the waters rise and pound the shore as the earth trembles and shooting stars blaze across the sky. Birds tucked into their nests for the night lift their heads and sing for joy, she is sure of it, and in the harbor, fish leap with happiness as the earth—the rocks and waters and wind—mix with the spirits of all living things, and the two halves of the world are one again.
CHAPTER ELEVEN
PYROLITHOS
Moonlight filters into the large cave through the sparkling veil of the waterfall, mingling with the flickering light of torches. Shafts of gold and silver light on Alecta's long red hair as she spins gracefully away from Pyrolithos, just out of reach. For a moment, the king is caught off guard by the sheer beauty of her movement. He lunges toward her and meets her shield; she sweeps her leg in an arc, pivoting and connecting with his flank. Pain radiates through him; Pyrolithos leaps to the side, out of her sword's reach, to regain his breath, watching from the corner of his eye as she lands on two light, deft feet. Her face is expressionless and ready.
He has been secretly training Alecta for weeks, training that Alecta takes back to the brothel and teaches the other women. Because, of course, no one must know that the new king of Dardania has been arming and training a group of prostitutes—the New Amazons, he calls them. At least, no one must know until he's ready to debut them as his new, most-trusted bodyguards. Then, he will lead troops out to capture surrounding territory. All his dreams are coming true.
It is, at best, however, an uneasy alliance. Though Pyrolithos never gave Alecta proof that he was, indeed, the representative of Queen Cynane, the shrewd redhead grudgingly agreed to the training. Prostitutes know better than most women that the world can be cruel, and men can be brutal, and there is no one to look out for them except themselves. Skill at arms could prove useful in any number of ways.
His plan was assisted when the bulk of the Aesarian Lords departed Dardania, headed, apparently, for Troy—called there on some important mission, High Lord Gideon said. He left behind a handful of men to help reorganize the army. Unfortunately, they were never able to ingratiate themselves with Pyrolithos by capturing Queen Cynane; no one has seen her since the night of the mad king's murder.
And none of them _will_ —until the time is right.
Pyrolithos circles to the right as Alecta crouches slightly, slinking like a cat around him. "How are last week's recruits faring?" he asks, as he asks every week, his voice sounding like a stranger's, mingling with the rushing water beyond the cave.
She smiles, and he feels something light up inside him. He has been combing the streets of the capital, the mountain villages, and the fishing boats for strong young women to add to their ranks—those dissatisfied with their lot in life, with abusive fathers and husbands, with little or no freedom, poor wages, and secrets no one must ever know. In the brothel, they can choose either to entertain men or do household work when they are not training.
Alecta lunges at him, and he raises his shield just in time, feeling all the muscles in his abdomen go tense. "As if you care about those women," she says, laughing harshly.
"But I do." He doesn't blame her; if he were her, he wouldn't trust himself, either. In fact, he's not sure he _does_ trust himself—at least not around her.
"What about the one from the mill?" He'd been worried about that one—too soft, too sweet to become a warrior. But he knew firsthand how that softness could streamline into strength with the right training. How the desire to please could be molded, carefully, into the desire to kill.
"The mill?" Their swords clang together, then separate.
"The one who uses lye on her legs." He couldn't get past the scent of it; even after bathing, it lingered. It was particularly strange for a mill worker who spent her days measuring grain into sacks, to bother with such hair removal tactics. But all women, he knows, have their little vanities.
Alecta lowers her sword and frowns. "What do you know about all that?"
Pyrolithos freezes. Men are not supposed to know the secrets of the women's quarters. "I...grew up with many girls," he says stiffly, hoping his voice sounds nonchalant.
"Hmm." Alecta narrows her dark eyes and raises her sword. Shifting light from the waterfall washes over her face, which is all angles, all skepticism. "Erika is doing well. She takes to the training, as if she has been waiting for it all her life. They all do."
"And the Persian with the swelling belly you told me about?" he asks, buffeting Alecta's shield with his own and leaping back. "I don't suppose she can do much for a while."
"She's stronger than she looks," Alecta replies, suddenly a blur of motion as a hard kick connects with Pyrolithos's elbow, sending his shield thudding to the ground. He curses under his breath. He really is a much better warrior than this, but he always loses his concentration when he trains with Alecta.
"It is a grave mistake to think that being capable of carrying a child makes us weak," she says, her voice steady as always, betraying no emotion, even when he wishes she would. "If anything, it makes us stronger. The Persian will do whatever it takes to be one of us."
Good. He knew that what he was doing would gain traction, but he had no idea just how popular it would be, or how quickly. Women from as far away as Athens have been coming to Aphrodite's Grotto for training and protection. It's getting so crowded that Alecta has to train the women in two groups—they can't all fit into the brothel courtyard at once.
"But to tell you the truth, the women have been...asking questions," Alecta continues, feinting to his left. "The attacks continue, and they seem to be heading west. It cannot be long before they reach us here."
He knows which attacks she means. Everyone is talking about it. In Eastern Greece, fishermen have been plucked off ships by flying creatures. Swimmers have been pulled underwater, their bones spit up on the beach at high tide.
"Last week, a woman asking for sanctuary with us told of a village in Paeonia she had come through," Alecta says, thrusting her sword at his arm. He casually knocks it away with his shield. "She was there when a shepherd ran screaming from his fields, saying his entire flock was gone. The villagers returned carrying bones with teeth marks in them."
Pyrolithos studies the outline of her body in the dimness of the cave, catching the stray, fractured moonlight as she moves. He's heard many of the stories from the east by now of creatures called Spirit Eaters, which were thought to have died out thousands of years ago, if they were ever real to begin with. Now people have begun to fear that the Spirit Eaters are not only real but are on the rise, spreading west, and heading straight toward them.
"First Byzantium, then Paeonia," he replies, lunging as she gracefully springs back. He can't deny that they seem to be headed west.
"Some of the Lords are fond of my girls," Alecta says. "As I'm sure you are aware."
He simply nods, deflecting a parry. It's true that before the Aesarian Lords left Dardania, a handful liked to visit the Grotto, much to Pyrolithos's chagrin. He hates the idea of his warriors-in-training entertaining those brutes.
"They've hinted to the girls that they knew something more about the attacks," she says, lowering her sword and stepping back until she's almost entirely hidden in the shadows of a recess in the back wall of the cave, slick with moisture and thick moss. "Filled their heads with stories of creatures from the Eastern Mountains of Persia, hiding quietly for centuries." Her voice emanates from the darkness, but he does not make a move to corner her. "They said if we knew the truth, we would be terrified. But no matter how drunk the girls got them, they never said anything more than that."
"Then we'll have to become even more terrifying than the Spirit Eaters themselves. Wrestle," he commands, dropping his sword.
With hardly a moment of hesitation, Alecta leaps out of the shadows like an arrow released from its bow, and then she is on him, crying out as she thrusts a leg between his, hooking it around his calf, trying to make him buckle. She's strong, but this body of his is much stronger than it used to be. Easily, he grabs her arms, forces her to turn around, and holds them behind her back.
"What have I been telling you?" he asks, his lips close to her ear. "You cannot fight against a man's strength. Stealth. Charm. Playing at fear and incompetence. Those are your best weapons."
Alecta stops struggling and hangs her head, her hair a golden-red curtain across her eyes.
Pyrolithos releases her. And she swings around, landing her fist at the edge of his jaw, sending him stumbling backward.
"Like that, you mean?"
Pyrolithos spits blood onto the dirt. Then he looks up, catching those flashing eyes. "Exactly like that."
She crosses her arms and stares at him suspiciously. "You have a remarkable understanding of women's bodies." She pauses. "How to fight with them, that is."
He makes a little bow. "Thank you."
"It's strange, though," Alecta continues, tilting her head. "How did you learn to train women?"
"I've told you before—I'm doing what Queen Cynane asked of me. I risk my throne every time I ride out here and give you my time and expertise. What do you think the nobles would do if they knew what we were doing here?"
"Hmm," Alecta says, shifting her weight. "Why _are_ we doing this again? Surely a new king could find pleasure doing other things at night..."
"Alecta, we've—"
"I want to hear it again. More slowly this time," she says, playing dumb, just like he had instructed her to.
He sighs. "I told you, Queen Cynane is being hunted. There is a reward for her capture, and she wants a new guard, one made up only of women. I, too, want such a guard, but I cannot reveal my new warriors until you are a well-trained, impressive force."
"And why do you want a female guard if you know the nobles will protest?" Alecta asks.
"Queen Cynane and I trust women," he replies. "We don't trust men."
"But you are a man."
He hopes she can't see his grimace in the shadowy light. "I can't help but notice, Alecta," he says, stepping toward her, the darkness suddenly too thin—nothing separating them from what they desire, "that in return for everything I do for you, all I receive are questions." His hands land on her upper arms and he easily sweeps her to the ground, straddling her.
He has never, he realizes, been this close to a woman's body before. Feeling her beneath him, he understands in a whole new way what it means to be female, to be both soft and firm at the same time. Tiny beads of sweat glisten on her face, and there is a smear of dirt on her cheek that he is intensely tempted to wipe away, or to lick. She looks up at him, expressionless.
It occurs to him that she must often find herself in this position.
Quickly, he rolls off her.
"I think that's enough for tonight," he says, rising. "It must be near dawn."
She brushes herself off and gets to her feet. "Thank you, your majesty." Her tone is suddenly brisk and distant. She moves toward the waterfall, about to pass through it, where, he knows, her tunic will become soaked through, but before she leaves the dark sanctuary of the cave, and its cool, minerally darkness the king has come to savor, she looks at him over her shoulder. "You will come to regret everything you've given me."
And then she is pressing herself against the side wall, slipping through the tiny opening protected from the water's spray, and is gone.
He has come to savor those parting words, the same every time. They seem to him both a threat and a promise. Even if it is the promise of betrayal, it keeps him coming back for more.
Approaching one of the burning torches, he summons the soot and ashes pumping through his soul and puts his right hand in it; he feels no pain. Then he pulls the flames out, like a cord, and twists them into a glowing bracelet around his other wrist, feeling the exultant rise of power inside him. The fiery glow subsides, and the bracelet turns into something hard and crystalline, like diamonds, just like the others he has made. Next, he calls on the torches for their smoke, and it billows forth, surrounding him as he coughs. Quickly, he channels it into a swirling column, walks around it, and, with a flick of his wrist, scatters it. He has been experimenting with his Smoke Blood, investigating what he can and cannot do. There is no one to teach him.
He paces the darkness for a few minutes, the smell of moss and river stone no longer enchanting, only cold and damp, until he is certain Alecta has made her way a safe distance. Then he moves through the slender opening at the lip of the cave. Splashes of icy water refresh his hot, sweaty skin. It's only the beginning of spring and the sun still has not risen, but he can tell it will be a warm day. The trees have begun bursting with green leaves and flowers. Grass and ferns have sprouted everywhere. Insects and birds appeared after their long winter hiatus, and now they rub themselves loudly, croaking at the coming dawn.
He squats beside the chattering stream, fills his waterskin, and drinks heavily from the flask. Through the thicket of trees, he can see the dark form of Alecta riding toward the road, growing smaller. Once she disappears on the strip of coastal road that rounds the mountain, he mounts his horse and follows.
There is little movement on the cliff-side roads, and the sky gradually lightens. He allows himself a moment of peace, sinking into himself, conjuring once more the stirring in his blood. He imagines himself a spirit, made of air and thought. The burn of Smoke Blood rises within him, coursing through his veins. His legs begin to tingle. After several moments, he realizes they have disappeared entirely, though from his hips up, he is still solid. He realizes he must look like a centaur from afar. He calls on the sooty magic within him to turn himself into a being of smoke, as Taulus was, but only his arms turn into gray wisps. His smoke hands still hold the reins; they could also hold a sword, he knows. But he doesn't have the power to turn anything more than a part of himself into smoke. It must take a great many betrayals indeed to do that. If only Taulus were still around, he could ask him for help.
But, of course, Taulus is not here. Without Taulus, Pyrolithos wouldn't have Smoke Blood. No—without Taulus's _death_. When his guilt becomes too heavy, Pyrolithos reminds himself that Taulus wanted to die, wanted Pyrolithos to end his agonizing existence. Otherwise, why would he have told Pyrolithos that, despite being a creature of shifting smoke, he still possessed a human heart, the last part of him to dissolve? When Pyrolithos plunged his sword into the swirling vapors of Taulus's chest, blood, red and hot, spurted all over him as his smoke dissipated.
He shakes his head as if to clear it from that atrocious memory and looks to the right, where breakers crash against the rocky beach; gulls cry and wheel above. By the time he reaches the city gate, he has regained his solidity, and the world has silvered itself awake.
He tightens his helmet strap—he's wearing the uniform of a palace guard to avoid recognition on the streets—and slows his horse as the gate guards wave him on. With his nosepiece and cheek flaps, they do not know their own king.
He picks his way around carts and pedestrians, past the many-pillared Temple of Zeus, and across the main marketplace where people sell fish, vegetables, and household goods. He takes a side street, past the walled town villas of the rich. Aphrodite's Grotto is on the edge of the wealthy section of town; the next street over, the neighborhood of artisans begins. He approaches the villa from its rear wall overlooking the courtyard and ties his horse to a tree, dropping a coin into the outstretched hand of Oikos, one of the brothel bouncers who guards the rear of the complex. Agilely, he climbs the tree, savoring the way his muscles pulse with use, and crouches on the corner of the wall, hidden by the leafy branches.
The brothel is closed to male clientele during training sessions, such as now. The women tend to use the hours no man would ever come seeking pleasure—those just before breakfast, when the drunks have staggered home to their own beds at last and the light has barely touched the sky.
Torches line the walls inside the courtyard, throwing flickering warm light onto the stones and the moving bodies within. Alecta is already on the far side of the pool, the splashing of its central fountain drowning out some of her words as she instructs some twenty-five helmeted women in the moves he taught her just today. Even the heavily pregnant one is there, awkwardly wielding a sword and shield.
They practice in stages, following Alecta in slow motion, and for a moment the whole vision from above looks to Pyrolithos like the incoming tide—all white tunic surf and flashing silver metal in the torchlight, as each one goes through the motions: raising the shield to pummel the opponent's chest, aiming it at the elbow of the opponent's sword arm to force her to drop her own, whirling left, then right again. He's not sure how long he's been watching them when something in the air changes. Every nerve in his body prickles with warning, and even the leaves unfurling on the tree surrounding him seem to freeze, the sea wind going still. The springtime warmth has vanished, replaced by a cold dread. The birds have gone silent, he realizes, like they do sometimes before a storm. But the women below don't seem to notice anything wrong as they circle and wrestle and spar.
Unnerved for reasons he can't quite say, Pyrolithos pulls on the branch and climbs over to stand on the wall for a better view. Clouds are moving across the sky in the distance, innocently enough. There's now a blazing line of orange gold on the horizon. Not a storm, then.
Before he has time to consider another option, an unearthly shriek rends the air.
Below him, the women look around and begin to point in alarm. At first, he thinks they are pointing at him—an intruder.
But no, he swivels just in time to realize they are screaming at something just beyond him.
Something dark is coming toward them from the sea, blotting out the just-rising sun. A dark smear dives toward the brothel, huge black wings outstretched. A thin howl turns his blood to ice.
In the swirl of movement, he cannot get a look at the thing, but it doesn't matter.
He knows what's coming.
"Get inside! Now! Run!" His voice, though strong and urgent, is muted by the women's screams and the slap and burble of the fountain.
Alecta, eyes wide and mouth open, catches his gaze. Their eyes lock for a moment. Then she pushes the women toward the many doors opening onto the courtyard. "Go!" she cries, shoving them hard.
The creature sweeps past Pyrolithos, the chill of it glancing off his body and sending an ache down deep into his core.
A Spirit Eater.
The creature dives into the panicked women in the courtyard, all the torches snuffing out at once in a billow of smoke. There is a terrible scream through the sudden smoky darkness as the creature dips and grabs onto one of the women, ripping her head off in one swift movement and spitting it out.
Pyrolithos has seen much horror in his life, but for seconds he is frozen in shock.
The creature swoops again and lands amid the chaos of the courtyard, flapping its foul black wings, then rips the bloody yellow intestines out of the dead woman's headless body and swallows them in awkward gulping motions, as a famished bird swallows worms. There is something shifting about its form, as if he were looking at it through a prism. One moment it seems to have two heads, then only one, and four wings, only to focus into two. He rubs his eyes.
"Swords out!" Alecta's voice—why hasn't she gone into hiding? "Sophonisba, throw us some spears!" she commands.
A tall dark-skinned woman begins tossing spears to Alecta and several other women. The creature raises its heads—no, just one head—from the bloody carcass, and sniffs the air. Its black eyes light on Alecta, standing at the front of the armed women, and Pyrolithos feels his heart trying to break out of his chest.
If his Smoke Blood Magic were more advanced, he could appear at her defense in seconds. Instead, he jumps down from the wall, draws his sword, and tumbles across the courtyard, swinging his shield off his back as he runs.
The creature swivels toward him, and his breath stops in his throat. He could swear there's something distinctly _human_ in the way it stares at him. A memory comes to Pyrolithos in that moment, swirling up and out of a past that feels too distant to be his own—a torture wheel. Axes and horned helmets and searing, constant, agonizing pain. Whispered conversations about witchcraft and magic—and Spirit Eaters. A murmured rumor—that they were once, somehow, impossibly, human.
The creature almost seems to smile, as if reading the king's mind, revealing sharp yellow teeth, dripping bright red blood. Hundreds of teeth in several heads... No, just one. It's a trick of the light.
Pyrolithos takes in the situation behind the creature. Most of the women have made it to safety inside. Alecta, Xanthe, Sophonisba, and a few others stand defiantly with spears outstretched.
"Get inside!" he manages to shout. "I can handle this." But it's a lie. For all his Smoke Blood, his powers are nothing against this ancient evil, this demon who reportedly ate the gods alive, ingesting all their magic and their knowledge.
An iron spear hurtles through the air and bounces off the enormous leathery wing as if it's only a feather. The monster hisses and turns back to the women. Its skin is like armor, Pyrolithos realizes in horror.
Two women run around the fountain to join Pyrolithos, surrounding the Spirit Eater. Alecta steps forward, sword raised.
Before Pyrolithos can scream at her to back away, the enormous wings send the women flying. Alecta is thrown into the fountain and rises, sputtering, still, amazingly, gripping her sword even as blood streams off her shoulder. One woman is thrown against the courtyard wall. Another hurtles straight into Pyrolithos, knocking him over.
The creature ignores the others, leaping with outspread wings toward the king. Pyrolithos has bravely fought many enemies larger and better trained than himself. But this is different. This causes his blood to become both ice and fire. He is no longer a body propelled by thought but pure muscle driven by animal panic. He scrambles to his feet and runs.
There isn't time to look back, but he hears the rustling of wings, hears the unearthly howl, feels the foul, freezing breath on the back of his neck. He crashes through the gate leading to the brothel's service courtyard, expecting agony to slice through him every moment as fangs and claws shred his flesh. Shouldering open a small door, he finds himself in what appears to be the brothel laundry and runs to the far side of a large bubbling cauldron in the fire pit, scanning around hurriedly for a weapon. A large wooden paddle rests among sheets in the cauldron; that would be useless. Then he sucks in a breath—the Spirit Eater has stopped in the doorway and stands perfectly still, blocking out the light, sniffing.
Hearing a muffled sob, he realizes there are two women huddled behind a pile of laundry in the corner: the pregnant Persian and a large middle-aged woman, who is probably the laundress. The creature steps inside the room and, now totally uninterested in the king, slinks toward the women.
Pyrolithos takes a breath and, with the razor-sharp tip of his sword, punctures the inside of his left arm. Pain radiates through him. Smoke, ashes, and flames roll through him as he coughs and writhes. He has no idea what he is going to do, only that it must be clever and it must be now.
As he straightens, his gaze lands on the fire beneath the laundry cauldron. He dives toward the fire pit and picks up ropes of flame, along with burning coals and feathery ash with his bare hands—it doesn't hurt. Instead, it seems to strengthen the smoke inside him, and he feels himself expand, not visibly but in power.
Using the full power of his concentration, he flings the flames and coals toward the Spirit Eater, who now seems to be a hydra of several heads, envisioning them forming a rope to lasso the creature, just as earlier this morning he played with making bracelets. His mind pushes the heat and light together, rearranging them just the way he has rearranged his own body.
A glowing orange lasso falls over the Spirit Eater's single head and shoulders. It turns, shocked, and stares at him.
Pyrolithos shudders with the magic flooding him now, dizzying and senseless and intoxicating. He can't stop, and he doesn't want to, funneling all his concentration into the burning rope as its many filaments become strands of heat and light. They crisscross one another until the creature stands in a _cage_ of fiery coals. It grabs the bars with its long clawed fingers, shaking them with all its might and roaring in rage. But the bars are stronger than iron.
Slowly, the fire cools to a glowing, hard, glittering white.
Pyrolithos blinks. He is staring at a diamond cage. Made by his own magic.
And within it writhes something miraculous and terrible and unthinkable.
He has caught a Spirit Eater.
He wants to open his mouth and laugh.
Just then, Alecta and the other women from the courtyard clatter into the laundry, wielding spears. For a moment, the only sound is their heavy breathing as they take in the astonishing sight.
"What are you waiting for, you fool? Kill it!" Alecta's command cuts through the general awe. She lunges toward the cage, but instinctively, Pyrolithos steps forward, blocking her path.
He shakes his head, then turns back in wonder to the powerful creature he has caught.
Once, he knew hunger.
He knew patience, too.
He has learned the hard way—when he was a she, when she was Cynane—that the way to gain power is not to destroy it, but to harness it.
CHAPTER TWELVE
HEPHAESTION
Fog lies thick over the battlefield, moving across enemy lines, revealing only glimpses of the battalions lined up across the green fields near the town of Chaeronea, a two-day march from Athens. Heph breathes in the cold morning air. When King Philip wants a war, he gets a war: the Greek city-states have rallied at his provocations, and now it's time to show off Macedon's finest asset. Namely: their army.
There was a time when battle wouldn't have been necessary, but while the king was loafing about in Byzantium and Alexander was in the grips of a spirit-possession, chaos had spread throughout Macedon's neighboring territories, and Heph knows the king is right—that there is no way to attack Persia without more allied forces. And if the forces won't come willingly, then Macedon will just have to make them.
Heph shivers slightly at the haunting wails of the salpinx, a four-foot-long bronze battle horn, as the sound floats over the armies like the shriek of some spirit wrenched straight out of the earth. Heph's horse, Ares, a dun-colored stallion, shifts under him, flicking his long tail and whickering softly, sensing the tension of thirty thousand Macedonians staring across the plain of Chaeronea at thirty thousand allied Greeks led by the forces of Athens and Thebes. The line of warriors stretches more than two miles, from the banks of the lazy Kephisos River just to Heph's left, to the foothills of craggy Mount Thurion on the right.
Next to him, Alex casually pushes a strand of blond hair back under his plumed, golden helmet. To others, he looks as calm as if he were planning on having a picnic out here on this fine spring day. But Heph can see the tension around his eyes, the nervous way he threads his fingers around the black mane of Bucephalus. This is their first great battle with King Philip, who has given his son the honor of commanding the left wing, while Philip commands the right.
In the far distance, a massive blue Macedonian flag with the gold sixteen-pointed star waves across the sky, raised high and then quickly lowered by a man standing on horseback. Soon after it, other flags begin repeating the motion until the battlefield looks like an undulation of waves crashing over one another. In front of Heph, rows upon rows of infantry lift their shields and lower their spears.
"Advance!" The mingled roar of thirty thousand men echoes through the fog as, across from them, the Greek allies blow horns, drop flags, and lower spears, too.
The foot soldiers on both sides launch toward the middle of the field, and within an instant they are clashing, screams of pain ringing out alongside the clang of a thousand spears. Moments seem like hours as Heph tries to see which side is advancing like a tide, and which side will cave.
It takes all his willpower to hold himself back, his entire body an arrow in a cocked bow.
The leather lining of his helmet grows slick with sweat. Ares flicks his tail and nickers in protest. The battle ahead has raised great clouds of dust; the heavy spring rains have not come this year.
"Telekles," he calls out. Telekles has the best eyes of any scout he knows.
Telekles raises a hand to shield his eyes. "All I can make out is dust and fog."
Phrixos, on Telekles's other side, leans forward and rolls his eyes. Even now, right before they race into a pitched battle, Phrixos's round, homely face has the look of someone about to tell a very good joke. "Is that all you can do, Telekles? I'm half-blind and I can see that much."
From somewhere deep within the yellow-brown whirlwind ahead of him, Heph hears three short bursts of a horn. _Advance!_ The Macedonian infantry has successfully punched through the enemy lines. Now it's time for the cavalry.
Alex turns to Heph and grins. "Battle!"
"Battle!" he replies, the word pumping a surge of excitement through his veins. He has already kicked his horse into motion and, sword raised, he plunges headlong into the fray, Alex at his side. It is good, so very good, to be doing this with his friend. There is no other feeling like it.
Together they roar into pockets of battling foot soldiers, stabbing those carrying an owl shield or wearing an owl breastplate, treading on the bodies of the dead and wounded.
Somewhere out here, Heph knows, is the Sacred Band of Thebes, a regiment of one hundred fifty pairs of warrior-lovers, each fighting not only for himself but for his lover, which gives added reason for victory. They have such a fierce reputation, they are called the Immortals. Heph is not afraid of any warriors in the enemy army, except the Sacred Band.
Heph and the Macedonian cavalry thunder quickly through the remains of the infantry and now face the Athenian cavalry galloping toward them. He picks his man from a hundred paces out. The graying beard hangs below the helmet, and the man looks ill at ease galloping on his horse. As Heph approaches, the Athenian raises his sword but Heph pounds him so hard with his shield that the man falls off his horse and the alarmed animal screams in fear and bolts. Two Macedonian cavalrymen push in front of Heph, and he doesn't see the Athenian he just unhorsed. Clustered all around him are knots of men fighting on horseback, the clash of arms, the staccato braying of frightened animals. Where is the prince? He searches desperately for the unique golden armor.
There, about fifteen paces to his left. Clinging to a rearing Bucephalus with one hand, Alex plunges a sword into an Athenian, who keels off his horse in grotesquely slow motion. Then the prince whips around in a tight circle, looking for more enemies. Out of nowhere, three mounted Athenians surround Alex, and Heph kicks his horse into motion. As he gallops around the trio, he stabs one through the back and slashes another in the arm. The first falls forward, and the next yells in pain and gallops off. While the third man, startled, looks for his comrades, Alex decapitates him in one clean sweep. His helmeted head flies into a nearby scrum of battling, mounted warriors, who scream and race off. The headless body slumps forward on the horse, which panics before tearing off through the crowd, likely trampling more of the enemy as it tries to flee.
Alex pulls up next to Heph, smiling. He reins in a nervous Bucephalus, who paws the ground anxiously. "We've almost broken through!"
An arrow whizzes past Alex's face, and his eyes widen as he pulls it back a few inches and continues, "Time to finish them off." He means the rest of the cavalry, and with arrows. He unhooks his battle horn from his belt and blows two long blasts. From across the Macedonian left wing, other horns echo the blasts.
Heph and the other archers sling their bows off their shoulders, nock them, and, guiding their mounts with their knees, race around the battlefield, looking for remaining Athenian cavalry. Heph spies an Athenian horseman on his left, sword raised, heading for a small group of Macedonians trying to carry a wounded comrade away on his shield. The horseman is so intent on his goal, he sees neither Heph nor Telekles, his waist-length golden hair thick with dust and sweat, approaching on his other side. Heph looses his arrow with a _thwing_ , and the Athenian falls to the ground as his horse bends down and nudges him. Heph gallops up and meets Telekles over the dead man, who has an arrow projecting out of each ear.
"My aim," Telekles says drily, leaning over his horse to study the dead man, "was a bit truer than yours."
Heph peers down at the dark-haired young man on the shield, his face white as a sheet, a blood-soaked bandage wrapped around his waist. "Get him to the hospital tent immediately. Guard these men on the way and then catch up to us."
"Yes, sir!" Telekles salutes.
As a shadowy mounted figure emerges from a cloud of dust, Heph nocks an arrow and raises his bow. He lowers it when he sees it's Alex.
"I think we've done our part," Alex says, wiping sweat from his eyes.
Three long blasts echo from the right. Heph's heart hammers in his chest. The signal for retreat. Philip's right wing of the army is collapsing, running away.
"Retreat!" Alexander begins shouting. "The king's side is retreating!"
The prince unhooks his battle horn and blows one short and one long blast. Across the field, others repeat it. Alex raises his sword in the air and shouts, "Macedon! Macedon!" as cavalry and infantry rally around him. When enough men have collected around him, he urges Bucephalus forward, Heph galloping at his side, the infantry running behind. They make a slow, wide horseshoe turn across the battlefield, killing enemy stragglers, and frightening many more who race away, throwing down their heavy shields to escape more quickly.
Several rows of men turn around to fight their surprise attackers with all the desperation of the doomed. Up front, some of the Theban cavalry wheel around and race back to rescue the rear. Then Heph sees King Philip's Macedonian flag pushing toward him through the opening. The Thebans are caught in between the Macedonian pincers, exactly as planned. Philip's retreat had been a feint to lure them into a trap.
With a whinny of pain, Ares rears, throwing Heph onto the hard ground. He sees his stallion race away, an arrow in its back. Scrambling, he finds his sword and shield and runs full tilt into the fracas. The Macedonians have the advantage tactically and emotionally. The Thebans, shocked and frightened, are being squeezed together in a death grip.
The fog is dissipating and reality setting in. Time slows. A drop of sweat trembles from Heph's eyelashes, for what seems like an eternity, before splashing onto his cheek in slow motion. A spear flies in slow motion toward his head. He pulls back agonizingly slowly, just in time to avoid his skull being impaled; the spear whistles only a hairbreadth past his nose. Feeling half-paralyzed, he pulls a dagger from his belt and throws it. It floats toward the man, head over hilt, enters his eye, and he drops writhing to the ground.
Now time speeds up as a panicked, riderless horse careens past him, heading straight for Alex on Bucephalus, who is facing the other direction, studying the remaining pockets of resistance. "Alexander!" Heph cries, but the warning gets stuck in his throat as he inhales more dust. He coughs and splutters as the frightened horse, large eyes rolling in panic, careens right into Alex, knocking him and Bucephalus to the ground in a horrible tangle of thrashing limbs. A cloud of dust wafts over them once more, and Heph can't see what is happening. He races toward them, crying, "Alex!"
"Hephaestion!" comes the reply. The dust settles, and to Heph's immense relief, he sees Alex nearby, standing tall, holding his sword and shield. Bucephalus is nowhere in sight.
"Are you all..." Heph begins but realizes that Alex isn't looking at him. He's looking ahead, his mouth open.
Out of the dust comes a group of some three hundred men carrying the double-headed Sphinx banner of the Sacred Band.
The Immortals.
They are the most enormous warriors Heph has ever seen, so tall and broad-shouldered they couldn't fit through an average door. Their tanned, oiled muscles glisten. The sun reflecting off their highly polished armor nearly blinds Heph. These are not men but beings of breathing bronze with death in their unblinking, flinty eyes. Several dozen Macedonian cavalry have amassed behind Heph and Alex, and the advance guard of the infantry has caught up, too.
He can't be sure how it happens, but suddenly Heph is in the midst of hand-to-hand combat so ferocious he expects to die every moment. He is surrounded by yells, grunts, screams, the metallic clang of swords on swords, the thumps of swords on shields, and dust. And everywhere, throat-clogging dust.
A muscular young warrior springs forward from the dust, sword outstretched. Heph meets it with his own, and the two swords do a ferocious dance. The warrior is an excellent swordsman; Heph has difficulty keeping up. The Theban's sword point slices Heph's biceps. He feels the sting and burn; hot stickiness slides down his arm.
Then the Immortal's sword swipes the leather strap of Heph's helmet, severing it. The helmet topples off him with a gush of cooling air. He groans inwardly. In battle, a head without a helmet is like a giant melon just begging to be sliced open. But the fresh air brings needed relief to his sweat-soaked head, and he feels a sudden burst of energy.
Ignoring the searing pain in his bloody right arm, he does a maneuver he learned from the Aesarian Lords last summer; he lifts his shield to pummel his opponent in the chest and at the same time plunges his sword underneath, stabbing the warrior in the thigh. The Theban cries out and backs up. Blood gushes from his leg as Heph advances, realizing he must have severed the artery. Still, the Immortal puts up a good fight with his sword. Heph has no idea how long it lasts, but finally, his opponent, weak from loss of blood, falls. Heph kicks his sword away and raises his own.
"You may not see an Immortal die," the man says, smiling despite the pain etching deep lines on his young face. Jerkily, he pulls his shield over his head. Heph raises his sword but hesitates.
"Damiskos!" a man shouts, rushing wildly toward Heph. The wounded man's lover—it must be. Heph has never experienced such a brutal assault. The other man is like an avenging Fury, full of venom and rage. He's so intent on fighting Heph, he doesn't see Phrixos casually walk up to Damiskos, kick away the shield, and send his sword straight through his leather breastplate, into his heart. There's a sickening crunch. When Heph leaps out of the way of yet another sword thrust, the Immortal sees what has happened.
"No!" Tears stream down the man's dusty cheeks.
Heph has seen grief on the battlefield as men find dead friends, brothers, fathers, and sons, but he has never seen such pain and anger before. The Theban's gray eyes blaze with hatred as he stares at Phrixos, who, for once, doesn't look like he sees any humor in the situation. In a flash, the Theban has kicked the shield out of Phrixos's hand and, Heph notes, probably broken his wrist. He swings his sword as Phrixos, left arm dangling like raw meat, stands gaping stupidly.
But before the sword can come down, Heph has fired an arrow into the back of its owner's neck. The Immortal falls and, with his last ragged breaths, drags himself toward his lover. Heph and Phrixos watch in horrified fascination as he crawls on top of Damiskos and dies. It is a sight Heph knows he will never forget in the coming years, no matter how hard he will try. He feels it has ripped part of his soul right out of him.
* * *
By the time the spring sun has reached its zenith, the great battle is over.
Heph and Alex wander through the carnage with attendants from the field hospital, searching for wounded and dead Macedonians. Greeks, too, scour the field, taking their casualties back to their own camp. Heph's arm wound throbs painfully; Alex bandaged it with the strip of cloth all soldiers carry on their belts to help their comrades stanch bleeding on the battlefield. But he said it was so deep the surgeon would have to sew it closed. Heph will wait a bit, though. The surgeons should tend to the more serious wounds of others before stitching this scratch.
Alex, despite having won ecstatic praise from King Philip, is grim, particularly when they search among the Sacred Band of Thebes; they find that many of the lovers died together. Periodically, Alex kneels to look at a face, feel for a pulse, or say a prayer.
"Gone," he says, unlatching his helmet and throwing it down in frustration. The feather is wilted, the patterned gold splattered with blood and dirt. "All of them, to the last man, dead. Courage, valor, devotion..." He spreads his arms wide as a tear trickles down his dusty cheek. "A waste," he says angrily. "Heph, all of this is a damnable waste. These heroes could have been our allies, could have helped us conquer Persia."
Heph hangs his head. "But the king was right—these same men would have invaded Macedon."
"Don't you think I know that?" Alex asks, his voice thick with emotion. He runs a hand through his stringy, sweaty hair and seems to come to a decision. "Be that as it may, these men will be buried together, and a monument to their courage built above to remember them forever. And in this way, the Immortals will remain truly immortal in human memory."
* * *
The sickly smell of charred human flesh still lingers in the night air as Heph and Alex leave the dinner in King Philip's tent and walk between rows of smaller tents lit by brightly burning torches. Heph notices that the pyres outside the Macedonian camp, on which some three thousand comrades were reduced to ash at sunset, still burn. Tomorrow men will shovel ashes into pots to be taken back to their families in Macedon.
He shivers. The temperature fell rapidly when the sun set, and the night is cool, almost chilly. Campfires burn as far as Heph can see, where soldiers huddle and relive the day's fighting.
Torches light the perimeter of the camp, where guards will patrol all night, ready to sound the alarm if anyone approaches. The Greeks, though they clearly lost the battle, suffered only about the same number of casualties, mainly because most of them ran off when they saw how badly they were losing. They could regroup and fight again tomorrow, though Heph doubts they will have the appetite for it.
They duck into their tent, where Basil, their servant, has lit oil lamps hanging from metal rods pushed into the ground. Heph and Alex rip off their shirts and wash off the filth and blood of the day in the large basin of water at the back.
As he plunges his entire head beneath the surface to get the dust and sweat out of his hair, letting the cold water roar into his ears and dull the ringing that has been in his head since combat, Heph holds his breath for a second, suddenly remembering floating in the ocean off Meninx, holding Katerina in his arms and kissing her for the first time. The way she moved against him with an urgency that surprised him, her reservations suddenly coming down between them.
He pulls himself up and grabs for a towel, gasping for air. This is the first time he thought of her today, which is unusual. Since they parted in the massive Byzantine flood some five months ago, he has found himself thinking of her the moment he awakens each morning—wondering if she, too, is awake somewhere, thinking of him.
Shirtless, he rubs his hair briskly with the linen cloth, allowing the night air to cool his chest, calm his tense muscles and racing heart. "What do you think Kat would have done in the battle today?" he asks, throwing on his sleeping tunic, careful not to disturb his wound-dressing. "Would she have had mountain lions race in to attack the Greeks or vultures dive down to take chunks of meat from their horses?" He blows out all the oil lamps except for the one between their narrow camp beds. When he looks up, he sees Alex, standing with his clean tunic balled in his hands. The single lamp casts a flickering light over his pale, taut muscles, which still drip from washing.
"You miss her," he says, finally getting dressed and pouring two cups of watered wine.
Every muscle in Heph's body is stiff and sore from the battle. He looks at his cot in the corner, with its fresh sheets and a clean woolen blanket—no bed has ever looked so inviting. But first, there's something he has to do—has to say to the prince. Something he has been holding in for too long. He has wanted to bring it up many times before, but somehow lacked the courage, or found that the moment wasn't right, or got interrupted after a few halting words. Something about today's battle, though—about bravery and fear, love and death—has made him realize he shouldn't hold anything back from Alexander, that doing so is a kind of lie between them.
Heph picks up his cup and sits on his bed. "On Meninx, I asked her to marry me," he says. His voice sounds weak to him, thin.
Swigging from his wine cup, Alex starts to cough. After a moment, he responds, his voice low, "What did she say?"
"She didn't really say anything. It wasn't yes. But neither was it no." He takes a deep gulp from the cup, his mouth feeling too exposed. "I was sure I could convince her in time, but then, in Byzantium, the waves parted us. It was almost as if..."
"As if Fate decided for you," Alex finishes for him, sitting down beside him, "and gave her back to Jacob."
Heph inhales roughly. Alex can finish his sentences for him. Word for word. Thought for thought. This, he thinks, is perhaps the true marriage. That's what Aristotle would have taught them: the marriage between two perfectly aligned minds and souls, the greatest connection one can seek in this life.
"And now they're in Troy," Alex says, looking so deeply into the dancing lamp flame it's as if he sees the smoking, broken towers of Ilium within it. Weeks ago, they received Kat's message that she and Jacob were living and working at the Temple of Asklepios, near the fabled ruins. "I can't say I understand it."
"Anywhere but Macedon," Heph says. "I shudder to think of your fath—the king—marrying her off to some petty princeling as part of a treaty. How many times has Philip asked you if you could locate her? Four?"
"Five," Alex says. "Though all this," he says, gesturing around them, "has kept him nicely distracted. For now."
The flickering lamplight casts moving shadows on Alex's face as he says, "I think such a marriage would kill her. Though, to be honest, Heph, I just don't know if Kat can be happy as anyone's wife. There is something about her that needs to remain wild and free, like the animals her mind can inhabit."
Heph wonders if that's true. Kat always seemed to be keeping a part of herself back. That was what had drawn him to her in the first place: the levels of mystery within her, the doors in her heart that he wanted to unlock, one at a time.
Heph lies back on the cot, staring up at the curved peak of the tent, swaying in shadow. "I think I could live with her marrying no one at all," he says, his voice scraping along his throat, "so long as I don't lose her to someone else."
Alex lies back beside him, their shoulders touching. When he rolls to one side, he looks at Heph with such compassion in his eyes that Heph feels suddenly raw, like an open wound. "Don't you remember what Aristotle always told us in school? Love is a kind of insanity."
Heph rolls toward him. He knows the prince is right, and yet... "There's something about her that I can't resist, that I can never forget. She's so... She's so much like..." He stops.
"Like me," Alex finishes, tilting his head and looking at Heph, his blue eye gleaming in the lamplight, his brown eye hidden in the shadows.
"Yes," Heph agrees, feeling heat rise in his cheeks, though he can't imagine why. "My loyalty is always to you. Always. I was an orphaned beggar when you took me to the palace and made me who I've become. It's... What I feel, it's...it's more than gratitude. It has always seemed that you and I...well, that we belong together." He rolls onto his back again. "And then I met Kat and, though it took a while for me to realize it, it was like she was a female version of you. And I loved her." He clears his throat. "Love her. Hope she'll come back to me. Even though my first loyalty will always be to you." He releases a breath, suddenly humiliated with how much he has revealed. "It's just that sometimes I think I am in an impossible situation."
"Whether she comes back to you or not," Alex says, putting a hand on Heph's shoulder, "the only truly important thing to me is that I never lose you."
Heph turns again to find Alex staring at him and sees the same sparkling intensity he often saw in Kat's eyes, the same expression of intellect and passion. It's almost as if Kat is lying beside him, a hand, warm and soothing, on his skin. Heat floods through his body at the thought and Heph sucks in a breath, deeply and rapidly. Suddenly all the nerves in his body are alive and tingling.
He has the strangest, most disturbingly real feeling that he and Alex are going to kiss, here, now, in the flickering lamplight of a war tent.
Then his head clears, and the moment passes with a gust of relief.
"I should sleep," he whispers.
Alexander rises. "Me, too." He moves toward his side of the tent, then stops, and turns back. "Hephaestion."
Heph sits up, breath trapped in his chest, feeling a little dizzy. Is he going to acknowledge what almost just happened?
But Alex just stares at him for a second, his face unreadable. Finally, he sighs, offering a slight smile. "I understand what it's like." He swallows. "To be in an impossible position."
Then he leans over, pinches out the flame on the lamp, and climbs into his own bed.
CHAPTER THIRTEEN
ALEXANDER
Alex wakes with a jolt. It's pitch-black inside the tent. His first thought is that Riel has returned, trying to claw his way back into his mind, and his heart flutters like a sparrow beating against cupped hands. Then he searches inside himself and realizes there is nothing unusual there. Just himself, roused from sleep.
He listens carefully for the snap of a twig, the metallic note of a sword being unsheathed, or men's low voices, signs, perhaps, of a sneak invasion by the Greeks. But he hears only Heph's regular breathing.
Just hours ago, he and Heph had the strangest moment, where it seemed, briefly, like their friendship might have turned seamlessly into something else—into something _more_. He thinks of the Sacred Band of Thebes, the many pairs of male lovers who died together yesterday on the field. He cannot deny the unexpected thrill of it, the _almostness_. But it terrified him, too, unsettled him. Went against everything he thought he believed. Nothing can change what they have, nothing can risk their bond.
He wants their friendship to remain the same.
Or does he? Is that the reason he doesn't want Heph to marry Kat, or anybody else, for that matter? Does he want Heph all to himself?
His muddle of thoughts is interrupted by the loud crunch of footsteps outside and two men arguing about a dice game—probably Macedonians sent to relieve the watch.
He punches his pillow and turns on his side. It's still dark out; he should try to fall back to sleep, to be rested for the challenging day ahead. But questions prick him. What will happen in the morning? Will the Greeks send emissaries to conclude a treaty? Or will they demand another battle? How will Philip negotiate? Alex worries that his terms will be so harsh that the city-states' pride will not permit them to accept. And more soldiers will die instead of swelling the Macedonian force to invade Persia.
If only Alex could still look into men's minds and use that power to persuade Philip and the Greek generals. Perhaps Kat, whose Snake Blood skills involve animals, could learn to travel into the minds of men. But she is far away in Troy. He wonders if there could be another Snake Blood in the Macedonian army, perhaps one who is unaware of his powers or hides them to avoid capture by the Aesarian Lords.
No, he can't sleep.
Quietly, Alex gets up and fumbles on the table for his flint set, strikes the iron against the flint, and watches as sparks fall on the tinder and blaze. Then he lights the wick of the oil lamp, squats down, and rummages in the trunk at the foot of his bed, careful not to make enough noise to wake Heph.
There, in the corner, wrapped in several layers of protective wool, is the Atlantean Mechanism designed to detect Blood Magic, manufactured by Heph, Kat, Aristotle, and Ada of Caria, on the island of Meninx.
In the Byzantine palace, on Philip's wedding day, Kat used it to find the Earth Blood who, according to prophecy, was meant to kill Riel, little knowing it would be her childhood friend Jacob. Heph had shoved it into the pouch on his belt as the waves came. Miraculously, he discovered it the next day in his soggy leather pouch. The water warped the wooden crank on the side—the part that gave energy to the spring beneath the dial—but Alex wonders if it still works. Was it the mechanism or Fate itself, as determined by the old prophecies, that led Katerina and Jacob to the Byzantine palace?
There's no doubt, at least, that the mechanism is unnaturally powerful. About the length of his hand, it gleams dark gold in the lamplight, as if lit by an inner fire. The little lever on the front is still set to a diamond shape, a symbol of Earth Blood. He winds the crank, which works more smoothly now than it did right after the flood, and watches the needle on the dial spin. It stops, pointing northeast. Perhaps that is Jacob in Troy, or some other Earth Blood in between. Then he changes the lever to the image of the snake for Snake Blood. The dial spins around again and points almost to the top of the device. Southeast. Alex pulls on his boots and throws his cloak over his sleeping tunic. He unpins the tent flap and ducks through, into cool night air, redolent of smoke, roasted meat, and charred flesh. The funeral pyres outside the camp have become a glowing, smoking crumble. High above him, a crescent moon scythes its way across the paling heavens.
He walks silently through the camp, following the needle on the gently humming mechanism. At the perimeter, where torches still burn brightly, he sees armed men patrolling and checks the device again. The closest Snake Blood is beyond the Macedonian camp, then, across the wide fields, though if he is one mile or a thousand, Alex does not know.
"No sign of the enemy?" he calls to the watch. Two men approach him, one holding a torch.
"No, my lord," the soldier says, his smile revealing several missing teeth. "I imagine them Greeks'll come to us, begging clemency soon as sunrise. Shouldn't be much longer."
The horizon beyond the river is a thin blue-gold line, a harbinger of approaching day. It was a foolish plan, he knows, to find a Snake Blood out here among the men, and he turns back toward the camp. But out of the corner of his eye, he sees movement on the dark road to Chaeronea. He turns back and squints.
In the distance, a tiny, solitary figure picks its way slowly toward the Macedonian camp.
"Someone is coming," he says, and the guards swing their bows off their shoulders and nock the arrows. "But it's only one person. Hardly the Greek entourage of negotiators and generals." They peer into the gloom. The dial on the mechanism moves in Alex's hands. By the torches' glow, he sees that it is pointing directly toward the traveler.
As the stranger approaches, the sky lightens. Something about his stride reminds Alex of... But no, that can't be. Heph said he returned to Athens, to his scientific experiments, unwilling to be mixed up in wars of magic and monsters. But as the first rays of the rising sun brighten the plain, Alex sees that he was right.
It's unmistakably him: the prince's old friend and childhood teacher. Aristotle.
Known throughout Greece as the wisest man in the world.
Sauntering straight toward a military camp as if he owns the place...
As if conjured by Alexander's need of him. Alex thinks fleetingly of Fate and its interventions—he has the uncanny sense that everything that has ever happened to him was determined infinite years before any of them were alive, a master plan, unseen by the eyes of men.
"Stand down," Alex barks at the guards, and he runs across the field, not even caring if the night watch sees his limp. The old philosopher, for his part, doesn't seem at all surprised to see Alexander racing toward him as the sun rises. His gray eyes twinkle as he stops walking and cocks his head. "Out for an early-morning stroll, my prince?"
"What are you doing here?" Alex grasps Aristotle's arms as if to be sure he isn't a figment of his imagination.
"All Athens could talk about was Philip and his young Alexander, come to conquer us." Aristotle smiles cryptically. "I thought you might need me."
"Why didn't you answer my letters?" How many had he sent the past several months? Three? Four? "Did you receive them?"
Aristotle nods, shifting the pack slung over his shoulder. "You have been through quite an ordeal." He scans Alexander closely, and, as always, seems to take in everything about him in a matter of moments. "Very well, then." He nods. "It's all right."
"What's all right?"
"What has happened to you."
All at once, Alex realizes he knows: Aristotle can see that his Snake Blood powers are really, truly gone, siphoned off with the horror of Riel's brutal spirit.
Aristotle looks at him pityingly. "Now you know firsthand the torment magic can bring. Better by far to rely on the power of your wits."
Alex shakes his head. "I'm not so sure. This morning the Greeks will probably send envoys to negotiate. Philip may refuse, may want to punish them. Athens fears he will turn them into a Macedonian colony. The war may continue, and who knows at what cost to our forces, or theirs, when the whole idea was to _grow_ Macedon's army, not weaken us."
Aristotle looks at him shrewdly. "Let us walk to the river over there and sit. I am an old man and tired from walking all night."
As they walk, shadows retreat like water draining into the hard-baked ground. The fields around them glow orange pink, and the Kephisos River looks like a sparkling band of beaten gold. They stop in front of a large tree that bends over the river, its long branches touching the dappled surface. Alex kneels slowly, realizing how stiff and sore he is from battle. It's always this way: to survive the next second, you call on muscles you never knew you had, and the next day they make you pay the price.
For a long moment, they don't speak. Aristotle unhooks a waterskin from his belt and takes a long swig.
"I don't want to lose another Greek or Macedonian soldier," Alex says, choosing his words carefully, "for two reasons. The first is they are men, Greeks, with families. Why waste their lives?" He looks at Aristotle, who wears an expression of polite curiosity, one bushy dark eyebrow raised.
"And the second?"
"We will need them all in Persia." He tries to slow down the tumble of words that want to rush out. "And we can't afford to lose any more time. The Spirit Eaters are on the rise. Surely in Athens you have heard of their predations."
Aristotle stares at a pair of swallows winging their way over the river, swooping and diving for insects. "And King Philip's thoughts?" he asks.
Alex makes a little grunt of disgust. "He doesn't believe they exist at all. He does want to conquer the Persian empire, though he is no great hurry. I agree it is wise not to leave enemies at our gates when we go. But we both know my father." He stops at the word, constantly finding himself entrapped by it. _Father._ His real father was a god.
His real father is dead.
He clears his throat.
Aristotle nods and says, "You're afraid the king will draw this out, that he'll dally with these negotiations instead of crafting an alliance and preparing to march."
He watches the two swallows—he thought they were mates, but now it seems that one is fleeing the other. "Tell me what we can do to get a treaty signed. _Today._ "
Aristotle turns to him and says crisply, "I don't believe you have the necessary abilities to pull off such a thing."
It's as if his teacher has slapped him. "What do you mean? Yesterday I led half the Macedonian army to victory."
Aristotle shrugs. "That's all well and good, but bashing people on the head requires far different skills than conducting complex negotiations among enemies to forge peace."
Alex had almost forgotten how infuriating the old man could be. "What are you getting at?"
"Show me how you would negotiate terms. Prove to me you can."
"How must I..."
Alex trails off as he sees a falcon swiftly flying toward them over the river, scaring off the swallows. It dives down before them, and to the prince's astonishment, it lands, transforming into a beautiful naked woman. She has tan skin; long, shapely limbs; and thick black hair threaded with strands of silver. She rises magnificently before him, and stares at him, unblinking. Her black pupils are ringed with white circles.
"Prince Alexander," she says, smiling. Alex is struck mute but manages to scramble to his feet.
"Prince Alexander, this is Ada of Caria," Aristotle says, removing his battered traveling cloak and draping it around the woman.
Now Alex understands why Kat was so enthralled with this royal Snake Blood. She exudes wisdom, strength, power.
"I have heard much about you," he says, bowing his head.
"And I of you," she replies. Her skin is as smooth as a girl's, except for the lines on either side of her mouth, betraying her age—she must be older than his mother.
Alex realizes he still has the Atlantean Mechanism in his hand and looks down at it. The needle now points directly at Ada. Of course.
"Ah," she says, reaching toward his hand and stroking the device. "Katerina wrote me that this fruit of our labors found the Earth Blood and caused Riel's death. For that we must be infinitely grateful. But that was only the first step. Now I need your help."
"My help?"
"My brother Pixodarus has chased me out of Caria. The people are terrorized. I need an army to dethrone him and make me the sole ruler."
"And thus," says Aristotle, smiling and rocking back and forth on his toes, "begin the treaty negotiations."
Aha, so this is some sort of test. It doesn't surprise Alex—he's used to Aristotle's cryptic riddles and unexpected experiments. He stares at the two of them. He has so much to do—how will he have time to drive out a petty tyrant in Caria? He has the uncomfortable feeling, as he considers the question before him, that Ada is reaching into his thoughts, prodding the way an army surgeon probes a wound. Unnervingly, it reminds him of Riel, sitting in the corners of his mind like a raven in the rafters, listening, watching.
The prodding vanishes, as if she has realized his discomfort and left as a courtesy.
"I can help you, once we have fought our way into Persian territory and conquered the Spirit Eaters," he says. "Though there will, of course, be a price to pay."
Ada's black arched eyebrows rise up questioningly and Aristotle's lip twitches.
"What is that price?" she asks.
"You are unmarried, I think?" he replies. "No children?"
She shakes her head.
"Make me your heir," Alex continues. "Make it publicly known to your people that I will be their king when you die."
Ada strokes her long black hair and purses her mouth. She looks up into the sky, now a glorious blue with white puffy clouds scudding across. "It is a tempting offer," she says silkily, "but my people are devoted to my royal house. They would never accept a foreigner, an outsider with no family relation to us."
Alex's mind churns. If she were younger, he would ask for her hand in marriage. "If you legally adopted me as your son," he says, "would your people accept me as their prince and future king?"
Aristotle chuckles; Ada simply raises an eyebrow, but the suggestion of a smile plays at the corner of her lips. "I suppose, in that case," she says slowly, "they wouldn't have a choice, would they?"
Alexander grins.
"So, you'll help me, then? If I officially adopt you as my son?"
Alex shakes his head. "There is one more thing I require of you."
* * *
"We will agree to form a league, but we will not agree to King Philip as its leader," says Chares, general of Athens, standing in a semicircle with the other generals in front of Philip's impressive field tent. The midmorning sun has chased away the night chill, but still Alex shivers, wondering if his plan will work or just make him look like a fool.
The Athenian continues, "We will vote on a leader. Or take turns with different leaders. What gives Philip of Macedon the right to rule us all?"
Philip, splayed in his ivory-inlaid camp chair, swigs wine from his skull cup, leans forward, and roars, "The right of conquest, perhaps?"
Alex, standing to the right of the king's chair, hides his smile. Though he usually hates his father's boorishness, at times like this it can be effective. He glances over at Heph, standing with Parmenion and the other Macedonian generals, and their eyes lock with just a hint of shared amusement.
When Alex returned to their tent this morning and told Heph what had happened, his friend was deeply disappointed he didn't get to see Aristotle—who is already on his way back to Athens—and Ada. But he was excited about the plan they devised to seal an alliance among these contentious warriors.
The red-faced General Theagenes of Thebes, whose army suffered the greatest number of fatalities in yesterday's battle, shifts from one foot to the other. "He has a point, Chares."
"Fools!" cries the Spartan commander, Agis. Dark-eyed, hook-nosed, and black-bearded, he seems as fierce as the reputation of his homeland's warriors.
"Why we ever came here to negotiate is beyond my understanding. You still have a strong army of twenty-seven thousand men! With Sparta's aid, you would be back to thirty thousand! Why let this one-eyed northern barbarian—he isn't even Greek, technically—rule over us?"
"Bravely said by a man whose forces didn't even fight yesterday," Philip sneers, his formidable bulk rising from his chair. "Who came to _observe_. And complicate the negotiations." Sun glints off his golden diadem and the rondels on his leather breastplate. Despite his battered face and coarse ways, in this moment, Philip seems every inch a king.
"Oh, we could give you another battle," he continues. "And kill more of your men. And march into Athens and Corinth and Sparta and burn your cities and rule them ourselves. But I am offering to let you keep control of your cities as long as you contribute to my army and follow my command."
He stands as tall and solid as an oak tree, hands on his broad hips. A brisk breeze picks up his gold-bordered red cape. "I have heard from spies that Persia is planning to attack Macedon. Do you think they will stop there? Once on the mainland, do you honestly think they would not rush down to grab the rich prize of Athens?
"Chares," he calls, and the lean, dark-haired Athenian commander snaps to attention. "Has your city forgotten what the Persian pricks did a hundred and fifty years ago? How they demolished your Parthenon, killed your men, and stole your women?"
"And who was it that saved Greece from the Persian Plague?" Agis counters, taking a step out from the group. "Was it not the Three Hundred Spartans, led by my own ancestor Leonidas, who blocked Xerxes's armies at Thermopylae until the other Greek armies could arrive? Where was Macedon to help us then? Living in caves and stealing each other's cattle!"
Alex sees a vein throbbing next to Philip's missing eye.
"If we form a league to stop Persia," Agis continues, gesturing wildly, "I say Sparta should lead it! We earned that right from the blood and glory of our men!"
"If that right is earned by blood and glory against Persia, then Athens should lead," says Chares, squaring his shoulders. "We stopped the Persian army during their first invasion, at Marathon. And sank their navy after Thermopylae at the battle of Salamis. They have not attacked us since."
"You pack of jackals, asses, and fools!" Philip jeers. "Who cares what your ancestors did centuries ago? Where were all your fine, glory-coated forefathers yesterday when I defeated your armies in battle? Ashes in pots and moldering bones in fields, that's where they were! Yesterday I handed you a few thousand more, and right now I would gladly add to that number!"
Red-faced, the Greek negotiators reach for their swords, forgetting they had to turn them in to the guards on the perimeter. Agis balls his fists, and Alex fully expects them to begin throwing punches. Heph shoots him an urgent, knowing look. It's time.
He's right. Though Philip told Alex to stand silent and impassive next to the throne and watch, he must intervene. _Now._
"It is for the gods to decide this," he says at the top of his voice, stepping in between the snarling generals and Philip.
"Sit down, boy!" Philip growls, pushing him aside. "I told you to stay quiet."
"Who is so impious he does not trust the gods?" Alex asks, holding his hands up to the sky. "You, Agis?" He points to the Spartan, who freezes. "You, Chares?" The steely blue eyes beneath thick dark brows meet Alex's, and Chares shakes his head.
"Is there anyone here who will not leave this to the gods?" he asks, walking in front of all the negotiators. No one says a word.
Phoebus, the red-haired Megarian missing a chunk of his nose, says, "Yes, we could ask the gods to decide, Prince Alexander. But how do they tell us the leader they want?"
Philip's single reddish-brown eye stares at Alex. "Yes, how?" he grunts.
"King Philip won the battle yesterday," Alex replies evenly. "He will sacrifice to the gods, and we will wait for them to give us a sign."
A few minutes later, Philip has sacrificed the lamb and placed its limp body on the oil-soaked pyre. Philip lights the wood with a torch. The fire quickly crackles to life.
"I have no idea what you're playing at," he whispers hoarsely to Alex as they back up from the heat of the blaze.
"Hold your hand up and call on the gods to confirm your right to lead," Alex whispers back.
Philip turns his scarred face to Alex and frowns. "What do you have planned, son?" The rising roar of the fire masks his words from the others.
Alex bows his head. "Trust me, Father." The exchange of terms: _father_ , _son_ , says everything.
"You'd better hope this works," Philip growls beneath his breath. He raises his right arm and cries in a deep voice, "Oh, gods of Olympus, show that you have chosen me to lead these armies against the foul barbarians of Persia! Zeus, Apollo, and Ares, give me a sign from heaven! A sign these idiotic Greek fools who rest on the honor of their long-dead ancestors cannot deny!"
Thick black smoke wafts all around them, making Alex's eyes water and his throat raw. Philip, his arm still raised, casts him a scathing look. But then Heph points toward the river as his voice rings out. "A sign!"
A falcon wings its way toward them. The Greek generals mutter and shout, shading their eyes with their hands to see better. The falcon flies right over them, through the smoke...and lands on Philip's outstretched arm.
There is something in its mouth.
The king throws his head back and laughs loudly. The bird steps sideways on long black talons and deposits on the king's head a branch of laurel—the symbol of victory and anointing of humans by the gods.
The entire area goes silent.
The other generals look at each other in awe.
Then, one by one, the negotiators take off their helmets and fall to their knees, heads bowed, praying loudly. All except Agis, the Spartan, who stands with his mouth gaping open.
Alex, too, kneels, his golden helmet tucked under his arm. Brightly polished, it reflects the early-morning sun, and a new red plume waves bravely from its center. Ada tilts her head and slides her sharp yellow gaze from the helmet, to Alex himself, then across the entire assembly. Her beak opens, as if she, like Philip, is laughing. Instead, there is a piercing caw. And then, with a flapping of heavy wings and a whirl of feathers, she flies up and out over the plain. Back to her fortress in Alinda, Alex knows, where she will wait for him to rescue her kingdom, as was their promise.
* * *
Philip sits behind his camp desk, muscular booted legs propped up on the table, and belches. He has eaten and drunk much at the feast to celebrate the new League of Corinth, a name less likely to ruffle Athenian feathers than the League of Macedon. After the clear sign from the gods in the form of Ada, the other generals—except the Spartan—agreed to a treaty, and voted Philip the strategos, the commanding general. This was enough: Sparta by itself is no threat against the combined forces of the rest of Greece and Macedon.
"Tell me how you did that," the king demands, throwing his crown on the folding camp table with a clatter and scratching his scalp.
"It must have been the work of the gods" is all Alex says, seating himself on a camp chair. The light from dozens of lamps flickers around the tent, and in the center, a small brazier of perfumed wood keeps the night chill at bay, its fragrant smoke drifting upward through the opening.
"You are a strange boy," Philip mutters, shaking his head. "And no boy of mine. If I didn't know it before, I know it now."
Alex feels his entire body go rigid, his blood turn to ice.
"I don't know how you did that thing with the falcon," the king goes on, "but you have always been...different. The way you tamed the wildest horse in the known world at the age of thirteen. I'll never forget it."
Alex cannot move.
Finally, Philip waves a large hand. "This doesn't change anything, of course." Now it seems as if it is the king who has the power to read minds. "Yesterday in battle and today in the negotiations, you proved yourself worthy as my heir. And, I suspect, your legacy will one day burnish my own."
Alexander sits back, dizzy with relief. And something else, too: something like real joy. Philip has never told Alex he loved him—has probably never said those words to anyone on earth—and his son never sought to ask it.
Now he doesn't have to.
Hatred and love are not, he suddenly realizes, opposing forces, after all. Perhaps they are folded into each other, inextricably tied up in the pain of wanting something you fear you may never receive.
Philip seems to feel the weight that has settled into the air between them. He lets out a rough laugh. "But your inheritance won't matter for a long, long time, boy, so don't get your heart set on having it anytime soon. I've got a lot of fighting years left in me!"
"May the gods grant it," Alex says with a smile. Then, after a long pause, "Now that we have an alliance, when do we leave for Persia? How quickly can you make arrangements?"
Philip swings his legs off the table and turns to Alex. "I will spend a fortnight seeing to the readying of allied forces," he says. "Then I will leave Parmenion to make the final preparations while you and I make a little trip west."
"West?" Alex says, much more loudly than he would have liked.
"What did I tell you about leaving enemies behind when you're gone?" Philip asks, his one eye twinkling with mischief. "They can take advantage of your absence to bite you in the ass."
Alex rubs his forehead. More delays. How can this even be possible? "What enemies do we have in the west, Father?" he asks.
"Oh, an asp of the most dangerous kind, especially now that Cleo is beginning to show signs of the new shape of things," Philip replies, leaning in to whisper in his ear. "Your mother."
Of course.
"The snake herself is currently nesting in Epirus with her snake brother, King Xander, who was never a great fan of mine." He gestures to a scroll on the desk. "Before we leave for Persia, son—" he smiles broadly, revealing yellowed teeth "—we need to solve the problem of these loathsome snakes once and for all."
Alex has a sinking feeling in the pit of his stomach. "She can't hurt you," he begins, but Philip gestures impatiently to silence him.
"Her brother has an army," the king points out.
"Epirus has never been allied to either Athens or Sparta," Alex counters.
"Epirus is currently not allied to Macedon, either," Philip retorts, "given the fact that I for all intents and purposes dethroned Olympias when I married Cleo. Father Zeus, I couldn't have those two women living in the same palace."
"What are you going to do?" Alex asks in a low voice. "Invade?"
"Oh, Hades, no," Philip replies, chuckling. "That would be a waste. I am going to convince Xander to join the League of Corinth in our invasion of Persia by using your mother's own weapons against her."
"And what are those?" Alex asks, uncomprehending.
The king laughs loudly. "You'll see. I enjoy little surprises as much as you, my son."
CHAPTER FOURTEEN
ZOFIA
A house of sin produces an incredible amount of dirty laundry, even on a normal day.
But one that secretly trains women in the martial arts is another matter altogether.
Zo can't help but grunt as she shoulders a basket of sodden sheets and exercise tunics through the brothel's service courtyard, a small, cluttered place housing the laundry, kitchen, gardening shed, and other work areas. Though separated only by a tall ornamental gate, this courtyard seems a world away from the luxurious pool and burbling fountain of the main courtyard, where women entertain clients by evening and refresh themselves with a swim in the cool of the morning.
No matter her level of exhaustion, every day since she arrived here in Dardania, home to the recently murdered mad king, the vanished queen, and this now-famed brothel, Zo has worked harder than she ever has in her life, despite the nightmares that leave her ragged at dawn, despite the unwieldy weight of her belly and how it makes her have to pee at the most inopportune times. She has to work hard—she knows she'd be kicked back out onto the streets if she didn't. More than that, she doesn't want to disappoint Alecta.
The day she knocked timidly on the front door, obviously pregnant, her faulty Greek bearing a heavy Persian accent, she expected Alecta to turn her away. But the redhead looked her up and down and nodded briskly. "I hope you're prepared to sweat" was all she said, beckoning Zo in. The relief and gratitude that flooded Zo then was enough to make her swear loyalty to Alecta forever.
But it's more than that. Working hard, throwing herself not just into chores but archery, spear throwing, and sword fighting—everything except for wrestling—helps her forget the heartache that tears at her from the inside. Helps her forget the fear of what may come, fear of the Assassins, of Fate itself, of the unknown.
Sometimes, on a good day, she pushes herself hard enough to burn away even the dreams that normally haunt her as soon as her eyes fall shut—dreams of the Spirit Eater, whose face morphs into Kohinoor's over and over again.
Yesterday she was practicing with a broadsword, and now, reaching up to hang the sodden sheets, one by one, on the clothesline, her sore muscles ache so much that her arms begin to quiver. She stops to catch her breath and rub her round belly. It has been just about nine months since Kohinoor told her she was with child, and Zo knows the baby must be coming soon.
Her heart snags in her chest. Cosmas, the father of the baby, is dead. Murdered. Kind, sweet, dutiful Cosmas. And yet— _what kind of person is she for thinking this?_ —it isn't the loss of Cosmas that hurts. It is the loss of Ochus, the man she loved after him, and loves still. The man who killed Cosmas. Abandoned her without word, without a reason.
The same man who lied to her, who held her captive, who led her into the mouth of death more than once.
And yet...
He is also the man who _couldn't_ kill her, who insisted on protecting her. And somehow, despite his roughness, despite how impossible it was to read his feelings, despite _all of it_ , she had opened up to him, had fallen for him, hating herself the whole time but unable to stop herself from seeing what was good and brave in him.
And eventually, he had confessed that he loved her, too.
Sometimes, in the lamp-lit night, when the other women gossip about the pleasure men seek from the brothel house, Zo can't stop her mind from traveling back to that starry night in Macedon, several months ago now, when Ochus whispered promises against her skin, when he held her so protectively she could have sworn he would never let go—and then her legs and belly tingle and her knees go weak from a sensation that is most certainly not tiredness.
That was the night before he left her with no message, no explanation at all.
Why did he do it—kill Cosmas and betray her? What did he stand to gain? Was he driven by jealousy or just plain cruelty?
These are the questions she cannot answer, and it torments her even more than her fears of being followed, even more than her nightmares of the threads of Fate itself twining around her ankles like vines and dragging her into the dirt, burying her alive.
There is so much to fear, even in this haven of safety and satisfying work.
And yet there is one thing that she cannot find it in herself to fear:
The creature in the diamond cage.
She can hear it stirring restlessly even now, in the storeroom off the laundry. She's not afraid—even though she saw what happened, saw what the creature did, how it swooped down on them, darkening the entire sky. How it attacked a former tavern keeper's wife, and when it was done, left only a smear of entrails, bones, and hair.
In the bloody chaos of those horrifying moments, Zofia had run into the service courtyard and hidden in the laundry. Huddling there with the laundress, she had witnessed nothing short of a miracle. King Pyrolithos called forth fire and ash to ensnare the monster in an enchanted cage. He is not just a king, she realized, but a man of incomparable magic power. The brothel women, who were suspicious of him and his purported ties to Queen Cynane before, now accept him readily. Protected by a wizard with powers like these, they would have nothing to fear.
Except, of course, the Spirit Eater.
Still, she isn't scared of it, only oddly mystified. Perhaps that is because whenever she approaches, it seems to go quiet, as if watching her closely.
Zo gathers another basket of wet laundry from the cooling cauldron in the wash room. But instead of going back to the clothing lines, she steps into the storeroom, squinting in the darkness at the monster. The bars of its cage seem to glitter with their own light, a kind of magic Zo has never heard of before, powers possessed by King Pyrolithos and no other.
The Spirit Eater itself is impossible to describe, even trapped as it is in this too-small, gleaming cage. The creature is all shadow and movement, all cold wind and hissing, all fang and claw and black leather tongue, forked like a snake's.
Zo has heard of terrifying creatures before, those that have risen from the ashes of myth and are rumored to still exist, like the Hellion—which, it is said, Queen Cynane of Dardania once tamed and rode, back when she was a mere princess in Macedon. And, of course, Zo has seen another such creature, less terrifying but just as awe inspiring: Vata, the white-winged Pegasus, whom she rode through the clouds to Persepolis.
But the Spirit Eater is unlike any such beast of myth. It is somehow fleshless and shapeless, constantly shifting, sometimes looking like more than one creature, making her wonder if something is wrong with her eyes or the light. Or perhaps the creature itself is made of light and shadow, constantly changing and moving. Yet it _must_ be made of flesh, because it devours flesh.
Then again, it is said that in the time before time, the Spirit Eaters also devoured the gods.
Now, staring at the creature, trying to take it in, to understand what she's seeing, Zo marvels at the strange synchronicity of fate. At Kohinoor's suggestion, she traveled far into the Eastern Mountains to find a Spirit Eater, in the hope of altering her fate, saving herself from having her blood mingled with that of Macedon's. She had never wanted to marry Alexander—she'd always yearned, simply, to be free to choose.
And yet, despite her unbelievable, excruciating journey, it is only _here_ , on western shores, that she has found not only a Spirit Eater but also a fate of sorts that she likes. Fate is strangely circular, she sees now, like a snake eating its own tail.
"Can you change mine?" she whispers. A forked tongue—or is it two?—emerges from the Spirit Eater's mouth and licks its black leather lips. If that is an answer, she doesn't understand it.
Despite knowing now that Kohinoor can't be trusted, she still wonders what the old woman intended when she told Zo to seek out these shadowy monsters.
She bends down for her heavy basket and cries out as a pain rips through her abdomen.
Someone is at her side in an instant, holding her up.
"Is it time?"
Alecta. Her arms around Zo, guiding her, taking on her weight.
Another cramp seizes her and she gasps.
Now that Alecta stands between Zo and the cage, the Spirit Eater has begun to lash out, Zo realizes, releasing its unbearably cold breath, slashing its claws out between the bars. If Alecta noticed it had been strangely calm when she first entered the storeroom, she does not say.
Gulping in air, Zo straightens up. "No," she says. "I...don't think so."
Alecta places her hands on Zo's shoulders. "It is too much, Zotasha," she says softly, using the mostly fake name Zo gave her on arriving. Anyone looking for the princess missing from Pella would be searching for Zofia.
Zo starts to protest, but Alecta shushes her. "Come, I will have Zara finish the laundry. You shouldn't be in here with the... _beast_...anyway."
Many of the other girls are terrified the Spirit Eater will escape and devour them in their beds, Zo knows. It is a reasonable thing to fear.
Alecta leads her to a door next to the garden shed, takes a key off her belt, opens it, and gestures for Zo to enter. It's a small room, lit only by the smoke hole in the roof. A girl Zo has never seen before sits at a table, chopping herbs with great concentration. She has a lovely profile, with smooth pale skin, thick dark lashes, and luxuriant black curls tumbling down her back.
"Yes," she says, not looking up.
"I have brought you the helper you requested," Alecta says. The girl turns and looks up expectantly. Zo gasps. The right side of her face has been ravaged by fire or acid. It's a mass of twisted red scars and lumps. The girl, staring hard at Zo, doesn't seem terribly impressed with her, either.
"Zotasha, this is Badri," Alecta says firmly. "You will do as she says. You will find this work less physically demanding."
Zo nods, and Alecta leaves them alone, locking the door behind her. Despite trusting Alecta, a shiver of uncertainty works its way up Zo's spine. Gentle as she was just now with Zo, Alecta could turn on her.
"Well, sit down and make yourself useful, if you _can_ ," Badri says curtly, in Persian, with a swift glance at Zo's belly.
Despite the girl's coldness, Zo finds it a relief, after all these months, to hear the elegant lilting music of her native tongue, even if it is spiced with nastiness. The Persian language is like a song, or poetry—medicine for her soul.
Badri pushes a small knife and a mound of dried roots toward her. Zo assumes the girl is preparing medications until Badri says, "Deadly nightshade."
Zo's heart skips a beat. Everyone knows deadly nightshade is one of the most poisonous plants on earth.
"Peel off the skin," Badri continues, "and once we have finished this pile, we will pound them with mortars. Don't touch your eyes or face. If you put your fingers anywhere near your mouth, you will die."
Zo stands there frozen for a moment. Then she swallows hard and pulls up a stool and sets to work, stripping the skin off the roots with the sharp knife blade. She works efficiently and wordlessly. Once she and Badri have finished skinning the roots, they mash them with mortars and pestles, creating a thick, gooey paste.
"What is this for?" she finally asks in Persian.
Badri eyes her with evident hostility. "Why do you want to know?"
"Well," Zo replies evenly, "I don't suppose I really need to. But the fact that Alecta brought me here means she trusts me. So, I think you can, too."
Badri blinks. "She might not know you for who you are, but I am not so stupid. Even Dardanians who speak Persian cannot tell a noble accent from one off the streets. They can't see that you are highborn." Badri scans her up and down. "Say something. In our native tongue."
"I don't know—"
But before she can finish, Badri cuts her off. "A high-ranking noblewoman is my guess. Maybe even a princess. So, tell me, what king or lord did not approve of his daughter's choice of lover?"
Zo can feel herself blushing furiously. "My story is none of your business." All the warmth she felt moments ago at the familiarity of the language has fled.
Badri continues to mash the deadly herb. Without looking up, she says only, "I imagine you are no stranger to fine jewels."
Is this another insult? Zo isn't sure. She doesn't like the way the girl seems to judge her. But then she notices a twitch at the corner of Badri's mouth.
"All of the women will be issued jewelry," she adds, still without looking up. She attacks her paste once more with the mortar. "Necklaces, bracelets, rings, and earrings, all with secret compartments."
"Secret comp— _oh._ " Zo recalls how often Alecta has said that even the strongest woman is unlikely to beat a man in battle if they are one-on-one. That they need other weapons to even the score: deception, surprise...and poison.
"We will also be given poison darts we will blow out of tubes made to look like flutes. If a man sees us pulling one out, he will think we are going to play music for him."
Very clever, Zo has to admit.
"So, where are you from?" Badri asks. "Near the west coast, from your accent, I think. Halicarnassus? Or no, Sar—"
"Sardis," Zo replies, putting down her mortar, astounded at Badri's ability to guess at both her rank and territory of origin in just a few minutes. The girl must be very studied, or—
"Don't stop!" Badri commands, glaring. "Not until I tell you to. The plant must be ground down while it is still moist."
Zo picks up her mortar and returns to mashing. They work in silence for a while, and Zo must admit to herself that this is easier work than the laundry, even if it does have the potential to kill her. When Badri judges the mashing to be complete, she and Zo spread the poisonous paste on wooden boards to dry, then set the boards on a table in the far corner.
"Over the next few days, it will turn to a powder," Badri explains. "Now, wash your hands in this basin," she commands, pointing to another table. "Very carefully."
After Zo has dried her hands, Badri hands her a painted clay cup. "Come, we have earned a bit of wine."
Zo looks into the cup, hesitating. Would the girl test her? Poison her just because she can?
"Personally, I prefer to be on top," Badri says.
Zo glances at her in embarrassed surprise, until she realizes Badri is commenting on the cup she handed her—on which a pair of figures are painted. The figures are very clearly having sex. Sometimes Zo is so exhausted she almost forgets that she lives in a brothel now.
Badri sets down the pitcher and starts to laugh, a pleasant, silvery sound at odds with her fierce demeanor and disfigured face. "That blush will get you in trouble, girl!"
Zo begins to relax. She sips the wine and happily bites into the honey cakes Badri offers, no longer worried about being judged—or poisoned. Badri becomes almost chatty, gossiping about Alecta and the other women she watches in the courtyard through a peephole in the door. Zo realizes that Badri is glad of the company—it seems she rarely leaves this room—and happy to speak her native tongue again. Zo begins to like her sharp humor.
"Where did you learn these skills?" she asks, gesturing to the poison drying on the planks.
"Ah," Badri says. "A long story. My family lives in Apasa. But my mother's younger brother, Kadmus, became a general in the Macedonian army."
Zo's heart skips a beat. _Macedon._ "Oh?"
"Great King Artaxerxes doesn't make all the decisions where our fine nation is concerned," Badri says, leaning toward Zo confidentially. "Given your status, you may have heard the whispers that he is under the thumb of other interest groups."
"Other interest groups?"
Badri rolls her eyes. "Have you really not heard of the Assassins?"
Some wine catches at the back of Zo's throat and she coughs.
"My uncle was made a spy against his will. If he didn't report on Macedon exactly as the Assassins demanded, well... My mother and I were their collateral."
Shaking, Zo pours herself more wine to calm her nerves—why does fate seem to twist around and around her no matter where she goes or what she does? How can this girl—hundreds of miles away from Persia, sitting across from Zo in her safe haven—be the niece of an Assassin? They say fate is a snake chasing its own tail, but this snake is swallowing itself whole.
"The Assassins took me as a hostage for my uncle's good behavior," Badri continues. "One of the higher-ups, Zand, forced me to become his lover."
"You mean he...he raped you?" Zo asks in a small voice. She still remembers her near rape on board the ship that took her to Macedon, by a foul-smelling sailor Ochus fought off her and then threw overboard. But if Ochus hadn't arrived in time...
Badri's eyes are distant. "He was ugly and cruel, but he had a soft spot for me. I grew used to him—enough to pretend. And, after a time, it worked; I became indispensable to him. He insisted they teach me a useful skill, the art of making poisons. I had been a skilled maker of medical remedies at home, you see. This work is much the same, only its purpose is to destroy rather than to heal."
Another sharp spasm passes through Zo's abdomen, like the harsh grip of a poison. She gasps, wincing. But Badri doesn't see it. She's staring fixedly into her past.
"It would have been all right, I think, if I hadn't fallen in love with someone else," she says softly. "Kazem. Tall and handsome and brave. We met secretly, when Zand traveled. But somehow, Zand found out. One night, as I slept, he painted my face with burning poison. I was two-faced, he said, laughing as I screamed in pain and tried to wipe it off. Now the world would see both my faces, he said. That same night, Kazem had an accident, it seems, and fell out of a high window."
Badri pushes her chair away from the table and clears the dishes. "I was closely guarded and had no choice but to stay on," she says, setting the cups and plates on a tray in the corner. "I worked hard making their poisons and never once complained about what he had done to me. Zand still wanted me. In some sick way, I suppose I should've been grateful for that. He said he was doing me a favor when he took me back, as no other man ever would."
Zo feels another sharp pain but grinds her teeth together to keep from crying out. Badri's story must be making _her_ sick. Rape, murder, poison, disfigurement. She wonders if she is going to throw up.
"And then one night, after a banquet, I realized that Zand and all the guards were drunk. I stole their coin pouches and fled, heavily veiled. I went first to Pella, to find my uncle. But Prince Alexander had stabbed him in a rage, it was said, and Kadmus had dragged himself off somewhere to heal. I couldn't discover where. How can a girl with half a face support herself? No one would hire me as a waitress or shopgirl. Not even as a whore. And then I heard of this place and came here. I know the Assassins, and I know Zand. They are looking for me. And a girl with half a face is talked about. These past weeks, Alecta has kept me in here, hidden and safe, making poisons for the New Amazons."
So that was why Zo hadn't seen her before today.
Zo's heart starts to pound as she wonders if Badri knows Ochus, knows, perhaps, where he is. Yet how can she ask without giving away her own identity?
"Once, in Sardis," she says, knotting her hands. _Did she wash them well enough? Is she dying even as they speak?_ She winces with another cramp, but tries to ignore it. "There was a sumptuous banquet at the palace. I looked down from the women's viewing chamber and saw the Great King. He was as tall and straight as a spear. I also saw his chief advisor, Darius, a small dark man. And Darius had a very handsome son striding around. I forget his name."
"Ochus," Badri replies, leaning back in her chair, still not observing Zo's discomfort. "Darius is chief of the Assassins Guild. Ochus works with him."
"Are they in Persepolis with the Great King?" Zo asks, hoping her voice is casual. She's not sure what answer she wants to hear. That Ochus died bravely? That he survived and was revealed to have been working in solidarity with his father the whole time?
Badri shrugs. "The last I heard before I escaped from Apasa was that Ochus was under suspicion for not following orders. That he had botched a mission and disappeared, obviously afraid of the Assassins' wrath. No one had heard from him. But I imagine he will appear soon," she adds, barking out a laugh.
"Why's that?" Zo asks, as something like hope flickers in her chest.
"For months, rumor has it Darius has been planning to assassinate the Great King, who has become a bit suspicious about what his advisor is really up to. If he succeeds, Darius will become the next Great King. And Ochus, as his only heir, will be brought back into the fold. I imagine Darius will explain that whatever it is Ochus did was, in fact, on secret orders."
Zo feels a warm flush spread through her—if Darius is king, surely Ochus, wherever he is—will hear the news and return. But what will it mean? Will he return to the umbrella of his father's protection? Become the new Chief Assassin himself? A pang of discomfort knots in her chest, making it hard to breathe. Will Darius, as the new Great King, force Prince Ochus to track Zo down and kill her once and for all, as punishment for escaping Macedon and ruining his plans?
The child within her gives a hard kick, straight into Zo's ribs, and she feels another stab of pain, this one unbearable—she cries out as hot water gushes from her.
Suddenly, without knowing how she got there, she is on all fours on the floor, panting in pain. Then it ebbs away, leaving her drained of strength. She tries to lie down on her side, but there are hands gripping her shoulders. "Lucky for you," a sharp voice says in Persian, "I am also a trained midwife. Let me help you over to my bed."
The next hours are a blur of grinding cramps slashing rhythmically across periods of utter exhaustion. Badri ties ropes to the top of her bed for Zo to hold on to when the contractions sweep over her. When she feels the discomfort start to rise, she grabs the straps and twists until it drains away and she goes limp as a rag doll. Her hair has become sopping-wet tendrils of sweat. Her skin is slick with it.
Sometimes, Alecta is there, mopping Zo's brow with cool mint water and offering words of encouragement. Badri gives her herbed wine to drink to keep up her strength, though once, when the pain becomes unbearable, she throws it up. At one point, lying motionless, waiting for more pain, Zo notices starlight filtering in through the smoke hole and flickering oil lamps set on the tables.
She has no idea how long this has gone on. Has it been days, or only hours? Each contraction leaves her weaker than the one before. The air smells thick and coppery. The sheets are soaked with blood. Is it too much blood? Is she bleeding to death?
She has been through so much during this pregnancy—captured and starved by slavers, riding hundreds of miles along the Royal Road as Ochus's prisoner, lost and wandering the Eastern Mountains, slashed by the claws of a Spirit Eater, falling from a Pegasus, drugged by Kohinoor—she wonders if it has all been too much. But despite it all, she has trudged on, endeavored to save herself so that she could save the child. Maybe it was all for nothing. Perhaps both she and the baby will die. Perhaps this time tomorrow their bodies will be consumed by flames in the funeral pyre, their ashes shoveled into a pot, and... She gasps as a new eruption of pain splays through her body like poisonous tentacles.
"It is time, Zotasha," Badri says, an urgent edge to her voice.
With a shriek, she finds the last vestiges of strength and pushes as hard as she can. Below her, Badri and Alecta wait, arms extended. "I see the head!" Badri cries. "Push harder!"
But Zo has nothing left to push with. Absolutely nothing. "I can't," she whimpers.
"Can we pull him out?" Alecta asks, her voice humming with fear.
"We'll have to do something, or he'll suffocate," Badri replies. Zo, eyes closed, feels prodding, tugging fingers. "Oh, goddess Anahita, he's stuck."
Waves of sickening pain thud through her, but they diminish as she feels herself sinking into unconsciousness. The voices around her fade, and she closes her eyes. She has never been so tired before. She will slip into blissful sleep. Nothing really matters anymore...
A bucketful of cold water slaps Zo's face. Sputtering and rubbing her eyes, she looks at Alecta's angry face.
"If you don't push now, your baby will die, and Badri will get her knife and pull it out in pieces. And while you will probably survive, it won't be for long, because I will personally feed you to the Spirit Eater," she snarls.
And yet it isn't fear that makes Zo push, but the knowledge of love and support all around her here, suddenly surging up within her, powerful, unstoppable. These women—total strangers—are fighting so hard for her. And her baby. Fighting with her _and_ for her, in a world where sometimes even mothers leave you—like Zo's did.
Gritting her teeth, she pushes, thinking of the anguish and anger of her mother leaving when she was so young, of not being loved when she needed it, of constantly seeking love, of birthing love now, raw and real and beautiful love.
"Good girl!" Badri cries, as the child passes out of her, feeling like some enormous, wet sea creature. She sees the tops of the heads of the two women—one black as night, the other red as a fiery sunset—as they turn the baby over and examine it.
Zo lies back, stunned. She can see that the women are looking not at a giant sea creature but a tiny, pink-skinned baby. She wants to weep. But then she realizes something is wrong. It's the silence. The baby isn't crying. Why aren't the women saying anything as they look at the baby? Is it alive? Oh, all the gods of Persia and Greece, please let it be alive. After all this—
"A healthy girl!" exclaims Alecta, wiping Zo's sweat-drenched face with a cool moist cloth as Badri slaps the child, eliciting a fierce cry of protest from the baby, piercing the air with its pure, high sound.
A brief sob gushes out of Zo.
"Good thing, too, Zotasha," Alecta says, a joking edge to her voice. "We don't allow boys to live here."
"Can I see... Can I hold her?" Zo asks weakly, pushing herself up. Badri washes the baby with white wine and vinegar and swaddles her in a white blanket, then hands her to Zo. She peers into the tiny, crumpled red face. Wide dark brown eyes stare up at her.
"Oh, my baby," Zo says, her voice shattering into a million muffled sobs of joy. She nuzzles her—this child she has made, this new life, this new chance, this untouched one. The girl who was fated to come into her life.
And for the first time in nine months, she is truly happy.
* * *
Days of exhaustion pass, blurring into sleepless nights of feeding and dozing and soothing and pacing, of aching breasts crusted in sweet milk. Slowly, her body begins to return to its former shape, or some semblance of it. One day, when the child is a little over two weeks old, Zo looks into the cradle at the beautiful bundle sleeping peacefully for once, and is shocked by the sense of peace that descends on her. Even through her tiredness, she marvels constantly at her child's perfection. The silken dark hair, the curling fingers with perfect miniature nails, the smooth skin and sweet baby breath. It is as if all the beauty and sweetness in the entire world resides in this tiny girl, as if Zo has never known joy before the wondrous moment when she first held her in her arms.
Over the past two weeks, the baby has grown and thrived. This lucky infant has dozens of mothers; all of the women in the brothel have offered to watch her, bathe her, and change her diapers.
In Zo's family, it is tradition to name the first female child after her maternal grandmother, so she should have called her Attoosheh. But the very name makes Zo's hackles rise in resentment. Selfish and vain, the widowed Attoosheh ran off to marry a Bactrian king when Zo was only a few months old, leaving her behind in Sardis. The only thing Attoosheh ever did for Zo was to return twelve years later, widowed again, clutching another infant, Roxana, Zo's beloved half sister. Attoosheh ignored Roxana, too, and Zo was more like a mother to her than a sister.
Zo considered naming the baby Roxana as a way of making her sister live again. But she realized it would be unfair to burden this new soul with such painful memories. That first night Zo escaped from the palace to find Cosmas, Roxana had followed her. The slavers who captured Zo also caught Roxana, but at six years old she was too young to do hard labor or work as a prostitute. They had taken her behind a stand of wheat and killed her. Her little sister's screams still haunt Zo, still make her heart ache with the loss.
Finally, Zo decided to call her daughter Mandana, the name of the old nurse who raised her for over a decade while her mother, Attoosheh, was away. Loving, generous, and warm, Mandana had nursed Zo when she was sick, played with her, told her stories, and listened to her problems. Zo has missed Mandana, too, and longs for her toothless grin and bright brown eyes alive with mischief.
"I can watch her now," says a throaty voice. Zo turns to see the stunning Ethiopian, Sophonisba, standing in the doorway. "Look!" she cries, holding up two tiny tunics—one blue and one green. "The girls have sewn some more!"
Smiling her thanks, Zo makes her way downstairs and across the courtyard. Her swollen breasts are painful at times. The rest of her is still somewhat sore and swollen, and she wonders if she will ever feel beautiful and young again. But it's been two weeks, and she has to get back to work. Her determination to contribute is no longer a silent appeal for Alecta to let her stay. It's a resounding thank-you to both Alecta and Badri for helping her through her labor and saving her baby's life.
As she approaches Badri's room at the far end of the service courtyard, she feels drawn to see the Spirit Eater. She opens the door, and sunlight falls on the blurring black form in the cage. But this time it no longer sits placidly when it sees her. It roars in rage and hunger, flinging itself against the bars of the cage, sending foaming flecks of spittle onto the floor. Its snarl seems to wrap around her, as if trying to pull her in and suck something out of her.
The natural fear that was absent in her before surges up now, and Zo leaps back, shocked. The creature has never done this in her presence before. What makes it so vicious today?
Could it be... It must be Mandana. All the times Zo came face-to-face with the monster before, she was pregnant. Today, the baby is upstairs in the house. Is it something about Mandana that tames the Spirit Eater? The creature continues to howl, shaking the diamond bars. Saliva continues to fling out at her in a sickening spray.
Zo stands perfectly still for a moment, stunned, trying to figure out why it has changed its behavior. Then, by some unspoken instinct, she turns on her heel, races back to her room, and takes little Mandana from Sophonisba's arms, causing both the woman and the baby to each give a startled cry.
Moments later, she is back in the dim room below, standing right in front of the Spirit Eater's cage. Sure enough, the angry movement, the hissing and lashing, immediately stops, and the creature seems to quiet, as if tamed. Zo, too, feels different with Mandana in her arms—safer, somehow, and calmer, too.
There can be no denying the change.
Nor its cause.
Mandana. The baby.
Somehow her presence has an effect on the Spirit Eater. It was never Zo's influence but the unborn child's that seemed to tame the monster. Mandana herself seems peaceful, curling into her mother and falling back to sleep with a soft buzz of breath.
Zo's heart races as she feels the tiny heartbeat of her daughter against her chest, held close. It _is_ Mandana who has calmed the creature. She's sure of it now.
Her mind runs in a million directions at once. This must have something to do with why Kohinoor so desperately wanted the baby. She remembers the ancient paintings in Kohinoor's cave in the Eastern Mountains, a winged infant holding a wax tablet rose above the earth as darkness descended from above...
Suddenly, Zo is shivering uncontrollably. Because she sees now that she was driven by fate to have this child, somehow. For reasons she still cannot fathom. She cannot know what will come, or how it will all take place, but neither can she shake the dawning conviction that the sweet sleeping baby in her arms may somehow carry the fate of the world in her tiny hands.
CHAPTER FIFTEEN
KATERINA
Torchlight flickers over the rows of pilgrims bedding down for the night on the cool marble floor of the temple. Kat bends down in front of Kephos, an old man with a palsy and, smiling, helps him tip his sleeping draft into his toothless mouth. Jacob and two priests attend to other patients, offering words of sympathy or a joke. This is the ritual every night in the Temple of Asklepios, a sanctuary dedicated to the god of healing, outside the ruins of Troy.
When they arrived from Byzantium, nearly five months ago, they found no sign of the Aesarian Lords. Just a huge mound of countless lumpy hills with trees growing out of them and heaps of fallen stones. Surveying this barren landscape, Jacob said, "Timaeus and the other Lords must have been delayed. We'll have to wait for them."
But _where_ to wait was the question. The midwinter wind whipping off the turbulent gray sea whistled right through their bones. And while they could have made a shelter on the fallen citadel, the shepherds who grazed their flocks in the area told them that at night the city was haunted, cursed. Demons and dark gods lived in tunnels beneath the destruction, they said. Those foolish enough to venture out at night saw smoke and fire rising from the crevasses, and heard unearthly shrieks and howls. Some people disappeared, never to be seen again.
And they believed it, in a way. This city is the site of one of history's cruelest wars, defined and marked by savagery and flame and loss, where it had once been a rich city of bustling trade and refined culture.
Troy was safe only during the day, when visitors sacrificed to Achilles, Patroclus, Hector, and the other heroes who fell here. As the sun set, travelers fled, abandoning its ghostly ruins to the wind and the memories still trapped and echoing among fallen stones.
The nearest village was a two-day walk. But one of the shepherds pointed them to the temple on a hill about a mile from Troy. It was one of several Asklepions, or houses of healing, in lands dominated by Greek culture, even here in westernmost Persia.
"People come from Lesbos and Byzantium for healing," the shepherd said, his weathered face an intricate map of lines and spots. "You could stay there for a while, if you help them with the work."
So they traipsed across the plain and up the well-worn track on the hill. There, at the top, stood the little temple, its mismatched columns of black and pink and white marble clearly plucked from the ruins of Troy itself. There was something both tragic and beautiful in their reuse: hope always rises from the ashes. People reclaim what they can and make something new.
The temple was the heart of a tiny village of sorts, surrounded by numerous other buildings: offices, stables, a blacksmith's, a kitchen, a henhouse, latrines, and a laundry. After offering to work for their board, Kat and Jacob settled into one of the many sturdy single-room houses in the back, most of them used by families of the sick.
When Jacob asked the priests whether they had seen any Aesarian Lords, they smiled and shook their heads, misunderstanding. "They do not bother us here," said Iolanta, a strong-jawed, silver-haired priestess, standing between the columns of the sanctuary and gesturing to the entire compound. "We have no magic blood." And here she cast an amused look at the two priests standing with her, who smiled slyly back. "We merely help interpret the gods' instructions."
But despite their denial of any magic blood in the sanctuary, Kat found herself fascinated by the temple's healing rituals and wondered if some of the priests were secret Snake Bloods. For one thing, her normally numb golden fingertip—the one Princess Laila of Sharuna gave her to replace the injury Cynane had done—tingled almost constantly, as it did when in the presence of Blood Magic. For another thing, the sanctuary was dominated by a statue of a bearded, muscular Asklepios, three times the height of a man, holding a staff, around which a huge snake coiled. Sacred snakes slithered throughout the temple and often wound themselves around the patients, staring into their faces as their forked tongues darted in and out. Then they returned to the priests and priestesses and seemed to talk to them about the person's illness, just as they would to a Snake Blood, Kat noticed.
Every night, bedding down inside the sanctuary, the patients were given a bit of opium infused in wine to help them sleep. The following morning, they reported their dreams to the priests, who interpreted them and, together with messages from the temple snakes, developed a healing regimen.
At first, Kat and Jacob helped the priests and priestesses with cooking and cleaning. They washed sheets and tunics, filled lamps with oil, swept the fine silky dust off the porticoes, and hauled water from the sacred spring for patients to drink and wash with.
Then, one day, Kat, who had been collecting food bowls from the pilgrims lounging in the sanctuary, found herself face-to-face with a large black snake in a patch of sunlight. Its head hovered right and left, and its unblinking yellow eyes stared at her, silently beseeching her for...something. She set down the bowls and sat cross-legged in front of it.
Breathing deeply, she closed her eyes and emptied her mind. Her longing for Alex, her guilt about Heph, the spot she wanted to scrub off Jacob's best tunic—everything drained from her. Then she became aware of warm stone under her scales. She remembered the joy of gliding in dark secret tunnels and the freedom of shedding a too-tight skin. She was still in her body but also inside the mind of the snake. She slithered across the sun-dappled marble tiles to Briseis, a wealthy middle-aged widow with a raw, flaking rash over half her body, sitting on her pallet in the shadow of the large statue of Asklepios.
The woman's eyes were sharp and clear, though crust and redness disfigured her eyelids. She set aside a scroll she had been reading and murmured words of welcome to Kat as the snake, extending her hand, for it is a good sign when the sacred snakes visit those in need of healing. Kat curled up in her lap, raising her head to look closely at the diseased skin. Flakes and runny red sores, torn skin and raw flesh. Then she sank out of the snake and into the woman's skin, into the blood beneath it, and traveled through veins into muscles and organs. She pulsated with the rhythm of the woman's heart. She became the intake and exhale of breath. She lost herself in the swirling bloodstream.
Briseis was...imbalanced. Something was...off. Not seriously, not anything that could kill. But something to irritate and disfigure. Images rose before her. Of undigested meat causing internal swelling and bad blood. This woman shouldn't eat meat. Of honey, golden amber, sweet and healing. Of tangy lemons for cleansing. Of sunlight, warm and clear. For years, Briseis had lived in the shadows, fearing what the sun would do to her skin.
Kat was no longer Kat, nor was she the black snake. She was pure Mind inside the woman, thought only, without form or voice. She was...
Lying sprawled on the hard, cold sanctuary floor.
"Katerina," said a kindly male voice. Confused, she pushed herself up to a seated position. Clonius, a bearded young priest, squatted beside her. "Are you all right?"
She nodded, rubbing her eyes. "Briseis," she said, gesturing to the woman on the other side of the sanctuary. The black snake had wrapped itself loosely around the woman's neck. "She must not eat meat. She needs to eat honey and lemons, and the honey should also be applied to her bad skin. Also, she needs some sun every day."
Clonius looked between Kat and Briseis, his eyes wide. Then he grinned. "Ah," he said. "I think I understand." He picked up the silver lotus blossom pendant hanging around Kat's neck and smiled. Lotus blossom helps Snake Bloods recover from their trances, bringing them back to themselves if they have been traveling in animals. "Have you had these experiences with animals before?"
Kat nodded, unwilling to say anything about them. "You are right to keep silent on such matters," Clonius says in a low voice. "There are those who would gladly take us east for execution if they knew. But if we are careful, claiming we have been chosen by the healing god to work with his sacred snakes, we will be safe. The Aesarian Lords respect the traditional gods."
The priests of Asklepios were secret Snake Bloods. Kat had been right. And yet Ada had never told her anything like _this_.
She looked at Clonius. "How—how was I able to do that? To go into Briseis's body to discover what was wrong with her?"
Clonius grinned. "These are special snakes with powers of their own, Katerina. For many centuries now, we have bred them in our sanctuaries to work with us in diagnosing the sick. They act as a medium between our own minds and the bodies of the ill."
From that day, she has worked with the snakes to help the pilgrims. Some of them were beyond healing, with growths curling around organs. But even then, she learned what foods and medications reduced pain and increased energy. It was striking, she thought, that Jacob had healed those with injuries in Byzantium—he could not cure those with illnesses—and she, as a Snake Blood, could heal those with illnesses in the temple—or at least suggest the right path toward wellness. She and Jacob were truly two halves of a whole.
Though Kat's feelings toward the snakes at first had been wary but curious, she now found herself opening up to this new side of her magic. Ever since learning that the forefather of Snake magic himself, the corrupted and vengeful god Riel, was her own father, Kat has struggled with her powers—the gift in her blood seeming more and more like a curse. On Meninx, she totally rejected them, until Ada encouraged her to embrace her abilities once more, and she tried. They had led her to Byzantium and to eventually confronting—and killing—Riel. To saving her twin brother. To reuniting with Jacob.
She should have been thankful, then, to her magic blood. It had, all this time, been showing her the way home. The way to do what's right. To live in her own skin. And working every day to heal the sick helped dispel the blackness wrapped around her soul ever since she had been mauled by the Spirit Eater.
And yet, awakening to her true potential was still fraught with fear and uncertainty and a longing to understand—what is it she is meant to do in this life, with these gifts, with these powers?
Every few days, she and Jacob walked down to Troy to look for signs of the Aesarian Lords, and every time they were disappointed. But they loved climbing over the ruins, speculating if this mound of rocks had once been the bedroom where the beautiful Helen had slept in the arms of Paris, the Trojan prince who had seduced her away from her husband, King Menelaus of Sparta. Or if that hillock was the tower from which Helen watched Menelaus fight the Trojans to get her back. As they walked over the flat land between the mound and the sea, they imagined where, exactly, the valiant Prince Hector of Troy fell, slain by the Greek Achilles, and pointed out the likely path of Achilles's chariot as he dragged Hector's body seven times around the city, punishment for killing Achilles's lover, Patroclus.
Sometimes, braving the cold, she and Jacob strolled on the wind-whipped beach, imagining the thousand Greek ships anchored here on the wine-dark sea. Was it here—or over there—that the Trojans found the great wooden horse, filled with silent, secret warriors, and foolishly took it inside the city? After the Greeks climbed out and opened the gates, after that horrible night of looting and raping and burning, the entire beach must have been filled with the newly enslaved women, chained and sobbing, doled out to victorious soldiers as war-prizes.
Sometimes Kat felt the tortured history of the city seep into her bones and settle there. For centuries, Troy had been a place of thriving prosperity, of palaces and fountains, of gold cups and silver platters, of gorgeous women bedecked in pearls and rubies, of music, dancing, and feasting. But in the space of a single night it had burned and collapsed.
So, what was the lesson of this painful past? To never trust again, to never let your enemies close, to never love? To suspect that any gift might, in fact, be a curse?
And yet...
Over the years, the wind off the plains had covered the ruins with dirt, from which delicate grass and tall, spindly trees now grew, swaying like young bodies in the breeze. The crisp smells of salt and sea filled the air, and the lack of human activity made the sky curiously clear of smoke and noise, made the whole place both ache with emptiness and whisper of space, of peace. Nimble goats chewed their way across the fallen city, their staccato bleating like laughter at passersby. As if to say what folly it is to be human, to be stuck in the past, to live on in painful memories when you could live in the moment, from root to root and blade of grass to blade of grass. When, stone by stone, you could rebuild.
Still, as much as Kat enjoyed their excursions to Troy, it was usually a relief to return to the temple, to the cheerful priests and priestesses and hopeful patients, to the snakes that spoke to her of healing. And to Jacob's warm, strong arms every night.
Days turned to weeks, and weeks to months. The sharp bite of winter lessened. The wind off the water no longer stabbed; after a time, it gently caressed. The battering waves calmed and lapped meekly at the shore. And thousands of green shoots sprouted on the lumpy mounds of Troy. The goats and sheep had babies that gamboled and leaped among the ruins. Birds came to land on the grave mounds of Achilles and Patroclus and many other noble warriors, buried with their armor and horses and rich grave goods. Sites of tragedy, scars of the past, they were now sunny spots where you could imagine spending a whole day lying out, thinking of nothing but the open sky—where you could forget about the dark truth of this place.
Where you could trick yourself into believing that magic—and life—was pure and good and easy, and no secret despair clung to your soul.
Jacob has never been so happy, Kat knows. She can tell by the smile always playing on his face, and the sparkling of his dark eyes. He could probably stay here forever, delighted to be healing and, most of all, to be living with her.
As for Kat, she, too, is happy. It is as if the wheel of her life has come full circle, right back to where she started, with Jacob. And she loves the healing work at the temple. So many in this world devote themselves to destroying, so few to healing.
And she _does_ begin to forget—what drew them here, what dangers may lurk beyond the walls of the temple, and to the east, where the monsters come from—the God Eaters, the Spirit Eaters, the destroyers of civilization that come with their fangs and their unending coldness and their hunger.
But forgetting, she knows, is a blissful and dangerous state. The Lotus Eaters on the island of Meninx wanted to forget, to suppress the true nature of their magic blood by ingesting copious amounts of lotus petals.
There are things Kat doesn't want to ever forget. Like Hephaestion, like their great journey to Egypt, like their time on Meninx, and the deep secrets they shared. Even though she never promised him anything, when she is lying close to Jacob at night, his arm loosely cast over her shoulders in protection, she feels as if she is betraying Heph somehow.
She misses Alex dreadfully, too. There is an aching hollow in her soul, as if some part inside her has been lost. But she can't go back to Pella—that much is certain—not with King Philip hoping to dangle her like bait to a royal suitor. Not for anything in the world would she allow herself to be married off to someone sight unseen. She'll never marry—this feels like a conviction that stems from some deep place in her gut, some place she can't quite understand and doesn't want to believe.
One thing she knows for sure: she loves using her Snake Blood to heal. Briseis returned home a couple of months ago, her skin smooth, a few pink marks the only trace of her hideous rash. Kat has healed many others, too, or at least made them more comfortable. She sometimes thinks she'd be content never to have children of her own—never to pass on the blessing and curse in her blood. Instead, the sick can be her children, and she'll remain content. She never has to leave this place.
And yet...
Tonight, torchlight washes across the brightly painted statue of Asklepios. Kat crouches down and pulls the blanket over a woman with the falling sickness. A small brown snake has curled up beside her, a very good sign of impending healing.
Jacob stands in the doorway, holding a torch. "Come," he says. "Let's go back."
They cook fish and vegetables outside their cottage. The air is warm; summer will be here again soon. It strikes her with fresh beauty, how—despite monsters, despite illness, grief, evil, loss—still, life pushes onward. Nature grows over ruins. We persist. We start again.
A crescent moon rises above a thousand glittering stars in the dark blue sky. Kat looks at Jacob across their small fire, admiring how the light plays with his face. The dancing flames illuminate his eyes, while throwing the rest of him into mysterious shadow. It's the perfect combination of familiar and new: the Jacob of her childhood; Lord Jacob, an Elder Counselor for the Aesarian Lords; and the all-powerful Earth Blood Jacob. Warrior. Healer. Lover. Friend.
He says little over dinner. Usually they talk about their day or reminisce about the past, before gradually curling in toward one another, discovering each other's bodies in the darkness, the places that make them cry out, shudder and gasp—the places in their hearts that are closed and tight but slowly, slowly, beginning to unfurl.
But tonight, Jacob is strangely quiet. The air pulsates with tension.
"Are you all right?" she asks, setting down her plate at her feet.
"I'm sorry," he says, his voice thick. "It's just... I have something I want to talk to you about. And I'm...nervous."
She scans her memory for something she has done or said that could have hurt him. He clasps his hands, then lets them hang, then places them on his thighs. "Those months, when we were apart," he says, his voice strangely deep. "I tried so hard to forget you. I figured you were in love with Prince Alexander." He laughs awkwardly.
"Good thing for me I never was!" she jokes, hoping to diffuse some of his nervousness.
"And then, to find you again, the day of the waves..." He shakes his head. "Surely the Fates wanted us to get back together."
Kat nods. She, too, has often thought the same thing.
"And then, since we've... Since our friendship has... _grown_ ," he says. Though she can't see his face clearly in the flickering firelight, she's fairly certain he's blushing. "Well, since then, I am more in love with you than ever."
There is a lump forming in her throat, impassable, and suddenly he is on his knees before her, gripping her hands in his. "Marry me, Kat. Let's tell the world that you belong to me. That I belong to you."
Her heart stammers, and for some reason she can't quite catch her breath. He asked her this question once before, on a late-summer afternoon, a hundred years ago—it can't possibly have been only ten months—after they had shared their first kiss in the pond outside their farm in Erissa. She didn't turn him down, exactly, just told him she needed to wait, but he took it as a rejection. The next day, he left the farm to compete in the Blood Tournament in Pella, and she followed him, and their lives changed. Jacob joined the Aesarian Lords, who soon after declared war on Macedon. When they were parted, she feared she'd never see him again.
So often, in the intervening months, she wished she had said yes to him that evening. That she had begged him not to go, told him he didn't need to become a king's guard for her to agree to marry him. That he could just be Jacob, the potter's son. His family would still be alive, and Kat never would have known that Queen Olympias, the woman Kat despised more than anyone in the world, was her real mother.
But no, that isn't right. It was her own fault—she is the one who followed Jacob to the palace, with dreams of revenge. She is the one who got everything wrong.
And if she hadn't gone, she never would have met Alex, her twin, or learned about the Snake Blood Magic fizzing in her veins. She would never have met Ada of Caria or Aristotle, or seen Persia and Egypt and Meninx. She never would have vanquished the Aesarian Lords, turning the tide of the fighting for Alex in the Battle of Pellan Fields. She wouldn't have killed Riel, preventing the Age of Monsters from destroying the earth. She would never have become who she is now, stronger, wiser.
She would never have met Heph.
His image appears unbidden before her, his tousled dark hair, his sparkling brown eyes, his long muscular limbs and slight swagger. She remembers that kiss in the sea off Meninx, when the water lifted them off their feet and they floated in each other's arms. She thought then that he would be her choice. Before they left Meninx for Byzantium, Heph, too, had asked her to marry him. And she had told him she wasn't ready to give an answer.
She wonders now—how many chances will she have at love? If she risks this one, will it be her last?
Jacob fishes for something in his pouch and pulls out a ring. It is heavy, old gold, with a round center of mother-of-pearl that gleams in the firelight like fish scales glinting in water.
"Please, Kat," he says, his voice cracking. "Wear this ring as a pledge that you will marry me."
She cannot move and does not resist when he slips it onto her finger. It slides smoothly over the knuckle and settles snugly below. She feels a whisper of something travel up her arm and into her heart, a muted cry of yearning for freedom, along with determination and persistence. But not resisting is its own kind of answer, she realizes. How much of our lives may be determined, she wonders now, by what we do not voice?
"It is beautiful" is all she says. "Where did you get it?"
He laughs self-consciously. "From Lord Poseidon, apparently," he says. "The same god who cast us back together in Byzantium. I found it on the beach one day while you were running ahead. It looks very old. I wonder if some Trojan woman slipped it off, not wanting her captors to get it."
She stares at it, thinking about that unknown woman, wondering if she will feel her pain if she wears her ring. "Of course," he says quickly, "if you want something new, we can go to Lesbos. I hear there are many fine jewelers in Mytilene."
"No," she says, holding up her hand, unsure how to respond. Somehow it seems right to wear this ring. Fate has many surprises in store for us all, for Trojan women and Macedonian princes, as well as peasant girls and boys from Erissa. The unpredictability of fate requires strength. This ring will serve as a reminder of that. Of strength and also patience. Life is much longer and more varied than she ever could have anticipated. The threads of Fate weave in and out of one another, forming a tapestry we can never fully step back and see until it is too late to change.
She raises her gaze from the ring to the hopeful face of the boy who always loved her and smiles. She still doesn't know about marriage, but it feels wrong to say no when her heart is alive and singing. So, she says nothing, only draws him close and kisses him deeply.
Soon, Kat is no longer chilled but flushed with warmth.
* * *
She wakes. A slice of moonbeam from the open window illuminates Jacob, breathing deeply, a slight smile on his face as if he is having a pleasant dream. He must be dreaming about her, about the two of them together. But Heph... She remembers him telling her his story of murder and loss, of the secret pain he concealed beneath his confident attitude. She knows he is waiting for her in Pella, hoping she will come back to him. He promised, like Jacob once did, that he would wait forever. Will he?
She can't bear the thought of causing him more pain than he has already suffered. And she can't bear the thought of losing him, either, of never seeing him again.
But if she _were_ to see him, would she be able to stop herself from wanting him?
Her heart is a coil of contradictions, writhing, snakelike.
There will be no more sleep for her tonight.
She grabs a light cloak and slips silently from the cottage. A walk will help clear her mind. She strides briskly down the hill toward Troy and turns to look back. The mismatched columns of the Temple of Asklepios glow silver in the starlight.
So far, Kat and Jacob have heeded the shepherds' warning not to wander about the ruins at night. But if she is stealthy, perhaps she can learn what is really going on beneath them. Most likely, there is nothing at all, and the story merely serves as ghoulish entertainment for bored shepherds around their fires.
Maybe she just needs to be reminded again of the epic love stories buried here, even if they all led, ultimately, to ruin. Even if all love, like this city, is doomed.
The mound looks completely different at night, heaps of jagged rock, silvery white in the moonbeams. She meanders around fallen stones, listening to the thudding of the waves on the shore, and the wind whispering ancient songs to the fallen towers of Troy. A small creature darts behind a rock. She breathes deeply, taking in the night air. There are no shrieks, no smoke and fire. She gazes around the ruins and wonders yet again what stories these walls could tell. That wall over there, standing upright about as tall as a man. Was it once part of a house? A gate? What lives were lived and lost there?
Suddenly Kat can make out a movement, as though the wall itself has rippled, or...
No, there is a form within the wall, or against it, in front of it.
No form: a girl, appearing as if emerging from the wall itself, or the air.
Kat blinks. Did the child step through an opening in the wall? Is she a ghost or phantom? Were the shepherds right?
Kat stands slowly, not wanting to frighten the girl, apparition or real. She half expects her to fade away, a figment of moonlight and imagination. But the child stands there, staring back at her, long dark hair framing a freckled face, round with youth.
"Hello," Kat says slowly, her voice sounding naked in the openness of the night. "Are you lost?"
The girl shakes her head. "Oh no," she says in accented Greek. "I live here."
Live here? Kat frowns. No one lives here. How many times have she and Jacob climbed over every bit of the ruins?
"What are _you_ doing here?" the girl asks, coming closer. She's a pretty child of six or seven, in a dark tunic. "No one comes here at night."
"I...couldn't sleep," Kat confesses. "I'm staying at the temple and decided to go for a walk. My name is Kat. What is yours?"
"Roxana," the child says, extending a small hand. A trill of fear moves through Kat and she freezes as the girl reaches out, as if testing to see whether _she_ is real. Then, in a voice that is wistful and distant, that makes her sound ancient, full of a sadness beyond her years, she reaches up to touch Kat's long hair and whispers softly, "My sister used to let me play with her hair."
"Your sister? Where is she?" Kat asks, hope rising that perhaps Roxana is not alone here, scrounging for food among the ruins. Her tunic is clean and fairly new, as far as she can tell in the moonlight. The child looks well-fed, her hair neatly combed.
"The mean men took her away," the girl says, her voice cracking. "They were going to hurt me, too, but I ran. And then other mean men found me and brought me here."
Kat has no idea what she's talking about. But her golden fingertip tingles. A sign of magic. Can this rag of a child be a Snake or Earth Blood?
"What are you doing here so late at night?" Kat asks.
"Practicing," the child responds.
"Practicing what?"
"Being the wall."
Kat shakes her head. "I don't understand."
Roxana laughs and goes back to the wall. And disappears.
Kat's jaw drops. Where did she go?
"I'm still here!" comes a laughing voice. Kat approaches, puts out a hand, and touches the girl's thin shoulder. Squatting, she realizes the child has altered the color of her skin to match the wall. She's a human chameleon.
Instinctively, Kat closes her eyes and stills her breathing. She senses powerful magic bubbling inside the girl, but it's not Snake Blood, for here is no kinship to other living things. Nor does she sense Earth Blood; there is nothing at all that reminds her of Jacob, and his oneness with earth and wind and water. This magic is dark as soot, burning as fire, cruel and hard. It is the opposite of life. It is...death.
With a chill, Kat lets go and backs away. Who— _what_ —is this girl?
Roxana becomes herself again and wriggles out of Kat's mental grasp, breaking her exploration of the child's magic. The little girl stands in front of a hill covered with scraggly bushes. And disappears. Kat walks over slowly. "Where are you?" she asks.
"Here!" the girl says, suddenly appearing, her skin shedding the disguise of grass and dirt in the darkness.
"Who are you?" Kat whispers, wondering again if the girl is some kind of vision.
"Well, I was a princess once and lived in a palace," Roxana replies.
Despite her wariness, Kat smiles. Every little girl considers herself a princess.
"Then, like I said, some mean men stole me and brought me here. Kids can fit into the narrow passageways below the city more easily and bring back gold and jewels."
Suddenly a ray of comprehension shoots through Kat.
There are no demons or ghosts howling below the city at night, but _human scavengers_. Slavers forcing stolen children to mine the ruins for valuables.
Her heart pounds hard in her chest as understanding washes over her in waves. The Trojan people, she knows, hid their gold, silver, and jewelry in basements, under floors, and inside walls during the long war with Greece. When the city collapsed in the inferno, the invaders took what they could find above and sailed back home. The howls and shrieks the shepherds have reported hearing in the night must have been the slavers trying to scare people away.
Though none of that explains the girl's magic.
But one thing is certain: if the slavers find them... Kat curses herself for leaving her sword, the one Ada of Caria gave her, back in the hut.
"I can help you get out of here," Kat says quickly, in urgent, hushed tones. "And the other children, too. There must be more of you. I am with a very powerful warrior who can—"
Roxana laughs. "Oh no," she says. "I've already been freed. All of us have. We aren't slaves anymore. We are the Chosen Ones now."
Kat frowns again. _Chosen Ones._ " _Who_ freed you? Where are they?"
Roxana takes her hand and leads her over a steep mound of fallen stones. They grab saplings to pull themselves up. The girl is as nimble as a monkey. She runs down the far side of the mound and stands still. A few feet away, Kat sees the flickering orange light of a torch and a wisp of smoke rise out of a hole in the ground.
"The Horn Men came and freed us all," she says. "They are down there. We sleep during the day and work at night."
So, the Aesarian Lords _are_ here—have been here, perhaps, all this time, but hiding out below the ruins. That was why Jacob and Kat never found them. She shudders. At night, the Lords work in the tunnels. During the day, when shepherds and visitors climb the ruins, they are as silent as the tomb, sleeping beneath them, or scheming, plotting, hoarding, and hiding children. Why?
Fear, cold and pulsing, moves through Kat. She creeps silently toward the slice of fiery light in the earth. "How do you get down there?" she whispers.
"Oh, there are places we know," Roxana says. "Staircases and openings we block."
"What's down there?" Kat asks. "Tunnels?"
"Tunnels?" the girl asks, giggling so hard she bends over and cups her hands over her mouth. Spreading her arms, she says, "There's an entire city below."
Kat gasps. An entire city. When the tall buildings collapsed, there could have been entire streets left intact, especially after centuries of clearing rubble below to look for valuables.
Then it occurs to her that the Aesarian Lords, as rich as kings themselves, wouldn't need children to dig jewelry from the ruins.
Which means they are using them for something else.
"What do the Horn Men make you do down there?" she asks, her voice shrinking in fear of the answer, her heart wrangling to understand, to help.
The girl giggles again. "They don't make us do anything," she says. "They teach us things, and we practice them."
Kat stares at the girl. Moonlight dances on her long dark hair. "What kind of things do they teach you?" she asks.
"Magic," Roxana replies. "They teach us how to become magic."
CHAPTER SIXTEEN
ARRIDHEUS ("RAT")
The setting sun turns the desert sand and the village behind him golden orange. Rat sits on a rock, playing with his little wooden puppets, working their strings so that they walk and talk, practicing his voices, and making himself laugh. Beside him, the Nile whispers softly; it has shrunk to a brown stream. But soon, Sarina says, it will be Inundation, and the Nile god will make the river rise and widen again, flooding all the fields. The temples and pyramids will be islands, reachable only by boat or raised walkway.
This year he experienced no winter at all. No smoky rooms with snow falling through the smoke hole and sizzling on the fire. No freezing days, when opening the shutter slats for a bit of light lets in the searing wind. In this strange land of eternal heat and blue sky, there is neither cold nor rain.
They have made him strong here, trained his tongue to speak and his body to use weapons. Still, he wishes he could go home. But he can't mention it. Or his real name—Arridheus, Prince of Macedon—or Yuf will beat him. But they cannot read his thoughts, and he often pretends Alex, his older half brother, is here with him, ruffling his hair and giving him sweets. And the nice beautiful girl he met shortly before he left his home, Alex's friend Katerina. She smiled at him so kindly, as if she didn't think he was stupid at all. And she was nice to his pet rat, Heracles, who liked her a lot, too. He hoped they would become good friends and always live in the palace together. But then the kidnappers took him away, to this hot, dry place.
At least he has Sarina, his beautiful dark nursemaid from that other life, the powerful priestess of Bast here. Sarina gave him the two wooden marionettes in his hands. They have jointed arms and legs and funny painted faces and wear little white kilts.
"You are ready now, my puppet," says Ramses, the bald doll with the wide smile. His voice is deep and loud—Rat likes making that voice.
"Ready for what?" asks the other doll, Sabu, in a high voice, his shoulder-length black hair and mouth like an O shape.
"Ready to kill Prince Alexander of Macedon," says Ramses.
"I won't!" cries Sabu.
"You will!"
"I won't!"
The puppets smash into each other again and again, screaming. Rat flings them on the sand, where they lie in a crumpled heap. He has to do it. He doesn't want to, but Sarina explained things.
_It has been prophesied that Alexander will conquer Egypt. When the Persians conquered, they killed or sold into slavery my entire family. My people cannot allow this to happen again, little Rat. You must kill Alexander in order to save thousands and thousands of innocent lives. You will be a hero forever._
He doesn't want to be a hero. He just wants to go home. Sarina said he could go home if he does what they want.
A shadow falls over him. Worried, he looks up, fearing it is Yuf. But it is Sarina. She kneels beside him, smiling, as her sandalwood scent wraps around him.
"It is dinnertime, little Rat. What are you playing at?" she asks, adjusting her sheer blue veil so that the little gold coins on it shimmer and clink.
"Nothing," he says, scrambling off the rock to pick up the dolls. Smiling, she extends her hand, and together they walk back to the village.
But it's not nothing.
After all, he heard _her_ saying it just this morning, telling Yuf and the village elders that the time has come. _The puppet is ready_ , she'd said to them. _The puppet is ready._
But Rat knows the puppet she meant was neither Ramses nor Sabu. The puppet is Rat.
ACT THREE
ATONEMENT
The gods, too, are fond of a joke.
—Aristotle
CHAPTER SEVENTEEN
PYROLITHOS
He's been king for only a few months, but already the people grow restive.
And none more so than the Aesarian Lords.
Where _are_ they?
Late.
Pyrolithos drums his fingers on the arm of his gilded throne and gazes down at the lovely women in whisper-soft gowns lounging on the marble benches along the sides of the throne room. Daylight streams in through the wall of open windows, along with fresh spring air tanged with salt, but nothing can reduce the tension coiled inside him.
The Lords have much to answer for. The king discovered that Ambiorix sent letters regarding diplomacy and treaties to neighboring kingdoms as if he himself were king. In return, Pyrolithos brought spies into the palace, a few of the girls most admired by the remaining Lords, to entertain them. So far, they have come up with vague information: that the bulk of the regiment indeed went east for reasons to do with the Spirit Eaters, and there seemed to be a plot developing between the Lords and General Georgios, perhaps to give him the throne.
The king has spent too much time with Alecta, that's the trouble—training her in the cave for several months, and then, after the attack of the Spirit Eater, training the women at the brothel himself. There seemed no point in secrecy anymore, at least among his New Amazons. Once word got out that he'd captured the Spirit Eater, the women eagerly agreed to follow such a powerful and magical protector. As far as the Dardanians know, however, the new king is simply spending all his time pleasuring himself with whores instead of running the country. Not exactly a good look, if he wants to be taken seriously.
The deepest secret, though, the worry that worms its way through him in the darkness of the night, is his Smoke magic—more specifically, his weak grasp on it, the fact that it seems to be fading, leaking out somehow. Lately, he could swear his face seems softer, more feminine. He hasn't had to shave in three weeks. Even his hands seem smaller, not delicate exactly, but not as powerful as they were before. And he knows what he must do—knows the curse of Smoke Blood—how it requires more of itself, endlessly.
He will need to commit another betrayal.
Fortunately, he has one in mind.
But first he needs answers.
Finally, he hears boots marching toward him. Lord Ambiorix strides in with the five others, all of them grinning broadly, as if they had just heard a very amusing joke. Their eyes scan the women on the benches, and they laugh out loud. He can only assume the joke had something to do with Pyrolithos's womanizing.
They stop in front of the royal dais and give Pyrolithos the most perfunctory of bows. "My lord," says Ambiorix. "You wanted to see us."
Pyrolithos sits slumped on the throne, glowering at them until he knows they feel uneasy. Finally, he drawls, "Yes. How kind of you to come."
Ambiorix turns red, though the king isn't sure if it is due to embarrassment or anger.
"As you are aware," Pyrolithos says, sitting up, "there have been attacks here in Dardania by monstrous creatures some call Spirit Eaters."
The Aesarian Lords look up, their eyes wide. "Yes, yes, we have heard of such attacks," mumbles Ambiorix in his guttural accent. "A handful only. A couple of fishermen, some shepherds..."
"It is my understanding that the Aesarian Lords know a great deal about the Spirit Eaters," Pyrolithos continues. "Their lair is not too far from your headquarters in Nekrana, in the Eastern Mountains of Persia, is it not?"
Ambiorix's light blue eyes flash. "Not that far," he says vaguely.
"Why have these creatures left their lair?" Pyrolithos asks, leaning forward. "People say they have lived quietly enough there in the Eastern Mountains for centuries. Why now are they spreading out and devouring living things?"
"I don't know," Ambiorix replies, tossing his blond braids over his broad shoulders. He has a shifty look on his face, and his gaze doesn't meet that of the king.
"Did High Lord Gideon take the regiment east to deal with the Spirit Eaters?" Pyrolithos asks.
"Our orders are secret, my lord," the Gaul replies, straightening and looking Pyrolithos in the eye. "We are your guests, not your subjects, and as such we are under no obligation to reveal anything to you. Remember Zeus's law of guest-friendship."
The king cups his chin in his hand and tries not to smile. He knows that the gods decreed that hosts may not harm their guests, or even question them too intently. Anyone who did so would be committing an unforgivable betrayal against divine law.
"Have you ever seen a Spirit Eater, Lord Ambiorix?" he asks.
The Gaul frowns. "No," he says, shaking his head. "We never went that..." He stops.
"That close?" the king asks.
Ambiorix says nothing.
"Well, in that case," the king says genially, "I have a little surprise for you." He picks up a horn from the table next to the throne and blows a loud, clear note on it. A moment later, the double doors at the end of the throne room open. Alecta and Xanthe, both tall and muscular, enter wearing gleaming armor.
But the Lords don't seem to notice the two female warriors. Their eyes are glued to the diamond cage on the cart the women pull.
Inside, the huge creature snarls viciously, its dark, fractured form, shifting around in tight circles, all heads and wings and claws. Pyrolithos feels its hunger, its outrage. Ambiorix and the other Lords unsheathe their swords and creep forward curiously. They pad quietly around the cage as the Spirit Eater thrashes against the bars and roars, a sound of endless, anguished hunger.
"Lord Ambiorix," the king says once he sees all is in readiness. "I command you to tell me all you know about Spirit Eaters."
The men turn from the cage to find that the two dozen women in the room stand holding bows and arrows pointed at them. Alecta and Xanthe have drawn their swords and swung off the shields on their backs. More swords and shields, plucked from beneath the benches, stand at the ready for the archers.
"What is the meaning of this?" Ambiorix says, his face turning crimson.
"Quite simple," Pyrolithos replies genially. "You tell me everything you know about the Spirit Eaters, or I will feed you and your men to this one. It hasn't eaten in weeks, poor thing."
Two Lords lunge at the women with swords. The king hears the soft breath of arrows, sees their blur, and hears them thunk into solid flesh. The two Aesarians are so riddled with arrows they look like pincushions and fall to the ground dead. The other four form a tight circle. "You will pay for this!" Ambiorix screams. "Aesarians killed by women! You have dishonored us!"
"So sorry about the dishonor," Pyrolithos says, studying his fingernails.
He looks at the women and nods. More arrows fly through the air. The four remaining Aesarians are impaled in the right wrist and drop their swords. Then they are hit in their legs, and fall to their knees, gasping in pain.
The king says, "Now, tell me about the Spirit Eaters, or you will become this one's next meal. If you tell me what I want to know, I swear a sacred oath that I will let you live."
Ambiorix, on the ground, shakes his head. With a grunt, he pulls the arrow out of his thigh and flings it angrily on the floor. "It is an Aesarian secret only those in the Elder Council know fully. I would rather die than reveal it."
"Would you?" Pyrolithos asks coolly. "Take him to the cage."
Alecta and Xanthe prod him with their swords, and the creature, sensing prey, howls and slavers, its form fracturing into several Spirit Eaters atop one pair of spindly legs.
"All right!" Ambiorix says, his face a mask of terror. "May the gods forgive me. I will tell you."
Haltingly, the wounded Gaul tells an ancient tale of a magical fountain of forbidden water, of villagers who drank from it and turned into monsters craving divine flesh. Of the Aesarian Lords' true mission—to feed the monsters magic wielders, thereby preventing them from fanning out from their lairs to eat mankind.
Pyrolithos feels that a lamp has been lit in darkness. He finally understands why the Aesarian Lords search the known world for soothsayers and Blood Magics to send east. Not to execute them, but to feed them to the Spirit Eaters. Now he knows why they tried so hard to discover why Princess Cynane's grievous wounds healed immediately after their torture sessions, and why they wanted her entombed in a casing of hardened ash. They intended to send her to the monsters—to be devoured alive.
All this enrages him even more against the Lords.
And yet, the Lords had to placate the beasts, as they had no powers to control them. Pyrolithos, however, does have such powers. He proved that by capturing this Spirit Eater. He returns to the bold idea he has been formulating: that he could learn how to control the one in the cage—much as you could tame a vicious dog with the proper techniques. He and his army could capture more Spirit Eaters, forging an invincible battalion of the creatures. With the Spirit Eaters at his side, no army in the world—not even the Aesarian Lords—could vanquish such a force. He and his army of women would become the most powerful warriors in the history of mankind. Ruling this backwater kingdom of fishermen and goatherds—which had, up until he captured the Spirit Eater, been his life's ambition—now seems contemptible, unworthy of him.
Ambiorix falls silent, his story spent. His face is pale, and beads of sweat have broken out on his forehead.
"Where, exactly, in the Eastern Mountains do you take your magic wielders to give to the creatures?" Pyrolithos asks, his excitement rising as vast new possibilities unfold before his eyes.
The Gaul shakes his head. "I only went on such a mission once," he says. "The high command took us up secret paths in the mountains. That is all I know."
The king sighs. It might not be easy to find the information he needs. With an impatient gesture, he signals the women. Three of them drag Ambiorix, bellowing with rage and pain, toward the cage. "You liar! You have no honor!" he cries. "You said I would live if I told you!"
Pyrolithos takes out his dagger and slices his forearm, the pain causing what little is left of his Smoke Blood to rise and curl throughout him. The magic is weak. He must concentrate intensely on the Spirit Eater's diamond cage, imagining the bars in the center of the front dissolving to form a hole. Alecta—who, with three other Amazons, pulls a struggling Ambiorix toward the cage—looks utterly disgusted, but Pyrolithos feels exhilaration. Revenge is sweet. This man was involved in trying to do exactly the same thing to him as Cynane—feed him alive to such a monster.
The women hold the Lord's head just outside the opening in the front of the cage. And then he is sucked inside.
The man's shrieks as he is devoured alive echo off the marble throne room walls. Some of the women turn their heads or close their eyes and put their hands over their ears. But not Pyrolithos. This is the man who smashed his bones with an iron rod when he was nothing more than a girl. Such a man deserves this death.
Fire, smoke, and ashes explode inside the king's veins. For here is a double betrayal: betraying the law of guest-friendship and breaking his oath to Lord Ambiorix to let him go if he told him about the Spirit Eaters. Power floods through him, racing down his veins and into his muscles and organs. Flush with incredible strength, Pyrolithos concentrates on restrengthening the diamond bars to keep the Spirit Eater contained.
"Take the others to the dungeons," he says. "And dispose of the bodies." The women clatter out, some with the prisoners, others dragging the two dead Aesarians, leaving smears of blood on the marble floor. The doors bang shut behind them.
Pyrolithos sits on the throne, watching the Spirit Eater make quick work of the huge Aesarian Lord, spitting out the leather, the metal armor, and the large helmet horns, but consuming the entire body except for a few chewed bones. Sniffling, it licks up the pools of blood beneath it with a long black forked tongue, then stares at him with beady dark eyes that rearrange themselves here and there on two—no, three—blurred, moving heads, until they focus sharply into one. Now it hisses, a sound so terrifying that every hair on the king's body stands on end.
For all the horror of being so near the monster, there is something about the creature that calls to Pyrolithos, a kind of tugging on his soul. Could it have something to do with what Ambiorix told him, that the Spirit Eaters want more than anything to devour magic beings, and he, a Smoke Blood, is now magic?
The king steps over puddles of blood and approaches the monster, who howls again, shaking the diamond bars with long black claws. He squats down in front of the cage, staring into the ancient flinty eyes, sensing its power, its hunger for magic blood, just as Cynane had been hungry for magic blood.
"You're still hungry, aren't you?" he asks quietly.
In response, the Spirit Eater opens its bloody maw and hurls itself against the bars.
The king holds his left arm directly above the cage. With his dagger, he reopens the slash on his left forearm. Blood, vibrant and bright red and full of Smoke magic, splatters between the bars of the diamond cage. The creature licks it up, ravenous. When Pyrolithos pulls his arm away, it howls for more.
A taste of what you crave will always keep you coming back for more.
After dinner, he slips back into the throne room. All the torches have been lit, casting shadows that seem to climb the walls and dance across the marble floor tiles, now cleansed of blood. He planned to study the Spirit Eater, to experiment with it, but a woman is standing in front of the cage, staring calmly at the creature who is curled up contentedly, like an old, well-fed family dog.
Pyrolithos doesn't know what surprises him more—that this woman isn't afraid of the creature, or that the Spirit Eater seems so tame. He's even more surprised when the woman turns toward him, and he sees she's carrying a newborn baby. It's the Persian girl. Zotasha.
"Why aren't you afraid of it?" he asks her. "The other girls keep their distance."
"Oh, at times I am afraid of it, my lord," she admits, and he notices the beauty of her large dark blue eyes. "But I am also oddly drawn to it. You see, a soothsayer once told me that these creatures can alter a person's fate, though I am not sure how. At any rate, I would like very much for it to alter mine."
Pyrolithos hardly has a moment to take in the girl's brazenness at speaking to a king so directly—almost as if she were royalty herself—because he has realized something. _That's_ what he's been feeling as he looks into the monster's eyes—his _fate_ being altered in some way.
"Wouldn't we all," he says now. "I want to capture more of them, learn about them, see if we can tame them and use them for our benefit." He doesn't know why he's sharing this with the girl, except that he instinctively feels he can trust her—an unusual feeling, as he has been trained all his life to trust no one. "But I don't think we can assume others will dive into our courtyards anytime soon. And I don't know where they live in the Eastern Mountains, which is a huge range.
"I suppose," he continues, walking around the diamond cage, "I could take the women east and explore. Persians might be suspicious of a regiment of foreign warriors but would surely let a man and his harem travel freely."
"I know exactly where they are," she says quickly, looking at him with intensity burning in her eyes. "I could lead you there."
His breath catches in his throat. "What?"
She shifts the sleeping baby from one arm to the other. "I have been to their cave," she says. "I have even licked that wall Lord Ambiorix was talking about, the one that trickles with drops of the diverted Fountain of Youth. At least I think it was that wall. I was so very thirsty, and one drop of that water gave me incredible energy, the energy I needed to survive."
Pyrolithos stares at her, suddenly taking in her many inconsistencies. There is something regal about her bearing and speech, despite her heavy Persian accent; clearly, she was not born in poverty. Yet she ended up in a brothel with an illegitimate baby, after wandering the Eastern Mountains. This girl—Zotasha?—must have an incredible tale to tell.
"I went to Persepolis," she continues, "and told Great King Artaxerxes about these creatures. I wanted to lead an army there to destroy them, but he didn't listen to me." She turns to Pyrolithos and says, urgency humming in her voice, "But perhaps _you_ will listen to me. Perhaps we could all go there together. And if you can't tame them—somehow I doubt you could—we can destroy them."
The king is amazed once more. Zotasha had an audience with the Great King of Persia, all the way in Persepolis. She knows where the Spirit Eaters live. And she wants to fight them.
"Who are you?" he asks, putting his hands on her shoulders. "Where do you come from?"
He can see a flicker of uncertainty pass across her eyes. She says nothing for a long while, clearly considering how much to tell him. Then, as she opens her mouth, the door is flung open and Alecta storms in.
"I want answers," she says, tossing her mane of long red hair.
Zotasha, looking from Alecta to the king, hurriedly bows and leaves, closing the door behind her with a little click.
"We will not live in this palace with a monster among us," Alecta fumes. "We have taken a vote. We will go back to the Grotto if you keep this thing here, and you can forget about your New Amazons."
Pyrolithos turns to her and asks coldly, "Do you wish to resume your lives at the brothel?"
Alecta narrows her dark eyes. "Anything would be better than getting eaten alive. But now that you have taught us to be warriors, I think our whoring days are behind us. We will find Queen Cynane and follow her."
"You are following Queen Cynane by staying with me," he says, crossing his arms. "I am her proxy."
Alecta scoffs. "So you say. But what have you done to her? We want to see her. Otherwise, we will no longer follow you no matter how many magical powers you have."
Pyrolithos looks at Alecta, then at the Spirit Eater in the cage. He didn't want to reveal the truth to Alecta, not yet, anyway. Will the truth ruin everything? Or will it make everything possible?
Finally, he stops in front of her, realizing he has no choice. For if he does nothing, she really will leave.
"Very well," he says. "I will bring her here."
Alecta looks around the room bathed in dancing amber torchlight. "Very well," she says, her voice hard but her expression softening. "Go get her. I will wait here."
Pyrolithos runs a hand through his thick black hair. "She is here already, with us."
Alecta rolls her eyes. "If you're playing some kind of a stupid..."
But she stops midsentence as the king, calling on the new Smoke Blood powers burning through his veins from today's double betrayal, feels his body shift and change. He grits his teeth, clenches his fists, and closes his eyes. And then, when he straightens, he sees delicate female feet in the now-overlarge sandals.
He stretches out his hands and sees the smooth, hairless hands of Cynane.
Alecta stares at him in shock. Her eyes are wide, her mouth open.
Then she draws her hand back and slaps Cynane hard across the face.
CHAPTER EIGHTEEN
JACOB
Moonlight streams silver white over the fallen stones of Troy, and Jacob could swear he can feel Kat's anger beside him, her frustration like a coiled heat, radiating outward. Last week, she came back from a midnight walk with tales of a magical child and an entire city hidden under the Trojan ruins. The Aesarian Lords were camped there and had been for months, she insisted, right under their noses.
Not only that, but she was convinced the Lords were teaching children a strange, dark magic. Jacob strenuously denied that this could be possible. Kat set her jaw in that mulish way of hers and said, "Let's find out, then."
Now, for several nights in a row, they have traipsed out here from the Temple of Asklepios, Kat carting the bag of Egyptian ushabtis, the statuettes of warriors she had taken from Princess Laila of Sharuna. "They could help us free the children," she keeps saying. "They have the spirit of a powerful ancient general inside them."
Every night, they have waited among the ruins, Kat squinting at shadows and walls, insisting it could be the elf-child, Roxana, practicing her magic. Jacob has begun to suspect that perhaps Kat only dreamed all of this, and tonight he is torn between impatience and worry. He has never known her to be delusional. Perhaps the overuse of her powers has put a strain on her.
"Time to go back soon, I think," he says, tapping the leather-wrapped lantern. "Unless you want to spend the night here."
Kat pulls away from him. Then, "Wait." She sucks in a breath and goes rigid. "Over there."
Jacob squints into the distance, across the thin pool of lamplight, to where Kat is pointing.
A small girl, standing in front of a wall.
She wasn't there a moment ago.
It must be a trick of the moonlight.
No, that really _is_ a child.
Kat is already moving toward her. "Roxana!" she calls out softly. "I am so glad to see you again! I've been looking for you."
Smiling a gap-toothed grin, the girl reaches up and touches Kat's hair. "I can't sneak out every night, you know," she says. "The Horn Men don't want us to go outside at all." Her gaze falls on Jacob and her smile fades. "Who's that?"
"That is my very good friend," Kat reassures her. "His name is Jacob." She takes the girl's hand and brings her over to Jacob, who smiles and stays seated. His height might frighten her.
"Hello," he says.
"Honey cakes!" Roxana says, pointing to the open basket he's carrying. "I haven't had those in ages. The Horn Men mostly eat meat."
Jacob has an idea. Long ago, in that other life, he had a surefire way of getting his little brothers to do things they didn't want to: feed the goats, pull the weeds in his mother's garden, sweep out the cooking hearth. "I could give you one," Jacob says, his voice playful as he reaches for the basket. "We could play a game."
"I like games," she says, nodding. He holds up a honey cake and she grabs for it, but he pulls it away as her small fist snatches the air.
"Well, let's say I give you a honey cake if you tell me a secret," he suggests.
The child considers this, then looks up at Kat.
"I think it would be a fun game," Kat says.
"All right," Roxana agrees. "What secret do you want to know?"
"Hmm. Well, I would like to know how you get into the city below these ruins."
Roxana looks hungrily at the basket of honey cakes. "But I would get into trouble with the Horn Men if I took you down there."
"We wouldn't tell them you showed us the way," Kat says quickly. "If they find us, we will say we found the way ourselves."
Roxana licks her lips. "All right."
She leads them around the fallen wall, over a hillock of stones and saplings, and pulls a covered lantern from behind a stone. After removing the leather cover, she and her light disappear into a crevice in what must have once been a turret. Kat bends over at the waist and squeezes in behind her. Jacob falls to his knees to push through. He angles his shoulders, the rough stone plucking at his tunic, and forces himself into a low, narrow passage, where his back scrapes against the ceiling. He removes the leather covering of his oxhorn lantern and holds it awkwardly as he crawls on three limbs.
A few feet ahead of him, Kat inches her way down the small tunnel. Finally, he sees the space open up. As he stands, the child, Roxana, runs down a narrow staircase, her own lantern a dull bloom of light in this rambling, pitch-black subterranean world. He and Kat run to catch up with her.
At the bottom of the staircase, Roxana stretches out her hand. "Honey cake," she whispers. He slides the bag off his shoulder, and before he has finished pulling one out, she snatches it from his hand and crams it into her mouth, eating greedily. Licking her fingers, she leads them on, down a dark corridor.
Despite his trepidations about seeing his brother Lords again after all this time, Jacob can't stop the tingle of warmth that flows through him upon entering the heavy embrace of earth and stone. Holding his lantern high, he realizes they seem to be in what was once a street, with elaborate doors and windows on either side. The wall on his left is made of finely hewed large stones. Was this once a home? A shop?
He places his free hand on the wall to draw in the power of the earth, but it seems he draws in something else: a whirlwind of sadness and horror that spins and ricochets inside him. He knows that Troy suffered ten years of siege, much of it in starvation. Men died on the plain below, Trojan sons and fathers and brothers. And then came that last awful night of fire and death. These are the facts that all men know.
It isn't the sadness of facts that has him in its grip, however, but the darkest possible emotions: heartbreak and rage, frustration and despair. The rock absorbs him and casts him down, down, spiraling into a place of such utter darkness he knows that neither light nor happiness could ever pierce it. The very walls ooze dark emotions. He is the wall. He is the sorrow.
He feels a soft hand on his arm and pulls away from the place of pain. It's Kat, her eyes shining with tears. "I can feel it through you," she whispers. "But don't take on this burden. This tragedy is not your tragedy."
"What are you doing?" Roxana asks, irritated and scampering back toward them. "Come this way if you want to see!"
He tears himself away from the black tug of the wall, of the song trapped in stone of loss and death, and puts his hand in Kat's before saying, "This is a place of ghosts."
Nodding, Kat squeezes his fingers and guides him forward. Silently, they follow Roxana through the underground streets. Sometimes Jacob pauses and holds his lantern up to an open door or window, but the pale light isn't strong enough to see more than a few feet. He catches only glimpses of brightly painted walls and mosaic floors. Periodically they see small dark openings in the walls, tunnels for scavengers.
"It's a labyrinth down here," Kat whispers. "Roxana, it must have taken you forever to learn these paths."
The child giggles. "I'm like Pegasus!"
"What?" Kat asks.
"Pegasus! She is never lost, you know. Pegasus knows the way."
"What do you mean?" Jacob asks.
"It's just something my sister used to tell me long ago." Suddenly, Roxana stops, turns to them, and thrusts out a hand. "Honey cake."
Jacob hands her one, which she eats only a fraction less quickly than the first.
The girl licks the crumbs off her hands, then puts a finger to her lips. "You must be very quiet," she says. "Not a sound." Jacob and Kat nod. They creep out onto a balcony and survey a scene below that Jacob doesn't understand, despite the light from dozens of wall torches.
The cavernous chamber holds perhaps thirty children in small groups. At a table, three boys stand before piles of sand, holding their outstretched hands over them. Two of them immediately change the sand into shards of highly polished metal; the third one takes a few moments longer, struggling, but finally transforms his pile of sand, too. At the next table, two children call smoke from a perfume burner into the shape of a small dog, then set it running across the room until it disappears.
Across the room, two girls practice camouflaging themselves. Jacob sees them disappear against the light brown wall, then walk, wall-colored, in front of a dark cupboard and suddenly take on the colors of the cupboard. He squints and looks closely at a little band of other children in the flickering torchlight. These children are not camouflaging; there is no movement and change of color as they walk from one background to the other. They are truly disappearing.
Jacob's heart leaps when he sees Lord Turshu, the little bow-legged Scythian, in a knot of five children. His initial reaction is to call out to him, to find the stairs leading down into the chamber and embrace his old friend. But then he sees Turshu giving a vial of liquid to a boy of about eight. The boy drinks it, grimacing. Then he reaches into a nearby cage and brings out a pearl gray dove. The boy feeds it some grain in his hand, and the dove nestles lovingly against him. Obviously, it is a pet.
Turshu smiles at the boy and then puts two fingers into his own mouth and whistles. The other children stop what they are doing and gather around. "Arkan," he says, "it is time."
The child strokes the bird with a trembling hand. Tears shine in his eyes. Then, in a swift motion, he snaps its neck and drops it to the floor, a sad, broken heap of feathers.
Next to him, Kat gasps. Jacob puts an arm around her, though his eyes are riveted on the scene below. He has no comprehension of what he is seeing.
"I was right," Kat mummers. "They _are_ magic."
Below them, the boy called Arkan falls to his knees, howling in pain and gripping his stomach as the other children retreat a few steps. After several long moments, he rises quietly, his eyes shining with unnatural brightness.
Kat, evidently, has seen enough. She gestures to Jacob and Roxana, and they walk back along the corridor through which they came.
"Roxana," she whispers urgently as she kneels to look the child in her eyes. "What are you? Some kind of Earth Blood?"
But Jacob knows even before the girl answers that she is no Earth Blood. He's never met another Earth Blood before—doesn't know if any others still exist—but he knows with his whole being that this is not Earth Blood. He could never change the shape of matter or transform smoke into whatever shape he chose.
Roxana looks up at Kat and Jacob, her eyes wide, her cheeks still holding a baby's curve, and shakes her head. "I'm not Earth Blood. I'm _chosen_."
Jacob glances at Kat, but she looks just as confused as he feels. Kneeling beside her, he asks, "What was it we just saw?"
Roxana lifts her chin. "The Horn Men choosing us."
"What do you mean?" Jacob presses. "How do they choose you?"
"They came in the night," Roxana says, playing with one curl. "They said we could be free if we drank the potion and cut off the hands of the masters."
Kat breathes in sharply, but Roxana doesn't seem to notice. "We said yes, and the Horn Men broke our chains, and then we took turns."
Jacob's mouth goes dry. "Turns doing what?"
"Cutting masters," Roxana says promptly. "That's when I became free and strong." Suddenly she is gone. Just not there in less than the blink of an eye. Jacob looks around for her, and she appears right in front of him, smiling.
A chilling understanding sneaks over Jacob. Strong and free...and _magic_.
"Have you ever heard of such a thing?" Jacob asks Kat. "Magic made, not born?"
Looking pale, Kat shakes her head. "No...but that does not mean it does not exist."
She looks the same way Jacob feels, as though her insides had been twisted, but Roxana seems unaware of the effect her story has on them. "I can do other things," she says, "turn into smoke for a few minutes, and hold my hand over fire for the longest time without getting burned." She prattles on, discussing her different strengths and favorite games, while Jacob's mind races.
He remembers snippets of Tim's letters. He remembers the Aesarians' ultimate goal: to keep the Spirit Eaters at bay. To keep them full.
_Magic made, not born._
What he and Kat saw back there weren't just children with unusual magic, but...
"A feast," he croaks out, staggering to his feet. "The children... _They're_ the feast."
He can practically see the word trickle through Kat and then the horror it unleashes as she realizes his meaning. The Aesarian Lords have turned innocent children to magic...to feed them to the Spirit Eaters and keep them placated.
How could his brothers commit such an evil, disgusting act? When Jacob took the oath of the Aesarian Lords, he thought he was joining a brotherhood of honor and courage. A community that swore to guard mankind from the monstrosity of magic. And yet, here they are, practicing senseless cruelty, planning to murder the very innocents they had promised to protect.
"I need to find Tim," Jacob says hoarsely. Though he loves to play the fool, Timaeus is the smartest man he knows. Clever and witty and kind—Jacob _knows_ this. It's why, late at night, he still sometimes feels the pinch of shame that he sent Tim to Dardania in order to protect his own secret. Jacob will tell him of the Elder Council's plan and, together, they'll find a way to stop this madness. But...
But how can he leave Katerina in an underground city full of Aesarian Lords, some of whom might recognize her as the one who defeated them at the Battle of Pellan Fields by catapulting pots of scorpions and snakes into their ranks?
Looking at her, he sees that her luminous eyes are no longer horror filled. Instead, they blaze with anger. "Go on, find Timaeus," Kat says, seeming to read his mind. "I can take care of myself."
"What will you do?"
Kat takes Roxana's hand. "Roxana, can you bring me some of your friends, secretly? Do you think they'll like honey cakes, too?"
The little girl's eyes widen, and she nods. "Yes," she says. "Oh yes!"
Kat looks around, her gaze fixing on a doorway down the hall. "I'll wait in there, Jacob." She slips the bag of honey cakes off his shoulder. "Meet me back there. Hopefully, by then, I will have learned more from the children, and we can find a way to set them free."
"I _am_ free," Roxana interrupts stubbornly.
"Of course," Kat says hastily. "Now, will you bring me your friends?"
The little girl smiles, sticks out her hand, and says, "Honey cake."
Worry gnaws a hole in Jacob's stomach as he watches Kat disappear into the dark room and Roxana skipping in the other direction, her lantern light disappearing around a corner. He trusts Kat, but still he wishes they had never discovered this horror—that they were instead curled by a hearth in the temple complex, limbs entangled, oblivious as they have been over the past few weeks. Thinking they were healing people, that they were helping, when all the while, right beneath their very feet...
He swallows hard. Right now he cannot be _that_ Jacob, the healer. He must be Lord Jacob, the youngest member of the Elder Council in the long history of the Aesarian Lords. Counting to ten, Jacob squares his shoulders, picks up his lantern, and, no longer seeking stealth, marches boldly through tunnels that smell of pungent mildew, dank earth, and torch smoke.
"Lord Jacob! Is that you?"
Jacob turns to see Lord Gaius, the tan, charming Roman. "I didn't know you'd returned!"
"Well met, Gaius!" Jacob says, smiling. "It wasn't easy to find you, even with the instructions Tim—Lord Timaeus—sent. I'm looking for the blacksmiths. I need to speak with him."
"It's been a long time since Lord Timaeus has worked the fires, Lord Jacob," Gaius says. "He's now second-in-command to High Lord Gideon."
Second-in-command? Only a year ago Tim was the fresh recruit given all the worst assignments, from sweating over the blacksmith's forge to cleaning out the latrines. He has risen high—and quickly—indeed.
"We thought you had died in the great earthquake of Delphi until Lord Timaeus told us a few months ago that you had contacted him."
Was that suspicion in the man's voice? No. The truth is, Jacob _has_ been gone a long time. Who knows how much has changed.
Jacob forces a smile. "I very nearly did. It's quite a story. I'll tell you all about it once I've met with Lord Timaeus."
Gaius returns the smile. "He is with High Lord Gideon, dining, I believe. Follow me. In these accursed tunnels, it's easy to become lost." Jacob sees the tunnel diverging into three paths. Gaius takes the left one.
"Did your sister marry that fellow?" Jacob asks. "Last time we spoke, you were worried the dowry negotiations would fall through."
Gaius's grin gleams white. "She did! And in a few months, she's going to give me my first niece or nephew!"
Jacob claps him on the back. "Wonderful news!" Though he's hardly listening—the thud of his heart threatens to drown out everything else. He must speak to Tim. He must understand. He must be wrong about their plans. He needs to hear it from Tim.
As they march down the corridors, Jacob hears greetings from his brothers as they recognize him. Lord Aethon, red-faced and barrel-chested, who can drink any man under the table. Melchior, the handsome Persian with the proud face of a hawk and a beautiful deep singing voice. Dexios, the Spartan, who can calm the wildest horse. As urgent as Jacob's need is to see Tim, he has to stop and greet his fellow Lords, men he has camped with, trained with, fought beside. Men who are, in a way, his brothers.
But even as his chest soars with something like relief to see their familiar faces after so many months, he asks himself how they could be complicit in what he suspects?
The murder of children.
He shudders. It simply can't be. Aethon has three boys of his own in Boeotia and spends a fortune on their tutors. Melchior has a beloved daughter by a beautiful courtesan in Sardis. Gaius is delighted that he is going to be an uncle.
There must be something he doesn't understand. Urgency courses through him. He must talk to Tim.
Finally, Gaius stops in front of a door. "They're in there." He smiles warmly. "It is good to see you back, Lord Jacob. We've missed you." Then Gaius continues down the corridor, leaving Jacob to enter the wide doorway alone.
He finds himself in a bright, torch-lit room. The staircase in the corner ends halfway up, but the walls are richly painted with scenes of gods and goddesses in white, blue, and red. The floor is of red marble, striated with white. Clearly, this was once a room in a rich man's house—nearly a thousand years ago. And in the center of the room is a table, where High Lord Gideon and Tim sit, dining.
There's a moment of silence while the two men take in Jacob, and then Timaeus leaps to his feet and strides around the table to embrace him. "By all the gods," he exclaims, grinning widely. "Jacob, you're back!"
Seeing Tim's comical face and bulging blue eyes, Jacob's heart leaps with unguarded happiness. After all these months of separation and danger, it is so good to see his best friend again. And Tim, he notices, has a new confidence, a spring in his step.
"Once we arrived here, I sent word to you in Byzantium," Tim continues, "to find us below the ruins. But I assume you had already left."
"Yes," Jacob says. "That must be what happened. I came weeks ago and have been waiting for you here ever since."
High Lord Gideon stands now, too, a bright smile lighting his dark face as he opens his muscular arms in greeting.
"Welcome, Lord Jacob," he says in his deep, rich voice. "I am so glad you have rejoined us at last. Eat, drink," he says, gesturing to the table. "And then you can tell us where you have been."
And while the scent of lentils and lamb is delicious, Jacob's insides twist. "I will, my lord, but first..." He hesitates. "I saw something strange in the tunnels. Children who seem...unlike other children I've seen before."
Gideon nods gravely. "The Smoke Bloods, yes. That was Lord Timaeus's discovery. Come now, sit, and Tim will tell you all."
_Smoke Blood._ Now Jacob has a name for it, but there seems to be no way around breaking bread with Gideon and Tim.
He joins them at the table while Tim spins a fantastical tale of ancient legends, a man of Smoke, the ambitious Queen Cynane, and a powerful potion. Of how, when he told Gideon of this new magic, the High Lord suggested they come to Troy, as the Lords had heard rumors of hidden, enslaved children here. There is something cocky in his behavior, something arrogant that Jacob has never seen before. In the past, Tim poked fun at himself. His foul jokes and curses could have stripped the hide right off a cow. Now he seems to be getting above himself, and Jacob realizes with a pang that he likes him less.
"In these mines," Tim continues, leaning forward, his pale blue eyes burning brightly, "are children who will never be missed when we take them east and, satiated with Smoke Blood, the monsters will return to their lairs for decades, harming no one else."
Tim smiles broadly, as if waiting for Jacob to congratulate him. But Jacob's heart sinks like a boulder catapulted into a pond. Tim isn't going to help him stop the Aesarian Lords. Tim is the _architect_ of this great evil.
The disgust that roils in his stomach is enough to make him want to cough up all the food he has just eaten, but he manages to hold himself steady.
He is already telling himself he must tread carefully when Timaeus reaches for something on the chair beside him and holds up a large iron torch, about the length of a tall man's arm. Cruel spikes adorn the long handle. On top of the handle is a basket of twisted pieces of iron. The Hemlock Torch. Jacob's heart skitters in horror.
"We have been testing the children," Tim says, and something in his eyes tells Jacob he senses his fear and revels in it. "And the torch has shown increasing levels of magic. As you know, when lighted, the torch burns green flames when Earth Blood is nearby," he emphasizes the last words knowingly and gives a sinister little nod that makes a shiver run up Jacob's spine. "It burns violet for Snake Blood, and red for soothsayers and other types of magic. But the torch burns black flames for Smoke Blood, beautiful swirling black fire with silver and gold sparks. And the flames rise higher every time we test the children."
Everything in him screams to get up and leave. Race from the room, find Kat, and get out of these haunted tunnels. The Hemlock Torch can detect them both. And Tim already knows his secret. If Jacob disagrees with his plan, will Tim reveal it—reveal _him_ for who and what he truly is? Yet how can Jacob allow him to sacrifice dozens, maybe hundreds, of children without even making a suggestion?
"It is a clever plan, Tim," Jacob stalls, "but one forged of desperation, don't you think? Are we really so low as this?"
Tim's watery blue eyes are cold as he slowly lowers the torch.
Gideon drums his large fingers on the table. "It is unfortunate we have come to this pass," he admits. "We know that. But we are forging ahead with this experiment, even as the regiments in Nekrana are working on their own solution. Supreme Lord Gulzar's engineers have created iron tubes that shoot long blasts of fire, which, they hope, will melt even the deathless soul of a Spirit Eater. They will lure the bulk of the monsters to a place with magic prey and then try to destroy them."
"We can only hope, then," Jacob says, "that the Supreme Lord's plan is successful."
Tim has taken Jacob's statement as an insult. His face has become as hard as the stone walls around them. "By sacrificing a few children, Jacob, we will save tens of thousands, perhaps hundreds of thousands of lives. The children understand that they have been chosen for a great purpose. Is there not dignity in helping to save mankind? Is not war a kind of mass murder all its own, especially war against the deadliest of history's monsters?"
Jacob's mouth feels dry. He is speechless at the horror of what Tim is suggesting, of what Tim has become. His old friend, a glimmer of dark pride on his face, continues, either oblivious to Jacob's horror or propelled by it. "Perhaps you, Lord Jacob, have another, better solution? Would you perhaps care to sacrifice...yourself?" He takes an oil lamp from the table and lights the Hemlock Torch.
Jacob's heart stops. Gideon, puzzled, looks between Tim, Jacob, and the torch, whose orange flames are starting to burn green. Jacob stands as his chair falls backward to the ground. His fingers are already around the hilt of his sword, when he sees the truth dawn in Gideon's face.
But even so, the large man surges to his feet, sword already bared. "A Blood Magic in our midst! Traitor!" Gideon bellows and hacks his blade down toward Jacob's sword—but the blades never meet.
A dagger suddenly blooms from Gideon's throat.
Timaeus's knife, thrown from the other side of the table.
Jacob's heart is racing in his own throat, and he desperately swallows back his shock as the High Lord's dark eyes widen in pain and surprise. Blood bubbles from his mouth. With an enormous effort, his shaking hands pluck the dagger out, and blood pumps obscenely from the ragged wound. A horrible, sick rattle rises from his throat. He covers the wound with his large hands but blood pours between his fingers. He falls forward, dead, knocking over his wine cup. Dark red wine flows onto the table, mixing with Gideon's dark red blood.
Jacob feels like he is going to vomit. He swings toward Tim. "By all the gods! What have you done?"
But Tim's eyes are shut. He grimaces and shakes. Something is happening inside him, Jacob realizes. He should kill him now. He raises his dagger and...can't.
Tim opens his eyes, which glow like twin blue flames. He seems different now, _stronger_. A slow, wicked smile spreads over his face. "A double betrayal is the best kind," he hisses, laughing darkly.
Jacob grips his sword tighter, but Tim doesn't attack. Instead, the man takes a whistle from his belt and blows it loudly three times. The signal for _a brother in danger_.
The truth rushes at him with a kind of stunning clarity: Timaeus has set him up. He will blame Jacob for Gideon's murder, and all the Lords will believe him. Jacob has been missing for months, while Tim is Gideon's trusted second-in-command, whose clever plan will save the world from the Spirit Eaters. And if any Lords were outside the door just now, they would have heard Gideon's bellowed accusation: _traitor_.
Rage rumbles deep inside Jacob. Tim has betrayed his best friend and his commander, has broken his vows of obedience, and has vanquished every vestige of ethics he ever possessed. Tim lunges toward him now, but Jacob slams his fist against the stone wall. Anger courses from his fist into the stones, and a shower of rock and grit falls between them as the earth trembles. Tim reels backward, his hands over his head, giving Jacob just enough time to run out of the chamber.
His heart hammers. He cannot believe what he just saw. He hears footsteps, urgent voices, sees the glow of torches coming around a bend in the tunnel. He runs into the dark shadows. His legs feel like lead. He can't see a thing, but his Earth Blood allows him to sense the right path—left here, right there, now straight.
Whistles and shouts follow him, echoing off the stone. Boots slap and thud. Finally, he sees a dull bloom of light from a lantern spilling from a doorway ahead and tumbles inside. She is sitting alone, her head against her arms, which are propped up on her knees.
"Where are the children?" he asks, but she shakes her head.
"They won't listen to me!" she says, her voice a cry of anguish. Tears track down her dusty face. "They don't want to leave. They keep saying that they are chosen to save the world! But, Jacob, what's happening? That earthquake, it was you, wasn't it? Where's Tim?"
"Timaeus...is not with us," Jacob says, not knowing how to tell Kat all that has happened in the last few minutes. "We need to get out of here, now!"
"But the children!"
Jacob grinds his teeth. Yes, the children. He has only hastened their deaths. With Tim now the head of the Aesarian regiment and knowing of Jacob's disapproval, he will most likely take the children east as soon as the sun rises in a few hours. This is their only chance to save Roxana and the others. But how? Even now, he can feel the stomping feet of the Aesarian Lords through the tunnels as they fan out to find him...and Kat.
"We would need an army to get out of here with all of us alive," he groans as the truth settles around him. He closes his eyes. This is the end. He has lived a life of impossibilities, but now he can perform no more miracles.
He feels a warm hand on his shoulder.
"Wait," Kat says, and he opens his eyes to see her bright with excitement. "We _do_ have an army."
Quickly, she reaches into her bag and pulls out the proof of her story: two dozen blue-glazed figurines shaped like tiny mummies, their arms crossed over their chests, their bodies stamped with strange signs. They look like normal Egyptian statues to him. There is nothing about them to suggest an ancient princess, a cursed city, and a powerful general whose soul split into countless pieces. How is Jacob supposed to transform the figurines into the warriors they once were?
He lets his fingertips brush over them and quickly pulls back his hand. "I do feel something powerful," he whispers. "But I don't know how to read those symbols. I can't do the right ritual. It's useless."
"Lord Gaius! Over here!" cries a deep voice from somewhere down the hall.
Kat's eyes sparkle in the semidarkness. "When I was on Meninx," she says, her voice tight with urgency, "I could no longer access my Snake Blood powers. But it was imperative that I do so to build the Atlantean Mechanism. Ada convinced me to swim into the ocean and not _try_ , but do the opposite of trying. Relax. Our powers are always present. Sometimes we just block them with fear."
Her head turns at the thud of feet against stone. They are about to be found out. And he _is_ afraid, so afraid—for himself, for Kat, for all those orphaned children—how could he not be?
Urgently, Kat grabs his hands, placing one of the doll-like sculptures into them. "Hold the ushabti and enter it, just as you can enter water and earth and rocks. The statuette is made of clay, of earth. And an Earth Blood created it. Your ancestor. You have his blood in your veins, his wisdom in your soul. You don't need Egyptian words or ancient rituals. You need only to relax and trust yourself. And _hurry_."
_Relax. And hurry._ Jacob wants to argue that doing both things is impossible, but he remembers his little brother Cal, all long gangly limbs and tousled hair, and how he couldn't save him from evil. How he would always regret not being able to save him. But there's a chance he can save _these_ children. He must at least try.
He closes his eyes and takes a deep breath, blocking out his anger at Tim, his horror at Gideon's murder, his fear for Kat and the children. He floats, disembodied, into the earthen statue in his hands and feels the molten heat of the earth inside it. He senses the spark of life, the hum of thought, and he opens himself up to it. He lets the thoughts and sensations flow over him. He smells lotus blossoms and hot sand and...rotting corpses. He sees, rising from the desert, a lovely walled city, which catches fire as people flee and scream. He feels warm wind caressing his face...and unbearable pain searing through him. Light and shadows, laughter and screams, life and death...
It is too much.
He feels his head will burst from the strange sights and sounds and smells. But something tells him to keep holding the figurine. After a time, the images fade.
_Blood of my blood, Earth Blood_ , calls a voice somewhere deep inside him. The voice is incredibly quiet despite its infinite power. It is ancient beyond measure and has always been inside him. He recognizes it as the center of his very being. And then he knows.
He sets the ushabti down, takes out his dagger, and pierces the inside of his left arm. A single drop of rich red blood falls onto the statuette and splashes. Kat looks expectantly from the ushabti to Jacob. The earth rumbles, and her eyes grow wide with fear as she looks around the room. But Jacob knows this is no earthquake that will destroy the underground city, trapping them all under tons of stone.
This is the rumbling of life coming into being.
There is a bang and a flash of light. Jacob, on his knees, looks up at a tall wide-shouldered warrior, his body made of cracked and flaking clay, the eyes in his broad dark face of orange flame. He wears strange armor, with a leopard skin thrown over his left shoulder, and a square helmet with long lappets on each side. In his massive right hand, he carries a heavy sword, and in his left a long rectangular shield.
He looks around, his thick neck muscles creaking, and his fiery gaze lights on Kat. "Princess Katerina," he says in a voice like thunder.
"General Wazba," she says, getting to her feet.
Now he stares at Jacob, who rises slowly. "You have awakened us," Wazba says to him, then turns to Kat, "though we were supposed to rest when Princess Laila destroyed her city. We toiled for her for centuries and deserved our rest."
Footsteps clatter in the hallway. A distant voice cries, "Turshu! He may be down there! Get torches!"
"We need your help," Kat says, "or the enemy will destroy us. Please, General, you served Princess Laila so faithfully. She thought enough of me to give me this." She holds up her golden fingertip.
He stares at it. "I remember. The princess was not known for her generosity once she was cursed. She must have liked you very much indeed."
"Can you help us?" Kat persists.
Wazba nods. "We will fight this enemy and then return to dust, to rest."
"Can you waken the others?"
Wazba tilts his enormous head. "My brothers and I are one. One soul, one spirit, one heart and mind." He raises his right hand, extending his sword, and mutters words that Jacob doesn't understand. The room explodes in bangs and bright light and is suddenly full of Wazbas.
"Down here!" calls a voice from the hall. "I see a light!"
"Kill the soldiers," Kat orders. "But save the children."
Wazba nods. He calls to his warriors, _"Ushbeth-ka-tark!"_ and gestures for them to follow. They rush from the room.
"Retreat! Retreat!" cries a panicked voice. Weapons clash in the corridor, metal on metal, metal thumbing on cowhide shields, grunts, and screams. Footsteps fading.
Jacob and Kat peer out. Two Aesarian bodies lie bloody on the ground. As Jacob approaches them, he recognizes Gaius, the Roman, so happy about becoming an uncle. And Phaedron, the Thracian, who bought each of his five brothers a large fishing vessel with his earnings from being an Aesarian Lord.
He feels his heart break. These were his friends, his colleagues. His brothers.
Kat, too, seems struck by the waste of it all. Tears well in her eyes, and she bites her lip. Then she deftly picks up Gaius's shield and hands it to Jacob.
"Jacob." Her voice cracks as she picks up Phaedron's shield. "I know this is hard for you. But we must make sure the children are unharmed."
Yes, she is right.
They run down the corridor and see more Aesarian bodies. Jacob forces himself not to look at the faces. They follow the sounds of battle to the little balcony overlooking the large room below.
The ushabtis and the Aesarian Lords battle viciously. The ring of sword on sword is so loud it is as if Jacob is in a forge with a hundred blacksmiths striking their anvils. He sees one Aesarian drive his sword deep into the heart of an ushabti. A stream of fire blasts out of the hole in the clay chest, burning the Aesarian's face. He screams, drops his sword, and covers his face with his hands as the ushabti, expressionless, runs him through.
Another Aesarian knocks the sword out of an ushabti's hand, then, in one clean sweep, slices off the baked-clay head. A spout of fire shoots toward the ceiling and the ushabti drops to his knees. He bends forward and blasts flame from his open neck, catching the lord's cape on fire. Howling in pain, he becomes a human torch, ricocheting off other soldiers.
It is an Aesarian who cuts off his head to put him out of his pain.
Jacob sees no children in the fray, thanks be to all the gods, but neither does he see Tim, who should be easy to recognize because of his slight size. What he does see is that, one by one, all his other Aesarian brothers fall, and each time his heart breaks until he wonders if the tiny shattered pieces can ever be put back together. Gone, so many brothers, so many friends—and it's his fault.
An explosion of heat, light, and clay shards forces Jacob and Kat to the floor, covering their heads with their hands. They rise slowly, looking down at the dust-filled room below.
Jacob turns to Kat and sees Tim standing behind her, his dagger against the back of her neck. Kat's eyes widen, and she straightens.
"So, I see you found your little lovebird," Tim sneers.
"Don't hurt her," Jacob commands. "She has nothing to do with this."
Tim's eyes blaze. "Everyone has always underestimated me," he says with an eerie calm. "Even you, who I thought were my best friend. You didn't trust me with your secret. You had me sent away to spy, to work as an acrobat and a fool. You were born with magic, and you don't want anyone else to have special powers."
The guilt that always sits in Jacob's chest finally splits open, slicing him in two.
"I am sorry for having you sent away." He hopes Tim can hear the truth in his words. "But, brother, this Smoke Blood has twisted your soul. The friend I knew never would have betrayed me with the torch or killed Gideon. Stop playing with this foul magic."
"Playing, is that what you think I'm doing?" Tim roars. "If I was as tall as you, would you say I was _playing_? Or would you say I was supervising? Experimenting? _Aspiring_ to a greatness you could never imagine?"
Jacob searches for the right words to say and can't find any.
"Little did you know," Tim continues, "that in sending me to Dardania, I would attain powers beyond anyone's wildest imagination. Queen Cynane discovered Smoke Blood there, and I merely spied on her to learn about it."
"Cynane?" Rubbing her golden fingertip, Kat turns around slowly to face Tim. Now his dagger is poised to thrust through her throat, but she doesn't seem concerned. Her eyes almost seem to glow with sympathy, and Jacob wonders if she is trying to use Snake magic to reach him.
"Timaeus," she says gently, "Cynane was never someone you—or I for that matter—should have trusted. You are better than this. Jacob has told me much about you. You are clever, funny, loyal, a good friend. You can be those things again if you let this evil magic fade from your veins."
Tim blinks. "Can I?" he says. "We shall see."
And then...he vanishes.
Jacob thrusts out a hand to see if Tim is still there, camouflaging himself, as Roxana can. But his hand meets nothing.
Tim is gone.
A silence fills the secret city.
There are no more screams, no more footsteps.
Cautiously, Jacob and Kat look back at the room below.
Dead Aesarian Lords are strewed across the floor. Jacob doesn't want to look. But he has to look, if only to honor them with his grief. He sees Ervin and Farbod, cousins from the far reaches of Persia with amazing archery skills. Asdrubal, from Carthage, an amateur magician who could pull a coin from your ear or a rabbit from under your cloak. Tutmose from Egypt, devoted to a god with the body of a man and the head of a crocodile. Jacob looks at each one of them—Greeks, Persians, Gauls, Scythians—and feels as if his soul will split into dozens of tiny pieces, just as General Wazba's did so long ago.
They are dead.
And everywhere, there is a fine layer of clay dust that smells faintly of an endless river churning through sand, mixed with the delicate scent of lotus blossoms. All that remains of the ushabtis.
But then...through the billowing clouds, small forms begin to appear, wandering into the hazy, blood-soaked room. Some of the children hold hands, while others sniffle.
"Katerina?" They turn to see Roxana creep forward, picking her way through the dust and bits of clay. Her small fist grips a honey cake. She looks around at the devastation.
"Is everything going to be all right?"
CHAPTER NINETEEN
ALEXANDER
As he paces up and down the small antechamber outside his mother's rooms in the Ambracian palace, Alexander feels a kind of trapped tension in his muscles—he is far better suited to battles on the field than in the home. What could his mother and Philip be talking about all this time, just on the other side of that thick door?
He should have gone to the guest quarters with Heph. But then, there was a whole other kind of tension there, in their closeness, in what almost happened but didn't in their shared tent near Chaeronea. Since then, they'd gone back to their old, easy friendship and had vigorously celebrated the signed treaty of the League of Corinth before packing up and heading here, to Epirus. Home of Olympias's brother, King Xander. But still, when things were quiet—in between feasts and strategy meetings and head counts and recruitments, in between arranging for ships to sail west and navigating not only the rough sea but also the storms of King Philip's moods—there were moments when Alex knew, in his heart, that things would never _really_ be the same between them, at least not for him. Something had awoken that he had hoped to leave dormant, curled, and sleeping peacefully in the deepest part of him forever.
Now he paces, wishing there were a place to sit, almost— _almost_ —missing the frivolous decor his mother always obsessed over back home at the Pellan palace. Everything he's seen of his uncle Xander's palace here in Epirus is old-fashioned and threadbare. When his entourage first rode into a courtyard and Alex saw the low-slung building with arrow-slit windows, he assumed he had approached the palace from the rear and was looking at the kitchens and stables. But it was the main entrance.
They disembarked in Epirus only an hour ago, and, after King Xander welcomed them, Philip insisted on talking privately to his wife. The two of them have been in her chamber for quite some time now, and it makes Alex nervous. The king has some plan afoot, a plan that he has hinted at but not revealed since the victorious Battle of Chaeronea.
Finally, the door bursts open, and Philip comes out, scratching his head. "I think her injuries from the flood have affected her brain," he says, then, possibly seeing the alarm in Alex's expression, adds, "for the better. I've never known your mother to be so calm. Pleasant, even. She doesn't seem to mind I brought Cleo, or that she is pregnant."
This last statement worries Alex even more. Either Philip is right, and Olympias's brain has been addled by her injuries, or Olympias is playing along with Philip because she, too, is plotting something and his plot weaves perfectly into her own. He wonders whether his mother will stay calm when his stepmother—who is only a year older than he is—fondles and kisses Philip in front of her, which she did with such nauseating frequency on the week-long journey from Athens that several times Alex wanted to jump over the side of the ship and swim to land.
"The oddest part was," Philip continues, rubbing his beard thoughtfully, "that she accepted my plan so quickly. I was sure she would fight it."
"What plan?" Alex asks.
Philip fixes him with his single dark eye. "Oh, you'll see very soon. You watch carefully and you will see a master strategist at work, boy." He beams broadly. "Well, it's your turn to see the witch. Enjoy yourself." And he marches off.
Alex inhales sharply as he places his palm on the door and pushes. How can he look into the face of the woman who both brought him into the world and almost took him out of it so she could be with her lover? Facing Olympias is more daunting than facing the Sacred Band of Thebes was. He squares his shoulders and steps inside.
He finds himself in a modest chamber with a low roof of oak beams, lime-washed white walls, and moth-eaten tapestries. Olympias sits in a chair by the open window, enjoying the warm spring breeze from the Ambracian Gulf. She turns to him, her face a strange mixture of delight and trepidation.
"Alexander!" she says, extending a thin white arm.
He approaches her, taking in how much she has aged—beneath her heavy makeup, her skin seems thin and wrinkling. A large pasty dollop cannot hide the thick red scar on her temple, the result, everyone thinks, of injuries sustained in the flooded Byzantine palace, but which Alex knows was caused by Jacob. He kneels for her blessing, and she puts both hands on his head. They feel like claws.
He looks up to find her studying his eyes, looking, evidently, for Riel's green ones. He forces himself to stare right back. She falters, looks away. Tears slide down her pale cheeks.
"Gone," she whispers. "Gone."
"Disappointed, Mother?" he asks, surprised that the tone of his voice is sharp enough to hew a tree.
"Do you know...everything?" she asks, her green eyes swimming with unshed tears.
He nods, clenching his jaw so tightly it hurts.
"I...didn't know," she begins, tears falling more quickly now, "that when I brought him back to human form, it would be _your_ form. I swear, I didn't know. And after it happened, I thought... I hoped that, with his powers, he could regain his old form and leave yours. I wanted you to come back. I've always loved you. Don't you remember how I've doted on you, all these years?"
Alex rises and looks out the window. Clustered at the bottom of the palace hill is a tidy village, and beyond that the sea. Is she telling the truth? Probably. She has always been fiercely devoted to him as an extension of herself and, as he grew older, as a means of wielding political influence. Her selfishness alone should ensure her devotion.
"He was slowly killing me," he says, his voice tight, strangled.
"How did you...come back?" she asks.
Alex laughs harshly. "You did it," he says. "That much I can thank you for. By scratching Jacob with that hairpin. The prophecy said Riel could only be killed by Earth Blood, and, it turns out, Jacob is Earth Blood."
Olympias's eyes widen, and a skeletal hand goes to her throat. "What?" she croaks.
"Yes," Alex continues, enjoying her discomfort. "And Katerina, the daughter you tried to kill, was in the palace that day, searching for the Earth Blood. She found the hairpin, coated with Jacob's blood, on the floor, and when Riel, in my body, attacked her, she scratched him on the neck with it. He died so I could live."
What he doesn't add: _a part of me died, then, too_. His blood powers. In that sense, Kat really fulfilled two prophecies that day. She _did_ kill Alexander, in a way, in order to free him, in order to vanquish Riel.
She sags into her chair. Alex expects her to pepper him with questions, but instead she just whispers to herself, "He's really gone, then. Forever." She's sobbing now. Large tears run down her cheeks, leaving white rivulets as they carry her cosmetics away with them. On the ship over here he had planned to yell at her, to punish her for allowing Riel to take over his body. But now, seeing this sick old woman crying like a baby, he cannot.
A small door next to the bed opens and a girl comes out, a look of concern in her large hazel eyes.
"My lady?" she asks, kneeling by Olympias's side. "Should I bring a calming draft?" Alexander recognizes her as a girl he had seen tending his mother's injuries in the Byzantine palace as soon as he and Heph returned from their horrifying night floating in the cold straits. Now, as then, he feels that there is something familiar about her.
"Yes, Patra," Olympias says, wiping her cheeks with the heels of her hands. "And when you return, light the perfume burner." The girl dashes from the room, and Alex finds his gaze following her slight figure until she has closed the door. Could Patra seem familiar because she reminds him of Kat?
"This palace is much worse than I remember." His mother sniffs. "Since Queen Heraclaia died, my brother has let it go. The mildew smell is awful. I've had a perfume burner put in your room, too, Alexander."
"Mother," Alex says, "mildew isn't the worst thing in this palace right now, I think. What is this plot you are hatching with Philip, or more accurately, against him?"
She smiles with such satisfaction that, for a moment, some of her former beauty returns, lighting up her face. "It's an alliance between Epirus and Macedon, Alexander. An excellent idea. I no longer have the strength to fight or plot. Philip can do what he wants." She looks at him with her old fiery intensity. "Except disinherit you for some brat that bitch bride of his gives him."
"He assured me he will not do that," Alex replies. "And I believe him."
"He assured me of that, too," she says airily, waving a thin hand so that her golden bangles clatter musically. "So, it seems we are all in perfect accord."
Words he has heard approximately never. Alexander shivers, uneasy, as he leaves his mother's chambers. Philip forced him on this side journey before promising part of his army to lead into Persia. Alex doesn't like an errand he doesn't fully understand. Shoring up allies is all well and good, but Epirus should be the least of his concerns right now.
Anxiety eats at him as he makes his way up the winding stairs toward his own guest room. So much has changed over the last few months, and even weeks. Two great threats loom in the east: the armies of Persia and a migrating mass of legendary monsters. Rumor, confusion, and fear have stirred up a kind of restlessness among the people he has not seen before.
Meanwhile, Alex has spent the last six months trying to make amends for his behavior when Riel possessed him: convincing people he was ill, not in his right mind, and winning back their trust and approval, all the time knowing that his birthright is one of darkness and evil. He spent many years hating the man he thought was his father—Philip—when he should have saved his loathing for the truth. And now, on top of all that, when he thinks of his closest and most trusted friend in the world—Heph—he thinks only of the ache buried deep in his chest, and the knowledge of what cannot and will not be.
Not for the first time, Alexander wonders at the insidious, snarled nature of fate, which shows us that our fathers are not our fathers, our fears are not what we thought, and our friends are the people who have the greatest power of all to cause us pain.
* * *
After the midday meal, Alex and Heph enter the surprisingly small, dark throne room behind King Philip and Cleo to find King Xander sitting on his throne—not on the dais, but at an ivory-inlaid table that must have been exquisite hundreds of years ago. Now some of the ivory is missing, the rest cracked or stained. Next to him, on the queen's throne, sits Olympias. Xander rises and bids Philip and Cleo sit on the thrones across from him. Protocol, Alex knows, requires no king to be higher than any other king.
Tall and gaunt, with Olympias's silver-blond hair and pale skin, Xander looks eerily like his sister, though his eyes glitter gray blue instead of green. To Alex, who takes his place with Heph standing behind King Philip, brother and sister, sitting side by side, seem like ghost monarchs from an ancient myth. The queen's handmaiden Patra stands demurely behind her throne.
It's a dim, ancient chamber with smoke-stained walls lined by old wooden benches. The fire pit in the center of the room is empty, and daylight streams through the smoke hole in the roof and small open windows on either side.
As Patra pours wine for the kings and queens, Xander's cold eyes linger on her lithe, graceful form, and then he and the others dash some wine to the ground as libations for the gods and ancestors.
"Congratulations, Philip," Xander begins, drinking deeply, "on your victory over the allied Greeks. My sister and I were so pleased to hear the news."
Philip nods. Cleo puts a soft, plump hand on his arm and squeezes it, looking up at him with big, simpering blue eyes. Her other hand rests against her swelling abdomen. Alex studies his mother. Her green eyes flash, but a slight smile plays on her ruby-red-painted lips.
"We need Epirus with us," Philip says. "We must show the Persians we are united against them."
Xander casts an appraising glance at Cleo, who has wedged herself so closely against Philip that they look glued together. "And yet, Philip," he says in his thin, reedy voice, "it seems you have discarded my sister as your queen and taken yourself a new one. Our old ties of alliance have, therefore, vanished." He snaps his long white fingers.
"I give Olympias all honor," Philip says defensively. "She is the mother of my heir, Alexander, who will remain my heir, and who has already proven himself in battle as a worthy leader. But we kings, who carry the burdens of our people on our shoulders, are allowed more than one wife."
"This is your eighth, I think?" Xander interrupts in an arch tone.
Philip frowns. "I have had more burdens than most," he says quickly. "And I still want ties of blood to bind Epirus to Macedon. It is time you, Xander, take another wife. You need an heir."
Alex stiffens at this. What young relative could Philip be offering Xander? He knows of no nieces or cousins of the right age.
Xander smiles, revealing long teeth that make him look like a rabbit. "Are you offering me some cousin of a cousin of yours?" he asks. "Because she would have to be very beautiful..." He leans forward with a passion in his face that surprises Alex, and says in a low voice, "Very beautiful indeed. And young. And a virgin. And sweet tempered. And I would have to see her before I agreed."
Philip leans back and laughs, slapping the table so that the wine cups shake. "By Zeus, Xander, you are more of a man than I thought. Yes to all those things. But the bride I offer you is not a cousin of a cousin." He examines a large iron ring on his left hand and says, "You have heard, I imagine, the rumors that the royal House of Macedon has a recently discovered daughter."
Alex's skin prickles. Has Philip managed to find Kat in Troy? Has he captured her and brought her here? Heph casts him a panicked, questioning look, but Alex does not betray his confusion, even as his mind races, trying to get a step ahead of Philip despite having clearly fallen two steps behind.
Xander casts a sideways glance at Olympias, who smiles congenially. Which makes no sense. Surely, she knows that a _new_ alliance with Epirus would only further usurp her own political relevance. She would become practically disposable. "I think everyone has heard that gossip," he replies evenly. "And let me tell you, I would never marry my own niece. I won't help you out of your problems with public perception, if that's what you're looking for."
"It is not gossip, brother," Philip continues, "unless such stories cast aspersions on the virtue of Queen Olympias." Alex watches his mother glow with pride; it must be the first time in ages anyone has called her virtuous.
"This daughter is not hers by another man, as is rumored by vile scandalmongers," he adds, "but is mine, by a palace-serving wench. No blood relation of yours." The queen smiles even more broadly.
"Oh?" Xander asks, sitting up straight.
"A lovely girl, only seventeen, with royal blood flowing in her veins."
Xander takes another gulp of wine. "Well, then," he says, licking his thin lips. "That is different."
Alex feels another chill. Is Philip really about to promise Kat's hand to this old king without ever having even met Kat or discovered her whereabouts?
Philip rises, pushing his chair out behind him with a scrape, and holds out his hand. "Patra, my dear," he says.
Everyone's head turns as, timidly, Olympias's handmaiden steps around the table and takes his hand.
Relief floods Alex. Not Kat, at least. But then he finds that his head is reeling in confusion. What in Hades is going on? What game is Philip playing? Alex looks to his mother, whose face is a picture of equanimity. Why is Olympias playing along with him?
Xander is speechless for several long moments as he studies Patra. Her unbound hair gleams golden red, her large hazel eyes are modestly downcast. The pale green gown reveals a slender, shapely figure.
"Sister," he says, turning to Olympias, "your handmaiden is King Philip's daughter?" His shrewd eye sweeps between her and Philip, calculating. "Why didn't you tell me the truth?"
Olympias looks at her lap and says, "We discovered her last summer, brother. King Philip made me swear on the vengeance of the Furies to tell no one that he had recently put his bastard daughter into my safekeeping until such time as he used her to craft a political alliance."
Xander rises from his throne and stares at Patra, whose pale freckled skin flushes becomingly. He walks over to her, drinking in every bit of her. He goes behind her, takes a long tendril of her thick hair and curls it around his finger, inspecting her as he might a prize horse. She has not raised her eyes and remains as still as a statue.
"Sister," Xander says, letting go of Patra's glossy curl. "What say you to this alliance? For I will do nothing that will dishonor or displease you."
"I am well pleased, brother," she says. "You need a queen and heir. This palace needs a mistress. And the future kingdom of my son needs an alliance. I do not resent Philip taking a younger wife. It is only natural." Here she beams at Cleo, who seems startled.
"And to prove my good will," Olympias continues, "I will bestow on Cleo rich gifts, worthy of a queen of Macedon, gorgeous robes, some of my best jewels, and other things. I understand she lost all of her bridal trousseau in the great flood of Byzantium, and I no longer need them."
Now Alex knows something is very wrong. He opens his mouth to say, "Mother..." when Xander interrupts him.
"Very well, then," the king says, his voice throaty with something like passion, and Alex marvels at what he knows he is about to hear—at the power of passion over men's capacity for reason. "I agree to take the beautiful Patra as my wife."
* * *
Bucephalus flies down the road toward the seashore, his hooves pounding the dusty earth into submission. As always when riding the big black stallion, Alex feels the horse's power as his own. He is no longer the prince with the weak leg, but a glorious being, perfect and strong, with the salty wind whipping through his hair and the soft golden light of the setting sun on his face.
Philip, thundering beside him, slows and veers toward a ridge overlooking the town below and, just beyond, the sun, like an enormous red ball, sinking into the choppy dark waters of the gulf. Alex, pulling up beside him, is mesmerized by the sight, and almost expects to hear the hiss of fire meeting water. The king pats his red mare on the neck and says, "It's good to be out of that dark old palace. An excellent idea of yours to go for a ride."
It is chilly out even though they are nearing mid-May—the capital of Epirus seems as inhospitable in climate as it is in personality. But the air _is_ refreshing. It clears Alex's mind. He looks at fishing boats returning to the harbor on a slate blue sea, white caps rolling onto a sandy beach, and the red tile roofs of hundreds of houses, briefly wondering what the place was like for his mother, who grew up here.
"Father," Alex says, turning toward him. "I wanted to get out of the palace so we could have this conversation unheard by others. What happened in there?"
Philip grins mischievously. "That, my boy, was a diplomatic coup."
"What do you mean?" Alex asks. "Is that girl really your daughter?"
"Hades, no," Philip says, laughing heartily. "At least, not that I know of. Olympias tells me she was the oracle of Delphi. Ran off during an earthquake and ended up in Byzantium."
That bit of news startles Alex. Now he realizes why Patra seemed oddly familiar when he first saw her in Byzantium. As Riel, he had spoken with her on several occasions. Some cloudy memory of those conversations must remain.
"I needed to make an alliance with Epirus that eradicates your mother's position as liaison, to take her power away," Philip continues, watching a seagull cry and wheel. "Xander was right. With his sister discarded, what connects us now? I was planning on using that real sister of yours, Katerina, but since you couldn't find her—" and here he turns to Alex, his eye narrowing a bit "—I had to improvise. The girl is young, beautiful, and devoted to your mother. And I thought perhaps Olympias might—after a huge row, of course—allow her new pet to become the queen of Epirus. But I didn't expect her to agree so quickly, to even suggest they marry immediately if Xander agrees."
Alex's mind whirls. Yes, his mother seems old, sick, and tired. But even if she was on her deathbed, thrashing in bone-shattering pain, he imagines she would still be plotting something. And her promise to give Cleo her jewels and gowns is like an alarm gong pounding in his head.
"Father," he says, his voice ringing with warning, "doesn't that worry you?"
But the king has already galloped off ahead of him.
* * *
Back in the warmth of his bedroom that evening, with the brazier blazing, lamplight casts soft shadows on the maps and charts littering the table. Heph tallies the numbers and says, "According to these latest reports, we will have five hundred and three ships. If the Athenians convince their allies on the islands to join, we might get thirty or forty more."
Alex nods. "Excellent. And we need to launch the invasion before the autumn storms can scatter the ships. Six more days here, Heph. To me, it feels like six more years. We should be in Athens now, not dillydallying in Epirus."
Heph straightens the papers into orderly piles. "Agreed," he says. "Though it wasn't a bad idea to obtain the Epirote alliance. That's three ships, eight hundred men, and two hundred horses."
It feels so right to be planning the invasion with Heph. Just like old times.
"Alex," Heph says, "Still no word on where King Philip wants to launch the invasion?"
Alex says, "Yes, he has decided to land where the Granicus River meets the sea." He doesn't add that the landing location is only a two-day ride from the Temple of Asklepios near Troy. Even though he no longer has his Snake Blood ability to read men's minds, he knows what Heph is after. Heph says nothing, but his face brightens.
"Let's give the king our new tallies right after breakfast tomorrow," Alex says. "Right now, I think we should get some sleep."
But hours later, the creak of a door pulls Alex from the darkness of slumber. He searches under his pillow for his phoenix dagger, dipped in the blood of the last phoenix. Its iron hilt is crafted in the form of a phoenix, its eye a glowing ruby, its long, pointed beak and wings raised skyward. The bird rises from curling flames of solid gold near the iron blade. It is a peerless heirloom, famed throughout the known world, and as soon as his fingers curl around the cold metal, his panic subsides, and he turns over to face the intruder.
His next thought is, _Heph?_ But no. Standing on the threshold is a thin, draped figure holding a lantern, the dim light from the lamp inside casting a hazy glow through the oxhorn panels.
He rubs his eyes, wondering if he is dreaming, or worse—perhaps this is some memory of Riel's from thousands of years ago. Then she closes the door behind her and approaches him. His whole body relaxes when he realizes this is no memory of the dead god but a flesh-and-blood girl.
"Patra?" he asks, pushing himself up in bed. "Is everything all right?"
"Yes," she says, with a sad little laugh, "and no. Forgive me for waking you, my lord, but I thought I would die if I didn't have someone my age to talk to, someone who might understand."
Alex slides off the bed and gestures to a pair of chairs. "Sit down," he says gently, as he takes the oil lamp from her lantern and lights all the other lamps in his chamber. He sits across from her, pulling his sleeping tunic over the snake-like scar on his left thigh.
"I remember you from the temple at Delphi. You were different then," she says, tilting her head. "Your eyes were green, and you were...harsh. Impatient. Arrogant. You are much nicer without him inside you."
Alex massages his forehead. "I remember very little of that time," he says.
She smiles sadly. "Funny, isn't it? I was born a slave, then sold into a different kind of slavery as the oracle of Delphi." She tells him of life in the sanctuary, watched over by sharp-eyed matrons and strict priests, of a constant stream of rituals and festivals where she, the lead actress, was forced to faithfully follow a script written by others. Alex listens because it's all he can offer her, really. He can't renegotiate Philip's dealings or expose his lies for the sake of a servant girl, no matter how much he may feel for her position. And perhaps, after all, to be listened to is all she wants.
"I escaped in the great earthquake, thinking I would finally be free," she says, toying with the fringe of her shawl. "I wanted them all to think I was dead, my body buried under a heap of rubble. You see, I was in love." She looks up at Alex, and he feels the weight of sadness in her large hazel eyes. "In fact, I think I still am. But he didn't love me. I hoped he would with time, but one night I realized he would always love someone else, that there would never be any room for me in his heart. And so, I left. Well, I tried to leave. Boarding a boat in the harbor, I found myself washed right into the palace when the waves came."
Alex frowns. How many people's fates were changed that day of watery horror? Not just those of Kat, Jacob, and Heph.
Patra continues, "It's as if the gods themselves don't want me ever to be free and have placed me in a situation where I now face a new kind of slavery. Marriage to an old man." Her pale face has a haunted look.
"He's not really...old," Alex says, fumbling for the words to comfort her.
"He's more than twenty years my senior."
Alex sighs, thanking the gods that Katerina avoided this. Yet it seems a hard fate for this sweet girl before him. "You will be queen," he says, in a cheering voice.
"I would rather be free," she retorts, pulling her thin shawl more tightly about her.
She seems so like Kat in some ways, he remarks again, not just her appearance. "Women are rarely free, Patra," he says gently. "Most of them must marry, and Xander is not ugly and not unkind. I believe he was already half in love with you when he thought you were merely my mother's handmaiden."
"How nice for him," she says sarcastically, "to be in love with me. What about my feelings?"
But they both know her feelings don't matter. "Cruel as heartbreak is, it does fade, with time," he says carefully. "Or so I have heard." She sighs deeply, and he's not sure if he really believes his word, either. What _can_ he say to cheer her up? Perhaps he should distract her. "Now, tell me," he says, "when you are queen of Epirus, what is the first thing you will do?"
She stares into the darkness beyond the lamps as if she is staring into her own bleak future. "It makes no difference," she whispers. In the glowing amber light, he sees in her profile the beauty of the woman she will become. And though his Snake Blood powers are lost, he has the strongest feeling, a kind of intuition, that she will be a beloved queen. A kind mother of many children. And, perhaps with time, a woman who loves her husband.
"No, really, Patra. Tell me. Will you go shopping for beautiful gowns? Have the jeweler make you a crown?"
Patra crinkles her nose. "The first thing I will do is clean up this palace. It smells mildewed."
"Yes," Alex says. "It most certainly does. My mother put a perfume burner on the table over there, near the window." He points to a terracotta figure of a woman's head wearing a tall headdress.
Patra rises and, taking the oil lamp from inside her lantern, lights the coals inside the headdress, opens the incense box on the side, and sprinkles several golden rocks of myrrh on the coals. She is careful to step out of range of the smoke, Alex notes, and with good reason, given the fact that she is an oracle. Soon the room is filled with a sweet, delicate fragrance.
"I'm sure my uncle will let you do what you want with the palace," Alex says, crossing his legs. "You can order new furniture, tapestries..." Patra's eyes are riveted on the scar on his thigh, now revealed by his hitched-up sleeping tunic. He quickly covers the scar.
"That mark," she says, pointing, "What is it?"
"Nothing," he assures her. "I've had it since birth. The physicians say my twin's umbilical cord wrapped around my leg in the womb. My real sister, Katerina."
A cool night breeze washes in the open window, wrapping thick plumes of incense smoke around Patra, whose eyes roll back in her head as she starts to sway.
"The threads," she says in a deep, throaty voice very unlike her own. "The threads of Fate. You struggled with them even before you were born, and that is their mark. Unbound. You are the one to unbind us all. By untangling the knot."
Alex is chilled. This girl speaks in the voices of the gods. It turns his blood to ice.
As Alex listens to the swaying Patra mutter about knots and threads, his heart hammers in his chest. He tells himself there is no reason for fear. Perhaps he should view this as an opportunity for information.
"Tell me about the Spirit Eaters," he says. "Are these monsters real?"
She flings her long hair around her and says in a voice of brass, "Oh, they are real. And they are _hungry_."
The hairs on the back of Alex's neck stand up, and he shivers. "Can they be destroyed?"
Patra stands in front of him, her chin on her chest, her eyes raised toward him. "Only he who is marked with the threads of Fate can destroy them."
"And who is marked by the—"
"You," she intones. "Only you, my king."
"I am not your king," he says, a kind of eerie dread filling his chest.
"No," she says slowly. "But you will be."
"How?" he asks, urgency thudding through him. "How do I destroy them?"
"Go to the Temple of Midas in Gordium," she says, pointing out the open window, to the sea.
Gordium? Alex learned about the city from Aristotle. It was founded by King Gordios of Lydia some three or four hundred years ago. Now it is part of the Persian Empire, though not nearly as far from Macedon as the Eastern Mountains. And there is a giant snarl of rope there, the famous Gordian knot. Long ago, a group of elders of the kingdom of Phrygia visited an oracle, asking for a new king. The oracle told them to return home and wait for the first man entering their town in an oxcart. He would be the man destined by fate to be their king. The first man to ride into town driving an oxcart was a farmer named Gordios. To memorialize the event, he declared the cart a holy object. To prevent anyone from stealing it, he tied it to a column with such a thick knot that no one thereafter could ever unravel it. He had to have used an ancient and forbidden magic, according to the stories—effectively, sealing his fate and that of his lineage forever to the spot.
"Gordium," Alex says,"is that where the hideous beasts have gathered?"
"No," she responds, her outstretched arm falling to her side.
"But—" Alex lets out a long, slow breath. Prophecies always fool you. They make no sense at all until after they have come to pass.
"You have only until the summer solstice," she goes on. "After that, their strength will wax beyond human means to destroy them. For, on the solstice evening, the new child of Fate will be theirs, giving them the power to devour the entire world."
"The new child of Fate?" he asks, trying to force this prophecy into clarity.
A man's low laugh bubbles out of her mouth. "Yes. Unless. Unless."
"Unless what?"
"Unless you slay the strands of Fate with the talisman of one who rises above mortal fate."
Alex rubs his forehead. She's talking in circles again.
"And only if you face them alone."
"Alone?"
"Alone, but not alone."
"You speak in riddles," he says, growing frustrated.
"You will need the help of one person only—a stranger. A stranger whose blood is destined to mingle with your own."
CHAPTER TWENTY
OLYMPIAS
Queen Olympias sits in her favorite childhood chair—battered olive wood, tainted with years of bad memories. It's still somehow welcoming, like the arms of someone who has hurt you but has come back to beg forgiveness. Patra stands before her, and she surveys the girl from the top of her gleaming golden diadem to the thick golden border of her yellow bridal gown. She is perfect—young, modest, beautiful—everything a jaded, middle-aged king like Xander could want in a wife.
Olympias knows Patra is a reluctant bride. The girl was horrified when the queen first told her of King Philip's plan to help Macedon and Alexander by marrying her to Xander. But her happiness does not really matter.
The queen could hardly believe her luck when Philip marched in four days ago and announced his plan. _Pretend your new handmaiden is my daughter_ , he said. _Marry her to Xander. Your favorite becomes queen and will let you rule the roost here, unlike another woman he might marry. And your son gets troops for our campaign._
_Yes_ , she said. _An excellent idea all the way around. Let's have the wedding soon. You must, of course, attend and dole out your blessings on the couple._ Philip was shocked at how quickly she agreed. He had expected a fight, knowing he was rejecting her forever as his wife and queen, and taking away any power she might have in the alliance between Macedon and Epirus.
For Olympias, this marriage offers a greater advantage than Xander getting a wife or Patra becoming queen. Greater even than Alexander gaining Epirote troops for his conquest of Persia. This marriage will, most likely, give Olympias the dearest wish of her heart. For the oracle of Delphi—the girl in front of her, oddly enough—prophesied with the voice of gods that King Philip of Macedon would die at the wedding of Cleopatra. And yet, he didn't die at his own wedding to that stupid tart, Cleopatra Attalida.
But if there's anything Olympias has learned, it is that fate, and the prophecies, work in mysterious ways, looping in on themselves, offering obstacle and opportunity at every turn.
So, he did not die as predicted. As she _thought_ was predicted. But he may still die soon. Perhaps they simply got _the wrong Cleopatra_. Perhaps instead, he will die at the wedding of _Cleopatra of Delphi_ to King Xander. And if that doesn't work, Olympias will arrange for him to attend as many weddings of girls named Cleopatra as necessary until he does die at one of them.
For Olympias is not naive. She knows Philip could always change his mind and make Cleo's child his heir, disinheriting Alexander. But once Alex is king, Cleo's child will be nothing. Except for short-lived.
No, it has always been clear to her: Philip must die.
Malia, Olympias's brisk gray-haired handmaiden, adjusts the golden clasp pinning the gown on Patra's left shoulder and asks, "Cosmetics, my lady? She is a bit pale."
The queen scrutinizes Patra and quickly decides. "No. Xander wants a demure virgin bride, not a temple prostitute. Pale is perfect. Here, Patra, sit."
Visibly trembling, Patra dutifully takes her place on the footstool in front of Olympias. For a strange moment, a memory—silken and sad—sweeps over her, and it is no longer Patra kneeling there but Helen, her childhood friend, her _only_ friend. Once. Helen was also an oracle, also modest, sensitive, and timid. Their friendship ended when both of them fell in love with Riel, so very long ago. She and Helen spent time together in this very room. There, against the other window, was Helen's loom, gone now. Like Helen.
But Olympias will not succumb to regretful thinking. Not after all that has happened. She did everything for Riel. Everything. And when he returned last year in her son's body, he didn't want her anymore. The natural aging of sixteen years, combined with the ravages of Lord Bastian's poison, had made her far less desirable. She denied her gut instinct, grasped at every compliment or kind word he sent her way, hoped the heavy makeup and false hair preserved the illusion of beauty. Alexander said Riel had come close to killing him, and Olympias wonders whether as king of Macedon he would have killed her, too.
She reaches to the table beside her and picks up the old lyre she found in a trunk. When her father gave it to her for her twelfth birthday, she had oohed and aahed at the shiny cow horns rising from the top of a polished tortoiseshell. It was dusty when she first pulled it out after her arrival here, the notes sour, and she struggled to tune the sheep-gut strings. Now her fingers move haltingly over them, teasing out the first song she ever learned to play.
Here she is, right back where she began. The same room. The same furniture. Over there is the table she hid under when her stepmother came to beat her. There is the bed in which she dreamed of her next rendezvous with Riel.
A sharp knock at the door interrupts her reveries, and Cleo enters, swinging her hips in an exaggerated way that makes Olympias's skin crawl. Perhaps it was seductive a few months ago. Now it is more like a waddle.
Cleo's black hair is piled in a tower of curls—much of it false, Olympias can only assume—and her heavy perfume is enough to choke an army. Little golden bells sewn on her robe tinkle as she walks as if saying, _I will not let you ignore me._ She's followed by a woman Olympias recognizes as her personal handmaiden, thin and olive skinned, her thick black hair coiled in a tight bun.
"Well," Cleo says boldly, a look of smug satisfaction on her painted face, "I've come for the gifts you've promised me." She clasps both hands over her pregnant belly.
"Ah," says Olympias, smiling warmly. "They are in that trunk over there, by the wall. You will find robes, jewelry, perfume, unguents, and cosmetics, including that red lip stain I gave you for your nuptials that always drove Philip mad with desire. I understand you lost it in the flood."
Like a greedy child, Cleo kneels before the trunk, throws open the lid, and starts rifling through the contents. Then she yelps with delight, holding up a dazzling silver-and-amethyst bracelet before thrusting it over her plump hand and onto her wrist.
"Sandalwood perfume from Egypt," Olympias continues, "an ivory comb from Ethiopia. A thank-you, Cleo, for treating my son as a prince and his father's heir."
Ignoring her, Cleo pulls out a shimmering blue-and-gold scarf and wraps it around her neck, then dives in for more. Watching the stupid girl, Olympias stifles a laugh. For months now—ever since Jacob injured her in Byzantium—she has felt weak and frail. But suddenly she feels as if she has been struck by lightning. Energy sizzles through her, sharp and tingling. She is alive again. _Alive._ Her life has an urgent purpose, even if that purpose is death.
She knows Cleo will test all her cosmetics for poison on her poor handmaiden, just as she did in Byzantium when Olympias first gave her the lip stain. But it won't make any difference, because the Persian mulberry pulp in the lip stain is harmless to everyone except Philip, who has a bizarre reaction to it. A hundred years ago, when she still loved him, she wore a skin cream scented with the berry's sweet essence. When he leaned in to kiss her, he began to choke; his eyes watered and his face broke out in angry red blotches. Palace physicians warned her that any further exposure to the berry could kill him. To Philip, it might as well be deadly nightshade, they said. She gave away the cream. But she tucked the memory away in her mental armory of potentially useful weapons.
Cleo slams the trunk lid down and leans on it heavily to stagger to her feet. She nods to her handmaiden and, without saying a thank-you, stomps out of the room, followed by the woman struggling with the trunk. Snorting in derision, Olympias's handmaiden Malia slams the door behind them.
"She doesn't bother you, my lady?" Patra asks, looking up at Olympias with concern. "She is not a nice girl, I think."
Now Olympias laughs out loud, a tinkling silver laugh that reminds her of being young and happy. "Not nice at all," she says, strumming the lyre. "But she has her uses."
She plucks a string hard, with purpose.
It snaps.
CHAPTER TWENTY-ONE
HEPHAESTION
Heph tips his goblet into his mouth, but only a single drop of wine falls on his tongue. He upends the oenochoe above the cup, but nothing comes out. His head is flushed with the warmth of the wine, which always has the strange quality of wanting more of itself. There's a meaning in that somewhere, he's sure, but he's too drunk to think of it.
Instead, he looks for another oenochoe on one of the dozens of tables set up in the palace courtyard. Around him, the wedding banquet is in full swing. Dancing girls in sheer costumes bend and swirl to the beat of drums and pipes. The gray-bearded man sitting next to Heph has already passed out. His head is on the table, his extended hands still grasp the bottom of his cup, and he snores loudly.
Two servants carry in a silver platter bearing a roasted peacock, its feathers sewn back on, deftly passing the krater, a bathtub-sized container where two other men mix wine and water with a paddle. But it's unmixed wine for Heph today. The bride, Patra, sitting on a throne next to King Xander's on the central wooden dais, reminds him a bit of Katerina. Who is not here. Who is with Jacob in Troy. Who is maybe in love with Jacob, maybe sleeping with him. Maybe in danger because she is Snake Blood and he is an Aesarian Lord, and Heph can't do a damned thing about it.
Frustration pounds in his head to the rhythm of the music. For months now, he has wanted nothing more than to leave Alex, go to Troy, and find her. Protect her. Convince her to love him, not Jacob. But he can't. Alex needs him. And he won't fail in his duty again, even if it means losing Kat.
King Philip's young wife, the appalling Cleo, approaches him and, to his horror, stops in front of his table. "Well, it's Handsome Hephaestion," she says boldly, smiling. He has drunk so much that she seems to be a blur of blue eyes, black hair, and dark red lips, moving into one another and then apart.
"Cheer up," she says, touching his cheek. He pulls back so quickly he almost falls off his bench as she laughs. "We'll find you a bride soon, Heph. We can't have you moping around the palace all...frustrated."
Is it so obvious then?
"If I weren't the king's most faithful wife, well..."
He's afraid she will stay and torment him, but she just laughs again and wanders off, thanks be to all the gods. While flaunting her position as the king's wife, she flirts with every decent-looking man in sight, even her own stepson. Even though she's pregnant. He and Alex have agreed that Cleo would never do anything to make Philip cast her aside; she values her position far too much. But she seems to want all men to desire her so she can turn them down, boasting about her own virtue.
Up on the dais, King Xander's narrowed, pale blue eyes are fixed grimly on Cleo, who is now laughing with a handsome young soldier near the krater. Then Xander turns to look at his own bride, the timid Patra, and beams. He leans toward her and whispers something that makes her smile shyly.
It could have been—should have been—Kat up on that dais married to the strange old man. The one thing Heph is grateful for is that she has avoided this fate.
A servant picks up the empty oenochoe on the table and sets down a full one. Heph grabs its slender neck, but a cool hand wraps around his wrist. "Not so fast," Alex warns.
Heph looks up at him questioningly. Alex's face seems to float above a sky blue tunic. "Where have you been?" Heph asks, trying to pull the pitcher toward him. Alex does not release his grip.
"All our lists of men, ships, horses, and equipment are on the desk in my bedroom," he says, straddling the bench beside him. "The list of battle formations, invasion plans, records of our meetings with Philip. It's all there, Heph. And the deciphered messages from our spies are behind the secret panel on the side of the desk. I showed you how to open it."
Heph laughs and pulls his wrist free from Alex's grasp. "Alex, this is a wedding, not a council of war," he says, pouring more wine. "You're a bit uptight. Should I pour some for you?"
Alex shakes his head and says, "I need to keep a clear head."
"I don't," Heph replies, taking a deep swig.
Alex puts a hand on his arm and says, "Heph, whatever happens, I know I can rely on you."
_Yes, even though my heart is breaking, you can._ But Heph just nods and runs a hand through his hair, which, he notices, is starting to get sweaty. Warm weather has finally arrived in this dreary outpost. The sky is a cloudless powdered azure, and the sun beats down mercilessly on the wedding guests. The breeze off the Ambracian Gulf is warm, hot, even, when it carries the heat from the fire pit near the gate, over which an entire wild boar roasts. The wine is warm. The people are warm. Heph mops his head and the back of his neck with his napkin.
When he looks up, Alex is nowhere to be seen. Cleo, however, is on King Philip's lap, passionately kissing him on a gilded throne set on a dais to the side of the newly married couple's. Across the courtyard, Queen Olympias, in a battered ivory chair on her own dais, stares at them, her eyes unblinking, her expression almost...eager.
Heph rubs his face and pushes his wine away. He's not seeing clearly. He will drink no more. He always found Olympias a hard person to like; at eleven, when he first came to live in the Pellan palace, he was terrified of this cold, selfish queen. But she accepted Heph as Alex's best friend and treated him decently. And, unlike Cleo, Olympias always behaved like a queen, in public, at least.
Phrixos and Telekles run between the tables up to the open area in front of the dais, pick up dancing girls, and swing them around to the beat of the music. Wedding guests clap and hoot. The actor Pausanias, wearing the grinning mask of Comedy, joins the dance. A tall, handsome man with riveting blue eyes and thick chestnut hair, he is Philip's latest male lover, according to palace gossip. Pausanias leaps and cavorts, the long multicolored ribbons on his costume fluttering wildly as he raps the dancers' rear ends with a beribboned stick.
The music, the heat, the smells of food and smoke wrap around Heph until he wonders if he is going to be sick. He pushes his slick hair out of his face and sees Philip, on his throne, quite red in the face, almost the same color as his scarlet tunic. He, too, must have drunk too much, Heph realizes, smiling. But then the king grabs the neck of his tunic and grimaces, as if he can't breathe.
Heph finds himself standing—a bit unsteadily—and looking around for Alex. The pale blond head is nowhere to be seen in the crowd. He looks back at Philip. Cleo has slid off his lap and stands beside him, bent over him and questioning. The king's face has broken out in huge red welts, as if he has the plague. He calls for something. Water, Heph thinks. As Cleo flags down a servant, the king, grabbing his stomach, keels over and vomits off the side of the dais.
Cleo screams, and the music stops. Phrixos and Telekles, still dancing with the girls in their arms, finally stop whirling and look around. On the central dais, Xander and Patra stand in alarm.
"The king is ill!" Cleo cries. "Where is his physician?"
Thrasybulus, tall and bald, pushes through the crowd. Now, in the silence, Heph hears how odd Philip's breathing is. How ragged and labored. Suddenly Heph's head is clearer. He leaps over the table and runs up to the dais. Philip has slid off his throne and is now on his back. His eye, lips, and face are grotesquely swollen. Thrasybulus opens the king's mouth and Heph sees an enormous swollen tongue.
"Help me sit him upright," the physician calls to Heph, and together they pull the wheezing monarch up to a seated position against his throne.
"Sire," Thrasybulus says, gripping the king's hand. "Sire, can you speak?"
Philip opens his eye—a dark slit in puffy flesh—and shakes his head. He's making strange, unearthly moaning noises that remind Heph of his near drowning in Byzantium, after Alex pulled him out of the water. Xander, ghastly white, has joined them.
"Is it poison?" he asks, his voice cracking with fear.
"No," Thrasybulus says, his long fingers on Philip's neck to check his pulse. "At least, I don't think so. The poisons I know attack the guts. His throat is closing up, suffocating him."
The king thrashes violently, a muscular hairy leg connecting with Heph's hip and nearly sending him flying off the dais.
"Hold him down," the physician says. Philip's favorite general, Parmenion, jumps onto the dais and helps Heph pinion the struggling king. Heph stares into the hazel eyes and battered face of the general, and they share a moment of pure panic.
Thrasybulus stands and cries, "I need a stick! Who has a stick?"
The actor Pausanias runs forward and hands the physician his smooth slender stick. Thrasybulus quickly tears off the many long ribbons, opens Philip's mouth, and thrusts the stick down his throat. The king gags, moves his head from side to side, and tries to push the stick away with swollen hands.
"I can't...get it...down," Thrasybulus says. He removes the stick and looks at Heph and Parmenion with a frightened expression on his face. "I need to open a passage for air. The king is suffocating."
Cleo starts to shriek like one of the Furies, bone-chilling sounds that stab the soul. "Take her away!" the physician commands, and two guards drag her out of the courtyard. Heph looks up and sees hundreds of shocked faces in the courtyard staring at the dying king on the dais. Every window in the palace is filled with faces. But none is Alex. His gaze slides to Olympias, who has risen from her throne and stands impassive and unblinking, her white face like that of a carved temple goddess.
The old physician looks at Heph. "Give me your dagger, son," he says through clenched teeth.
"Dagger? Why..."
"I need to cut an airway in his neck," Thrasybulus replies. _"Now."_
Hands shaking, Heph slides his dagger out of its sheath. The physician grabs it and, to Heph's horror, cuts an opening at the bottom of Philip's neck. Blood spurts wildly out of it. Air moans through it. Thrasybulus thrusts a finger inside, then withdraws it.
"His lungs aren't functioning," he says. "There is nothing I can do."
Helplessly, Heph watches the king twitch and go still. For a long time, no one moves.
Thrasybulus lays his ear against Philip's chest. When he pulls it away, half his face is smeared with the king's blood. "He is dead," he whispers. People in the courtyard gasp, cry out, and utter prayers to the gods. Some of them kneel, rip their tunics, and throw dirt on their heads, the ancient gestures of mourning.
The physician picks up the king's wine cup from the table between the thrones, sniffs it, dips a finger in, and licks it, then runs a finger over the rim and looks at it. "I see no trace of poison," he says, "but to be sure, let someone bring a dog." Parmenion races into the palace.
Heph scans the crowd again. Patra has rushed to Olympias for comfort and cries against her shoulder. Xander's expression is grim, and his servants look terrified. They know they may be tortured to death to find out who poisoned King Philip. There is absolute silence, except for, somewhere, inside the palace, Cleo's echoing shrieks. Heph must find Alex.
Alexander is now king of Macedon. And doesn't even know it.
But somehow, Heph cannot leave the horrific scene on the dais. His body is heavy, numb. He wonders if this is a bad dream and he will wake.
The crowd parts to let Parmenion through, leading a scruffy yellow dog on a piece of rope. "The cook's dog," he says. Heph sees a plump man following the soldier, tears running down his cheeks.
The physician descends the dais steps and sets the king's wine cup before the dog, who laps up the wine greedily and licks his chops. He looks around eagerly, hoping for more, while the cook throws his arms around him. They all wait.
The dog wags its tail. Its pink tongue hangs out and its bright eyes look around, evidently happy at all the attention. Meanwhile the physician cuts open Philip's tunic and examines the inside. "Though the king's skin has erupted in welts, I find no trace of poison on his clothing," he says. "Meanwhile, let us keep watching the dog for a time and feed it food that the king was eating."
"Where is Alexander?" comes a cool, clear voice. Olympias. "Where is the new king of Macedon?"
Heph pushes himself up to his feet. "I will find him, my lady," he says and races into the palace, his mind whirling. What did Alex say to him after telling him the location of all the invasion documents? _Heph, whatever happens, I know I can rely on you._ It's almost as if Alex was planning on running away. But no, he would never do that without letting Heph know. Heph will find him in his chamber. Alex, who doesn't enjoy drinking and feasting very much, is probably reading the latest dispatches.
But when Heph throws open the door, Alex is not in his chamber. All is tidy, and the prince's battle armor and sword rest on a table near the window. Perhaps Alex is with Bucephalus in the stables. But as Heph turns to go, a strange feeling scrapes at the back of his neck. Something isn't right. He throws open Alex's trunk and starts rifling through his tunics. The blue one is missing, but he was wearing it at the wedding. The green one is also missing. And the burgundy. And his dark red cloak. Heph sits back on his legs and runs a hand through his hair. Could they be in the laundry?
He glances around the room for some sign of where his friend has gone, if indeed he has. Then he sees a scroll on the desk with his name on it. Trembling, he unrolls it.
_Heph,_
_It has been prophesied that if I am to defeat the Spirit Eaters, I must go to Gordium and do so without you. The Persian invasion must, at all costs, go forward. Work closely with my father. Tell him I have gone ahead to Persia to fulfill a vow to sacrifice to the Greek gods there. Take your time and focus on the invasion. If fate directs the Greek armies along the path to Gordium, meet me there. Hopefully, by the time you arrive, I will have fulfilled the prophecy. Heph, I know even a few months ago, you would have left Epirus immediately to search for me. But you have changed, grown in wisdom, loyalty, and duty. That is why I am trusting you with knowledge of my destination. I know I can count on you to do the right thing for Macedon and for all of us._
_Alexander_
No.
_No._
Heph slams his hand against the desk until it stings. He shuts his eyes a long moment, then reads the scroll again, hoping it will say something different this time. It does not. He picks up the helmet, mystified. Why did Alex leave his armor behind? Perhaps he hasn't gone yet. Perhaps Heph can stop him.
Still clutching the helmet, he races along the dark upstairs corridor, down the winding steps, and across a narrow hall. Bursting into daylight, he passes the laundry, where tunics and sheets hang on lines drying in the sun, and flies around the corner to the stables. A skinny boy of about fourteen sweeps hay into a pile and looks up. "Is it true, sir, what they say?" he asks. "Is the king..."
"Have you seen Prince Alexander?" Heph asks, grabbing the youth by the shoulder. "King...Alexander?" he corrects himself. The words sound strange together, but he knows they must be said. The passing of the throne to the heir is immediate upon the king's death—soon, there must be a coronation, but fate does not await ceremony.
The boy's mouth drops open. "Y-yes, sir," he stutters. "Sometime ago. He went for a ride. B-but he did not take his usual horse."
Heph pushes past him into the corridor between the rows of stalls. There, in the largest stall at the end, stands Bucephalus. His enormous black head hangs over the gate. His huge nostrils widen as Heph approaches, and he whinnies and rears. Then he turns, and his back legs kick the gate. He seems to know Alex rode off on another horse, and he's furious about it.
But why would Alex take another horse? He never goes anywhere without the stallion. Except... Heph's heart sinks when he realizes Alex took another horse because he didn't want to be recognized. And everyone throughout the world knows that a blond young man riding a gigantic, wild black stallion with a white star on his forehead can only be Alexander of Macedon. And he left his unique golden armor behind for the same reason.
He sinks onto a low wooden stool in front of Bucephalus's stall, staring at the helmet in his hands, its intricate scrolls of goldwork, its jaunty red feathers. He's not sure why he puts it on his head. Does he want to be close to Alexander? Or does he want to hide from the world for a few fleeting moments before he takes it off and finds himself exposed and helpless in the face of all that has happened this morning?
He feels like he is teetering on the precipice of despair. At any moment, he can fall into its bottomless depths. He can't let that happen. The prince...no, the _king_ has placed his trust in Heph to see things through. He expected Heph to work with Philip on the Persian invasion. But now Heph must do it by himself. Though how can he without Alex? Who is Heph, a nameless orphan, to invade Persia in the name of Macedon? Think. He has to _think_.
Running footsteps rouse him. "Lord Alexander!" cries Parmenion, breathless. "Thanks be to all the gods I have found you! Something terrible has happened!"
Heph stands slowly and raises his hand to stop the general, but he rattles on about King Philip's death, clearly thinking that the person wearing Alexander's helmet and standing next to his horse must be Alexander. Heph takes off the helmet. "It's me, Parmenion. Alexander is not here." The general's face falls, but without another beat of hesitation, he runs off to continue his search.
Heph, however, doesn't move. A plan starts to formulate in his head. A crazy, ridiculous plan that could never work. Or...could it?
In the _Iliad_ , Homer's great song of war, love, and loss, the hero Achilles refused to lead the Greek allies against Troy, and the demoralized army lost battle after battle. Achilles's lover, Patroclus, begged him to return to the field and, when he refused, asked to wear his unique armor to pretend to be him and rally the troops. Achilles agreed, and Patroclus led the Greeks to victory that day.
Heph remembers the words every Greek schoolboy must learn by heart.
_At his words, Patroclus began to clad himself in gleaming bronze. First, he clasped the shining greaves, with silver ankle pieces, about his legs. Next, he strapped Achilles's ornate breastplate round his chest, richly worked and decorated with stars. Over his shoulder he hung the bronze sword with its silver studs, and then the great thick shield. On his strong head he set the fine horsehair-crested helm, its plume nodding menacingly. With this, he put heart and strength into every man, and they launched themselves in a mass at the Trojans._
It didn't end well for Patroclus, of course. In the heat of battle, he forgot his plan and chased the enemy too far instead of retreating when he should have. But the new Patroclus Heph has in mind is the calmest person he ever met in a harrowing situation. He must see if his Patroclus is willing to take the risk.
Helmet in the crook of his arm, Heph walks slowly back to the courtyard. Thrasybulus, bent over the dog, says something to the cook, who gleefully picks him up and marches smiling back to the kitchens.
"No poison, then?" Heph asks the physician.
Thrasybulus scratches the back of his neck. "It could be a new kind, something from Persia perhaps. But I don't know how the king ingested it. Certainly not from his wine or food. The guards have arrested the actor Pausanias."
"Pausanias?" Heph asks, incredulously.
The old physician nods wearily. "Apparently the king and the actor got into a terrible fight last night. And he was found to have in his possession the king's gold signet ring. Well, if you will excuse me, I must prepare the body for burial."
Two soldiers carry the king's body on a shield, his fine red leather sandals poking out from below a tablecloth.
"Did you find the king?" Xander asks, his pale brows knit in worry. Heph looks back at the sandals disappearing into the palace.
"King _Alexander_ ," Xander adds.
Heph pulls himself up to his full height, squares his shoulders. "King Alexander left during the wedding celebrations to fulfill a vow to the gods and sacrifice at the tomb of Achilles in Troy." It is not _all_ a lie, anyway. "The invasion will go forward as King Philip planned. We will leave in two days for Athens, pick up the allies, sail for Persia, and meet the new king at our first camp."
Xander nods and follows the body into the palace, with Heph close behind him. He has an urgent letter to write.
_To Katerina of Erissa and Macedon at the Temple of Asklepios near Troy, Greetings from Hephaestion of Macedon._
_The allied Greek army will sail soon for Persia. Alexander has gone ahead on a secret mission and wants you to help us. You will find our camp a two-day ride north from Troy, where the Granicus River meets the sea. Come to the camp cloaked on the night of the last half-moon before the summer solstice. Give the guard at the perimeter the enclosed pass._
On a strip of parchment, he writes:
_Allow the bearer of this pass to enter the Greek camp and provide an escort to the tent of Lord Hephaestion of Macedon._
He stamps it with his signet ring in ink, wraps the letter around it, and seals it with red wax, which he also stamps with his ring.
In the courtyard, the scene of so much celebration and tragedy, Heph watches the messenger stir up a cloud of yellow dust as he rides to the port. He feels a strange envy of the letter, which soon will touch Kat's hands.
Unwilling to go inside the palace, which echoes with ritual wailing, he sits on a bench as servants silently clear the tables. He feels the sun on his head, the breeze on his face. The same sun and same breeze he felt moments ago.
But now the whole world has changed.
ACT FOUR
MONSTERS
It is in our darkest moments that we must struggle to see the light.
—Aristotle
CHAPTER TWENTY-TWO
PYROLITHOS
Rage. It is a glorious thing, feeding his veins, making him strong and alive. He hadn't realized just how trapped he'd felt in Dardania until he left, until he and the New Amazons took to the sea, bound for the roiling shores of Persia.
But Pyrolithos knows the truth: that without that rage, he is riding on a wind that could leave him plummeting at any point.
No, not wind—smoke.
He leans back in his chair, itching to move. Eating has become perfunctory. He lives now for movement, for the open road, for the skirmishes and the righteousness and the look of sickened shock in men's eyes just before they die.
For the blood—the spray of it against his skin, the way its heavy stench fills the air in his wake.
Alecta kicks at the leg of his chair. "Not eating this? I'll have it, then."
He watches the fig in her fingers as it moves toward her mouth, her lips as ripe and succulent as the purple fruit. She bites and a chill runs through him. Alecta raises her dark eyes to his and spits out a seed.
He never should have shown her the truth. Ever since he revealed himself to her as Queen Cynane, instead of the awe and gratitude she should have shown, she has berated him for lying to her. For lying still by not explaining the strange magic that transforms a woman into a man. Trust, she says, is more important than courage. Truth is the only real honor. Lies are badges of shame worse than cowardice. It is strange, Pyrolithos thinks, that a whore has a greater sense of honor than all the kings and queens he has ever known.
But she is wrong. Courage is more vital by far. It is not truth that will save your life, he knows, but the willingness to lie, to steal, to cut, to survive at any cost. To come out on top. It is ruthlessness alone that lets us win where others wither, that lets us vanquish instead of fall, that ultimately brings freedom.
He will not be trapped again. His power, his freedom—these are everything.
And besides, he enjoys Alecta's snark, her harsh words. He has known no other kind of affection except that of the men who succumb easily to seduction, women who become flustered and light-headed and lose themselves completely. He will not be like them, and he can never respect someone who is.
A warm breeze rolls in through the small window, barely dispelling the tavern smells of old wood smoke, fresh bread, and sour wine on the not-too-clean earthen floor. The heat of the day has passed; it is time to go. They have far yet to journey.
And the Spirit Eaters are waiting.
Waiting to be tamed.
So far, the New Amazons have cut a swathe of violent justice across central Greece and have at last reached Persia, pushing their way through the vast and varied territory, heading east all the while. And they have shown no mercy. Men who beat, rape, and abuse women have had their heads stuck on pikes in public squares at night. The heads sport paper crowns that say _Justice Courtesy of the New Amazons_. Their bodies have been fed to the Spirit Eater in the diamond cage.
Some of the women were afraid to travel with the monster, no matter how hidden it is kept. But they desperately want what Pyrolithos has promised them—taming all the Spirit Eaters to command an invincible army and rule the known world. And even if they _didn't_ share his vision, they know better than to cross it.
A waitress sets down a plate of bread. She's skinny and whey-faced, with lank, ash-colored hair. As she turns, a stream of sunlight catches her face, revealing a puffy black eye.
Pyrolithos is suddenly struck with hunger—or something akin to it. He grabs her slender wrist. "Who did that to you?"
"No one," she says nervously, looking over her shoulder at the door to the kitchen. "I...fell." She tries to twist out of his grasp but to no avail.
"Your husband?" He looks into her small light brown eyes. "Your father?" And then, with his voice lower, "I can help."
"No one can help me." It is more a moan than a statement. Her breath catches, raggedly. "The tavern owner. He calls me slow. But I have nothing. Nowhere else to go. Please don't make trouble, sir."
Fury lights Alecta's face. "We _can_ help you," she says in a low voice. "You can come with us."
The girl looks around the tavern eagerly. There is a long table of loud men at one end, but all the others are occupied by women in modest robes eating quietly. "Who are you?"
"Travelers," Pyrolithos says, shrugging. "A man and his... _extended family_."
"Oh."
She looks over her shoulder with a flinch just as the door from the kitchen bursts open and a fat, red-faced man strides up. "Efimia!" he cries, slapping her so hard across her face that she falls. "We have orders in the kitchen waiting to be picked up. Lazy, good-for-nothing piece of goat dung! Get back to work!"
He kicks her in the shin and Efimia whimpers as he disappears back into the kitchen.
Efimia picks herself up and holds a hand to her cheek. When she removes it, Pyrolithos sees red finger marks.
His eyes meet Alecta's, and they both nod.
They have their next mark.
"Go back to work, Efimia," Alecta says softly. "We will take care of things."
With a worried look in her eyes, the girl scurries back into the kitchen. Moments later, she shuffles back out holding an enormous tray of bread, meat, and salad, and sets it down on a long table of loud men nearby.
"Well, now, that took long enough," says a balding man with a long nose and droopy eyes. "I see your boss knows how to discipline you. I wouldn't mind a turn at it, you know."
Efimia bites her lip and quickly sets out the plates.
"Careful, Cyrus," says a bearded man with a thick thatch of black hair. "Or the New Amazons will come and get you." The men all laugh.
One man wipes beer foam off his beard. "I heard last week they cut off a man's head in Mittania."
The thin one, barely out of his teens, sneers. "Check your ears. Women are made for one thing only, and I'm sure it isn't that!"
The muscular man at the end of the table slams down his flagon of wine with another loud laugh, revealing missing front teeth. "I heard one of them carries a baby strapped to her back," he says. "Sounds more like a nursery on the loose!"
Fingers going tight around the base of his wine cup, Pyrolithos grits his teeth, glad he left Zotasha at the camp with the other women. Looks like there will be more work to do today than just the tavern keeper.
"Girl!" Pyrolithos pounds his cup down. The girl turns a frightened face his way. "Bring us an oenochoe of your finest wine and fresh cups! I want none of this filth." He knocks his cup to the floor with a clatter.
A cue, if you know how to read them.
And the New Amazons do—he can see their smiles of understanding, as, one by one, a few of them slowly take off earrings and rub their ear lobes as if they are sore. One removes her necklace as if to check the clasp. Others examine their golden rings or play with their bracelets. Moments later, he pours bubbling ruby liquid into the cups held by his companions.
"Gentlemen!" he says, walking to the men with the drinks on a tray. "I have recently received good news. How about some of the house's best wine on me?"
The six men gladly take the proffered cups, congratulating Pyrolithos.
One of them eyes him warily. "And what news?" he grumbles, taking a gulp.
"Oh, you'll know in a moment!" Pyrolithos says cheerfully.
They drink his health, clapping him on the back.
Sure enough, a few minutes later, the youngest staggers up from his chair and vomits all over the floor. "No," he moans, on his hands and knees like a dog.
Gripping his belly, Cyrus stands up, white-faced and sweating. "What the—" He looks around at the others. One of them pukes into his soup bowl.
"What's in the wine?" Cyrus cries, his hand on his dagger.
The dozen women around Pyrolithos push their chairs back and stand all at once, throwing off their cloaks to reveal breastplates, swords, and small shields on their backs.
Cyrus and two of his companions pull out their daggers as the Amazons place long wooden tubes in their mouths.
The fat one says, "Are you going to rob us while you play the flute? Well, I'm not so sick that—"
Darts fly across the room into the men's necks and faces. The fat man falls to the floor before he can finish his last word, gasping and writhing.
This seems to galvanize the rest to action. Several of the men are desperately trying to pull the darts out of their skin, and still others leap up. Benches and stools go flying. One man keels over with a sword thrust through his chest.
The skirmish lasts seconds.
All of a sudden, the remaining five stop fighting and fall to the floor nearly in unison. Dead. The women step back, lowering their swords. A few of them readjust their jewels—reclasping the hidden compartments where their poison is stored.
The wine—and darts—have done their work, and efficiently, too.
The tavern owner rushes out of the kitchen wielding a butcher knife. Alecta throws her dagger. It hits him in his right eye, and he falls forward as a rush of wet blood floods his face. He utters an unearthly wail, then goes still.
For a moment, there is silence.
"Take as much as we can carry," Pyrolithos says quietly.
The women set about gathering up food and wine into their packs. A moment later, Sophonisba comes out of the kitchen dragging a gangly straw-haired boy of about twelve. "What do we do with him?"
Pyrolithos sees pure terror in the boy as he gazes around the tavern at all the dead bodies. Alecta puts her boot on the tavern owner's face, withdraws her dagger from his skull, and wipes the blood on his apron before sheathing it. Calista quickly rifles the corpses, her curtain of long blond hair hiding her theft of their coins and rings.
"What is your name, son?" he asks the boy.
"M-M-Mikkos," he stammers. His whole body shakes. Pyrolithos considers a long moment, then rejects the idea. He's too young, and there's something about him that reminds him of Prince Alexander, when he was younger. Not that they were ever close. How could they have been? Alexander was lauded and celebrated, and trained in the military arts, while Cynane was ignored and neglected, and laughed at when she picked up a sword.
But time and distance have made Pyrolithos more levelheaded. Lately, he has found himself wondering what has become of Alexander, one of the few people in the Pellan palace who never did Cynane wrong.
"Well, Mikkos," Pyrolithos says now, taking a step toward the boy, who is shaking visibly. "We will not harm you. We will tie you to that pillar over there, see? Someone will come here soon and rescue you. And then you must tell them what I say, all right?" The boy continues to shake, too afraid to speak. "You will tell them the New Amazons were here, dispensing justice to brutal men. Will you do that?"
Wide-eyed, the boy at last nods.
Moments later, Mikkos is gagged and tied. Alecta, Agatha, Calista, and the others stream outside with hoards of food and skins of wine. Pyrolithos is ready to leave when he hears a whimper and looks beneath a table. It's Efimia, curled into a ball. He had already forgotten about her.
Well, she will serve his purpose.
"I promised I would protect you," Pyrolithos says, leaning over her. "Do you trust me?" The girl nods timidly, reminding him of a mouse.
The New Amazons have no need of a little mouse-wench, though.
"Come out," Pyrolithos says. Haltingly, the girl crawls out and stands.
"Close your eyes," he says. She squeezes them shut.
Yes, he feels confident of it: mice must be fed to the cats. He plunges his dagger into her heart.
Her eyes open in surprise, and then she falls to the floor.
"No!" Alecta, standing in the doorway to the yard, rushes forward and kneels beside the dead girl. "Why?" she asks. She rises and shoves Pyrolithos hard.
But Pyrolithos couldn't answer her even if he wanted to, because the flaming power of Smoke Blood is roaring through his veins, temporarily drowning everything else out. The ashes curl around muscles and organs. When the rising Smoke magic settles, he looks up, but Alecta has already stomped outside to the wagons.
The wind does little to cool his hot skin as he and the New Amazons ride back to their packed-up camp.
Even later, as they saunter back to the Royal Road with their renewed disguises, to all the world a small group of men riding beside several harmanaxas, he still feels the hot wave of power that moved through him at the death of the girl, and hardly notices Alecta's coldness, the way she rides at the far edge of the group, closest to the trees that dot the sides of the road.
Toward sunset, they veer off the road and make camp in the woods, erect their tents, and roast the stolen food, flames leaping high over all the pilfered flesh. Once again, though, Pyrolithos isn't hungry.
After making sure the first watch patrols the perimeter, he returns to his tent...and finds Alecta there, sitting cross-legged on his sleeping mat.
His heart skips a beat.
"Why?" she asks simply—the same question she demanded of him in the tavern.
"Don't question me," he says.
"You are strong and brave, Cynane, and yet you rely on deception and cruelty to lead."
That name. That former self. He bends to his knee, planting his hand around her mouth. "I've told you before not to call me that."
Using a move he taught her, Alecta flips him onto his back. He feels his Smoke magic flare up in a pillar of anger, but he controls himself, lets her stay on top. "Didn't we create the New Amazons to fight injustice?" she asks, straddling him, hissing close to his ear. "Didn't you commit a huge injustice to poor Efimia today, ending her life for... What? Some sadistic feeling of power? Some of the girls are talking among themselves, you know, asking if you will kill them next for the sport of it. This isn't what they signed up for."
Not for the first time, he wonders what Alecta would say if he told her about killing Taulus to obtain the Blood of True Betrayal. About using his own admission—that when everything else about him had turned to smoke, only his heart remained mortal—to kill him. The hot spray of bright blood that exploded out of Taulus's smoky form, even as he dissipated into the night air.
How disgusted would Alecta be if she understood his _need_...
That he _must_ keep betraying innocent people?
Would it be enough to cause her to leave him? And would the Amazons follow her?
She's looking at him with flat dark eyes, waiting for an answer.
"You're right." A ball of tight emotion throbs in his chest. "I've always had to be deceitful. I've always had to be cruel." And it's true—even before Smoke magic. Long before it, in fact. "I don't know that I was born that way," he adds, trying to still the tremor in his voice even as visions of Audata fill his mind, her milky skin in the bath, her body floating in her own blood. "I think my life has made me cruel."
Alecta's air of ice seems to thaw somewhat. Her eyes soften. "You can stop, though," she whispers.
"No..." His voice breaks, and all the pain and injustice he ever suffered seems to bubble up from somewhere deep inside his chest.
"Cynane," she whispers, pulling him up to sitting, pulling him back into that old self, that old knowledge, the name like a surge breaking a dam.
He lets out a shuddering breath and finds himself telling Alecta the story of Cynane: how she was the daughter of Philip's lesser wife, Audata, whose authority in the palace had been completely stripped away when he married Olympias. How Cynane had always known of the rivalry between Olympias and Audata, and how it had made the more powerful queen vicious toward her. How she'd only been eight years old when she found her dead mother's body, black tendrils of hair waving in the red water like the tentacles of a sea creature. How, many years later, the Aesarian Lords had captured and tortured her before her desperate escape. How Olympias had laughed in the ruins of Knossos as her guards hustled Cynane onto a ship sailing straight for Dardania, to become the unwilling bride of a madman. How King Amyntas had proven even more terrifying and insane than anyone knew, hissing threats in her ears day and night.
How she had always, always been alone and had to learn to look after herself.
This is Cynane's story, and it is _his_ story, his past, his truth.
_There it is_ , he thinks. _The truth that can ruin you._
Her hand warms his cheek, then wipes away the dampness there. His heart stutters, and he experiences a violent urge to pull away. To scream, _How dare you? How dare you touch me like that? How dare you touch me with kindness and threaten to undo everything I have built up to protect myself?_
But he doesn't pull away. He stays.
Because no one has ever made him feel like this before. And this isn't just a game like what he's played before. He no longer knows the rules.
And then comes a sick realization—a temptation both thrilling and foul.
If he truly needed to commit a great betrayal, one with the power of sustaining his magic for years and years...
Wrestling with the thought, Pyrolithos looks up, his eyes meeting hers. He grabs her and his hands meet her throat, feeling the gentle throb of her pulse, the allure of it, the power in it, the heat of her...
He leans in, not knowing what he wants—the power of betrayal or something else—the delicate, secret thing she offers him that terrifies him more than any Spirit Eater. That kindness, that care.
That word he has never once in all his life allowed himself to believe in: love.
It sickens and chills him but it moves him, too. Everything he comes close to loving dies. _You are not meant to love in this world. Love is a weakness. Love kills._
His hand presses harder against her neck, and she inhales a short breath.
And then, before he knows what he is doing, and before he can stop himself, he leans in even closer, breathing in her scent, dragging his lips against the edge of her jaw, before finally they meet her mouth.
The kiss overtakes him. She doesn't push him away but neither does she succumb—instead, the kiss becomes a kind of sparring, a teasing that makes him want to fall into her more—forever. His arms are shaking, as though the powerful feeling Alecta is emitting is greater, even, than Smoke Blood. Impossible, and yet that is how it seems, as she takes control, pulling his hair and biting his lip and running her hands over his shoulders, pushing him back. She is taking him. Pyrolithos wants to dominate; even as Cynane, he was used to dominating, and can feel the spring and tension in his muscles. But it is impossible. He, who has always sought power, has found the greatest rush in the world: the sacrificing of it, an utter surrender. It is he who submits, inhaling the sweet scent of her skin, feeling the silken waves of her hair brush his face, making his mind and all its yearnings for greatness go slack, until all there is is this: heat and musk and the sweetness of two bodies in the darkness, skin against skin, souls opening into one another endlessly, until he is no longer Pyrolithos or Cynane.
He—she—they are no one.
CHAPTER TWENTY-THREE
KATERINA
Sometimes the fate of a nation rests solely on the actions of one person.
In this case, it rests on the actions of one horse: Bucephalus.
If only Kat can get him to behave. Everyone knows, the demon stallion listens only to Alexander. But Alex isn't here—which is the whole reason Kat has come instead.
When she received Heph's letter in Troy and agreed to join him in battle, she expected something different. She'd felt stirred and excited—more than she had in a long time—by the idea of joining him and the rest of the Macedonian army, fighting side by side, like she had in the Battle of Pellan Fields. She felt her spirits lift out of the deep well of sadness that had been her constant companion since the attack of the Spirit Eater at King Philip's wedding in Byzantium. Perhaps, she reasoned, this sickness of the soul wasn't doomed to haunt her forever. She could get better, be her old self again. Feel joy unencumbered by the knowledge of death and suffering all around her.
She and Jacob had secured safety and healing for the rescued children with the priests in the Temple of Asklepios. The ushabtis had killed all the Lords beneath Troy, except for Timaeus, who had disappeared, and without their training, the children's magic drained from them like water into parched earth, until nothing was left.
And then, of course, there was the other cause of her excitement...the knowledge that she would see Heph again. It kept her up the night she received the letter and most of the two nights since then as she and Jacob rode to the allied Greek camp north of Troy, through hilly country with green fields and stands of olive trees, the salt breeze of the sea in their hair.
Then, last night, they heard the whispering of the Granicus River and followed it until they saw the torches and campfire of the Greek camp spreading out from the beach. Her face hidden by a large hood, she gave Heph's pass to the first perimeter guard she met, a tall man who, looking Jacob up and down suspiciously, recognized him as a fellow Macedonian guard who last year had become an Aesarian Lord, and Macedon's enemy. Reluctantly, she left him there, promising she would soon return with Heph to bring him into camp by order of King Alexander. The guard guided her through the chaos of soldiers and weaponry, forges and outhouses and fires, to Heph's private tent, decorated in the Macedonian royal colors.
Seeing him, she sucked in a breath as all the things that had happened between them in the past came flooding back: those long days on the island, working on the mechanism, growing closer to an answer for how to defeat the Spirt Eaters and growing closer, too, to one another. How he had taken her in his powerful arms and kissed her, told her everything about himself—all the fire and fury of his past. How he'd asked her to marry him.
How she'd never really given him an answer.
As soon as she stepped into his tent, Kat felt heat flood through her. She couldn't meet Heph's eyes but felt them blazing against her skin.
"You came," he said, his voice low and breathy.
"Of course, I did," she answered, wanting to step toward him but afraid to—afraid of how it might make her feel—aware that even one inch closer to him was one inch closer to a betrayal of Jacob's love.
"Be my Patroclus," he said.
"What?" She was thrown off. There were millions of things he could have said to her after their months apart.
"You know the story," he prodded.
And she did—Patroclus had been Achilles's lover, or so the myths told. Was that what Heph wanted from her?
She took a step back, uncertain how to respond.
"You remember what Patroclus did," he added, clearing his throat.
"He led the Greeks to victory wearing Achilles's armor," Kat answered. "You...you want me to wear your armor?"
Heph laughed, and she couldn't help it this time—her eyes were riveted toward his, and the heat of seeing him again flashed through her once more, dizzying.
"Not my armor, _his_. Alexander's. Don't you see? He couldn't be here in battle. But you can—in his place. His twin."
Kat balked.
He took a step toward her, reaching out to grasp her hands. She let him. "It is the only way we can keep the alliance together until Alex returns."
What he was proposing finally sank in. "We could never get away with it," she protested, her voice low and soft.
But Heph explained his plan. In mourning for his father, the newly crowned king wouldn't show his face or speak to anyone other than his most trusted advisors. The helmet's elaborate cheekpieces and nosepiece would hide her face. When she pointed out that she was several inches shorter than Alexander, Heph said she would only be seen riding Bucephalus.
It was a wild, dangerous thought. And yet, she had to admit, she wanted to say yes. It was politically necessary; she could see that. The soldiers would never follow Heph. They needed their leader. But it was more than that, too. It was as if Kat had been born to do this—as if she and her brother had been born twins for this very reason, and it was somehow her sacred duty not just to be there for him but to _be_ him, when he could not.
And to be, even for a few days, a king?
Was it not every girl's dream?
* * *
And so, here she is, in the early-morning light, standing at the makeshift stable housing Bucephalus, Alexander's trusted steed. A beast no man has ever been able to control but him.
The Macedonians gather around her. Kat checks the tightness of her helmet strap, squares her shoulders, and approaches the horse. Jacob holds Bucephalus, who snorts and jumps, eager to get away. All around her, Macedonians stare at what they believe to be their new king in his golden armor. The king who, after sacrificing at Troy, rejoined the army and now stands ready to lead the Greek invasion of Persia. Kat senses the horse's outrage—at being stuck in a small ship's cabin for days, at waiting for his master to return. She keenly feels his distrust of her. While she might be able to fool the Greek generals and their men, she could never for a moment fool Bucephalus. Now, sniffing her, he brays his disapproval loudly, sidestepping away from her.
Hot panic races up her spine. This animal is a tense, dangerous muscle, ready to explode in fury at any moment. Before she can mount him, she pours all her Snake Blood Magic into calming him. She tries to settle into his mind—no easy task, as he resists at first, thrusting her out as easily as he'd toss an unwelcome rider off his back. Finally, she seeps in, just the thinnest strain of her thought, and begins to soothe him, sending the creature the same calm he gets when he smells the approach of Alexander. The horse seems not so much to have accepted her message as to be considering it. She wonders if he will throw her, proving to the entire Greek army that she is not Alexander.
But she must risk it.
She opens her eyes and swings herself deftly on top of the huge stallion. He shakes his head, nickering. Then Kat turns him toward the gathered soldiers. Sensing freedom, smelling moist earth and green grass, Bucephalus bolts like an arrow launched from a bow and gallops through the Greek camp. It's all Kat can do to cling to him. Men setting up tents, unloading wagons, and building fire pits salute their general, cheering wildly.
Around and around the camp she rides, then back to the wide white beaches, eyeing a sea that is black with ships as far as the eye can see.
Finally, sensing that Bucephalus is calmer, she finds the royal tent, its cupola dyed with Tyrrhian purple and embroidered with gold. Jacobs waits out front for her and takes the reins as she dismounts. He nods to the guards on either side of the opening, men sworn to allow only Alexander, Jacob, and Heph inside on pain of death.
In the cool darkness barred with light, she takes off her helmet again and finds her hair slick with sweat. Her stomach feels as if a horse has kicked it, and her head throbs with a nervous headache. Here she must stay, speaking in low tones only to Jacob and Heph, emerging only when needed for battle, in her golden armor, hot and sweaty in the summer heat, to ride the demon horse, fearful of discovery at any moment.
Unless...
Her body may be trapped in the tent whenever she is out of her armor, but that doesn't mean her spirit must be trapped.
A falcon flies over the enemy camp—and within the bird flies Kat's mind, analyzing, strategizing, observing. The Persian camp is like no Greek army she has ever heard of. To be sure, there are plenty of warriors—mounted and on foot. And a baggage train with tents, provisions, and weapons.
But at the center of the camp is a kind of traveling palace. Perched on a high tree branch, for hours she watches the Persians set up luxurious steepled tents and fill them with valuables. She sees servants carry enormous silver platters, golden chalices, and inlaid tables into the banquet tent. In the gods' tent, bearded priests in orange robes and conical headdresses set up gem-studded altars and gilded incense burners.
Then there is the elaborate bathing tent. Whereas Greek soldiers wash in a leather bucket of water—or not at all—the Persians unload silver-plated bathtubs, gilded buckets, painted amphorae that must contain fragrant oils, and mounds of finespun towels. Next door, they set up the harem tent, fill it with rich carpets, beds with precious hangings, and plump tasseled pillows. Gorgeous women with heavily lined eyes and spangled veils wander in and out, seeing to the disposition of huge wardrobe chests.
Now an official emerges from the tent that holds wooden chests so heavy it took six men to carry each one using sturdy poles. Soldiers wait in front of a man at a little desk. The official opens a chest, and her falcon eyes are so dazzled by the sun reflecting off gold that she is momentarily blinded. She twists her head away, blinks, and then looks back. The official—who must be the paymaster—gives each man who comes forward a gold coin. If all those trunks she saw stored inside the tent also contain gold coins, it would be enough to buy a small kingdom.
Overall, there is good military order, she notes, but seemingly no worry among this army that they are facing a superior force. Perhaps they are overconfident they will chase away the barbarians from Persian soil, which the Greeks could use to their advantage.
She has seen enough. Beating her wings, she rises vertically from the branch, then thrusts herself forward, winging her way back to the Greek camp, cheered by the hours of soaring flight, golden sun, and bracing air high above fields blooming with early-summer wildflowers. She releases the falcon from her control with a heartfelt thank-you and feels the bird tear away from her, happy to be free once more as it soars into the impossibly blue vault of the sky.
With a start, she finds herself back in her body, sitting cross-legged in Alexander's tent. Jacob holds a vial of lotus blossom oil under her nose.
She blinks and looks around, confused. It is always a shock to be flying in the sun-gilded air one moment, feeling the breeze ripple through feathers, and the next moment waking in a solid fleshly body of arms and legs, heavy and earthbound. At first, her eyes—so used to scanning the distance as the falcon—cannot focus. She sees the blur of Alexander's camp bed and portable desk. Sunlight streams in through the open flaps high on the sides of the tent. Her vision sharpens as she looks at the two men who love her—and hate each other—sitting on low stools in front of her.
"Are you all right?" Jacob asks, concern etched on his broad face.
"Fine," she croaks. Her voice is always hoarse when she returns from voyaging with an animal host.
"What did you see?" Heph asks, his dark eyes flashing with interest.
"The Persian camp..." She swallows and coughs, and Jacob jumps up to fetch her a cup of watered wine, which she drinks gratefully. "It's about ten miles from here," she says, "along the banks of the Granicus River. It must be one of the coastal defense armies our spies reported. It's not the army of Great King Artaxerxes. It's not that big—maybe only fifteen thousand men—and among the leaders I didn't see a strangely tall, thin old man with a long white beard. I don't think he has had enough time to get here from Persepolis."
"Even if it is a coastal defense army," Jacob says thoughtfully, "I wonder why they didn't attack us on the beach. Their own scouts must have told them we landed."
"I think," Kat says, as the images of what she saw—the harem and bathing tents, the general feeling of the Persians being on a kind of holiday, "they are not terribly worried about us. They must think that we are barbarians they can squash like flies in their own territory."
"How much cavalry? Infantry? Catapults?" Heph asks.
"Maybe two thousand cavalry," she replies, her voice starting to resume its normal timbre. "Six large catapults, unassembled, left in pieces in their wagons for the moment. But the most amazing thing was the treasure wagons." She tells them exactly what she saw.
"I have heard of this," Heph says, nodding and looking at her, but not at Jacob. "The Persian armies do not travel as we Greeks do—light, flexible, and mobile. They believe their luxuries increase morale. And the soldiers fight not only for their homeland but also to guard their treasure and the commanders' women."
"All right," she says. "So, what would Alexander do?"
Heph runs a hand through his unruly dark hair and considers her question for a long moment. "He would divide our forces into three parts," he says at last, confidence blooming across his face, making him more handsome than ever, though Kat pushes the thought away. "The first part will act as a diversionary force, drawing off a good part of the Persian army. The second part will attack the treasure wagons, drawing in those Persians who remain to guard it."
"And when that battle is fully underway, the third part will move in and finish them off," Kat concludes.
Heph nods.
She turns her gaze to Jacob, whose chin rests in the palm of his hand as he contemplates. She knows Heph would never ask him his opinion. The tension between them is thick. Ever since they reunited, the two have barely spoken a word to each other. At first, Heph absolutely refused to even bring Jacob back from the camp perimeter where she had left him the night she arrived. An Aesarian Lord, he said, an enemy of Macedon, could easily betray them, betray _her_ , no matter how well he had taken care of her in Byzantium. Because this was no longer about personal feelings, but war, victory, and the rise of empires.
"He will not betray Macedon or me," she argued, knowing, without proof, that it was true—that if she couldn't trust Jacob, then she couldn't trust anything, even herself. "Besides, he's no longer an Aesarian Lord." She gave him a brief outline of how Jacob had left the brotherhood during the earthquake of Delphi, resolved to find and kill Riel on his own, and how he had ended up killing almost his entire regiment in the tunnels beneath Troy to save the children. "But what he learned from them will help our army. The Lords are the deadliest fighting force in the world."
Even still, it was only when she threatened to refuse his offer altogether that Heph consented.
"What do you think, Jacob?" she prods, rubbing her eyes. She's still adjusting to the dimness of the tent after her flight within the body of the falcon. "What would the Aesarian Lords do?" She can't help but notice his arm muscles flex in a tense pulse in response, and she can tell it is going to require all her diplomatic skill to keep these two from punching each other.
"The plan should work," Jacob finally says, and she can tell by a tightness in his voice how hard it is for him to admit. "But it will be better if we can trap the army in one place, taking away much of their mobility when we attack."
"How can we do that?" Heph asks, irritation sharp in his voice.
Jacob looks coolly at Heph. "As you know, I have certain abilities. Kat says they are camped on the river's edge. Just before dawn tomorrow, we will encircle the Persian camp. I will cause the river to rise, trapping everything and everyone in the mud. The treasure wagons will be hopelessly stuck. Those that remain to defend them, we will pick off with our arrows. It will be like spearing fish in a barrel."
"And how do we prevent our own men from getting stuck in the mud when we attack them?" Heph's question ripples with disdain.
Jacob's dark eyes flash, but his face remains impassive. "Easy. I channel the water only to their camp." He shrugs, a small movement but it doesn't go unnoticed.
"Hmm," Heph says. "Last time you caused water to rise, you killed hundreds, maybe thousands, of people. You almost killed Katerina."
Jacob's mouth sets in a hard line, and he practically spits his words. "I had just learned that my entire family, including my three little brothers, had been horribly butchered. I hardly think the incident is indicative of my capabilities."
"All right," Kat says, hoping her words can slice through the tension in the air. "So, we have a strategy."
"Even a good plan can be improved upon," Heph says, determined to have the last word. "Let's shoot flaming arrows at the tents. Between the rising water and the flames, they will not know what hit them."
* * *
The following morning, shortly before dawn, Kat leans forward on Bucephalus, peering into the darkness. Across the field, the torches on poles around the perimeter of the Persian camp flicker weakly, illuminating now and then a soldier patrolling in strange garb—polka-dot trousers and checked tunics. Something in her twists. Because of her, these men will probably die in the next few minutes. Do they have children? Parents? Brothers and sisters? She thinks of little Roxana, whom she left at the Temple of Asklepios to continue her healing along with the other children. "First my sister left me," the girl said, when Kat told her she must leave, "and now you." Pain, loss, and unbearable sadness wrap around Kat, weighing her down like a wet wool cloak.
She had been so excited to fight—to be driven forward by a purpose greater than her own. To get that much closer to facing off with the Spirit Eaters. She thought the sadness she had experienced was gone.
Why now are these thoughts—these sympathies—invading her mind and destroying her focus? Is the Spirit Eater venom still inside her, dragging her down? Will it never go away?
Suddenly, there's a drastic shift within her, and battle no longer seems as simple as brawn and strategy and victory. As with the divide in her own heart between two men, she sees now that there can never be a clear winner, never one side more deserving of victory than the other.
Beneath her, Bucephalus shifts his weight in frustration. He wants to race across the field, and it is all she can do to keep him in check. He galloped the ten miles from the Greek camp, his noble heart reveling in the freedom to run, and he wants more. _Soon._ She pats his moist neck. _Soon._
The stars in the night sky are fading. A smudge of orange streaks the horizon. It is time. High up in the gnarled, knotty pine tree on her left, Jacob must be calling the Granicus River to overflow its banks and pour into the Persian camp. The thought of him like that—curled deep within himself in order to call up a power far greater than himself—sends a chill of terror and awe through her. Jacob understands her. Jacob is _like_ her. There was a reason Helen raised her so close to his family—so that, one day, she might discover that she belonged somewhere, that she was not alone.
A bird trills, greeting the coming day, and all around her, others reply.
Now voices echo across the silent fields, punctuated by screams and loud orders in Persian. Soldiers run along the perimeter, lighting fresh torches, which they carry inside the camp. Horses whinny. The guards have all completely disappeared. After a time, Kat sees something large move out of the camp and stop. Hears the crack of a whip and the bellowing of oxen.
"It must be the gold wagon," Heph whispers, leaning in from his bay mare. "They would want to get that out first. But it must be so heavy, it's already stuck in the mud."
"Time for fire arrows," she says. Ignoring the rising horror inside her, she takes the bird whistle on the thong around her neck and blows into it. Up and down the Greek line, the sound is repeated. Heph and the other archers open their containers of glowing coals, insert a tar-tipped arrow, and bring it out burning brightly. Moments later, flaming arrows arc through the dark blue sky. They stick in the proud turrets and domes of the distant tents, which burst into flame. High-pitched screams erupt, followed by shouts and orders. Flames leap from tent to tent, and from this distance, it all seems like a strange fairy tale come to life, the bright colors of the camp igniting in a burst of fire and dawn.
Horses shriek. Kat feels their panic as if it were her own fear—instead of her own doing. Moments later, dozens—no, hundreds—of horses run out of the camp, their long legs sticking in the mud. In the shimmery rose-silver light, she sees them twisting, struggling to free themselves, heads raised, huge teeth exposed. Panic—fueled by the roar of flames and the thick haze of smoke—helps extricate them. Once outside the circle of floodwater, they hit solid ground, pound their way in different directions across the fields, and disappear.
People, too, stagger out of the camp. Kat sees women holding hands, helping each other navigate the goo, and servants carrying silver platters that reflect the rays of the rising sun, flashing and blinding. More horses appear, but these are saddled and carry soldiers. The Persians are launching a counterattack. But the weight of the riders makes it more difficult for the horses to pull their legs from the muck. A new round of Greek arrows sinks into the soldiers, who scream and fall off their mounts. Kat wants to scream, too, but clenches her jaw so tight it hurts as she watches the horses, freed of the burdens, struggle to safety and vanish.
Some of the horsemen, however, make it to dry land, followed by a steady stream of others. Kat nods to Phrixos, on her left, who raises the battle horn and blows. The signal for the attack. Kat kicks Bucephalus, who jolts forward almost as fast as any arrow. Horses' hooves pound, men shout, and people in the camp scream.
She raises her spear. She is supposed to be King Alexander of Macedon, after all, and must show no weakness. Beside her, Heph fires arrows one after the other. And then the two sides meet in a furious clash of metal and leather and flesh.
Ada taught her to use her Snake Blood to feel what is coming—the sword blow, the arrow, the spear—and veer away from it a moment before. It doesn't work if she's distracted, which is what happened last summer in the Battle of Pellan Fields, when an Aesarian Lord stabbed her. Now, instead of feeling fear or aching despair, she just feels what is there. An arrow is whizzing toward her, though she can't see it through the billowing smoke. She swerves Bucephalus and hears the arrow pass a finger's breadth away from her left ear.
The smoke clears, and Kat sees a Persian horseman riding straight for her, straight for King Alexander of Macedon, he must think, curved sword raised, a malicious gleam in his cruel dark eyes, his mouth open in a laugh or shout. Kat raises her spear—Alexander's gilded spear—and urges Bucephalus into a gallop toward him. But just before they meet, an arrow enters the Persian's mouth. Blood spurts from his throat. He tumbles backward off his horse, which runs away in a panic.
Heph pulls up next to her, his bow in his hand. "Excellent target practice," he says coolly. But she can see by the flush on his face that he is not as cool and collected as he pretends. Behind him, Kat sees a Persian headed for him, spear raised shoulder high, ready to throw. She throws hers first, using her mind to send it straight and true into the man's chest. It pierces right through, a good foot of it coming out of his back, and as he falls to the ground, Kat feels part of herself falling with him.
The Greek armies, waiting on dry ground, quickly dispatch the Persian soldiers spilling from the camp. Kat notices that Heph stays by her side, to protect her, of course, but probably also to gall Jacob, who is still stuck up in the tree, channeling the water. She signals him to follow her, and they ride to the river, where hundreds of heads bob, desperate to get away from the flames and enemy swords. Probably any Persians who could swim have jumped in, paddling for their lives. Greek archers send arrows zooming toward the heads, which hit them as if they were bobbing melons used for target practice. Horsemen on the banks throw spears at them, laughing as they hit their mark or people dive under the water. This killing of unarmed, fleeing men and women is a game to the soldiers. Kat is so disgusted she can barely move as the lifeless bodies float downstream.
When the flames have burned out and the battle is over, it is time for her to blow three long wails on the battle horn: the sign for Jacob to drain off the water. But she finds she doesn't have the air in her lungs to do it and hands the instrument to Heph, whose blasts echo loudly across the field. Even after the water drains, there is still mud, of course.
She and Heph pick their way through the remains of the once-magnificent Persian camp.
The tents, rugs, furniture, and chests are ashes, once-bright colors now caked in dirt, dust, still-wet blood. Glorious fabrics shredded and limp, like flags of surrender, sway in the breeze, forming the only movement in the camp.
But the silver bathtubs, mounds of gold coins, and jewels are intact. In their first battle, King Alexander has not only won a decisive victory, but a huge pot of plunder for the Greeks to distribute to their men. If only Kat could be pleased about it.
Carting the valuables out of the camp and rounding up the fine Persian horses takes the rest of the day. When the sun has set, she and Heph ride into the Greek camp, which is alive with singing and feasting on rich Persian food and wine. She hasn't seen Jacob all day. She wants to thank him for his amazing feat of Earth Blood, but also to confide in him the sinking despair settling into her bones.
To tell him about the daydreams that threatened to drag her to her death out there on the field today—fleeting visions of being back on Meninx, an uninhabited island of wind and waves, trees and birds, with no bloodshed, no war. A place where magic, once thriving, was put to sleep forever, leaving in its wake an emptiness, a kind of relief.
"Come to my tent for a drink," Heph says in that charming way of his. People were surprised young Lord Hephaestion no longer shared Alexander's tent, but he put it about that this, too, was part of Alexander's mourning ritual for his father. "I have an amphora from the Persian general's tent that I want to share with you and only you. The finest Persian wine, made in Babylon, I hear."
She leans toward him and whispers, "I should always stay in my tent when not exercising Bucephalus or leading the armies," she says. "That was what we agreed."
Heph yanks off his helmet, runs a hand through sweaty dark hair. "I know, but Jacob is always there. His tent is closer to yours than mine. I bet he is waiting for you now. You and I haven't had a moment alone. Please come. For old times' sake. For Egypt and Meninx and all our adventures on the sea. Please. Just a few minutes."
He smiles hopefully at her. How can she say no? It would seem cruel, childish. And perhaps she can open up to him about the dark feelings caused by this battle, the death and suffering. She confided in him on Meninx, didn't she? And he confided in her.
"All right," she says quietly, as if whispering makes it less a crime.
_No, there is no crime here_ , she tells herself—or if there is, it is only inside her chest, where no one can see it.
They slide off their horses in front of Heph's tent and tie the reins to a tent pole. Kat slips between the canvas flaps and finds a camp bed, camp table, a couple of folding chairs, and a trunk. It's smaller than hers—Alexander's—with no luxuries, no plush carpets, no gilded perfume burners, and no large desk for maps. But his servant has lit the lamps, which hang from poles driven into the earth, and it looks tidy and comfortable.
"It seems I am doomed to always miss those I love," Heph says sadly, pulling out the two chairs from the wall and opening them. "I miss you, even though you are here. And I miss Alexander. Wonder what he's doing. Pray that he is safe."
"I do, too," she says, sitting. She unstraps her heavy golden helmet and removes it. The feeling of cool liberation is much like taking off a sweaty, blister-causing boot. It is nothing less than sweet, blessed relief. She rubs her hands through her damp, tangled hair.
"Can you...feel how he is?" Heph asks, pulling the amphora out of its circular stand and releasing the wooden cork. He pours two cups of wine, hands her one, and sits across from her. "Many twins, I hear—without Snake Blood—have a kind of connection. There were two old women in the Pellan palace who were twins. One was a cook, another a laundress. One day the cook ran screaming through the palace that her sister was dead, and everyone laughed at her. They had just seen the sister walking to the laundry with a basket of soiled sheets. But the cook was right. Her sister had had a fit and fallen over dead. They later found her in the garden. How did the cook know?"
Kat doesn't answer. She drinks deeply. The wine has a spice to it that she has never tasted before, something exotic and eastern. It slips easily down her throat, sore from smoke and soot. "There was a similar case in Erissa with a miller and his twin sister," she says at last. "But as far as Alex and me, well, I do sense that he is alive. Or maybe it's more that I know I would sense if he were in trouble or...or no longer..." She pauses. "I would feel the loss, the emptiness."
Heph seems to understand. "Well, I'll take that over nothing."
"And after our victory today, I think," she says, "the Greek allies will praise the strategy of young King Alexander. The fire, the flood—it will all seem as though even the gods are on our side." The words prick at her and she flinches.
For all his bluster and pride, all his charm and passion, at least he notices. "What's wrong?" he whispers.
"Ever since I was attacked...by the _thing_ , the Spirit Eater, I mean." Dark waves, clawed and fanged. The shuddering cold, more soul crushing than the chill of the freezing waves. "I've felt..." But how can she tell him? How could he—or anyone—possibly understand?
"When we were on our summer campaigns with King Philip," Heph says quietly, "the prince and I would stay up late forming battle plans until we fell asleep together." He looks at her, his eyes suddenly dark with something she cannot read. But if she had to guess, she would say it was longing.
"I wouldn't mind doing that with you, my king," he says, his voice a whisper now, so low it rumbles through the air, almost like a caress.
For a moment, she feels hot and chilled at the same time. She leans closer to him. But then, just as quickly, anger, hot and bright, flares up in her, forcing her back—anger at herself for feeling tempted these last few days whenever Heph is near. Anger at his inability to really understand her, the pain she has gone through, the loss she has felt—even when they were back on the island of Meninx together and she first learned the evil of her true father. Both then and now, it's as though some inner piece of her has broken into shards that even love—his or Jacob's or both—can't seem to fix. An indefinable lostness that she'd hoped to drown in battle but instead it has become only more piercing and more pure.
She sees it now—even before the attack in Byzantium. How she has been floating, at a loss, with no direction. Ever since learning the truth—that her mother was the evil queen of Macedon, the very person she'd sought revenge on her whole life until that point. That her father was Riel, the Last God, a terror to the very earth.
She sees now how she floated, lost, even on Meninx, trying to bury that sense of directionlessness in Heph and their quest to create the mechanism. How she'd floated again, just as lost, with Jacob during the past weeks they spent in Troy—helping the sick but ignoring the sickness that tugged at her own heart, twisting up inside her like a pit of snakes.
How even coming here and pretending to be her brother—how just last night their plan had seemed euphoric and thrilling but had then come crashing into reality this morning as nothing more than another disguise.
Who is she now?
No man can answer that.
And if they cannot, then the last thing she needs is to get lost in either of them.
She stands abruptly. "This was a mistake," she says, setting down her goblet and picking up her helmet. "I'm going."
"No," Heph says, grabbing her arms. His flirtatious charm has vanished, replaced with an urgency that almost frightens her. "We need to talk, Kat. And we can't with that—with Jacob always hovering around you. All I ask is honesty. What happened between you and Jacob all those months in Byzantium and Troy? Did you... I mean... I know what you once were to each other. What I must know is what you are to each other _now_. I deserve that much, Kat."
"It's none of your business," she flings back at him. Her whole body is shaking. With fury. With the lack of answers. With hatred at herself and the confusion that has overwhelmed her ever since they launched the battle this morning.
"It is my business to ask the question," he counters. "You know where I stand, Kat. Nothing has changed for me. I still—" His voice breaks and she wants to run away but can't. She wrestles with the heavy helmet in her arms and the emotion welling up in her throat, threatening to choke her. "I still feel the way I did on the island. I still...love you." His voice, wavering, drops back to barely more than a whisper. "And I will wait for you, if you let me. But..."
He reaches for her but she holds her ground. They both stare at his hand, floating in the space between them.
"But while you told me Jacob gave you that pearl ring you always wear, you won't tell me what it means. Tell me, Kat. If you have chosen Jacob. Just one word, and I will back off forever." His hand goes to his heart as if in vow.
Kat opens her mouth to speak but can't find words. Because the fact is she hasn't chosen Jacob. Not really. Fate pushed them together, yes, and after he had done so much to heal her, it felt right to be with him. It had always been the two of them together since she was six years old. Still, she can't bring herself to say it, or to lose Heph—or if she must lose him, then it would be more than she could bear to lose him by her own doing.
"You want honesty, and that is a fair request." Desperately she tries to steady her voice. "The truth is I just don't know. Until I know who I am and—and what my true purpose is—I _can't_ know, and all I can do is play these games, breathlessly trying to keep up, to seek meaning where there is none, forcing myself to see victory where I know there is only loss—everywhere. On both sides."
Heph shakes his head. "Does Jacob know how you feel? Because if so, he doesn't show it. He acts like he owns you. Maybe you should tell him he doesn't."
"There are more important things on my mind than which man to choose," she says angrily. "Today I commanded a victorious army. I am probably the only person in the world who has killed a god. Why are you pushing me, Heph? Jacob is patient with me, understanding. He knows I need time to...to figure things out."
Heph's dark eyes narrow, becoming slits in the lamplight. "Those soldiers out there might think you are Alexander, but you are really nothing like him. Alexander is decisive. If you had his spirit, you wouldn't dangle Jacob and me for months, torturing us both. You would make up your mind and be done with it."
"As we both know, I am _not_ my brother!" she cries, hating how thin her voice sounds in the thick, hot air of the tent. "Maybe you would prefer it if I was?"
Heph's face floods with pink.
She crams the helmet back on her head so she can run outside and get far away from Heph. But as she turns, she glimpses a man's startled face peering between the tent flaps.
She cries out. The face has vanished.
"What is it?" Heph comes up beside her.
Kat stands rooted to the spot, panic gnawing her stomach. "Someone was standing there, looking at us," she says, her voice trembling.
"Jacob?" Heph smirks. "Are you worried it was him?"
"No! Not Jacob," she snaps, her nostrils flaring. "I wish it had been. It was a servant, perhaps. Or a soldier. I couldn't see his face clearly. Fairly young. Clean-shaven." Her gaze slides from the doorway back to Heph. "Don't you see what this means? Someone saw me standing here in Alexander's armor, putting on his helmet. Someone knows Alexander isn't here leading the armies. That the person riding his horse and wearing his armor is an imposter. And not just any imposter. But a girl. If word gets out to the armies, they will probably go home in anger and disgust. We just won a battle, but now the whole war is at stake—at the mercy of our ruse."
Quickly she stuffs her long hair under the helmet and dashes out of the tent before he can contradict her or tangle up her thoughts once more. _Stupid, stupid._ If they had argued in her own tent—Alexander's tent—surely the guards would have prevented someone unknown from entering. Was the person she saw a servant or soldier hoping to speak to Heph? Seeing Kat in Alexander's armor, did he become afraid and run away? Perhaps, fearing for his life, he will never tell anyone. Or perhaps it was a spy from one of the allied armies following "Alexander" to find out what was really going on.
As she mounts Bucephalus, she thinks of the other generals and her heart sinks. Chares, the Athenian, lean and dark, was reluctant to be led by Philip, even less so by his teenage son. The Theban leader, Theagenes, red-faced and quick to anger, will probably never forgive Macedon for killing his Sacred Band of Immortals. Either man would pay generously to learn the real reason that no one can talk to King Alexander on this campaign or even see his face.
Kat walks the horse around Heph's tent and sees five men sitting on logs around a fire, roasting an entire lamb, courtesy of the Persian army. Three of the men are bearded. The other two don't have the right features to be the man in the doorway. Who was he? Panic fills her head, thudding in her ears.
By the light of a torch, two men play golden dice on a silver game board, another bit of Persian plunder. One of them could be the man she saw. Couldn't he? They don't even look up as she passes. Frustration nearly explodes inside her. About half the clean-shaven young men in the army could be the intruder.
A strong hand grabs the reins and Bucephalus rears. Calming him, Kat looks down into the worried eyes of Jacob. "What's wrong, sire?" he asks. "You look upset."
She gazes around the camp in despair. Men sitting around fires, polishing swords, sewing ripped tunics, playing dice, carrying wood and water. Because of one mistake, she could have lost everything for Macedon. For Alexander. She wants to kick Heph for inviting her into his tent. Mostly, she wants to kick herself for going, for wanting to be near him, for needing _either_ of them.
"Come with me," he says. "I can see you don't want to talk here."
A few moments later, Jacob has led Kat through the phalanx of guards and into Alexander's royal tent. The lamps are lit, making strangely shaped shadows on the walls and roof.
"What is it?" Jacob asks. "Tell me."
"Someone saw me take my helmet off," she hisses, flinging the helmet onto the carpet. "Someone saw that I am not Alexander."
"What?" Jacob asks, aghast. "Who? How?"
"A few minutes ago. I don't know who. I just saw a face."
"But why didn't the guards..."
"There were no guards, Jacob. I was in Heph's tent."
Jacob crosses his muscular arms. His face sets in a mask of control. "And why were you there?"
"We were...talking. About Alexander. I didn't think, that one time, anything bad would happen. But that's not the point. Don't you see—"
"It's clear he wants you, Kat," Jacob interrupts. "And you're not being fair to him or me. You haven't told him you and I are together. Have been together as man and wife. That we have a future together. You accepted my ring, after all. That was a promise."
Was it? She didn't think so at the time.
But clearly Jacob did.
"Seeing him again has confused you," he continues. "I understand that. You still have feelings for him. But let me tell you this. You will _never_ be first with Hephaestion, Kat. That much I know."
"What do you mean?" she asks. Why are they even discussing this? They should be searching for the man she saw. Or strategizing what to say to the Greek commanders if they come barging in, demanding to speak to Alexander.
"Hephaestion will always put his relationship with Alexander first, especially now that your brother is king," Jacob replies.
"We all put Alexander first," she says angrily.
"I don't." Jacob looks at her steadily, his brown eyes sad. "I put you first." She pulls in a shuddering breath as his words tumble over her, the truth of them. "I always have, and I always will. If you and Alexander were trapped in a burning building and I only had time to rescue one of you, it would be you. Do you believe that?"
Her head is suddenly so heavy she fears she may fall, her chest so full that it is hard to breathe. "Yes." It comes out like a whimper. How can someone's love feel simultaneously like joy and like a crushing weight? What does it mean to feel that way, to love him back and yet to yearn for a freedom she can't even define?
"Do you think Heph would do the same?" he whispers.
She tries to picture it. Smoke, flames. She and Alex both injured, unable to move. Heph racing through the fire to grab one of them before the roof collapses on the one he leaves behind. The fact is, she doesn't know which of them Heph would save.
"What a silly question," she says, waving her hands as if to physically push the thought away. "That will never happen. You're not talking sense."
"I'm just saying it's what you deserve. You deserve someone who will always choose you first. Who lives for you and would be willing to die for you. I am that man. And Hephaestion is not."
"I need—" She stops. What she needs is Alexander, her brother, her twin. But he is not here. "Jacob. A man saw me. We must do something about that first—before...before anything else can be discussed. We need to find him, and I don't know how, and I don't know what to do."
"Then do this," Jacob says, his hands warm and strong on her shoulders, his dark eyes intense. "Keep leading the armies to victory as we follow the Royal Road east. Today's plunder has made every soldier in this army a comparatively wealthy man. If you keep giving them more, they won't care if Alexander is really someone's maiden aunt." And then, as if reading her thoughts, Jacob says softly, "Perhaps we will find Alexander in Gordium. It is only an eight-or nine-day march, by all accounts. Our scouts say there is only one Persian cohort in our way, the Third, just outside the city."
Eight or nine days. To Kat, it feels like eight or nine years.
But if she finds Alex at the end of that march, it will be worth any price she has to pay.
She just has to make it that long. While her body fights battles in her brother's armor and her mind soars in the sky above, Kat herself just has to float, like she has been doing all this time. Float and wait and pray that answers come.
CHAPTER TWENTY-FOUR
DARIUS
Darius urges his horse forward to come up alongside the imperial harmanaxa. Sunlight reflects off the gold-plated winged bulls on its red leather sides. The wheels are plated with gold. The leather traces and bridles are studded with golden rosettes, and the horses' bits are solid gold. The whole contraption has the effect of a giant gilded beetle traversing the sparse and dusty landscape—desert as far as the eye can see.
He signals to the driver, perched on a small chair in front, to stop the eight white horses, and the carriage grinds to a halt. He dismounts and hands the reins to one of the numerous bodyguards riding alongside. Another guard opens the door, and he climbs inside.
Great King Artaxerxes reclines on a heap of brightly colored pillows, his long white hair and beard a stark contrast to the shades of scarlet and purple all around him. The sweet, cloying scent of incense rises from a perfume burner on the wall.
The eunuch Bagoas is already there, of course, sitting cross-legged on a rich carpet next to the king's heap of pillows. A block of light—coming from an open panel in the ceiling—illuminates him, as if he is the lead actor in a theatrical production, while the Great King is a bit player in the shadows. The sight disconcerts Darius, the brief flash of some divine omen. But then the impression is gone. This is simply the way the Great King's carriage allows light and fresh air to enter while preventing prying eyes and ears from discerning what happens inside. In the shadows lies freedom and invisibility—this much Darius knows better than anyone. It is in the shadows and not the broad light that the slyest and most effective strategies play out, after all.
Darius and Bagoas nod curtly at each other, keeping up the pretense of their old hostility. Artaxerxes must never know they have combined forces.
The harmanaxa lurches into motion, and Darius folds himself neatly onto the carpet in front of Artaxerxes.
"My king," he says.
The black glittering eyes fix on him, aglow in the dim recesses of the cabin. "News, Darius?" he asks.
"I have received word that the Greek forces have achieved a victory against General Gardashasp's army on the coast. They burned the camp and have taken all our treasure and our horses."
"Did they now?" The king's thin lips spread into a smile under his long hooked nose. Ah, so he has already heard from his own spies. "My messenger tells some tale of wizards who conjured the river to flood our camp. As for me, I think it is an excuse for a poor defense. Gardashasp burned to death, I hear, trying to save the treasure. But I will execute his lieutenants if they are found."
The Great King snaps his long crooked fingers, and Bagoas unstops a flagon, pours some wine into his own cup, and then some into the king's. Bagoas drinks first, swishing the wine around his mouth, then swallows, and nods—no poison.
Artaxerxes drinks deeply. "This early defeat could, actually, have a positive outcome for us. The easy victory will give the Greek forces overweening confidence, more than they should have taking on the Great King of Persia. We will arrive in Gordium in eight days and prepare to give battle there. We are a much greater force than that under the command of Gardashasp. I am curious to see if the Greek wizards will vanquish me. Especially since I already have my priests working on counterspells." He drinks again.
Darius steadies himself for a moment, takes his time, listening to the crunch of wheels, the rhythmic clomping of horses' hooves, and the voices of soldiers, an unassuming backdrop for the astonishing news he is about to share. "I have placed a spy in the Greek camp as a servant. As you may have heard, the young king Alexander has been behaving mysteriously, staying in his tent, emerging only in his armor to ride his horse and give battle."
Artaxerxes rearranges the pillows behind his back. "The boy pretends to mourn for his father, I should think, to disguise the bloodlust that drives him on. Perhaps he even killed the old king himself. Is that the theory you were hoping to surprise me with, grandson?"
"There is more to it than that, my lord," Darius says, leaning in. "King Alexander did not lead the Greeks against Gardashasp. It was someone else masquerading as Alexander." He lowers his voice. "A girl, no less."
Now that catches them both off guard. The king starts, spilling some wine on his bejeweled purple robe. Bagoas's wily dark eyes open in shock.
"A girl?" The king blusters. "Perhaps one of these New Amazons terrorizing their way across the empire. Are you sure your man hasn't succumbed to the Greeks' wizardry himself?"
"I assure you my men are reliable."
The king leans back on his pillows and looks up at the ceiling, staring at the square of light as it jolts and sways. "Some say King Philip died of poison. Is it possible someone poisoned the prince, too, and is pretending to rule in his name? Pity none of our spies know where the _real_ Alexander is, then—or if he is even alive."
"The young king is on his way to Gordium," Darius says, his voice smooth and confident.
The Great King raises his eyebrows but says nothing, clearly not wanting to reward Darius with his surprise this time.
"Gordium?" Bagoas asks, speaking for the first time. "What's for him there?"
Darius shrugs. "The impersonator spoke of it. Perhaps Alexander himself is on a spy mission. Perhaps he wants to try his hand with the famous knot."
Darius continues, "Anyway, Alexander's purpose doesn't matter, really. If your majesty agrees, I will send word by our fastest messengers to all our men to search the environs of Gordium for a young blond Greek with a slight limp, a snake birthmark on his left thigh, and eyes of different colors." But he has already sent word to all his Assassins to fan out across the Royal Road to search for the new king. _He_ must find him first.
Artaxerxes tugs at his beard thoughtfully. "Send word not to kill Alexander if they find him, but to bring him to us alive. If we can prove to the Greek armies that we have their leader—that he is not leading their troops in his golden armor—their morale will plummet and their ranks will be ours."
"Yes, my liege," Darius agrees. "And we must send word to the Ninth and Twelfth Cohorts to surround the Greeks on the road. They have a large force, but they are on our soil, and when we collect our forces together, we will have more. We can crush them before they get too much farther into our territory."
Artaxerxes smiles. "No, Darius. We will let them come deep into Persia. Too deep to turn around and flee. Let them come east. Our great empire will wrap them in its snare like a cobra, squeezing and squeezing until every last one of them falls prey, and we are fattened on the taste of Greek meat."
Darius fumes silently. Why let the enemy come so far when they could stop them near the coast? "Yes, sire," he says smoothly. "But I would also like to suggest we send an advance force to lay waste the land through which the Greeks march. Prevent them from foraging for food, water, and wood. Poison the wells. Evacuate the villages and burn them. Destroy the crops and orchards. These Greeks may be much stronger than we imagine, and we will weaken them through hunger and thirst. Even powerful wizards need food and water to survive."
Artaxerxes waves a hand, his rings glinting in the golden block of light pouring in the opening above. "The farmers and villagers will evacuate to the fortified cities, but we will not burn their homes or poison their wells. Do you know how long it takes to cleanse a poisoned well? As for food, I imagine the Greeks will pluck the harvest from the fields and the fruit from the trees as they pass, anyway. Let us not destroy the orchards entirely. Next year our people will need that harvest. How many years does it take for an olive tree to produce good fruit? Twelve?"
Anger pumps through Darius at the old man's stubbornness. The very survival of the empire is at stake and Artaxerxes hesitates because of olives. "Very well, my liege. But I have crafted a plan to surround the enemy with fire in battle. If their wizards use water, we can use fire."
"We need not concern ourselves with Greek wizards," the Great King replies wearily. "I have notified the Supreme Lord of the Aesarians in Nekrana about the purported magical Greek, and I expect him to move his men west to search for him. The Lords are already fanning out across the empire, desperately looking for even one powerful magic being as part of a plan to destroy the monsters. This supposed wizard will certainly interest them."
He takes another swig of wine from his glistening golden chalice studded with emeralds the size of robins' eggs. "That map you gave me, Darius," the old king says, "the one your Macedonian spy, Kadmus, sent you that gives the location of the Fountain of Youth, near which the monsters live..."
"Yes, my lord?"
"I sent a regiment there to investigate for ourselves, and it never returned."
Darius is not surprised. "We must hope the Aesarian Lords are able to stop the creatures, Great King. For, in the end, the Spirit Eaters are far more dangerous than the Greeks, who are, after all, only men."
Artaxerxes rubs his eyes. "Amazons. Spirit Eaters. Wizards. Invading Greeks. A missing king. It is too much for an old man like me."
Darius locks eyes with Bagoas. This is one point on which they both agree with him.
The king looks up. "Bagoas, I want a woman. Fetch me that Olbian redhead."
Bagoas raps sharply on the ceiling with his ivory walking stick, and the carriage rolls to a halt. As Darius remounts his horse, he realizes that he will have to move to the next part of his plan sooner than he thought.
King Artaxerxes imperils the entire empire with his hesitation. He is too old, too careful to rule a nation with so many disasters facing it at once.
And Darius's plan is not just for Persia, but also for himself. For two decades now, he has been waiting, plotting, and planning. The richest, largest empire in the history of the world is just a finger's breadth beyond his grasp. Only one thing stands between Darius and his magnificent destiny.
The beating heart of Artaxerxes.
CHAPTER TWENTY-FIVE
RAT
Rat trudges from the river into the camp, two heavy buckets of water suspended on either side of the yoke around his neck. Yuf, he knows, is somewhere nearby, watching him.
It was Sarina who decided Rat and Yuf would join the great Greek army Alexander was assembling to invade Persia. Rat and Yuf would work as laborers, would become familiar faces to the soldiers and the king's guards. They hired on in Athens and joined the Greek transport ships sailing east. After the victory near the coast, they have been marching through empty fields and vacant villages as people who are afraid of the army run and hide.
But it has been hard for Rat to get near King Alexander. Ever since he joined the army on the Persian coast, the king has appeared only in his golden armor for the long marches east. Rat cannot stab through armor. The rest of the time, the king spends in his tent mourning his father.
_Father._ Rat has the image of a burly, one-eyed man with a red-brown beard and a loud voice. But Father was never kind to Rat and always called him bad names because he wasn't smart like the others.
No, Rat is not sad that King Philip is dead.
But he will be very sad when King Alexander is dead.
Even if it means that it will be _his_ turn at being king. The Rat King. It has a kind of ring to it.
But Alexander always was nice to him...
Several times Rat tried to carry fresh water inside Alexander's tent, but the guards in front of it merely took the buckets from him and sent him on his way. The only ones allowed to enter the tent are Hephaestion, who was also nice to him, and another man they call Jacob.
But Yuf is getting frustrated at the delay. _You must kill the king soon_ , he says. He has threatened to kill Heracles, Rat's pet rat, when he gets back to Egypt. Rat left Heracles in the care of Sarina.
Today, Rat has a new idea. Sarina and the Egyptian physicians healed his mind to make it more like those of other boys, and now he can come up with smart ideas, too.
He passes the medical tent, where doctors treat not only battle wounds but also the stomach ailments and fevers of an army on the march, and around the pens of baaing, grunting livestock, where the butchers prepare meat for the men. The wooden yoke, even though it is padded, cuts into the sores on his neck, and he sets the buckets down a minute, then picks them up, and walks unsteadily past the blacksmiths, who hammer damaged armor back into shape, the black smoke from their portable forges heavy in the hot summer air. Going around a large tent, he stumbles into a merchant rolling a cart filled with weapons, capes, and boots for sale, and falls to his knees. He thanks the gods the buckets didn't tip over. The merchant curses him, but a pretty young laundress, sets down her basket of soiled tunics and helps him up with a kind word.
Finally, he approaches the royal tent of King Alexander and sets down his burdens, removing the yoke from around his bruised neck. One of the guards calls back through the closed tent flap, "Lord Hephaestion! Fresh water for the king!"
The guard turns back to Rat and says, with a dismissive flick of his hand, "Well, boy! Be off with you!" But Rat doesn't move.
The flaps part and Heph comes out, eyes on the buckets. Rat cries, "Heph! Don't you recognize me?" The guard pushes toward Rat with the shaft of his spear. "Get out of here, you little—"
Heph stays the guard with a hand on his shoulder, looks at Rat, and frowns. Rat realizes he must look very different. He is tanner, leaner, and taller than he was last year when he was kidnapped. He is also dirtier, his hair unkempt and stringy, his tunic ripped and soiled, a far cry from the pampered little prince of Pella.
"Arridheus?" Heph says, his eyes opening wide in disbelief.
Rat has to force himself to remember the name. It feels heavy and foreign, like a suit of armor he has not worn in years and has outgrown. "Yes! Yes, Heph! It's me!" Rat says at last. Just looking at Heph, he remembers so many good times in that other place where he always had enough to eat and his body wasn't sore and throbbing.
Heph puts his arms around Rat's narrow shoulders. "Arri, what happened to you? We looked for you for so long!"
"I was...taken by bad men," he gasps. "I got away. I heard Alexander was marching here and joined the army as a water boy. But I couldn't get near him. I tried so many times."
Heph stares at Rat. "Arri, I don't believe I have ever heard you talk so much at once."
Rat beams. "A priest healed my mind and tongue and made my body strong." He frowns. Has he said too much? And he shouldn't be happy. He has come here not to be happy, but to kill Alexander. "Can I see my brother?"
Heph frowns, as if something is wrong, and Rat wonders if he knows what Rat is planning. "Wait here, Arri. I will tell him we have found you, at last." He turns to the guard. "Borus, don't let this boy out of your sight. He is indeed the king's younger brother."
The guard looks at Rat with new respect. In the hot sunlight, Rat shivers.
He hears low voices in the tent and, after a time, Heph parts the flaps and leads Rat inside. Rat feels the dagger in his belt. Yuf makes him sharpen it every night.
The sun was so bright outside that inside he cannot see for a few moments. It is cooler, too, and his dirty bare feet feel a soft carpet beneath them. Alex stands at the far end of the tent, in his golden armor, his arms outstretched.
"Brother?" His voice sounds thin, muffled by the helmet.
Rat runs toward him and feels strong arms wrap around him.
It is so good to be in his brother's arms again. But how is he going to stab him in his golden armor? Then he sees the back of Alexander's neck, soft and vulnerable, below the helmet. Slowly, he takes the dagger from his belt and puts his arm around Alex's neck. He will push it in very hard and run.
He _has_ to do it.
But he cannot, for it feels so good to be hugged, to be loved again.
No, he _must_ do it. What will happen to Heracles, otherwise? He cannot let his pet down, or all the people who helped him, even if they were not very kind about it.
He positions the dagger just above the soft flesh of the neck, but he hesitates. He can't do this. He can't. Alex pulls away from him, and the moment is gone...but his helmet gets caught in Rat's arms. Rat's dagger falls, and the helmet falls...and a tumble of long golden-brown hair falls.
Rat stares in astonishment. This is not Alex. This is someone else.
Alex's friend Kat. Who was always nice to him and Heracles.
Kat looks stricken, and he watches her neck as she swallows. The neck he almost cut with his knife, which now lies useless on the floor of the tent, staring up at him like an accusation.
"Ah, Arri," she says, his real name sending a shiver through him, a current of some old feeling starting to reawaken. She bends to her knees. "Now you know our little secret."
But Heph pins Arri's arms behind his back so hard that pains shoot up into his shoulders. "You were holding a knife!"
"Heph, let him go!" Kat cries.
Rat feels the mean hands let him go. He hangs his head.
"Arri," Kat says, "you weren't trying to kill me, were you?"
"No," Rat says. Kat casts Heph an accusing glance. Then Rat adds, "I was trying to kill Alex."
* * *
It takes Rat a long time to tell them—and the man called Jacob, who joined them—about everything that happened to him. About the puppets he'd been given and how he'd also become one. He didn't know people could be puppets, too—that was a thing he'd learned in all this.
How he had been kidnapped on the road from Pella and taken to Egypt, how the priests had cured his mind, how Yuf had trained his body but also beaten him, and how Sarina had wanted him to kill Alex to save Egypt from more bad men and promised that one day he would be king and a kinder ruler than any of the others had been before him.
Heph, Kat, and Jacob talk a great deal about this, all at the same time, until it hurts Rat's head. Then Heph runs out of the tent, while Kat gives Rat roasted meat and bread. When Heph comes back, he says, "The man called Yuf has been arrested. We are questioning him."
"You are safe, Arri," Kat says.
"My name is Rat," he says. "I am not allowed to be Arri anymore. Yuf said so."
"You are Arridheus, Prince of Macedon," she says firmly, "Brother to King Alexander, who will be here soon. But you must tell no one that he is not already here. Do you understand?"
He nods, wondering if Yuf will come in the tent and beat them all. Though these two men look strong, Rat has the feeling that only Alex could save them from Yuf. "Where is Alex?" he asks.
"On very important business," Heph says. "Secret business. The army must think he is here, which is why Kat puts on his golden armor."
Rat nods and eats an olive. He hopes Alex will be here soon. Maybe Alex can get Heracles back. He misses his little rat, who never hid anything from him and never pretended to be anyone he wasn't.
"We can't trust him," the big man named Jacob says. "You say he was never right in the head. And those people twisted him. How do we know he will keep our secret? How do we know he won't kill us all?"
Heph pushes his hands through his hair. Rat looks between the two young men, both strong and broad and handsome, men who use all of the most impressive words and stand as though the world is theirs. Men that Rat will never grow to be like, no matter how hard he tries. Some things just aren't fated for all of us.
That was another thing he learned about in his time away. Fate.
"Agothacles, the blacksmith's assistant, is deaf and has a way with children. We will put Arri in his care when we are not looking out for him," Heph says.
"Heph," Kat interjects. She nods gently in Rat's direction. "There'll be others to watch for, surely?"
"Sarina and her clan don't know this one has failed," he says. "Eventually, someone will have to go to Egypt and disarm her one way or the other. But we can't worry about her now."
Rat tugs at Heph's sleeve. "When you see Sarina, will you get Heracles from her?"
Jacob asks, "Who is Heracles?"
"His rat," Heph and Kat say.
"Zeus help us," Jacob says. And Rat hopes that Zeus will.
That night, after a hot bath, Rat slips into a clean, soft-woven tunic and tumbles onto a soft feather mattress in Heph's tent. For a moment, he feels a shiver of fear, remembering that he failed in what he promised Sarina. But then he remembers that he did the right thing. So perhaps he is not a failure but a hero, after all. The one thing he has managed to learn about fate is this: you can never know what yours is.
So tomorrow, he will awaken as Arri. There is no more need to be anyone else, and that in itself is a wonderful thing.
CHAPTER TWENTY-SIX
ALEXANDER
Gordium doesn't sprawl outward, as most cities do, but upward, walls and towers curling around a massive gold-gray rock at its heart. As he approaches it, Alexander feels a profound sense of relief. The first part of his quest, at least, is nearly at an end. And he has two days left. Two days until the summer solstice. After that, no one will be able to defeat the Spirit Eaters, if what Patra spoke was true.
After many delays due to storms, last week his ship at last landed in Apasa, on the Persian coast. There he rented a horse at a posting station and set off on the Royal Road—a marvel of the modern world—sixteen hundred miles of hard-packed, well-maintained thoroughfare, wide enough for two carriages at once, with ferry crossings over rivers. He has spent every night in a comfortable room in posting stations, fortified hotels with baths, a tavern, and stables, situated every fifteen miles along the road. One day, when Alex becomes king, he would like to copy this system of moving goods, soldiers, and messages quickly and efficiently throughout his lands.
_When he becomes king._ According to tales he heard on the road, spread by royal couriers from the coast who ride like the wind, he might already _be_ king. Philip was dead, they whispered in posting station taverns, from poison, from apoplexy, from a knife wound. No, he was alive. The death had been that of his brother-in-law, King Xander of Epirus. Or Philip's new, young wife, poisoned by his older one. Though Alex questioned everyone he could, all he could ascertain was that someone important had died at the wedding, but the retelling of tales had jumbled it.
And then there were Patra's words to him before he left. _My king._ When he'd said he was not her king, her reply had been, _Not yet._
If Philip really is dead, all their work in assembling the allied Greek army would be for nothing. Only Alexander himself could lead them now.
But the summer solstice cannot be forestalled.
Looking at the city rising before him, he wonders about the object he needs to destroy the Spirit Eaters, the talisman of one who is above mortal fate. The fate of all mortals, he knows, is death. Who doesn't die? A god? A spirit of some sort? Where can he find this object? And what if he doesn't? Not only that, but the oracle said he required the aid of a stranger whose blood was destined to mingle with his. Will it be a woman with whom he will have a child? Will he even know when he has found her? He has looked carefully at every person on the road, in the posting stations, in the towns he visited, expecting some sign telling him, _That's the one._ But none has come.
He never imagined making this journey alone. Always he had envisioned Heph by his side. A year and a half ago, when he and Heph were studying at Aristotle's school in Mieza, they rode out exploring and found a cave with an altar of sorts at one end, and above it, gleaming from the light of their torches, a large painted eye with a bright blue iris. On the altar was an ancient vase, and inside was a scroll of thick parchment. Out in the sunshine, they unrolled it and discovered it was a map. The ink was faded, the language archaic, but they eventually deciphered it.
In the province of Asia in the Persian Empire was marked, _Fountain of Youth, Well of the Gods, Providing Physical Healing and Spiritual Power to All Who Drink of It._ As soon as Alex read about the fountain, he felt hope rise inside him, utter joy at the thought of healing his weak leg and becoming a perfect king for Macedon. Perfect people don't have physical disabilities, which are punishments given by the gods to those who deserve them—or so everyone says, and sometimes, what people think matters more than what is true.
Next to the fountain were the words _Spirit Eaters_ , though he had no idea then what it meant, and he and Heph had joked about it. They had planned to find the Fountain of Youth, using the bag of gold Heph would win at the Blood Tournament. But Heph, against all odds, hadn't won. A peasant named Jacob had won, and a girl named Katerina showed up at the palace, and suddenly all their plans changed. Heph should be by his side now, but the oracle prophesied that to defeat the Spirit Eaters, he must do so without his friends.
Alexander isn't used to putting this much trust in the unknown; he is a planner, a strategist, a deep but decisive thinker. And yet here he is, putting his journey into the hands of Fate itself, and the flimsy few words spoken by a young oracle who might have simply been choking on too much incense when she uttered the prophecy that he ought to come here. And yet, the prophecies about him, Kat, and Riel—even those that seemed to conflict—ended up being absolutely true. He can no longer shrug off oracular predictions. His very existence is a testament to their validity.
Truth be told, it isn't only saving mankind from the monsters that has driven him forward. There are private, personal reasons, too. For one, he wants to prove that he is not a weak boy with a crippled leg, but a man who is stronger _because_ of his weakness. For another, there is Heph himself—the space Alex needed from his friend, and from the confusion of intimacy that has been brewing in his chest, threatening to unravel his clear thinking and good sense.
And there's something else. It's as if, stripped of the magic that colored the way he saw the world all his life, everything seems starker and simpler to him now. There seems to be a knowing deep inside him of what to do. To trust the prophecy. To go east. To vanquish the Spirit Eaters.
* * *
He nudges his horse into a gallop and races toward the rocky citadel ahead. All around, as far as he can see, giant rocks rise from lush emerald plains and smoky blue-green woods. He passes a temple carved out of living stone, with stairs, columns, and a pediment—perhaps the tomb of an ancient king.
Just outside the city gates is a posting station, where he turns in his horse and receives his deposit. He approaches the steep, crowded city on foot. At the gate, two burly, black-bearded guards block his way.
"Your business?" asks the taller one, gruffly.
"I am on a pilgrimage to the Temple of Midas," he says in his best Persian.
The guard grins, revealing sharp white teeth. "There is no Temple of Midas here. In Gordium, the Greeks go to the Temple of Zeus to worship their gods. By your accent, that is where you should go, too."
The other guard sees Alexander's confusion and adds, "But there is the Arch of Midas up on the old citadel, by the knot. Perhaps that is what you seek."
"An arch?" Alex asks, hope fluttering in his chest, even though it is not what Patra told him to find. She said he must go to the _Temple_ of Midas.
"Yes," the first guard replies. "An arch put up many centuries ago by King Midas, next to the oxcart of his father, King Gordios, founder of this city. Perhaps you have heard of the famous Gordian knot?"
Alexander nods, remembering that Aristotle told him some men had lost their minds trying to solve the puzzle of the knot. They starved to death or were dragged away unconscious.
"It is a strange courtyard," the second guard says, stroking his black beard, "reportedly haunted. The townsfolk keep away from it, and it stands empty other than visitors who try to untangle the knot and children who dare each other to walk through the arch. I have done so many times, and believe me, it is just an arch." The two guards chuckle.
A cart groans to a halt behind him. Alex turns to see an oxcart filled with huge, bulbous amphorae. "What do you have there?" barks the first guard, moving toward it.
"Olive oil" comes the fatigued reply. The driver wipes his sweaty brow with a dirty piece of cloth and pulls out a wax tablet to show the guard. "My master already calculated the tax."
"Ardashan! Come and help!" shouts the first guard.
"You'd better go," the other guard says to Alex. "The arch you seek is in the Courtyard of the Winged Genies, off Leather Street, high up on the citadel. It's no temple—at least I don't think so—but you can pray there if you need to fulfill a vow."
Perplexed by the strange statements about the arch, Alexander passes through the city gates and begins to follow what seems to be the main street, curling upward toward the palace high above. The noon sun reflecting off the golden palace roof is blinding. He learned in his schooldays that everything inside the Midas palace of Gordium is covered with pure gold—tables, chairs, beds, floors, and walls.
Either the stories of Midas's magic touch are true, or he had even gaudier taste than Olympias.
Legend has it, he even turned his _daughter_ into gold—by mistake, it is said. It's a cautionary tale, though Aristotle argued there are too many men in this world who believe a daughter of gold more valuable than a living one.
The streets are crowded with pedestrians, ambling riders, and donkey carts of merchandise. Dozens of shops boast carpets, jewelry, furniture, and weapons—enameled, curved swords that dazzle as he passes by—and little marketplaces burst with booths offering cheese, wine, and olives.
The street opens onto a square, the Temple of Zeus at its far side, painted in blue and gold stripes. Like everything else he has seen in Midas City, it is small and centuries old, but beautiful. A pink-cheeked boy marches past rolling a hoop, and when Alex asks where Leather Street is, the boy points to a narrow lane off the square.
Alex walks past shops and booths displaying sandals, boots, belts, bridles, and pouches. It is stuffy on this tiny street, and perspiration trickles down the back of his neck and into his tunic. Finally, the street ends at a tall set of stairs, cracked and crumbling. Dangerous, but he climbs them. At the top he sees a small square, bounded by a wall on one side, on the others by abandoned buildings. All the walls are engraved with the figures of muscular winged gods in profile, their long beards plaited. Persian genies. In the center stands an arch.
And on the side, tied to a column in a colonnade along a once-grand building, stands a rotten oxcart, its front pole and yoke hidden by a thick tangle of knotted rope. It's a decrepit thing, useful only for a few minutes as fuel to warm those sitting around a bonfire. Is this really the famous Gordian Knot? He approaches it and runs his hand over the tight jumble of rope, feeling a spike of something that makes him withdraw his hand immediately. It's as if the ropes contain fear, frustration, anger, and sadness all balled together.
He steps back in shock. Then he squats down to examine it from below. He can't find an end, and even the knots have knots.
He turns his attention to the arch in the center and walks slowly around it. It is a doorway without a door: a triangular entablature over two flat pillars flanking empty space. The Arch of Midas. But not the Temple of Midas. He walks over to the edge of the little square and looks out over a winding street in the city below. The residents have beds, chairs, and tables set on the roofs of their houses to enjoy the fresh air outside of stuffy little rooms. On one roof directly below Alex, a man sweeps the floor. Next door, a woman sits carding wool under a striped canopy.
A cool breeze wafts up from the valley, washing away some of his heat but none of his frustration. He had expected something here. A mysterious woman, perhaps, holding out an ancient amulet of some forgotten god. If only he still had a vestige of his Snake Blood Magic, he would know what to do. But it is gone, sunk into nothingness with Riel, the father-god who had given it to him. Without it, he is nothing but a silly schoolboy sent on an adventure by the wild riddles of a girl in a trance. He's a man now, perhaps even a king. He should be with the Greek army doing a man's work, not out here on this fool's errand. He needs to get another horse and ride straight to the coast. He turns to go.
_No._
The voice is inside his head, but loud and clear. For a moment he panics, wondering if Riel is back—Riel, who used to sit in the rafters of his mind like a watchful crow, cawing out his comments from time to time.
_Stay here. You can do what needs to be done._
But it is not Riel. Nor is it Snake Blood Magic with its visions of others' lives. This is something else. Something new. Some deep part of him that he has never known.
What had Aristotle said? That men relying on magic lose the chance to test their own abilities, find their own strength.
_How?_ he asks himself. _How can I do what needs to be done?_
He feels a tug—he can't explain it as anything else—and turns around. It is as if the arch is pulling him toward it.
He slowly walks closer and studies the carved stone again. It is, as the guard said, a normal arch. Then why does he feel he must walk through it? He will find only himself on the other side, here in the hot, deserted square near the top of the citadel. Now the feeling is stronger than a tug. It is nothing less than a demand. He takes one look around him, up at the heartbreakingly beautiful canopy of cloudless sky, and over at the yellow cliffs and green hills. In the distance, a cloud of large black birds wings its way toward him.
Inhaling deeply, he steps through.
And into a temple.
Light from a hundred torches dances off golden columns and walls. The paving stones are covered with gold and edged with rubies. At the far end is an altar of sorts, though he cannot see the god honored there, because an old woman is standing in front of it, pouring libations of wine into bowls on a golden table.
Alexander looks behind him. There is no door to the outside, just a wall a few feet beyond the same stone arch. There are no windows. Panic races through his veins. Is he in a cave? Has he slipped into a different world? Is he trapped here? How can he ever get back out? On either side of him, in between tall glittering columns, are looms of solid gold instead of wood, with threads of gold instead of wool, and loom weights of gold-speckled rocks.
"What is this place?" he asks in Persian, as he approaches the altar.
The old woman turns. Alex sees stringy gray hair, a crooked toothless grin, and cloudy violet eyes.
"This temple," she says in Greek, in a voice that sounds like wooden wheels rolling over gravel, "was built by King Midas and is accessible only to those born with Blood Magic. Anyone else passing through the arch will simply walk out the other side." She turns back to the altar, feels for a bowl, and pours wine into it.
Now Alex understands why only a few individuals could enter the temple.
He has rarely, if ever, heard anyone speak so openly about Blood Magic.
"But my—" He stops himself from admitting the truth: his magic is gone.
He hears something like the repeated slamming of a door and realizes the crone is laughing. She turns to him again. "You will always have a kind of magic all your own." It's as though she has sensed the truth—or read his mind. Perhaps _she_ has Snake Blood.
"What did King Midas know about Blood Magics?" He approaches her cautiously.
The old woman goes back to pouring the libations. "Midas was an Earth Blood who used his abilities for his own gain, sniffing out minerals in the mountains and earth. After his death, the legend developed that everything he touched turned to gold, which was not quite true."
"And who, if I may ask, are you?" he asks. It is a relief to speak his native language again, to feel the words sliding from his tongue without worrying about tenses and grammar and sounds his mouth can't quite make.
She sets down her oenochoe as a smile lights her wrinkled face. For an instant Alex thinks he can see that she must have been very beautiful, a thousand years ago, when she was young.
"Merely a servant of Fate," she says, shrugging crooked shoulders. She seems like a bundle of ancient bones beneath her rough-spun gray robe. "Since fate decided you must come east, it was I who put the ancient map of the Fountain of Youth in that cave back in Mieza, for you to find. I have been waiting for you ever since."
Alex's heart judders and skips. _She put the map in the cave?_
He asks quietly, "Why? Why was I supposed to come east?"
She turns from him, arranges the bowls of wine, and bows to the figure above the altar. It is, he sees now, a winged baby holding a tablet. He has never heard of any god or goddess—Persian or Greek—taking such a form.
"It is your destiny, young king." She twists her head to look at him, her lips curling into a smile.
King? His heart sinks. Philip is dead.
"Yes," she says, her violet eyes shining strangely in the dim light. "Your father died the day you left on your quest. Before you were even out of the city."
Alex feels tears stinging his eyes. He last saw his father striding around the Epirote palace in excellent health, issuing orders about the royal wedding, bellowing with hearty laughter about dry, bloodless King Xander taking such a luscious young wife.
"How?" His voice is a ragged whisper.
"He received the kiss of death," she hisses.
"Mother," he says, but it is more a groan than a word. Olympias found a way to kill him. Only she would arrange for him to be killed with a kiss. At a wedding. He knew it. He knew she could never truly accept Philip's rejection of her for an obnoxious new wife half her age.
"The queen, yes, in such a clever way no one suspects her."
Alex's mind reels, memories overtaking him. When he was seven, Philip took him to the tallest tower in the Pellan palace on a cold, clear night and pointed out the stars, including that of Ares, god of war, the deity all soldiers must worship. When Alex shivered, the king took off his own thick cloak and gently wrapped him in it. It was a simple gesture, and yet, at that moment, Alex had never felt so convinced of his place in this world.
Philip had always been a larger-than-life figure, hurting him, irritating him, and, in his way, loving him. There is such a huge, empty space in Alex's world without him that Alex is at a loss at what to put there.
He should go back. Macedon is without a king. The Greek armies are without a leader. Urgency pulses through him like a hammer ringing on iron.
"Has the alliance fallen apart?" he asks. "What is happening with Macedon and the Greek allies?"
"You have good friends," she says in a singsong voice, "who have seen to everything as you would desire."
Heph, yes. Who probably sent for Kat as soon as he landed in the empire. Yet the Greek allies would never follow Heph and Kat into war against Persia.
"And yet they have," the old woman says, evidently reading his thoughts. "And won a great battle. Still, your twin is in terrible danger from enemies and allies alike."
Alex winces. He needs to be in two places at once. "I cannot go back until I defeat the Spirit Eaters," he says through gritted teeth.
"Are you _sure_ you want to do that, young king? After all, all those who have grappled with Fate have failed. And the Spirit Eaters are the children of Fate."
"What do you mean?" he asks, his head beginning to throb. "I don't understand."
The woman hobbles toward him. Her head comes only to the middle of his chest. She seems so thin and frail, and yet he senses a frightening power in her. If he grabbed that knobby wrist, she would probably fling him across the sanctuary.
"Midas's father, King Gordios, was also an Earth Blood," she says, staring not at him with those blind eyes, but _through_ him. "A descendant of the god Brehan. Unlike his son, Gordios cared nothing for wealth but wanted to battle the goddesses of Fate. Men and women, he said, should make their own fates, and not, like puppets, have fate determine the course of their lives. Gordios failed, and succeeded only in hopelessly tangling the strands that connect fate to mankind."
Alex frowns. "You mean those ropes on the oxcart out in the courtyard? He tied those there to prevent anyone from stealing the thing that gave him his throne," he says.
"In this case, the myth is not nearly as exciting as the truth," the woman says, giving him a sinister smile. "The knot came when he fought against the threads of Fate. He wanted to leave the palace and take the cart back to his farm, for he was far happier being a farmer than a king."
Alex's blood throbs with impatience. "What does that have to do with me?" he asks.
_"Everything."_ The deep voice rings throughout the temple, bouncing off golden walls, penetrating his mind like daggers.
The tiny figure before him smiles, blinks, and adjusts her gray head shawl with bent fingers. "You and Gordios have much in common, young king. You have already grappled with Fate. In the womb."
Alex just stares at her.
She reaches toward him and he steps back, alarmed for reasons he can't explain.
"The mark," she mutters, staring not at him but at his pants—at his leg.
At his scar. As though she can see right through the fabric to the scar that winds its way around his upper thigh.
"The mark," she says again.
"M-my twin's umbilical cord," he stammers, unsure why he has become so unnerved. "It was wrapped around my leg when we were born."
"A lie," she croaks.
"What?"
"That mark. It was caused by threads. The threads—"
He shivers, suddenly cold.
"You," she barks out, and he is rooted to the spot by the power of her low voice. "You wrestled with Fate. Before you were even born. You wrestled with the threads of Fate. In the womb. It damaged you, didn't it? Made it hard to walk like other men." Her words send an icy chill through him, seem to choke him from the inside.
"I did what?"
She ignores him and goes right on. "Why would you want to grapple with Fate again, boy?" she presses. "Go find your friends and resume your battles. You and I will meet again."
After the summer solstice, it will be too late. He can't lead the army against the Persians, he can't help Heph and Kat, until he is finished here. _Two days._
He wonders if he can get out of this place, with its windowless, doorless walls, wonders if he is trapped in here, considering that the stone arch is next to a wall. He has a sudden need to get out, to feel the sun on his face, the breeze in his hair. But first, he needs answers. "Where is the stranger whose blood will mingle with mine?"
"Close," the crone says. "Closer than you think!"
Anger pumps through his veins. He is tired of riddles. "Where is the talisman I need?" he asks, more loudly now.
"Closer than you think!" she repeats gleefully, her laughter echoing off the walls.
His head is spinning. She won't help him. And Philip is dead. Even though he heard the rumors along the road, he still can't really process what she has told him. That he is now king.
"And the Spirit Eaters?"
She laughs, her blind eyes rolling upward. Smiling, she pushes an oily lock of white hair away from her face and says, "When the time comes, you will find them...where all the knots of fate are tangled together."
That much, at least, is clear. "I thank you, servant of Fate," he says, turning toward the arch, urgency pumping through his veins now. He's got to get out of here or he will suffocate. He needs fresh air. He needs to think.
"Don't thank me yet, young king," she croons, her eerie laughter ricocheting off the golden walls.
He hurries under the stone arch in front of the rear wall...and onto the windswept little square at the high edge of the citadel.
Clouds have filled the sky. How long was he in the temple? There is something in the air that is...different. He walks to the wall and looks down into the little street below. What he sees is so unreal he turns away and rubs his eyes. Surely his time in the golden Temple of Midas has addled his mind. The news of becoming king. The strange words about fate and its threads.
He turns back. No, this is real. The furniture on the roofs below is scattered, broken, the mattresses and pillows shredded. The woman who carded wool is a heap of bloody rags, a hand, a foot. The man who swept is a skull with dark hair and some ribs.
_Spirit Eaters._
He reels, gripping the stone wall to prevent himself from collapsing. This is their work. It must be.
Heart in his throat, he dashes down the crumbling staircase. More bones and body parts. Doors and window shutters have been ripped off houses. Roofs and walls have been torn open. He stops to look through the gaping hole in the wall of a house. The furniture has been thrown around and broken, and the floor is smeared with blood.
Down and down the curling streets of the city he runs, frantic, in search of a single living soul...someone he can save, someone who can tell him what happened...
Until, nearly out of breath, he hears something behind him, soft as a whisper, chilling as an old wrinkled finger tracing up the back of his neck.
He swivels around.
No one's there.
A flash of darkness.
One of the creatures, left behind? Or something else...
A shadow.
A figure?
He could swear there's a human darting between the fallen rubble and remains.
He pivots again, but before he can see what has been following him, a sack goes over his head and a heavy weight tackles him to the ground, turning the whole world black.
CHAPTER TWENTY-SEVEN
ZOFIA
Zofia has a choice.
She could choose to believe that she is helpless. That all of the desperate decisions that have governed her life have only entangled her further in an unseen web, a preordained plan over which she has no control. She can choose to believe that Fate is a trap, a curse, a slow-unfolding terror into which she has birthed an innocent being, who now, too, will become inextricably entangled.
She can choose fear.
Or.
She can choose, instead, to believe in something else. She can choose to believe that her choices matter, that evil like the Spirit Eaters can be conquered.
That humans can prevail, even if the gods could not.
That love can take many forms, that it can surprise us again and again, and when we are at our lowest, offer us a hand and a way out.
Zo has chosen hope. It is the only way to go forward.
But as they've traveled the Royal Road, unreeling before her like the snake of fate, bringing back eerie memories as though from another life ago, though they are shockingly recent, it has been hard. She has clung close to Mandana, always keeping her strapped to her body, not just for the ease of nursing her while traveling, but because feeling that tiny heartbeat against her own is sometimes the only thing that keeps her from slipping into the dark thoughts and fears that threaten to overtake her at every stop, at every cruel memory.
But when the New Amazons had settled into their hidden camp this afternoon, not more than a mile from the Royal Road, the air was filled with terrible, wrenching screams. Zo's heart nearly stopped, and for a moment, she forgot to choose hope. Fear came rushing back in, cold and splintered. Ugly and strangling. Trembling, Zo emerged from her tent, holding the baby. Those who had been stacking firewood or feeding horses had stopped and looked fearfully at the sky.
For Zo, the screams—the sheer horror of them—were not unfamiliar. They brought back the terror of that day at the brothel...
The day the Spirit Eater descended.
The day they lost a sister to its fury, its unfathomable hunger, its reckless, violent power.
So when they turned to their leader, perhaps they should not have been surprised by his response. Stepping from his tent, alongside Alecta—who had taken up sleeping with the warrior king every night despite the censure she received from the other women for doing so—Pyrolithos scanned their camp, then the sky. Then he looked each woman in the eye, and Zo knew what he was going to say before he said it.
"It is a sign. We have come closer than we thought."
The Spirit Eaters had come again. Somewhere very near.
And so, in stark silence, they mounted their horses and rode straight toward the direction of the horrific sounds and blackened sky: a city rising high into the air, curled around a steep and massive gray rock.
The city of Gordium, famed, Zo knows, for once having been the home of King Midas, a center of uncontained greed and power. A place, too, known for drawing out the foolish pride of men who traveled there to try to prove their prowess by taking a turn at untying the Gordian knot, though none has ever succeeded.
_They'll probably keep trying until the end of time_ , Zo thinks. Men are all slaves to their pride, and it will forever be their weakness.
The same could be said of Pyrolithos, who leads their party from several horse lengths ahead. He does not look well of late, and Zo must wonder what illness is eating away at him. An illness, surely, of the heart and not of health. For he has been even more ruthless than anyone had imagined when they first set out.
They all have. They've had to be. Fierce and unforgiving. That is the whole point of their band. And yet. Pyrolithos has been more than ruthless—he has been blood hungry and vacant. And now that darkness seems to be taking its toll. He is less animated, narrower somehow, where he used to take up whole rooms with his presence. It occurs to Zo that Pyrolithos looks far too weak and ill to create any more diamond cages for the beasts. Yesterday, when she brought Mandana to calm the creature before the Amazons went to sleep, it seemed the once-glowing adamantine bars had turned to iron or even—in some places—smoke. She wondered whether the creature would break out and kill them all in their sleep.
Yes, some angry fire inside Pyrolithos is burning dangerously low, and that gives Zo almost as much trepidation as the thought of facing the Spirit Eaters.
* * *
When they arrive at the base of the city, though, it is clear the damage has been done, and they are too late. There is no time for relief in it: the women stare in shock and disgust at the disaster that lies before them, towering along the sides of the rock city, and Zo tightens the straps of her sack, pulling Mandana in closer.
It is as if a great fire has ravaged the place—a fire with teeth.
Heaps of bones—human and horse, as far as Zo can tell—litter the road near the gate. She shudders, looking at the tall city before them. No, it is no longer a city, vibrant with people living, working, buying, and trading goods. It is a tomb full of chewed bones.
Everything in her screams to protect her child, to flee. But if she fled, she'd be considered a traitor, and possibly beheaded if caught. She has little choice but to hold her ground, to refuse fear even as it rattles through her.
Alecta looks nervous, too. As for Pyrolithos, it's impossible to say.
"We have to get supplies, anyway," he mutters to Alecta, just loud enough for the others nearby to hear.
And it's true that they're running out of food, though the idea of pillaging this haunted, tortured place is sickening. Zo can't help but wonder if it's not their need for food but the king's desperation to find another Spirit Eater that drives him to dismount, stepping over the ripped-open corpses that must have once been the city guards, and shove open the gates himself.
All the women, including Zo, reluctantly follow, their backs straight, admitting no fear, though Zo knows they are feeling it as much as she is. Their horses, however, smell death and roll their heads in protest rather than enter the city. They struggle to turn around, neighing loudly as their hooves dance sideways. Only with difficulty are they forced forward, over piles of bones and puddles of blood at the city gate.
They ride up a street circling the outer city, past ravaged shops and homes, their nervous mounts stepping over body parts. The coppery scent of blood is nauseating. The silence is so loud that it hurts Zo's ears. There is no sign of life. No barking dog. No crowing chicken, bleating goat, or crying baby. And memories of the village she passed through in the Eastern Mountains flood back to her, causing her whole body to shake and baby Mandana to mewl softly against her chest.
Zo notices up ahead something shining on the street. A broken wooden casket has spilled its contents—dozens of thick gold coins—into the street in front of one business. Next to it is a skeletal arm.
Pyrolithos nods, sliding off his mount.
Alecta, too, dismounts, and the two scoop up the gold and throw it in their saddlebags. Then they enter the shop nearby—a moneylender's.
Zo slips off her horse and stands in the shadows, straining her ears for any indication of Spirit Eaters. But all she hears is the sounds of Pyrolithos and Alecta throwing furniture around in their search for loot.
The other women boldly disperse, after more of the same.
Zo doesn't want gold. Or wine. Or food. She wants to leave this place with her baby. The revolting smell of blood and organs on hot stone rises in the humid air. Then, for the first time in this city of death, she hears some sign of life: the buzzing of flies that have come to drink dead blood and devour dead flesh. She puts her hands over her mouth, willing herself not to be sick.
She moves apart from the others, trying to get a view from one of the city walls, a reminder that the rest of the world is still thriving, that there is green and hope and life beyond this terrible place.
She takes several turns down abandoned roads when she hears something else. She stops abruptly.
Men's voices, loud and arguing, coming from around the corner. Survivors! Her heart leaps with hope and she kicks the sides of her horse.
But as she trots closer, a dangerous feeling winds up her throat. What kind of men might they be, and why are they arguing so violently? More looters, probably. Like them. And potentially just as lethal.
She should call for reinforcements, prepare to attack.
First, she needs to know for certain. She cannot stand the loss of more innocent lives on her hands.
She ties her horse to a post and looks for a place to hide Mandana, who has drifted to sleep in the eerie quiet. Across the narrow, curved street is a destroyed shop, whose mattresses and pillows for sale have been torn to shreds. Silently, she removes the baby from her back and sets her down on a soft heap of feathers. Mandana moans softly, yawns, and falls back to sleep.
Staring at her for a moment, Zo considers whether it is safer to carry her. No, she'll be fine here just for a moment or two.
Zo darts back out of the shop and creeps closer to the men, flattening herself against the corner of a stone house, and holds her breath.
"On what grounds?" a man says in Greek-accented Persian. "Unhand me! Do you even know who I am?"
"We know who ya' are," replies a deeper voice, a Persian voice. "There's not many who look like ya' in this land."
A higher, edgier voice joins in. "We been trailing you for two days. Ran away from your army, did ya? Left a girl to take your place in battle."
"What?" the captured man says, sounding truly flustered. "I know not of what you speak."
"You've a funny way a running your country," says the first man. "Like no king I ever heard of. First, you kill your father. Then, you put a girl in charge of your pathetic little mishmash army."
Zo wonders which king could be the soldiers' prisoner. There are dozens of Greek kings. Every island has one.
"I've done no such thing. If I am a political prisoner, then I demand you take me to the Great King."
One of them laughs wickedly. "Oh, we don't work for the Great King—and yer hardly in a position to demand anything of us."
"Ain't you heard of us, then?" says another, snickering.
The first shoves him and the man lets out a grunt.
"What? He don't even realize he's reckoning with Assassins, does he? Some king."
"Shut it," says the first man.
Zo doesn't hear the rest because, with a gasp, she steps back and away from the wall.
_Assassins._ The word wrenches through her with a slicing fear. The sinister group of murderers who have tried for the past year to kill her: unbelievably, they are here, in this remote, devastated city where everyone else is dead.
Another trick of fate, coiling itself ever tighter around her neck.
Steadying the tremor in her fingers, she removes her blow dart tube and three poisoned darts from the pouch at her belt. Two of the darts are tipped with deadly nightshade, the poison she and Badri made that kills almost instantaneously. The other is laced with scorpion venom, a toxin that paralyzes the limbs and relaxes the mind for several minutes before killing, often causing its victim to tell truths they might otherwise have wanted to hide.
Holding her breath, she tiptoes back to the wall and peers around the corner. Three men dressed like civilians—two Persians and an Ethiopian from the looks of them—are holding a handsome young blond man. One of the Persians has tied the Greek's hands and holds a long length of rope. The others hold curved daggers.
She inserts the truth-telling dart into the tube, aims, and blows hard. The tiny winged dart strikes exactly where she wanted it to—in the neck of the Persian with the weapon.
He slaps at it as if it were a stinging insect, pushing it more deeply into his neck. He drops the dagger, pulls out the dart, looks at it, and screams. Panicked, he flails his arms, sending his rust-colored cloak to the ground as he topples beside it. The Persian holding the prisoner's ropes drops them, pulls out his dagger, and races toward Zo, along with the Ethiopian. She has readied the second dart, the killing one. She blows hard, and it sticks in the cheek of the Persian. He shrieks, pulls it out, and keeps coming for her. Behind him, she sees the blond prisoner pick up the first man's fallen cloak with his bound hands and throw it over the Ethiopian from behind, tackling him to the ground. Straddling his neck, he wrenches it hard with a sickening snap.
The man coming toward her is more than twice her size, with murder in his dark eyes. His face is beet red with rage. She blows a second dart at him; it hits him in the shoulder. He raises his dagger—it seems like it is in slow motion—and she wonders how long before the poison fells him. Too long. She turns to run, feels the tug of his knife on the loose material at the back of her robe, hears the sickening rip of linen, and feels the soft air on exposed skin. She is going to die.
A thud resounds behind her. She can't stop running. She needs to... At the top of the tiny street, she realizes there are no footsteps behind her. She stops, turns, and sees her pursuer flat on his face, a dagger in his back. The young blond king has his foot on the man's shoulder. He looks at her, pulls out the dagger, and wipes off the blood on the dead man's tunic.
"I won't hurt you, you know," he calls to her, in Greek-accented Persian.
Slowly, still breathing heavily, she approaches him. When their eyes meet, she realizes that his are unusual, one sky blue and the other dark brown, which reminds her of...
But no, that couldn't be. Though, he would be about the right age, seventeen or eighteen, and athletic. His wavy hair is the color of sunshine, his skin a golden tan.
Silently, she cuts the ropes binding his wrists, marveling at how well he fought bound. What could he do with both hands free? "I don't know what a beautiful girl like you is doing here alone," he says, "armed with poison darts in this city of bones, but I am lucky you are."
Averting her gaze from his, she kneels beside the Assassins she shot with the truth-telling dart. The Greek comes over and helps roll the man over. He is still breathing.
"Hey!" She shakes him by the shoulders. "I have questions."
"Huh?" he asks sleepily, his eyes shut. "I will never reveal..."
"You will answer me," she insists, knowing he will not be able to fight the poison coursing through his veins. He will tell the truth before he dies. Badri, mistress of poisons, said they all do.
"Tell me the location of Ochus, the Chief Assassin's son," she says.
The man laughs feebly, his eyes still tightly closed. "Ochus," he mutters in disgust. "No one knows where the traitor is. He disappeared after killing many of his brothers in Macedon."
Kneeling on the other side of the Assassin, the Greek tenses at this. His strange eyes meet hers with questions. She refocuses on her prisoner. "What brothers did he kill in Macedon?"
The man smiles. Clearly the poison is having the desired effect: that of intoxication. "When the Persian princess was sent to Macedon," he says, slurring his words, "Ochus was supposed to kill her and make it seem as if Alexander did it. To start a war. We wanted war with Macedon, not a marriage alliance," he says and stops. Zo wonders if he has fallen asleep.
She shakes him again. "Yes?" she says. "Then what?"
"But his father, Darius, didn't trust him," the man continues, his voice sleepy and hoarse. "He had Assassins among the Persian guards that escorted her. Even among the Macedonian guards at the..." He is drifting off. "At the palace." He falls silent and, a few moments later, starts to snore.
She slaps him. "What did Ochus do to them?" Though she suspects she already knows.
"Killed 'em," he murmurs.
Zo suddenly understands what happened that horrible night when she lost both men she loved. Ochus didn't kill Cosmas. The Assassins did.
In retaliation—and to protect _her_ —Ochus killed them all.
Ochus didn't abandon her on a whim. He had to disappear beyond the reach of the Assassins, including the most powerful one of all, his own father.
If what she suspects is true, it means that Ochus was acting out of loyalty. _To her._ He really did love her. The knowledge is like a sudden spring after the most bitter winter—a crisp hope inflates her lungs.
The Assassin is snoring loudly again. She pinches his nostrils and puts a hand over his open mouth. He winces and shakes his head. "What are your instructions now?" she asks. "Why were you arresting this man?"
He purses his lips. "Instructions...in the hidden pocket...of my cloak."
The young Greek king helps her roll the man off his long brown cloak, and together they examine it, but they cannot find a pocket.
"Where?" the prisoner says.
"Button." The man gasps. "On the...back."
The Greek holds up the cloak, and Zo sees a decorative button on the pointed bottom of the back. The thought occurs to her that Ochus's cloak, which she left back at the camp, has one, too, which she thought was merely sewn on as a decoration. But when she looks more closely at the button, she realizes the cloth of the border has been slipped around the button. Slowly, she unbuttons it. And finds a tiny pocket. Inside is a scroll with Persian writing.
She unwinds it and reads aloud. _"'Find King Alexander of Macedon and bring him alive to the Chief Assassin. Blond. One blue eye, one brown eye. Slight limp. Snake birthmark on left thigh.'"_
She looks up, the truth of the situation dawning on her.
"Are you..."
He nods. "I am. And I will arrange for you to receive a royal reward, when I can."
So her hunch was right. The man standing before her is none other than Alexander of Macedon. The man she was supposed to marry last summer when she ran off to be with Cosmas instead. The man she hoped, last fall, to convince to come to Persia with her, to seek and destroy the Spirit Eaters.
"You are king now?" she asks, her voice a whisper.
He nods, sadness shining in his eyes. "I've only just learned of my father's death."
She gapes at his beauty. The golden hair and golden skin. The straight nose and square jaw. The perfection of his slender yet muscular body. If he does limp, she has not noticed it. And the mismatched eyes—which she thought of as a deformity—are arresting, hypnotizing. This is the creature she worked so hard to avoid marrying, this tall, strong, gloriously handsome person. She was enslaved by rogue slave traders, and nearly died from starvation and a fall from a Pegasus, all to avoid marrying... _this_.
She starts to laugh—a long, loud, uncontrolled bellow. Her belly aches, and she bends over it, unable to stop herself.
A kindly hand touches her arm. "Are you...all right?" he says.
"Yes." She wipes away tears. "Yes."
"I would like to know the name of my savior," he says, "so that in future I can suitably reward her."
"Zofia." She gasps for air. "Princess...Zofia of Sardis."
He tilts his head and blinks in confusion. "Surely you can't be... Are you the Persian girl I didn't want to marry?"
"I didn't want to marry you, either!" It comes out like a cry of glee. "Last summer, I ran away from my family in Sardis to avoid it. And when a soothsayer said it was fated for my blood to mingle with that of Prince Alexander of Macedon, I did everything I could to change that fate."
He inhales sharply. "What did she say exactly?"
"Just that—that my blood was destined to mingle with...with yours." She has stopped laughing as his face goes pale and serious.
"An oracle told me something, too, recently. That I was to accomplish something great, something unthinkable—but only with the help of a stranger _whose blood is destined to mingle with mine_."
She eyes him. "What thing? What great thing are you—are we—destined to do together, then?"
He pauses, as if trying to read her face. Finally, he opens his mouth and says something she never expected. "Defeat the Spirit Eaters."
They stare at each other in wonder. Then he says, "But you did come to Macedon to marry me, last fall."
"I wanted to tell you about the Spirit Eaters. I know where their lairs are in the Eastern Mountains. I've been there. I hoped you and I could lead an army there to destroy them."
"Have you seen them?"
"I—" Where to start? How to tell him?
"Yes, last year. Now I am with the New Amazons. Out of Dardania. Perhaps you have heard of them. Of _us_."
Alexander's eyes open wide in surprise. But before he can respond, a cry resonates through the air.
The baby.
Zo is already racing around the corner as panic slices through her. Mandana.
Wordlessly, she runs to the mattress shop.
Mandana has awakened, wriggled out of the blanket, and lies, clad only in her diaper, on a heap of feathers. As Zo starts to pick her up, she freezes, staring down at her child.
Coming up beside her, Alexander sucks in his breath. "By all the gods," he says.
"What?" Her voice comes out in a harsh whisper. Is he seeing what she's seeing?
"The baby...with wings."
Slowly, she nods. Mandana gurgles and coos. All the mattress feathers sticking out from behind her back look just like the image she saw on Kohinoor's cave wall—like _wings_.
"I saw a statue just like that in a temple I was in not an hour ago," the young king says.
"What temple?" Her voice has gone tense with horror.
"Here, in the city, but to most people it looks like an arch. Only Blood Magics can enter."
Zo wrenches her gaze from the baby and stares at him in wonder. "Are you a Blood Magic?" Of all the strange tales she heard of Alexander of Macedon, this was not one of them.
"I...was," he says. "But I still have something that allowed me to get in. At least, that's what the old woman who worked there said."
"What old woman?" she shrieks, her heart pounding. "What did she look like?" She bends down to pick up Mandana, letting the feathers fall away as she holds the baby to her chest. Fear and panic and _knowing_ rise in her like bile. She needs to get away from here. Far away.
"Ancient. Stringy gray hair. Cloudy purple eyes. Blind, I think. A voice like an oxcart rolling over gravel."
Now Zo's heart beats so fast she wonders if she is going to faint.
Kohinoor is here.
Kohinoor wants her baby.
The baby with wings.
"We have to get out of here. Now." She can hardly get her breath. As Zo quickly hurries out of the shop and around the corner, she explains over her shoulder, "That old woman's name is Kohinoor, and she has been stalking me for a year now. I don't know why, but she will do anything to get her hands on my baby." They must find the other Amazons. Surely the army of women would protect them from Kohinoor, from other Assassins tracking them.
"What do you mean she's been stalking you?" the king asks, following her. "If—"
"There's no time!" she cries, panic rising inside her. "We need to find the others!"
"The others..." he says slowly, hesitation in his voice.
"You cannot stay in this city of the dead," Zo says, making the decision on the spot. "We'll find the New Amazons. They're here, looting. We will find our leader, Pyrolithos. You will come back with us to our camp. It is less than a mile from here. And if we are truly fated to...to work together...then we must not be parted, not yet."
Alexander pauses, looking up at the highest part of the city. Then he lowers his gaze to Zo, and she feels the burning intensity of his mismatched eyes. A blush rises hot and red on her neck and cheeks as she stares at a crack in the ground.
"All right, then," he says at last. "Let's go."
CHAPTER TWENTY-EIGHT
CYNANE
Pyrolithos can feel the heat of Alecta's gaze on his skin, on the back of his neck, on his sweating, tensed muscles, as together they pick their way carefully through the abandoned Midas palace crowning the citadel of Gordium. But he doesn't turn around. Can't look her in the eyes. Not right now.
The other women have scattered throughout the ruined city, gathering and pillaging what they can. But the palace is eerily silent, save for his and Alecta's footsteps. The building itself is ancient, the rooms smaller and ceilings lower than any palaces built in the last two hundred years. Yet every surface is plated with thin sheets of hammered gold—almost blindingly so. Though he can appreciate a spectacle, Pyrolithos has never been one for gaudiness. Still, he can't help but feel temporarily dazzled by the patterns gleaming in the gold—here rosettes, keys, meanders, waves; there palmettes, ivy, hanging lotus buds. The furniture, too, is wrapped in gold and studded with emeralds, agates, and carnelians—in stark contrast to the singed bones that lie scattered about in pools of blood and bile.
He enters what must have been the king's bedchamber, brightly lit, now, by sunshine streaming in through the broken roof. At its center sits a golden pillared bed studded with amethysts and covered in dark silken sheets. Next to the bed stands a life-size golden statue of a girl of about eight years old. She is so realistic—including the look of surprise and pain on her pretty round face—that the question springs from his lips without pause. "Could it be her? The daughter Midas touched and..."
"Turned to gold," Alecta finishes. She walks around the statue slowly, a flicker of sadness playing across the angles of her face. "It must be. Though perhaps it is just a reminder and not really _her_."
What would it have been like for King Midas to know that he killed the person he loved most?
Pyrolithos's palms are sweating; he drags them across his pants, lets out a breath.
He hadn't _planned_ on doing this today.
He's been waiting for the right opportunity, as his strength subsides, and his body softens, and the bars of the Spirit Eater's cage weaken...all the while hoping it would never come, and yet knowing it had to come.
The leaden realization that today is the day causes his heart to stutter and his head to throb.
If Alecta can tell, she makes no show of fear. "Should we take her?" She sets a hand on the little golden girl's shoulder.
"No," he says quickly, putting a hand on Alecta's arm, then pulling it away again. He could not bear this reminder of what he will have done. "We have enough."
This is it. He must do it.
The ultimate betrayal, to obtain the ultimate strength. He has to keep his goals in mind. The promises he's made to the other women. If he takes the right steps, they could become the most powerful fighting force in the world.
Make the Aesarian Lords cower before them.
Before him.
And all it requires is a simple movement of his hand, the flash of his dagger.
Mere seconds. Glimmering silver. The heat and pulse of blood when it's first released from flesh. The look of shock that will fix in her eyes. The hurt. The knowing.
He has only been truly known once: by Audata. But she died a terrible, forgotten death.
And he is no longer a child, no longer a woman, no longer weak.
No—he won't think of his mother, won't let the memory sink him, drowning his will.
Love comes and goes or turns to hatred and disgust. Glory stays forever, even long after men die. A thousand years ago, the proud warrior Achilles was given a choice by the gods: a short life and everlasting glory, or a long, happy life, with friends and family, and ignominious obscurity after death. He chose the former.
Which is why we know his name.
"We are alone," Pyrolithos whispers. _And I will be more alone, soon, than I have ever been in my life._
"That we are," Alecta says, her voice carrying a current of emotion he finds hard to interpret. There is too much of it. He wonders, sometimes, how she can feel so much, so powerfully, and still stand.
How she can take it.
Her hand slips into his, pulses once. Then, gently, she pulls him away from the child statue. And though he knows exactly what he must do, what he _will_ do, he allows himself, one last time, to be tugged slowly, gently, downward.
The royal bed, laden in gold and silk, catches them as they fall.
* * *
Alecta's dark lashes curl from her pale cheek. He decides to remember her like this, warm and content, one lean arm flung out over her pillow. He stares at her a very long time, and she stares back.
Then slowly, quietly, he reaches over for his belt on the floor and takes out his dagger, honed to a deadly point. There will only be a little pain. He gazes at the gleaming iron. He tests it first on the pad of his thumb. A drop of red blooms there.
"What is it?" Alecta asks. Her fiery hair spills over the sheets as she sits up.
"Do you remember, along the way, those innocent people I killed?" he asks, his voice shaking. "The farmer in Paeonia, the waitress in Pessinus, the fisherman in Athens..."
She pulls a sheet up to cover herself.
"I am not cruel by nature. I wasn't born this way. But nor was I born with...power. Not like this. Not magic. Do you understand?"
She shakes her head, but by the flicker of fear in her eyes, the sudden stiffening of her muscles, he guesses she _does_ understand.
"I needed to do it," he whispers.
"No."
"Yes. To keep my powers. To stay like this—a man. To keep the Spirit Eater contained in its cage."
Alecta stares at him through narrowed eyes. "Then stop. Stop pretending to be someone you're not."
"And the Spirit Eaters?" Pyrolithos asks, his voice breaking with desperation as he turns to look her in the eyes again. "Without Smoke Blood, we will never be able to have an army. I could never control them."
"We are already formidable," Alecta says, her voice holding steady. Perhaps he was wrong—perhaps she is not afraid, only prepared to face him, to fight if necessary. The thought sends a shiver down his spine. Does she not fear him because she trusts him? Or because she knows exactly what he intends to do to her, and is still willing to stare him down, unblinking?
"We don't need them," she says, touching his arm. "The beasts. The creature can go. I would rather have you. _You._ "
Gently, he shakes off her grip. "You don't understand. All my life I wanted power. I wanted to lead. And I needed to be a man to do it. Now I have everything within my grasp. And everything to lose..."
He can do this. He is strong. He can be cruel when necessary. He has been practicing for as long as he can remember.
For a moment, she is silent.
"Very well, then," she says at last.
He turns again to look at her, to study her features, so glorious, all hard angles and mournful eyes and angry mouth and wild hair.
"If all you want in the world is this—is power—if you choose that over me, then you have made _my_ choice, too." Slowly, she lets the sheet drop away from her body, revealing herself to him.
He is overcome by the power of her body—not the desire that awakens inside him all over again, and not her fighting strength either, but the simple miracle of her existence taking form before him, so human, so unfathomable. His and yet not his. No one's.
Free.
All her life, Alecta was a prostitute. And yet she is the freest person he has ever met.
Her hand pulls at his—the one holding the dagger. "I choose," she says again.
Pyrolithos holds the knife so tightly that his knuckles turn white. His hand aches. Everything he has ever wanted is at his fingertips.
He remembers the lost, motherless little girl everyone laughed at for wanting to learn the use of arms. Of all her dreams of commanding an army and ruling a nation, dreams that could never come true.
How can he hesitate? How can he disappoint the girl he was, the girl who did everything she could, sacrificed everything, to be where he is now?
He forces himself to push the tip into the side of Alecta's neck, just over the large vein. A pearl of blood blooms on the pale skin. Alecta's full lips are parted. Her eyes stay focused on him, the calm flooding her features so profound that it shakes him.
With a growl, Pyrolithos flings the knife across the room.
It falls to the floor with a clatter.
He is shuddering uncontrollably. The magic rages up in him, hot and smoky and choking, craving more of itself. Hungry. So hungry. Never satisfied. His head hammers with a beating, relentless sound of screaming and tearing, as though the Spirit Eaters have found him and are sucking out his soul through his skull. The Smoke magic wants what it wants. He must betray. _He must betray._
No.
He must resist. He can resist. He _will_ resist.
He doesn't know how it happens, but somehow he is on the floor, curled in on himself, panting for breath, arms over his head to block out the glare of the setting sun through the windows, ricocheting off all that gold.
Gold, gold, gold.
Light, light, light.
But no. The fire, the smoke, the blackness, the hate, the hunger. These things keep rising up and demanding the last of his strength. He can't even pry open his eyes.
He understands now the truth: that if he doesn't betray Alecta, he has chosen to betray himself instead, to betray his magic. And the magic is angry. Unrelenting.
How much time passes, he cannot say—he is feverish, shaking, sweating, and sick.
Strong arms wrap around him. Maybe they have always been there.
"Shhh" is all she says. "Let it go. Let it all go. The hate and pain and anger."
How does she know? How _can_ she know? His mind is alive with torture. His body is dying from the recoil of the magic. He won't survive. He made a mistake and now it's too late. He is weak. He is nothing. He will surely die.
"I am here," her voice says, and a sob opens up inside of him, a well of sound and water and fear he knows will drown him—of sadness, loss, and hurt that has been held underground so long, swelling and falling, he had forgotten it was there.
Tears run down his face.
"I will never leave you," her voice says.
Her voice is a rope. Pulling him up out of the waters, out of the depths, out of the darkness, out of the hurt.
_I will never leave you._
In that moment, she is his mother and his lover. She is his own self, past, present, and future. She is everything. She is the only way forward, the only way up.
Her words, such simple ones, really, change everything. They cut through the smoke.
He can breathe again. Months—no, _years_ —of rage are draining out of him. He is weak, but now it is a good kind of weak. Tired. Spent. But cleaned out. Purged. Renewed.
Steadying himself, he revels in the release of tension as even his body begins to change. The convulsions are akin to what birth pains must feel like—racking and whole. And yet, euphoric. He is birthing himself anew.
Alecta gasps, pushing his sweaty hair out of his face.
"Cynane," she whispers.
And that's when he knows. He is no longer a he. He is no longer Pyrolithos.
He— _she_ —is Cynane again.
Her hands find their way to Alecta's shoulders, her fingers graze the cut where her knife barely punctured Alecta's skin, just at the base of her neck. She wants to weep. How could she have done that?
"It's all right," Alecta whispers.
And this time, when she kisses her, the kiss is a whole other kind of experience. Sweet and salt. Slow and intentional. Cynane sees now how touch isn't just a replacement for love, or even a trigger for its release. How, in that moment, touch _is_ love. They are forming it from nothing, like gods.
The miracle of it is so consuming that for a few seconds they don't notice the way the walls and floor thrum with vibration, as though the world itself is changing beneath them and because of them.
And then the rest of the New Amazons are pouring into the room, and the women pull apart. There are gasps and murmurs as the Amazons see what has become of their leader. Still in the clothing of Pyrolithos.
Now Cynane.
One by one, they kneel down before her.
Zo kneels, too, bringing her child down to the floor with her.
The person beside Zo is no woman warrior, though.
The last person standing is a man. And not just any man.
Cynane's brother.
Alexander.
CHAPTER TWENTY-NINE
HEPHAESTION
The forest here is different from the ones back home. The trees are scrubbier and more densely packed—and the scents: rich and cedary and so sweet. Everywhere, fat bees hover, abuzz with the warmth of coming summer. Hephaestion, too, is buzzing with something—urgency, or perhaps fear—fear of losing Kat. No, it's humiliation, or frustration, or perhaps just a cloying, syrupy impatience. Kat is so near and yet so far. It has been nine days since he asked her the question he could hardly bear to ask—nine days since he demanded the status of her feelings for him. Nine days since she once again told him nothing, gave him no assurance. Is he crazy to think he still has a chance?
He rode this far into the woods, at first, because he needed to be alone. Needed the cool of the forest shade and the time to clear his head, to escape the heat of her always watching him with those big, unreadable eyes. If she doesn't want him, so be it. But the torture of not knowing is too much.
Only once he was surrounded by the hum of so many low hives, throbbing and spinning with active bees, did the idea come to him: if Kat wants someone big and dopey and thoughtful like Jacob, Heph can do that. He can impress her, bring her back some of Cimmera's precious honey. There is a saying about honey, isn't there? That it's the best way to catch a fly.
Kat is no fly. And yet she is a thing he cannot catch. That much is true.
She is more like one of these bees. Innocent, hardworking, full of impossible sweetness, and only a sting of pain away.
The harder he tries _not_ to think of her, of her sweetness, of her sweet mouth, of all the parts of her that are infinitely, unbearably fragrant with the scent of her, the less he is able to stop himself.
He slaps away another stinging bee.
This idea to gather honeycombs came to him in a heady rush, as though he had tilted his head back and swallowed a rich, golden mead—it left him dizzy and elated. Something soft and sweet for her. Something to remind her he is more than just Alexander's right-hand man, more than a soldier, more than a fling. To show her he is steady and that she can trust him, that he's not going anywhere.
Of course, however, he _has_ gone somewhere. He has ridden so far from their camp, in fact, that it takes him the better part of the morning to find his way back, and by the time he does, his victory over the bees seems to have intoxicated him completely. In fact, he is light-headed with victory.
Something isn't right. He is _too_ light-headed, he realizes, as he dismounts, stumbling into Kat's—the "king's"—tent. The world tips around him, the basket falls from his arm, and he collapses onto all fours like a dog. He is sick—somehow his desire for her has driven him to madness. That must be it!
No, that's nonsense. He must have a fever. Heatstroke.
He was stung many times—could that be it?—but bee stings never bothered him before.
"Heph!" He feels her long hair tickle his face, and from deep within him, a demented laugh bursts out.
"He's drunk, the fool," Jacob growls. Heph hates Jacob, his too-broad shoulders, his too-wide smile.
"It's not yet noon. Heph doesn't drink unwatered wine in the morning," Kat says, her voice tight with worry.
Heph rolls onto his back, heavy and helpless.
"Then explain this," Jacob says.
Heph would like to explain it with his fists, but he can't seem to sit up.
A thin boy's voice cuts into the dimness of his mind. "Is he sick? Egyptian priests can heal him." Arridheus.
Cool hands touch his face. "Heph," Kat whispers. He half opens his eyes, and her face goes in and out of focus. "What happened?"
He gestures to the basket he dragged in with him, now on the floor. "Honeycombs. I was in the woods and found so many honeycombs. I brought you some, Alexander." He chuckles and closes his heavy eyelids.
"Don't touch it, Arridheus!" Kat commands.
A bee lifts out of the basket, hovering nearby. Heph shudders, suddenly cold.
"Hold on," Kat whispers, and then goes quiet.
Heph wants to know what she's doing but he cannot keep his eyes open or his mind working, and gradually his world goes dark.
* * *
When he rises into consciousness through the pounding of his head sometime later, orange light slants through the tent flaps and glints off Kat's golden breastplate. She sits beside him on her bed, dabbing his forehead with mint water. Outside, he hears men issue orders, horses whinny, carts creak and roll by. Is the army moving camp? At this hour?
"What happened?" he asks, his voice a raspy whisper.
"Our salvation, I think," she replies, "even though it comes at the cost of a splitting headache." A cup of cool water meets his lips. He drinks greedily.
"The honey..." he says, wiping his chin.
"Yes, the honey. It took me a while to sift through the bee's mind—hectic and...and frenzied. But then I saw what he was after. The pollen from a flower with a beautiful purple bloom."
Heph rises to his elbows, and the world swings heavily to one side. He slips back down onto his back and groans.
"I think their honey is highly intoxicating to humans." She wrings out the cloth in a basin. "Though the bees themselves have gotten used to it."
He opens his eyes again. He needs to get up, but somehow his body doesn't follow his instructions. Why is Kat talking about bees? They have to craft a strategy to battle the Persians camped just a few miles away. After the battle of the Granicus River, their army was left unmolested as they marched east along the Royal Road. And now, only half a day's march from the city of Gordium, their scouts have told them that the Third Persian Cohort waits just ahead, blocking their path.
A man's voice just outside the tent calls out. "We'll pack King Alexander's tent soon, Kaunos! We're just waiting for his signal. Bring the cart around."
"Are we breaking camp?" he croaks. "Moving ahead? I thought the Persians were blocking our path."
"We're retreating."
Retreating? His head throbs. "Why?"
"Listen, Heph. I instructed the bees to vacate the hives."
"Instructed?" He knows better than to question her.
"Coaxed. They will do it." There's confidence in her voice. Her magic has evolved, even since their time on Meninx.
"Jacob has already led the men to collect the honey. Hundreds of baskets, Heph. The entire Cimmerian forest was filled with hives."
"I know," he says, his throat dry as a husk. "But why gather the honey when it is obviously poisonous?"
"We gathered the honey _because_ it is poisonous," she says.
Of course. Now he feels like a bigger idiot than before.
"Not lethal, but highly intoxicating. As if you had drunk several oenochoes of wine."
"I can attest to that."
"So we're retreating a couple of miles, leaving behind half the tents, some of the treasure, and all the food..."
"Including the honey," he fills in.
"Exactly. The Persian army will think we became afraid and ran off. After they've gorged themselves on our leftovers, that's when we come in for attack." She tucks her helmet against her side and rises.
Heph forces himself to a seated position. His vision swirls, then steadies. Her shining green eyes bore into him. The last rays of the sun strike her golden-brown hair, setting it alight, setting his heart to pounding. "Beautiful," he says, appreciatively, though he is not sure if he is talking about Kat's plan or Kat herself or, most likely, both.
* * *
The next morning, as dawn brightens and warms the sky, Heph and the Greek army edge their horses back toward the very camp they vacated only yesterday. The sounds of laughter and singing echo from the site, long before it comes into view. Someone is banging a drum; others play flutes.
As the advance force approaches, he sees Persian soldiers, who have thrown off their armor, doing cartwheels and somersaults. Several dance wild jigs. Many have passed out among the tents. There must be hundreds of them acting like drunken buffoons at a wedding.
The honey has worked its magic.
Kat, in Alexander's golden armor, sits low on Bucephalus.
"What's wrong?" He tries to gentle his voice, as if approaching a skittish horse. "Don't you see? Your plan is working."
"Yes," she whispers, but there is no victory in her voice. Finally, she looks at him. "I thought coming here had cured me of...of this darkness. But now, seeing those men... Knowing what's going to happen... How can we fight them like this? It will be like slaughtering lambs in a pen. It's not fair somehow."
Heph frowns. What is she talking about? Wasn't this the whole point of their strategy?
"Yes...my lord," he says carefully. "But they would gladly slaughter _us_ if they could. This is, after all, a _war_."
Jacob, on the other side of Kat, looks at her with compassion in his eyes. "We must crush this army before Artaxerxes comes with his much larger one. If they join up, we will have no hope, and we will all be massacred. Your sadness—that...that darkness you speak of. It, too, will pass, Kat."
Heph doesn't understand what sadness Kat is talking about, nor why this girl who reveled in winning the Battle of Pellan Fields a year ago is suddenly afraid of hurting a fly. Finally, a deep sigh breaks from her as she nods and raises the battle flag. Behind them, the eerie notes of the salpinx rise, announcing imminent attack. A few of the Persian soldiers look up and laugh. Most do not even notice. And perhaps it is for the best. Heph wonders if he'd rather be boldly awake when he dies or half in a dream, already on the verge of surrendering to the endless sleep.
But before he can ponder the question, the command is sounded again and he urges his horse into motion, every thought burning away but one: _win_.
The Greek army thunders into the camp, spears outstretched, swords raised. Heph cleaves off the head of a Persian still dancing a jig. His horse crushes the bodies of countless slumbering soldiers. A soldier in baggy Persian trousers staggers out of a tent, trying to lift a spear, and Heph impales him with his own. It's a bloodbath, a massacre. Heph's arms blaze with heat and power and movement. He is all weapon, all focus.
Several dozen soldiers race toward him now from the other side of the camp, weapons at the ready; clearly, these men must not have come near the mad honey. But they are outnumbered by the Greeks, who ride systematically through the camp, killing every Persian they meet. Hundreds flee into the woods, and the Greeks gleefully hunt them down and butcher them as if they were stags.
* * *
As Kat, Heph, and Jacob return to their camp, General Chares, the Athenian commander, strides out of his tent and blocks their path, spooking Bucephalus, who whickers angrily and bucks.
"I demand a word with King Alexander," the general says. Under thick black brows, his bright blue eyes flash in a way that Heph finds unsettling. "We must discuss the equitable distribution of spoils from the Persian loot. The Boeotians have taken more than their share of gold."
"Speak, then," Heph says.
"In private." Chares grins. "Or is there a problem with that?"
"You know the king has taken a vow to mourn his father..."
"Has he now?" Chares asks, taking a step closer. "What mourning vow would prevent us from getting a glimpse of his face?"
"We will discuss this later," Jacob says gruffly, allowing his horse to take a few steps toward the general. "Move out of the way, Chares. We are exhausted from the battle."
"I say I want Alexander to take off his helmet now!" Chares pushes toward them. "Because if this man is not the king, then _I_ will take command of this army. And _I_ will distribute all the rich booty we have won." Greed and ambition gleam in the Athenian's eyes as he slowly unsheathes his sword.
Heph's jaw clenches and his muscles, exhausted from battle, go hard and ready. The moment has come, the challenge they knew they'd eventually meet.
"Chares," he says calmly. "Step. Back."
But just then a man pushes his way through the crowd, crying, "King Alexander! Another regiment is coming!"
Heph whips his horse around to face the man. The men they just slaughtered, he knows, were an advance brigade of the army of Great King Artaxerxes, a much larger force.
It's impossible the full Persian army—and the Great King himself—have already reached them. His mind races. He will have to line up the infantry...
"No, my lord! Women!"
The crowd parts as he, Kat, and Heph ride to the other side of the camp, their men following. A line of Greek archers, arrows nocked, stands poised to shoot at a group of about fifty well-armed, mounted women surrounding a cart bearing a howling black beast in some sort of dully gleaming cage. A chill goes up Heph's spine. These must be the New Amazons, a group of ruthless, highly skilled female warriors he has heard about. Have they allied themselves with the Persians?
The long-limbed, black-haired woman who appears to be in charge slides off her horse, as does another tall, athletic woman beside her. Their faces are hidden by the nose and cheek plates of their helmets, but the woman beside the leader has a few locks of pale, wavy blond hair escaping her helmet. Beside Heph, Bucephalus strains against Kat's reins, nickering loudly, twisting his huge head and pawing the ground. It is all Kat can do to keep him from bolting toward the visitors.
There is something strangely familiar about both women. Their posture, perhaps. The way their bodies are knit together. The way they ride a horse.
The leader removes her helmet. Her dark eyes sparkle with merriment as her handsome face slides into a wide, white grin.
Despite all he's seen in his life, Heph's breath catches for a moment.
It's Cynane. Once his lover. The girl who tried to kill Kat and succeeded in cutting off her fingertip. The girl who acquired strange and horrifying powers when married to the mad king of Dardania, before killing the lunatic and escaping.
Seeing her now, tall and tan and smiling, Heph's heart jumps, though he has no idea if it is fear or joy at such an unlikely reunion. As trustworthy as a snake, he must remind himself.
He must keep her away from Kat at all costs.
"Greetings, Lord Hephaestion." Cynane strides up to him, spreading her arms wide to show she bears no weapon in her hands. There is something different about her. She beams with a deeply rooted confidence he has never seen before and has certainly never felt himself. She almost glows with it.
"I, General Cynane of the New Amazons, Princess of Macedon and Queen of Dardania, have just learned of your presence here and have come to offer the Greek allies my military support." She looks at Kat in Alex's armor, grins again. "Dear brother, I would recognize you even without your armor."
The Greek commanders nearby murmur to one another.
"I would like a word with my brother," Cynane says.
"I think...not," Heph replies archly. "Say what you have to say here, in front of us all." He gestures to the well-armed men around him.
Cynane shakes her head. "The information I have is for the ears of King Alexander and his most trusted advisor, Lord Hephaestion, only."
Kat turns to Heph and nods. "Let's hear what she has to say. If these Amazons are only a fraction as ferocious as reports indicate, they could be an excellent addition to our army."
Heph leans in. "She tried to kill you. Betrayal was her weapon of choice even before she delved into that pit of darkness."
Kat considers this and studies her golden fingertip. "I sense that there is a little magic there, but not much. And if she wished to disappear and kill us, she could have attempted so before now."
Heph can't argue with her logic, but he doesn't like this. He can't forget how Cynane drugged his wine and seduced him in Egypt, then raced to Kat's room to murder her. With Smoke Blood powers, however weak they might be, she must be infinitely more dangerous.
And it's not just her newly dark magic—it's the power she has always had over him, to make him lose himself, to make him forgo his honor with the arch of an eyebrow.
So it is against his will that Heph turns to Cynane and says through gritted teeth, "All right. In the king's tent. With one more of his advisors, Lord Jacob." He eyes the other women. "You will come alone."
"I must bring my lieutenant." She gestures to the oddly familiar blonde woman beside her. The woman is tall and muscular. She walks like a man.
Heph hesitates. That will be three of them, armed, and their guards right outside the door, against two unarmed Amazons. "All right. Just the two of you, then."
Moments later, they dismount in front of Alexander's tent, Bucephalus rearing and kicking as two grooms lead him away with difficulty. The women hand over their weapons and enter.
"Say what you have to say, then," Heph orders, hand on his sword, wary of any sudden move.
"First, let me introduce my lieutenant," Cynane says.
The blonde woman next to her removes her bronze helmet.
It is no woman at all.
Heph utters an embarrassing sound of shock.
Alexander. Safe. _Here._ He wants to throw his arms around him, but Kat has already thrown her helmet on the ground and leaped into her brother's arms. The two of them swing each other around as they laugh, and for a moment, the ground seems to shift beneath him and Heph feels as though he is standing on a boat, already setting sail away from the two people he cares most for in the world. But at least Kat will be happy now. Surely whatever darkness she was talking about will be gone.
Now Alex is in front of him, his hands on Heph's shoulders. "Dearest friend. I have much to thank you for. And we have much to do. Quickly."
"Welcome back, sire" is all he says.
CHAPTER THIRTY
DARIUS
An abandoned orchard is not, necessarily, a fruitless one. Even in a land believed to be cursed.
Decades ago, this area of Persia thrived, known for its figs—until a plague forced the farmers to leave. Fearing the spot to be ill-fated, no one has returned since. Now it is abandoned, overgrown. Darius pushes his way through the high summer grass as Bagoas, at his side, points out a variety of beautiful old trees that have somehow survived without tending, growing twisted and wild. Great King Artaxerxes nods calmly, all the while, as if the orchard had sprung up purely for his own pleasure.
"Ah, sire, that is a most excellent specimen!" Darius plays along, pointing to a leafy tree, wider than it is tall, its branches heavy with round dusty-purple figs. They trample a narrow path toward it as rows of bodyguards make their own tracks on either side. Smiling, the Great King wanders around the tree, searching for the most delectable fig, then plucks a large, glistening one. Darius and Bagoas twist off smaller ones and bite into them. Over their heads, the sun beats down. The trees are too stunted to cast much shade.
"Pity that we leave fig country behind tomorrow," Artaxerxes says, biting heartily into the fleshy pulp of the fruit. Juice dark as blood runs into his white beard.
Yes, today is the last day Darius and Bagoas can execute their carefully constructed plan. This is the last day of fig trees. And in a few days, this army will meet the Greek invaders for the first time. They must do so with fire and fury. Not with the plodding, old-fashioned strategy of an eighty-year-old king. The empire needs a new king—a younger, more ruthless king, willing to burn crops, poison wells, do whatever is necessary to repel the invaders, and then launch a punitive invasion of its own in Greek territory. A king who will make these barbarians feel the agony of Persian wrath.
A fat fly buzzes around his head, darting in and out of the branches of the nearby tree. Bagoas swallows, wiping his hairless chin with the back of his hand. "We will have some packed for you, sire, from this very tree." His tone is obsequious, as always. The king does not seem to notice any smugness in the man's smile.
Darius spits out some seeds and takes a second bite of his fig, savoring its sweetness. It tastes rich—like the future. "Four days until we meet up with the Third Cohort."
"Even if the Greek army attacks before we arrive," Artaxerxes says, throwing down his fig and twisting another large, shiny one from the tree, "the Third will keep them at bay."
They stroll around the abandoned orchard for a time, discussing their strategy, until Artaxerxes winces and grabs his stomach. "I must go back to camp," he says, striding purposefully forward. Sweat glistens on his tan forehead. Suddenly he stops, leans forward, and vomits. Then he falls on all fours, moaning like a wounded animal.
Bagoas's eyes lock on Darius.
"Call...my physician," the Great King says, his voice threaded with pain.
"I think—" Darius kneels beside him in the hot sun "—that there is nothing your physician can do, sire."
Artaxerxes turns his head, a knowing look in his glinting, obsidian eyes.
"How?" His voice has gone ragged. "How did you do it?"
"We knew your propensity to choose only the fattest fruit on the tree. Last night, our men crept out here and painted only the biggest figs with poison. We knew you wouldn't have your tasters test fruit on the tree."
The king grimaces and clutches his stomach, panting heavily.
"Don't worry," Darius says smoothly. "Persia will be safe in my hands. Safer, actually. Perhaps if you had only listened to my advice. We must meet the Greeks with deadly fire. But you refused. You are too old, my king. You have become soft. It is time to pass the scepter on to stronger hands."
Artaxerxes collapses completely, whimpering and writhing.
"Bagoas and I have already planned your funeral. A very grand one in Persepolis, once we have trounced the Greeks. We will build you a huge mausoleum."
The king whispers something through tightly clenched teeth.
"What?" Darius leans in. "I didn't hear that."
"The gods punish traitors and cowards," Artaxerxes manages to hiss out. "You have written your doom—and that of Persia."
He closes his eyes, shudders, and is still. Darius looks impassively at the face as finely carved as a cameo: the noble brow, the hooded eyes, the hooked nose, and high cheekbones.
As he turns, a pair of viselike hands grabs the neck of his robe and pulls him down with astonishing force for a dead man. "You...will...never...be...safe," Artaxerxes whispers, his face contorted. Darius tries to pull the hands away from him. He pries them loose and flings the king back. His dark eyes are open as air rattles out of his throat and his body goes limp.
Darius stands and claps the dirt from the front of his black robe.
Bagoas prostrates himself on the ground in front of Darius. "All hail the King of Kings, Darius of Persia!" The words echo all around them as men throughout the orchard—secret Assassins working as soldiers and servants—take up the refrain.
At last.
And now this land can burn.
Now he will show the Greeks what they are really made of.
Fire.
CHAPTER THIRTY-ONE
KATERINA
Images flash before her closed eyes. A lake of shimmering, magical water that doesn't slake her thirst but increases it, changing her human form over time. Wings black and wide as night. The glowing golden flesh of gods—fang torn. The devouring, all-consuming hunger of a canyon, an abyss that can never be filled. Animal fury, but human memories. And everywhere, ropes of light, harnessing her, yanking her, entrapping her in a massive unseen web.
"What do you see?" Cynane whispers from close beside her.
Kat flings open her eyes, and the sensations fly away, dissolving in the light.
They are at the rear of the Greek camp, guarded by a wide perimeter of Macedonian soldiers, spears at the ready. They believed Alexander when he said the Spirit Eater was a deadly weapon against the Persians, a sure means of victory, and completely under control.
Alex lied.
Jacob and Heph flank her sides, ever protective.
The sun has just risen, spreading a lovely rose-amethyst glow over the land. And before her, beyond the bars of its glimmering cage: the hard, dark eyes of the trapped Spirit Eater.
She can't answer Cynane. She's still overcome by the beast's ravaging hunger for magic blood, for her blood. She cannot understand what makes it seem to shift, to expand and contract as the light hits it. Sometimes, it looks like several creatures crammed into the diamond cage; a moment later, only one.
She shakes her head, unable to fully escape the murky pull of the creature's mind.
Jacob holds a vial of lotus blossom essence under her nose, and as she breathes in, her head begins to clear.
"It was human once," she says slowly.
"Did you sense anything that would help us to destroy the rest?" Alex's arms are around his little half brother, Arridheus, who stares in awe at the monster. "Today is the summer solstice, Kat. At sunset, we've run out of time."
"I'm not sure," she says slowly. "It was more like... I felt something. Ropes..."
"Ropes?" Cynane asks sharply.
"Threads. Something binding me. Invisible. But I could feel them."
"That hardly helps us," Cynane says quietly. "And I am getting weaker by the hour. I won't be able to contain it forever. And we can't exactly rely only on _her_." She nods toward the baby in the arms of the Persian princess.
Mandana is a beautiful baby with wide dark eyes, a fringe of black hair, and dimples. She reaches a chubby fist toward her mother, and Kat wonders what it is about the child that calms the Spirit Eater. Looking at the lovely girl who saved Alexander's life in Gordium, Zofia, Kat is struck once again by how strange the workings of fate can seem.
"There is only one thing to do." She sighs, steeling herself for what lies ahead. After all the sadness she suffered after the Spirit Eater's attack in Byzantium, what will this do to her? Will it sink her even deeper into horror and despair? Still, it's the only chance they have.
"I will become the Spirit Eater."
"Kat." Alex takes another step toward her.
She glances up at him. "It's the only way. I must enter the creature fully this time, so that I can control it. Cynane can dissolve the cage. Then I will guide it, and we will fly away, as far as I can, before..." She trails off, knowing how dangerous, how ill-advised the suggestion is. Her personal darkness aside, what if she isn't able to control or persuade it? Every creature is different in how it responds to her Snake magic.
Silence hangs heavily around her. "What if it won't release you, Kat?" Jacob puts a hand on her shoulder. "It's not...natural. Not like a falcon or a fish, or..."
She doesn't know what to say. He's right.
"It's our only choice," Cynane says, and her eyes trail to the cage itself. They are all aware of how dull and faded the diamond bars have become, no longer quite solid. It is as if they are becoming smoke and, with a gust of wind, will blow away, liberating the ravenous monster within.
Kat stands. "I can do this. I'm ready," she lies.
From the still place inside her, she slides once more into the creature's consciousness, before anyone can stop her.
Another onslaught of images. A deluge. A screaming, howling pain.
She is dizzy with it.
A frenzy of muscle and movement.
This body is only part body, part something else, made of spirit or smoke, or somehow cut from the fabric of evil itself.
And yet. Her mind is drawn toward something at the core of its being, something burning but solid, like an orb.
She imagines holding the glowing orb in her hand.
And suddenly, she can feel what has happened, can feel the transference click into place as her body becomes the creature's body, its darkness _her_ darkness, its torment hers, too. She can see through its eyes—the familiar faces of those she knows, grown distant and strange, cloaked in expressions of disgust or fear.
Her first feeling is that of sheer rage at all of them for keeping her cooped up in this too-small space for so long. Her second feeling is that of ravenous hunger for the magic blood so near, the blood that has been tantalizing her with its delicious smell. The blood of the warrior man-woman, who lassoed her and put her in this cage. The blood of the broad-shouldered warrior man, and the others nearby.
A force inside her commands her to tear open their soft bellies, to devour their organs, to taste the raw magic flesh, to make the blood her own. A roar rips through her, and the bars around her begin to shake. The worried faces around her retreat. Swords and spears point at her. She laughs, but it comes out as a howl.
A glimmer of Kat comes back to herself, enough to realize this joining is nothing like those she has had in the past with jellyfish, birds, whales, and scorpions, when she slipped inside unopposed, easily experiencing the creature and herself at once, merging together into one wish, allowing her to both guide and observe. This being is powerful beyond measure, perhaps impossible to control. She fights to dominate the roiling spirit of the creature, to assert herself as master. It is as if she is wrestling with a large black demon in a place of utter darkness, writhing and twisting for the upper hand. Her weapons are not fists and feet and teeth, but fear against love, despair against hope, beauty against horror. She is sprained and wrenched and clawed, her Kat-ness dripping out of her wounds like blood, and so exhausted it is tempting to just sink into the darkness and let the monster take control.
A mere breath of herself remains, but it is enough to know that she must fight. She focuses on all the love she has ever felt—for Helen, her true mother; for Jacob's generous family, who took her in; for Jacob himself; for Alex and Heph; for Aristotle and Ada of Caria. She calls on all that love to surround and protect her, spiritual armor against the ancient horror of the Spirit Eater.
And in the darkness of the creature's soul, something golden gleams, and the rage of the monster recedes—just enough.
Just enough.
She calms herself, aware that even now she is not fully in control. The creature's fury stirs just below the surface, and she senses an unendurable longing to be out of this prison, to be free, flying in the clean crisp air, searching for prey, since she knows the invading spirit inside her will not permit her to eat this prey standing around the cage.
The golden-haired male approaches her, staring intently. She cannot understand his words—Spirit Eaters long ago lost the use for such clumsy things—but in his tumble of sounds, she hears one thing over and over again. _Katerina? Katerina?_
The humans exchange more words, then each lights a torch from the campfire nearby. They reform in a circle around the cage, holding a spear or sword in one hand and a torch in the other. The black-haired man-woman closes her eyes and utters a deep sigh.
The darkness around her seems to melt and sigh, too, and then the shimmering bars of the cage begin to flicker, turning to dust, to smoke.
The cage is gone.
She is free.
Her brothers and sisters are nearby; she can feel them.
She flies.
It is unlike any flight she has ever experienced before. The flight of birds is calm and focused. It is as grounded as walking is for a human, or swimming for a fish. But this. This is something else altogether, as though the sky is folding and unfolding around her, as if all of time and space are not fixed things but morph as she moves through them, vulnerable, as all life is, to influence, to change.
Below her, treeless hills push up from the earth, and patches of woods, then a silver stream sparkling over plains of grass.
In the distance, she sees them: dozens, perhaps hundreds, of the others, wheeling over golden cliffs in a dark mass.
Far below, an enormous white-winged horse is tied to a pole with a thick rope.
Pegasus.
A magic creature that could sate the hunger for many years.
The Pegasus rears and bucks and whinnies, rolling its huge black eyes and pulling its lips back to show large white teeth.
Its fear makes her hungrier.
All around the horse gather men with horned helmets. The sight brings Kat to the fore as she pushes back down the hunger of the Spirit Eater. Because these are Aesarian Lords. They've come, most likely, from the headquarters of Nekrana to the east. Her glance lingers on the faces. One of the men has long iron gray hair and a helmet with golden horns, and she wonders if this is Gulzar, Supreme Lord and Commander. He is talking to a short Lord beside him.
Timaeus. She shudders. After he left the underground city of Troy, he must have sought out the Aesarian headquarters and joined a new regiment.
Her gaze shifts from Timaeus to the Pegasus. All around the animal, the men have positioned long metal tubes, engines of some sort, atop wheeled carts. Men pump enormous bellows attached to the end of each tube, and near the front is a cauldron of fiery pitch and sulfur.
Ah, so the Pegasus is a lure. She understands now. A lure for her, for all of them.
The little Lords are trying to outsmart the Spirit Eaters once and for all.
The Pegasus twists and shrieks as the monsters spiral down, ready to tear out its living flesh, and flames roar out of the tubes...
She inhales sharply.
She is lying on the ground, familiar faces gathered around. Zofia is massaging her temples while Alex holds the lotus scent beneath her nose.
Tears spring to her eyes.
She realizes her whole body is shaking. _Her_ body. She is Kat again.
She has shaken off the Spirit Eater.
And yet.
The darkness of it still tugs at her, a turbulence stirred up somewhere deep inside of her. Months ago, its bite had infected her with despair. And now her journey into the soul of the monster has magnified the sorrow. Black tendrils wrap around her heart, squeezing.
Thunder rumbles nearby. Raindrops hit her forehead, as if the gods are weeping for her, or perhaps trying to wash away the filth that has befouled her soul. But nothing can wash her clean.
"Where did you go?" Heph asks, kneeling beside her, caressing her hair.
She knows he means, where did she leave the Spirit Eater? Where did _it_ go? To the mountains. To the others. To the Pegasus—bait.
The Aesarians. Their plan.
Yes, she knows what he is asking. But all she can think is, _Where did I go?_
What happened to _her_?
No, she will never be Kat again. But there is a battle to fight.
* * *
Mere hours later, still weak and exhausted, Kat sits on the hill overlooking the plain of battle. Below her, the Greek forces are massed, tens of thousands of men on foot and horseback, archers and spear throwers and swordsmen. Every regiment has catapults on wheels. Facing them are tens of thousands of Persians, mounted and unmounted, with catapults of their own.
The three men she loves are down there: Alex, Heph, and Jacob, leading the Macedonian army. Alex was torn. He wanted to go back to Gordium, to the knot, the place where he was prophesied to defeat the Spirit Eaters on this—the last possible—day. Would Katerina take his place once more, he asked himself, to fight the Persians as she had already so ably done?
_No._
She can't stomach more bloodshed, more slaughter, see the light fade in the eyes of good young men on both sides, hear the screams of the horses, smell the blood. She thought she had shaken off the worst of this melancholy by pretending to be Alex, to help her twin by keeping the alliance together until he could join them. Then, seeing him, knowing he was safe and she could stop the nerve-rattling pretense, she was even happier.
But the journey she had just taken into the soul of the Spirit Eater brought all the darkness back and more. The dead and injured at the Battle of the Granicus River weigh heavily on her once more, but it's the unmitigated slaughter of honey-mad Persians at Cimmera that haunt her like vengeful ghosts. Even now she sees them dancing...and dying. She wonders if she can ever be happy again, knowing she did that.
Knowing the army would never fight without him, Alex grimly nodded and said only, "We will do what we must."
Even if Kat couldn't take Alex's place, there were unique and vital ways she could help him. She has already told them the detailed battle formation of the Persian army, gained from entering the mind of a kestrel and flying over the enemy lines. And she will continue to do so throughout the battle, waving flags from the hill to signal a change.
The rain, barely a sprinkle, left the air thick and misty, though the occasional wind from the valley below cools Kat's cheeks.
The ghostly moan of the salpinx rises over the battlefield, along with the echoes of men's cries as the armies race toward each other and clash. There is a scrum of warriors and horses, the flash of sunlight on swords and shields. She cannot see clearly what is happening, even from the hill. Down on the plain, the Greeks will be lost in the fog and chaos of war.
Calming her breath, she sinks into herself, transferring her thoughts to a raven flying along the side of the fighting. The bird was flying home to her young, a stomach full of worms to regurgitate into yawning beaks, but she relinquishes control easily to Kat, who flaps broad black wings to arc back over the battle. On either side of the melee, lines of Persian infantry are running around the Greeks.
She opens her eyes, rubs her forehead. "Zofia! Pincer movement. Blue flag."
Zofia picks up a blue pennant and waves it back and forth for several minutes.
Kat looks for the raven, but she is gone, home to her chicks. A sparrow is flying almost directly overhead, however, and she enters it, dipping lower to take a closer look at the flanking Persians, who seem to be kneeling.
Flames catch in the dry grass. A line of Persians has flanked the Greeks and are setting fires.
She snaps out of the sparrow, gasping.
"What is it?" Zofia asks, kneeling beside her.
"Red flag," Kat says. "Fire. Circling us from behind. We will be trapped."
Zo picks up the red flag and waves it in a semicircular motion, first one way, then the other, to indicate the fires being set around them. Suddenly she cries out and falls to her knees. An arrow has pinned her hand to the flagpole.
A Persian archer must have spotted them.
"We need to get back to the trees!" Kat pulls Zo away from the front of the hill as the wounded girl gasps and clutches the flagpole with her other hand. Arridheus, who had been watching Mandana in the safety of the trees, steps out of their shadows. "What is it?" he asks.
"Get back!" Kat cries. When they reach the trees, Kat yanks the arrow—and Zo's hand—from the pole as Zo shrieks. The tip has gone clean through the girl's palm. Kat takes out her dagger and chops the arrowhead off the shaft, then pulls the shaft out of Zo's hand. She wraps it in the bandage all soldiers keep rolled in a bag on their belts.
"Arri," she says to the wide-eyed boy, "look after her. Both of you, stay back here."
Throwing a shield over her back, Kat crawls to the front of the hill and sees, to her horror, flames licking a giant U shape around the Greeks, wafting waves of smoke over the embattled soldiers. She looks right. Left. Where is the Persian archer? Behind a rock on that hill over there? In that tree?
She sees movement below, three Greek soldiers climbing the steep path up the hill. "Watch out for archers!" she cries. One of them takes the shield from his back and holds it over them. Heph and Jacob clamber over the crest, supporting Alex.
"Get down," she says, momentarily grateful that the first clouds of black smoke hide them from the archer. She coughs—and then her heart stutters as she realizes Alex is bleeding profusely from a head wound. "Oh, great gods, what happened?"
"I'm all...right," he mumbles, his eyelids fluttering.
"A Persian sword came down hard on his helmet, splitting it in two," Heph says, examining the wound with deft fingers. Alex's pale golden hair has turned scarlet. "Cynane killed the Persian who did it. I think it's just a flesh wound, but he's disoriented. I thought Jacob could heal him if we had some peace and quiet, a place where he could concentrate."
"Drag him back to those trees," Kat commands. "And, Jacob," she says as he casts her a searching look, "help Zofia, too."
Sprawled out on her stomach, Kat studies a moving plain of men, fire, and smoke. Catapults lob flaming, tar-dipped stones through the air. Smoke obscures her view of the plain at times, then clears.
"Kat!" Zofia's voice reaches her through the din and smoke. Kat crawls between the trees. Zofia, cradling a fretful baby, holds up her blood-slick hand. "Jacob healed the wound," she says. "Alex's, too, but he's still dizzy."
Kat approaches Jacob, Heph, and Arridheus, who kneel beside a prone Alex.
"I'm really fine," Alex says woozily, his eyes closed. He puts his bloody hand on his bandaged head.
"I think that's the best I can do," Jacob explains. "It will take some time for the dizziness to fade. Right now, I should be dousing the fires."
"Yes." Alex sighs. "Douse the fires. We must save my men. And women."
Their shields slung over their backs, Jacob and Kat creep to the front of the hill. Now smoke billows across the battle below, wafting over the hill and making the two of them cough. "See that stream? I'm going to channel that water to the fires." She nods.
He lies absolutely still, but she can see the effort it takes by the set of his jaw and the lines in his forehead. Below the shifting clouds of smoke, she can see the water spreading toward the roaring fires. Some of the flames, those farthest away from the hill and nearest the stream, go out.
Jacob groans. "There's not enough water here."
Kat sees that now the stream is empty. But then she sees something else, something that strikes fear into her heart like a blade of ice. Black figures wheeling over the battlefield, diving down to pick up struggling soldiers, eating them in midair. Their sharp, piercing screams make her want to cover her ears.
Spirit Eaters.
They've come.
"My men," Alex says. He's crawled out to join them, as have the others. He's white as a sheet, his hair matted with clumps of blood.
Heph grabs his arm. "You couldn't make it through the fires at this point. Look, they have almost encircled this hill."
As Kat looks down, she sees two Greek soldiers leaping over flames to the hill, then scrambling up. Cynane's head appears first, followed by Alecta's.
"What's going on down there?" Alex asks, rubbing his forehead.
"Total chaos," Cynane says. "The Spirit Eaters have caused a rout. The Persians have fled, the Greeks and the New Amazons running after them as the fires have blocked their own retreat."
"Cynane," Jacob shouts urgently. "In the tunnels below Troy we saw Smoke Blood children taming flames and smoke. Can't you do that, even a little bit?"
"Don't you think I tried?"
Alecta puts a calming hand on her shoulder.
The wind shifts, and thick clouds of black smoke cover the hill, choking them.
"Get down!" Heph commands, coughing. "It'll be easier to breathe."
Jacob puts his arms around Kat as they lie flat on the ground. "I'm going to try again to draw out water from under the ground. There has to be some nearby. An underground spring, something." He closes his eyes. His jaw tightens. After a time, beads of sweat form on his forehead and a little vein throbs in his temple. But water does not flood the burning plains. All is smoke and the screams of men and wheeling black figures diving and shrieking over the battlefield. Now a baby's cries join the cacophony. Next to Kat, Zo arches over Mandana, shielding her from the smoke with her body.
"Jacob will put out the fire," Kat says, wishing she were more certain. "Don't worry. He'll..."
A piercing shriek rends the air, echoing over the plains below. It comes again and again, bloodcurdling, unbearable, as those on the hill put their hands over their ears. Is it a cry of pain? No, Kat realizes, seeing the black shapes take to the sky in a rain of blood, flapping away in the same direction, toward Gordium. The monsters are answering a call.
Just then, a dark winged shape flies over the hill, obscured by the thick clouds of smoke.
Cyn draws her sword, taking up a position next to Zo and Mandana, with Alecta right beside her. Kat and Heph also leap to their feet and draw their swords. Kat's eyes sting and water, and her throat is raw from the smoke. Alex tries to stand but is too weak. Heph bends to help him up.
The figure wheels around and around the smoky hilltop. Kat begins to sense it is not a Spirit Eater, not something hungry and evil. It is...
"Pegasus," Zo breathes before her. "She is never lost."
"What?" Kat shouts, as a huge winged horse thunders through smoke onto the hill and gallops in a circle until it stops, neighing angrily, its sides heaving with exertion.
The same creature Kat saw tied as bait for the monsters, but soot has blackened its hide, the feathers on its enormous wings are singed, and a nasty red burn oozes on its shoulder.
"Pegasus is never lost," Zo repeats, holding Mandana close to her chest. The phrase reminds Kat of something. The little girl she met in Troy—Roxana—had said the same thing.
Zo is approaching the restless creature, and now she puts out a tentative hand. "Vata," she says, as if calling it by name. And sure enough, the horse whickers a warning, then sniffs at her. Recognition enters its large dark eyes, and its breathing calms.
No one moves for a long moment as clouds of black smoke waft over the hilltop. Kat hears the crackle of flames now and looks over the side. Fire is licking its way up the dry brush on the hill, inching toward them.
When she looks back, the beast is kneeling, and Zo, tying Mandana firmly to her back, climbs on top. "Please understand," she says, "I have to save my baby."
Before any of them can respond, Heph drags Alex toward the creature. "He's wounded," he pants out. "The king is wounded."
Alex has barely been slung onto the back of the Pegasus before, neighing in anger, the creature flaps its wings. Through wisps of smoke, Kat sees Zo grab the burned length of rope around its neck and quickly wrap it around Alex's waist behind her. "Hold on to me!" she shouts, as the Pegasus bolts in circles around the hill, then leaps over the edge, into smoke and fire. Kat's heart skips a beat as the horse and its passengers sink, but then they rise, climbing above the smoke and into a clear blue sky.
But the smoke and heat of the flames still comes ever closer, and there is no path for the rest of them to make a run for it now.
They are trapped here, by a wall of fire.
"We should be fighting—like warriors—not waiting, like lambs for the slaughter." She turns to Alecta.
Alecta nods, then clasps Cynane's hand.
And then, together, chins down and shields raised, they run headlong—
Into the flames—
Only, instead of screams of agony, instead of disappearing into the thick smoke, something happens.
Kat gasps as the smoke seems to part, separating around the two women, forming a billowing black wall on either side.
Then it separates again, now into individual columns.
No, not columns. Enormous soldiers of smoke, three times the height of a man, all of them holding smoke shields and swords.
A small column of black smoke glides toward them, and it slowly forms into a man. A small, wiry man wearing a horned helmet.
Timaeus.
CHAPTER THIRTY-TWO
JACOB
Jacob immediately unsheathes his sword, though part of him wonders if this is some trick of the smoke, some fantasy borne of his exhaustion from trying to draw water.
"Ah, some old friends, I see." Tim walks up to Jacob and makes an exaggerated bow. "Lord Jacob. Savior of the children of Troy. Ah, and here we have his childhood love, Katerina."
He turns to Heph, who eyes him as if trying to remember where he saw him before. "Don't remember me, Lord Hephaestion? It was I who bested you in the Blood Tournament, tumbling out of your sword's path, hitting you in the head with a slingshot pebble. Oh, how the crowds roared with laughter to see a funny little acrobat from a flea-bitten fishing village beating Pella's noble champion!"
Comprehension dawns on Heph's face, but he says nothing.
"And here we have Princess Cynane. Pardon, my lady, _Queen_ Cynane," Timaeus continues, bowing before her. "Blamed for murdering her husband, the mad King Amyntas of Dardania, an honor that should really have gone to me."
"In that case," Cynane says, pulling herself up to her full height and looking down on him, "I owe you a debt of gratitude, Papari."
Jacob can see the anger ripple like a smear of oil across Tim's face. Before he can react, Jacob steps forward. "Why are you here, Timaeus?" he asks.
"Since you killed my entire regiment, I journeyed east to headquarters," he says, shrugging. "I am already rising high with the Supreme Lord. It was my idea to capture the Pegasus as bait for the Spirit Eaters. And I was the one to catch her as she was grazing. She never noticed the figure of smoke approaching her."
"But why are you _here_?" Jacob asks.
Tim grins. "You mean you really don't know?"
And then, of course, he does.
Tim wants another betrayal—a great one.
"Take me," Jacob says. If Timaeus takes the bait, at least the others might be spared.
"You can't trust him." Cynane takes a step toward Jacob. "No matter what promises he makes to us, he will break them."
Timaeus smiles. "She's right, of course. But you are all going to die up here, anyway. So it doesn't seem like you have much of a choice. Besides..." He glances sneeringly at the others. "I have no use for the rest of you. Or this little war of yours. Perhaps some of you would like to leave?"
Jacob looks at the towering clouds of choking black smoke taking the form of enormous warriors, and the racing flames charring the brush along the edges. "Kat..." he begins.
"I'm not leaving you." Kat has stepped up beside him, slipping her hand into his.
"Please." Heph reaches out toward her.
She shakes him off. "Don't touch me."
"This is no time to be a martyr. Even the enemy has fled the fire." Both enemies—the Persian army and the monsters.
"He's right." Jacob clears his throat, unable to look into Kat's eyes. For the first time ever, he and Heph completely agree on something. "Kat, go with him."
"No," Kat says, low and hard.
"You would choose to die with him rather than live with me." Heph's face is unreadable, smoky, ashen wind billowing his dark hair into his eyes, his mouth set firm.
She doesn't answer, but her grip on his palm tightens.
And for Jacob, that small signal is everything.
She _has_ chosen him.
The knowledge of it, that she has done it, has finally chosen him, here, at the end of all things, makes everything suddenly settle into clarity inside his chest. Resolve burns clear and bright, and he knows he can do anything—even vanquish an impossible enemy.
"A lover's quarrel," Tim says, eyeing them. "I won't wait forever. I don't need to, you see." He gestures to his army of smoke and darkness.
Heph says at last, "I won't leave you without a fight. Your brother would never forgive me."
For Heph, it will always really be about Alexander.
But Kat says, "Alex will always forgive you, Heph. I know it."
"But I won't. Like I said, we don't go down without a fight." Cynane's sword is drawn, and before Jacob can stop her, she attacks Tim.
He could turn to smoke and vanish, but he parries her blow, disappears only to reappear behind her.
Alecta joins in, and Tim releases his hold on the smoke army. They descend on the group below, embracing them with sooty arms. Smoke coils around Jacob, forcing itself up his nostrils, into his mouth. His eyes water and sting, and he can't breathe. Through the smoke he sees Tim fighting Cynane, their swords ringing out, then dancing over to Heph. Tim is enjoying this, like a cat toying with doomed mice.
Anger rises in Jacob, fury that his friend, one he loved and trusted, is doing this to him, Kat, and the others. The earth trembles with his anger, but Jacob channels it out of the earth and into the air. Gusts of wind scatter the forms of the smoke soldiers for a few moments, enough, at least, for everyone to cough out the soot and breathe clean air. He looks for Kat. She is nearby, her face blackened, gray ashes in her hair. She wipes at red runny eyes, then meets Jacob's gaze, and nods grimly. Tim is fighting Alecta, who stabs him in the shoulder, in the groin, her sword merely pushing through smoke as he laughs at her frustration.
Now the smoke returns to the shape of soldiers, black and churning. And Jacob realizes that his wind has merely fanned the flames burning the dry grasses around them. The conflagration is unbearable. Heat and flames pour forth from all around the little hilltop—a fire flung with deadly swords, a fire stoked by the breath of hate and rage. Heph is on his knees, coughing. Alecta has fainted. Cynane crawls on top of her, convulsing.
Wind. He needs the hilltop to become a vortex of wind to push the flames and smoke out and away, over the plains, so those here can breathe. Exhausted as he is, he must do this, or they will all die in the next few minutes. He thinks of Tim's betrayal of his friendship, of Olympias killing Jacob's family in Erissa, and fresh rage pulsates through his veins, out of his body, and into the air, swirling in rapid circles, pushing away the fiery fingers of flame, the poisonous clouds of black smoke, causing even the smoke soldiers to dissipate.
But one figure of smoke does not dissipate. Timaeus marches up to Jacob, fury in his pale eyes.
"Why are you doing this, Timaeus?" Jacob asks, gulping in the fresh air. This is only a respite, he knows. He is drained of energy, drained of Earth Blood powers. Tim is far stronger. Tim will kill them all. If only he can reach that part of Tim he used to know, his humor, loyalty, and good nature. Surely, that must still be there, somewhere, folded into all the smoke and darkness?
"Power" comes the response. "But I think you know that, already, don't you? You, who were so desperate for it yourself."
Jacob shudders. "Promise me you will let her live."
Tim's sword sparkles like crystal.
"Promise me."
Tim's sword plunges into Jacob's abdomen—tearing through flesh and muscle.
A searing pain.
Blood bubbles up, soaking his tunic below his breastplate. He drops his sword. Both hands clutch the gaping wound.
He would heal himself, would save himself, but all of his powers have gone to form the whirlwind that has kept the others alive and fighting.
"Jacob!" It is Kat's voice crying out to him through the darkness and chaos.
He can't see her. He tries to reach out but crumples to one side, sitting lopsided on his legs and arm, pressing one hand against the wound, slick with hot blood.
"Open your eyes, good friend. There is one last thing I want you to see."
Jacob tries to squint through the blaze of heat and choking black smoke. A Timaeus of smoke stands before him, holding Kat by the wrist. "You will see her die, slowly and painfully." Tim shoves her to the ground and conjures a ray of dazzling light.
No, not a ray, a rope. A rope made of fire and soot, which lashes out and twists around Kat's neck. Jacob wants to pick up his sword and save her—she is so very close—but as he rolls onto his knees, he falls forward, dizzy from blood loss. He cannot reach his sword. He cannot save her.
He cannot save her. The others are unconscious, or dead.
He can only turn his head and watch Timaeus choke the life out of her slowly as she gasps and writhes on the ground. With his last strength, Jacob reaches out with one hand toward the sword that lies between them. It seems so far, and his hand is so heavy. He picks it up, his hand trembling with the weight of it, and reaches toward Kat.
"No," she growls, trying to slip her fingers under the unforgiving rope around her neck. "I will not...let you...do this." She pulls the length of rope hard, bringing Timaeus close to her. He's clearly enjoying this, laughing in her face. "I battled a god last year," Kat says, between coughs. "My own father, and killed him. And believe me, you are no god!"
With her free hand, she takes the sword from Jacob and plunges it into the place where Timaeus's heart would be. Bright red blood explodes all over Kat, some of it splattering onto Jacob. The smoke face is a portrait of utter shock, eyes wide, mouth gaping. The smoke dissipates, all except the protruding eyes. They look at Jacob, and then they disappear, too.
And as the smoke of the man called Timaeus blows away, up into the sky, it sucks with it the smoke warriors surrounding them, and all the smoke and fire on the plains below.
A stunned silence follows.
A vortex of cold air swirls in circles, lifting the soot and flames higher and higher until they disappear into dark clouds.
Watching it, Jacob wonders if he is dead.
And then, it begins to rain.
He slides against wet, charred grass as someone drags him slowly—one step at a time—under the protective canopy of the trees, still smoking with soot and embers.
Soft hands are pulling at his tunic. He hears a gasp, then the ripping of cloth. Someone is stuffing bandages into the gaping wound as he grits his teeth.
Kat.
"Channel your Earth Blood now, Jacob. You are tired, wounded. But it is still there. You can do this."
"Kat," he says, his head spinning, "I don't think I can."
"You have to," she says, quiet and determined. "Don't give up! We have a life, a future together, children..."
He feels a smile play against his face as the images move through him, and the pain in his abdomen redoubles. His smile turns into a cough. His whole body shakes with the effort.
"It's true," she insists. "Because you were always right, from the very beginning. That last day in Erissa, when you kissed me in the pond, do you remember?"
He nods, though he can barely move his head. He could never forget. He finds the strength to open his eyes to look at her. There's not much more time to look at her. He doesn't want to waste it.
"And you asked me to marry you," she continues, her eyes shining in the rain, her hair, undone now, pressing damply against her face and neck. "I should have said yes. I should have begged you not to fight in the Blood Tournament. But I wasn't ready. I needed to..."
"Kill Queen Olympias," he finishes for her.
"For killing my mother," Kat says. "Little did I know, Olympias was my real mother. The woman I hated more than anyone in the world."
"I tried to kill her, too," Jacob says, closing eyelids that suddenly seem too heavy to hold up. "And yet, she lives."
"I don't even care anymore," Kat says, holding his hand tightly. "She can be queen of the entire world for all I care. The only thing I really care about now is you, Jacob. You always put me first, you were right about that."
"What about..."
She knows. "Hephaestion will always put Alex first. You were right about that, too. Alex needs him by his side to advise him and protect him. They have a love that can't be broken. And I need you by my side. Stay with me, Jacob. Don't leave me here alone."
It is getting so dark, Jacob notices. He had forced his eyes open a bit so that he could see the light for the last time, but there is very little.
"Kat," he whispers. "I—I don't know if I can." She is flickering, like Tim did, only it is not because of any Smoke magic. It is his own vision going, he knows. It is so hard to see, to think, to feel.
Her voice comes to him in the ebbing light. "We will return home, Jacob. To Erissa. Can you see it? Hunting and farming. We could build our own farmhouse, free from prophecies, kings, and battles. Jacob, can you see it?"
He does. He sees it all now. The courtyard of his family farm back in Erissa. He's confused, at first, because someone told him it had burned down. But here it is, just as it always has been. His little brother Calas chases a chicken around the well. He hears the clatter of pots and dishes as his mother prepares food inside the house. The mouthwatering smell of freshly baked bread mixes with the smoke from his father's kiln in the field beyond.
He is so happy to be back here, with them, after all the terrible things he thought had happened. But his happiness is complete when he looks down the lane and sees Katerina, her long limbs sun bronzed, racing toward him beside a gazelle, her wavy golden-brown hair flying out behind her. She smiles, waves at him, and runs toward him with her arms wide-open. And he feels love, such pure love, that the pain fades, and he's not afraid anymore. Because wherever he is, if there is love such as this, there is absolutely nothing to fear.
ACT FIVE
THE MUSES
No evil can happen to a good man, either in life or after death.
—Plato
CHAPTER THIRTY-THREE
ALEXANDER
It is cool up here, near the clouds. Far below him, hills roll through the dappled light of the setting sun like a distant ocean, and to either side spread the wide white wings of Pegasus, as if in a dream.
That's it. This must be a dream. That would account for the dizzy feeling.
But no. Alexander's head is still spinning from his wound, from the brisk air, from the incredible height.
His arms tighten around Zofia's waist, the baby between them. He has no idea where they're going, or how long they've flown—only that the others he left behind are facing certain death, and he should have stayed to help them or die in the attempt.
Heph.
Kat.
All his men.
After the Spirit Eaters routed both armies, leaving both sides fleeing, hiding, being devoured alive... Will this mean the end of the war, or that it is only just beginning? He withdraws one arm from around Zo's waist to feel the comfort of his sword. But its sheath is empty, a useless shell, like a snakeskin without the snake. Then he remembers. He dropped it when the Persian nearly split his head open. His hand fumbles for his dagger. That, at least, is still there.
He closes his eyes again, hoping to quell the sickness rising in his stomach. After a time, it seems to him that Vata's wings angle downward. They are descending. With a clatter of hooves, the Pegasus hits the ground at a gallop. Alex opens his eyes and realizes the horse is circling a familiar plaza, high up on the citadel. There, in the center, stands the Arch of Midas. Chained to a column of the arcade, along the side, is the Gordian knot, lengths of tangled rope wrapped around the pole of a rotting oxcart.
_Pegasus is never lost._
Against all odds, he has made it to the place where he can defeat the Spirit Eaters. The realization is like spring after a bitter winter, like balm to his soul.
The giant winged horse kneels, and Zo carefully slides off, then takes Alexander's blood-smeared hand to help him down.
"My lord," she says, looking at the bloody bandage on her own hand. "Our blood."
"The prophecy."
She nods.
_The stranger whose blood is destined to mingle with your own._ Perhaps not in the way they'd thought.
Yes, he thinks, feeling a bit dizzy again. First one part of the prophecy has come true, then the other.
Zo's dark blue eyes slide from his to the square. "Where are we?"
"Back in Gordium," he says. "Remember I told you about a temple with an old priestess? One that only Blood Magics can enter? I entered through there."
As he speaks, a small, bent figure creeps out of the arch, appearing where once there'd been nothing, only a vacant opening between stones, revealing the sky.
And indeed, it is the same priestess he saw within the impossible, hidden temple, only now she holds an enormous golden chalice in her small hands. A toothless grin stretches across her face as she shuffles toward them.
"Welcome, beloved ones." Her cloudy violet eyes peer out between loose locks of gray hair. "We have been waiting for you."
Zo takes a step back, and Vata whinnies nervously. There's a coldness in the air—and a shadow, a piece of moving darkness, comes into view along the skyline, muting the golden rays of the setting sun.
A Spirit Eater.
Silently, the shifting creature lands on the roof of a house across the square, nearly doubling its size as it hunkers down, seeming to fold in on itself. He and Zo stare in awe and terror as others follow, serenely calm as they land, until all the roofs and the wall overlooking the city below are filled with the monsters. They cast long shadows on the paving stones.
"Do not fear," Kohinoor says. "They will not hurt you or the Pegasus. They are under my command." Vata, however, has backed up into the shadows of the arcade, whickering in fear.
"Our command," says another old figure, emerging from the arch.
As the second one steps forward, Alex shudders and blinks. It is as if Kohinoor has doubled, the second woman a nearly exact replica of the first.
"They cannot be destroyed," says a third, coming behind her sister. "They will always be."
He stares. There are three of them. Three Kohinoors. Eyes and voices and small, hunched forms each exactly like the other.
"Who are you?" His voice commands the air.
The women laugh, a series of soft rustling sounds that scurry together, impossible to locate, as if their laughter is made of the wind itself. A new wave of sick dizziness descends over him. Perhaps the question he should have asked is not who they are but _what_.
"Fate," one of the women says.
"Fate," the others echo.
"Fate, fate, fate." A dark, whispery chant.
Zo clears her throat, coming to stand beside Alex bravely, though he can see that she is trembling. "What do you mean? We do not understand you. What do you want?"
"We are the Fates," they say, their voices misaligned with one another just enough that the words come in rippling waves.
Zo's face has gone as white as parchment.
"And these?" Alex sweeps his arm toward the hovering Spirit Eaters, forming their own layer of night laid down atop the city.
One Kohinoor smiles, waveringly. "Everything comes at a cost, my dear boy."
"Fate is beautiful," says another.
"Beautiful," echoes the third.
"But," says the first, "all beautiful things must have their price."
"Enough riddles," Zo says, her body going rigid beside his.
"They," the second Kohinoor explains, gesturing above, "are the byproduct of Fate's elegance. Fate cannot exist without its other side. Without chaos. That is what they are. Chaos embodied."
"Chaos," coos the third priestess.
Alex swallows, trying to focus, even though their words have a lulling effect over his mind. "How can they be destroyed?"
"They can't," say the old women, gesturing to the setting sun. "The solstice is upon us!"
The first Kohinoor steps even closer. "Here," she says, looking into the goblet. "See for yourself."
He begins to move toward her, but Zo pulls his arm back. "Don't," she warns.
Kohinoor ignores her, her blind eyes glittering at Alex. "Here lies the greatest fate we have ever woven for any man."
She holds up the sparkling goblet. "Drawn from the Fountain of Youth. Drink, and your body will be made whole."
_Made whole._ His scarred leg. His imperfection. This is it. This is what Alex has wanted every day of his life since he was old enough to know he was not quite like other boys. Without knowing how, he has come even closer, and now he stares into the broad goblet. The water shimmers like glittering diamonds. He is suddenly so parched he wants nothing more than to slake his thirst with the cool, sparkling water.
But there is always a price to pay. They said so themselves.
"What do you want in return?" It is a whisper, as if to speak more boldly would be to break the enchantment.
"The baby," she coos, so gently Alex almost doesn't hear her.
The other priestesses stretch their gnarled fingers toward Zofia and the small bundle still on her back. "The baby, the baby."
"Give us the baby. Then you can heal yourself with this life-giving water, take the princess, and fly away from here on her winged horse, on to your great destinies, for we have woven one for her, too. It is not much to ask, is it? A tiny infant, a mere girl. The princess will have others."
"Why?" Zo's voice breaks through Alex's reverie.
The violet eyes look kindly at her. "She is one of us," Kohinoor says.
"The newest."
"The next."
"Through her, our powers will wax strong."
And understanding dawns in Alex, just as it must in Zofia. "No," she says, low and hard.
"Fate must go on," Kohinoor says. "It must be woven by the fated fingers of the future. It is her. _She_ is the fate of Fate. We do not live forever. We will raise her, train her. When the time comes, she will become us. She will become three."
"The fate of Fate," echo the others.
"Why, though? Why her?" Zo's voice is throaty with anger.
The Kohinoors whisper-laugh. "We pick our replacement at the moment of her conception," says the one in the middle.
"We were there that night," says the one to her left. "We came to you. We felt the powerful pull of you—of your desire."
"Your desire to change fate. To weave your own."
"Do you recall," asks the third, "the black cat in the palace basement, where you met your handsome lover?"
Alex has never seen an expression of such horror as Zo now wears. A dark red blush. Or is it the last rays of the setting sun?
"We arranged for your capture," says the third one.
"The slave traders," Zo whispers in shock.
"We tried to watch out for you on your journey."
"We sent the Spirit Eater to chase your Pegasus that day in the Eastern Mountains," says the second one, "to cause you to fall, so that we could find you there and nurse you back to health. We caused the Pegasus to escape the Aesarian Lords' fire and the Spirit Eaters' hunger today, so she could bring you here, to us."
"Give us your baby willingly," the third one says, "and we will make sure you will find the man you love again. He is waiting for you even now. And the two of you will, quite soon, rule over all Persia."
"No," Zo says, backing up toward the nearest column. "Not if it means giving you Mandana."
A question has been forming in Alex's mind. "Why do you need to ask Zofia to give you her baby if it is fated for her to become your replacement? Why do you need to ask me to drink from the Fountain of Youth? Why don't you just make it happen?"
The Kohinoors look at each other questioningly. Finally, the first one says, "There is, among the countless threads of Fate, a tiny breath of..."
"Of what?" Alex asks, thinking he knows the answer. The old women stand uneasily, looking at one another. Is it his imagination or do they seem to flicker, just slightly?
"Of free will." It is Zo who names it. "That's it, isn't it? All of us have free will, and if we choose to fight against the Fates you have woven for us, maybe..."
"We can break them," Alexander finishes.
A sunbeam breaks through the thick clouds, illuminating the three women, who seem to shift in and out of the threads of golden light, bringing to mind what Aristotle once told him. _We are all connected, and controlled, by invisible strings of energy. We cannot see them but if we are wise enough, we can feel them—and alter them._
Above the old women, the Spirit Eaters shift, restless.
"Many men before you have tried to undo fate," warns one of the Kohinoors.
"But they cannot untie the threads. You are bound to us forever. All of you."
_Untie the threads._ Patra's prediction returns to him. _Unbound. You are the one to unbind us all._
Suddenly, the truth flows into him so directly he wants to laugh with the simplicity of it, of why they've come here, to this legacy of Midas, the Earth Blood with the touch of gold. Home of the Gordian knot.
"Men have tried," they repeat, "but it rarely makes any difference, in the end."
"It does with me," Alex says, stalling.
He must untie the fabled knot.
It is as easy as that.
As impossible as that.
For no one has ever been able to.
The world still spins around him. He needs to think, to figure this out. He's not ready when the first Kohinoor walks toward him, smiling. "Look into the water, young king," she says, "and see the glorious destiny we have woven for you."
He cannot help it. Though he knows he should not look, the temptation is too great. He moves toward her as if he himself is a mere puppet, controlled by invisible strings, and looks into the sparkling crystal liquid, dappled in the scarlet rays of the setting sun. At first, he sees only the reflection of his face surrounded by crimson. But then the liquid swirls, and when it clears, he sees himself at the head of victorious armies, crossing mountains and rivers, deserts and fruitful plains, and sailing the seas in battleships. He leaves behind shining cities in all his new lands, all of which he calls Alexandria so they will remember his name.
Brides sit beside him on queenly thrones: sloe-eyed Persians, shapely dark-skinned Ethiopians, golden-haired Gauls with sapphire eyes. And playing at their feet are strong sons who will grow to be kings in their own right, and lovely daughters who will marry kings and sire yet more kings. He sees Hephaestion and Katerina by his side, helping him rule his empire as the three of them grow old. And Princess Zofia, a new friend, but one he knows he will grow to love, sits on a great throne, allied to his, beside a handsome, amber-eyed warrior.
He sees that all of mankind, thousands of years from now, will know his name, will call him great. Warmth spreads through him like hot spiced wine on a snowy day. All his dreams will come true.
"Drink it now," Kohinoor croons sweetly, pushing the cup toward his lips, "and you will be healed. Then all that I have showed you will come to pass."
He hesitates, remembering what Aristotle said about his birth injury. _You have strengthened your leg with exercise. You have run with weights on it in the face of tremendous pain. Would you have tested yourself so, mentally and physically, if I had uttered an incantation and miraculously healed it?_
And the answer, clearly, was no. It was through his weakness that he found his strength. Aristotle said something else, then, that stuck with him. _The only true magic is human ingenuity. Search for answers to your problems within yourself, Alexander. For in yourself all problems and all answers lie. Not outside. Magic is an illusion. So is perfection. So is power, unless it is the power an individual wields over himself._
He pushes the cup away. "I don't want it."
It is only part a lie—he does want it, but not like this.
"It is your fate to drink it and be healed," the old crone shrieks. Gone are the strange smiles and gentle crooning. Her eyes glow with dark purple fire. "Drink the water."
"No." Remembering he has no sword, he draws his dagger.
"How dare you hesitate, you ungrateful wretch! Yours is the greatest destiny we have ever woven!" the third Kohinoor says, her face twisting in rage.
But he turns from her, an eerie calm descending over him. "We live and we die," he says to all of them, "and in our brief spark of light, we deserve to make human choices. Human mistakes, too. We deserve an Age of Man." And didn't the ancient prophecy state that he could bring such an age into being?
He approaches the oxcart, examining the knot, understanding now why he felt such sinister energy when first he touched it.
In a flurry of wings, a Spirit Eater flaps down from a roof and onto Zo, who screams and falls as it plucks her baby off her back and swoops over to the Kohinoors. Mandana howls her fear, wailing high and loud. Zo scrambles to her feet and lunges toward them, but Alex holds her back. He needs time to think. To figure this out. He knows the monster won't harm the baby.
The Spirit Eater, who seems to be three laughing heads now, squats down, holding the child in front of him to face Alex and Zo as he spreads his black wings. Alex blinks. It is a similar image to what he saw in the Temple of Midas, a winged baby. Like the first time he ever saw Mandana in the ruined mattress shop, with feathers splaying out beneath her arms and shoulders. He feels suddenly dizzy again.
Mandana looks uncertain, but she has stopped crying. One of the Kohinoors hobbles forward and hangs a tablet around her neck. There is strange writing on it, symbols of stars, the sun, a crescent moon, rain, and lightning. "Let her fate be written," the old crone says, "that she will become Fate."
"She will become Fate! She will become Fate!" cry the others. The baby opens her mouth and shrieks in protest as the old women cackle. Zo reaches futilely toward the child and cries.
No, he cannot allow this to happen. What did Patra say? That on the summer solstice, the new child of Fate will be theirs, giving them the power to devour the world. And the only one who can stop it is he who is marked with the threads of Fate. Alexander himself.
The prophecies buzz in his head. He stands in the city where Patra said he could destroy the monsters, and at the place where the strands of Fate are all tangled together. He has by his side the person whose blood has mingled with his. All he lacks is the talisman. Well, he can't wait any longer. He fingers the cold steel of his dagger, ready to brave the Spirit Eater's claws and fangs and take the baby back by force before the sun dips below the horizon. A ray of orange sunlight glints off the glowing ruby eye of the phoenix on the hilt of his weapon, off the curling flames of gold that surround it.
And then he knows. He knows exactly what he must do.
Because the phoenix doesn't die as mortals do. It has no mortal fate. It bursts into flame and rises from its own ashes. The talisman he needed was with him all the time, in his belt. _Closer than you think_ , the Kohinoor in the Temple of Midas had said, laughing.
He turns to the knot, and with a roar, he raises the dagger.
A white light radiates from the cart, so bright he can barely make out its shape. The three Kohinoors scream out. The dark shapes of shrieking Spirit Eaters swoop around him. He can feel their cold, killing breath, their rage and hunger— _chaos itself_ —even as his dagger slices down on the knot before him.
Heat and light and screams erupt all around him. He is blown back from the cart, landing hard on the pavement, his dagger sliding away from his hand.
When the blinding light fades, he finds himself looking up at a sky dark with clouds. Rains falls gently on his face as a baby's sobs rend the air. He scrambles to his feet and sees that the Kohinoors are gone, leaving behind three heaps of gray rags. The Spirit Eaters, too, seem to have vanished. The cart is gone, blown to splinters and a few fibers of rope. And the arch has fallen, its golden stones scattered and cracked.
Zofia is crawling toward Mandana, who is on the ground, kicking and screeching in outrage. "My baby," she says, picking her up and comforting her. The tablet Kohinoor hung around the baby's neck has disappeared.
And in the distance, just as the sun sets on the horizon, Alexander sees the Pegasus, winging its way toward the horizon.
SIX MONTHS LATER
CHAPTER THIRTY-FOUR
KATERINA
Ada, the new queen of Caria, leaps at Kat with her curved sword as Kat parries her blow and twists out of her way. A crisp winter breeze off the sparkling, white-capped harbor of Halicarnassus cools the sweat beading on Kat's brow. She is tired. No, more than tired. Stifled. Ready for whatever's next.
Alexander's army conquered the kingdom of Caria and freed it from Ada's unstable brother, Pixodarus, a week ago, fulfilling Alex's vow to Ada and carving away another prosperous western territory from Great King Darius of Persia. It was an easy battle, as the whole city rose up to aid the conquerors, eager to be free of the cruel king's tyranny.
Kat throws her sword onto the ground in a haze of dust. "No more." She walks across the lawn to the little beach, looking across the water from the island palace to the hilly city of red-tiled roofs. Halfway across town rises the Mausoleum, a stone tower twenty times the height of a man, with an enormous temple on top. Sunlight glints off the golden figures on the roof. It is the tomb of King Mausolus, built by his wife, a poem, in stone, of love and loss.
"You were doing so well." Ada comes up beside her, sheathing her sword. "What's wrong?"
Kat shakes her head. The world of men is one of war and power. Always, someone must fall so that someone may conquer. Life is predicated on death. She knows that. And yet...
The pearl ring Jacob gave her glimmers on her finger, and she wraps her other hand around it to feel the love it will always contain.
"I am afraid I have played a role in all this tragedy," Ada says, watching a merchant ship ply its way to an empty pier. "It was I who sent Helen to Erissa to hide from Queen Olympias. But she was an oracle, and her child was prophesied to be a Snake Blood. I thought it would be somehow safer—sweeter—to be near a family of Earth Bloods, even though Jacob's family had little idea of the extent of their abilities. Olympias found her anyway and killed her. And now Jacob's death has broken you."
"I am not broken, Ada," Kat says, rubbing her thumb over the smoothness of the pearl. "Though I am changed. And it's not just losing Jacob that hurts. It's all those whose deaths I was responsible for. The Persians whose tents we burned. Those I drove mad with the honey so our troops could slaughter them."
Ada stares at her. "That's war, Katerina."
"Yes, it is. And every one of those who died was a son or brother or husband or father. Their faces float before me in my waking hours and haunt me in my dreams."
Ada is silent. "Well," Ada finally says, "if this life isn't for you, then what will you do?"
"That is the question, isn't it? Alexander has no need of me. He will make his mother—our mother—his regent. It is what Olympias always wanted, and she is suited for it. I've already told him."
Ada's dark eyes glisten as she touches Kat's silver lotus blossom pendant. "Alex has had a disappointing day indeed, then. I hear Princess Zofia has turned down his offer of marriage."
Kat looks at Ada in shock. Over the last few months, she has grown fond of the pretty Persian princess and her beautiful baby—and it had warmed her to see how much time she and Alexander spent together. "Why?"
"Here she comes now," Ada says, nodding toward the palace. "You can ask her yourself. As for me, I have an appointment to train young Arridheus in how to wield a sword. I'm looking forward to a more eager pupil, even if he will never be one of your skill." Her voice carries a mix of disappointment and levity—and Kat knows that she has been forgiven for pulling away. Though Ada may never understand how she feels, she will always respect her.
Zofia, in sparkling dark blue robes, smiles at Ada, who disappears into the palace. Zo greets Kat with the same warm smile. "Well, I finally have a moment to find you," she says. "Mandana has just agreed to take a nap."
"Let us walk," Kat says kindly, curious what the girl will have to say. They march along the little beach, toward the stone bridge connecting the island to the town. Oxen slowly pull a cart of amphorae toward the palace.
"Ada tells me you turned down Alexander's offer of marriage," she says, after a moment of quiet.
Zofia smiles sadly and looks to the distance. "Yes."
Kat kicks a shell on the sand. It falls into the water with a little _plop_. "I am sorry," she says, "because, of all the women I've met, I think you would have been the best wife for him. You are so alike, so brave and clever and good."
Zofia's face is contemplative. "If I had met him before all of this, yes, I think we could have been happy, even though he's not in love with me."
Kat raises an eyebrow.
"Oh, he likes me well enough. And perhaps with time..." Zo trails off. "But I am in love with someone else. Someone who risked everything to save my life."
This is news. Zo told Kat about running away from Sardis to avoid marrying Alexander and to find Mandana's father, who was later killed. But she has never told her of loving another man.
"Do you know where he is?" she asks.
The princess's eyes sparkle. "I have an idea." She reaches into her pouch, takes out a small scroll, and unrolls it. Kat can't read what it says; the letters are Persian.
"The last night I ever saw him," Zo says, running her fingers over the writing on the scroll, "he asked me where, of all the places in the world, I would like to go. And I said the royal library of Babylon." She looks up at Kat and her smile radiates across her whole face now. "I was pregnant, and wanted my child to learn more than I ever had. I was a foolish, silly girl who never paid attention to my tutors. And I thought, if I could go to that center of learning in Babylon, I could understand science, astronomy, philosophy, engineering, so many things to teach my child. Even women can study there."
Kat studies the strange little symbols on the scroll, like trees with lopsided branches. "What does it say?"
"It says, _'Meet me at the Great Library of Babylon. I will wait until you come, no matter how long it takes.'_ He left this for me hidden in his cloak, though I didn't find it until just before the Battle of Gordium."
"Do you think he will still be waiting?"
She runs a hand through her hennaed dark hair. "I don't know." She pauses. "I can't be sure what will happen, only of what I must do. Now that Great King Darius is retreating into the heart of the empire, I think I may eventually be able to safely travel along the Royal Road again."
Kat feels a wave of warmth for this girl. "So you are leaving, then," she says, grasping her hands. "I wish you weren't. I wish I knew that you would always be here, waiting for Alexander. But I understand why you will not. And I—I admire you. You have conviction. You know what you want."
"Yes," Zo says. "And, Kat? I understand you, too. I know the pain of loss. A beloved little sister. She was so young—just a child—and she died trying to rescue me and bring me home."
"I'm sorry. I didn't know."
Zo's eyes fill with unshed tears. "Roxana. She was only six and followed me out of Sardis the night I ran away. The slavers who captured me took her into a field of wheat and..." Zo pulls a hand out of Kat's grasp to wipe away a tear.
Kat is speechless. Transfixed with shock. With hope. A trickle of realization moves through her. The pretty little girl in the ruins of Troy. Disappearing in and out of the wall.
Her name was Roxana, too.
Kat can still remember her saying, _The mean men took my sister away. They were going to hurt me, too, but I ran._
_I was a princess once and lived in a palace._
Kat hadn't believed her. All little girls like to pretend they are princesses.
"Zofia." Her voice cracks with emotion, with the impossibility of it. "Roxana. I—I may have found her."
Zofia's expression is blank. She blinks several times. "Wh-what?" she asks. "But I heard her cry out. The slavers said..."
"She got away!" Kat says, jubilant now, more sure than ever that she is right, that it was fate that brought her to Troy and to Roxana.
Perhaps it has always been her destiny to help instead of hurt. Perhaps that is the lesson she has learned about herself. After so much of her life spent plotting revenge against the queen who turned out to be her own mother, seeking a path full of darkness and anger, violence and blood. Perhaps, in that way, she was far more like Olympias than she ever realized.
Only, now she has a choice. A chance to be different. To be more like Helen, who was a hero in her own right, though she never fought a battle in her life, never likely even lifted a sword. She didn't have to. She served others through humility and patience. Through personal sacrifice.
"But...how can you be sure? What if you're wrong?" Zo asks, clearly unwilling to have her hopes raised and dashed.
"It's like you said, sometimes you can't be sure what will happen, only what you must do."
Zo nods, her face full of wariness and hope.
Kat puts her hands on Zo's shoulders. "Roxana told me once that Pegasus is never lost. Like you, she has conviction. I know that you two will find each other again. I'm going to help."
_"Pegasus is never lost,"_ Zo whispers. Then she puts her face in her hands and sobs.
* * *
Alexander paces around the council chamber table filled with maps, his hands clasped tightly behind his back. "It is a day of goodbyes," he says, his voice tight. "First Cynane, just when I was beginning to like her for the first time."
Kat smiles. She, too, liked the new Cynane, to whom Alexander granted the satrapy of Lydia, where she will rule in his name with her army of Amazons. They left this morning, riding like the wind out of the gates of the city.
"And Princess Zofia," he adds, sighing. "I hope she finds those she seeks."
He stops in front of her and puts his hands on her shoulders. "But it is parting from you, Katerina, sister, twin soul, that breaks my heart. Why do you want to leave me alone?"
"You won't be alone." She hates to hear her own voice breaking. "You will have Hephaestion. The love you two have for each other is far greater than any feelings you will ever have for someone else."
Alex pounds his fist onto the table. "Are we only allowed to have one person we love, then? Why can't I have you both at my side?"
She looks out the window. A Greek military trireme, its red sails round bellied with wind, heads toward a pier. Sunlight glances off the helmets and spears of soldiers crowded on the deck. "Those who stay by your side must constantly live with battles and wounding and death."
After a long pause, he says, "Please, just tell me where you are going."
"I will let you know."
"And not just where you are, but if you are happy."
"Always."
A tear slides down his cheek. "I have a feeling we will never meet again."
She, too, has that feeling. "We are in each other's hearts and souls, my brother." She smiles through tears. "We can never truly be parted in life or death."
* * *
She finds him in the stables courtyard, currying his dun-colored stallion, Ares. Briskly, he brushes the yellowish hide, taking care to avoid the thick scar from an arrow wound on the horse's back.
She hesitates in the shadows of a doorway, giving herself time to drink him in, knowing she must make this memory last a lifetime. The messy dark hair he pushes impatiently out of his face. The lean, muscular body that bends down agilely to dip the brush in a bucket. A sob threatens to rise in her throat. A moan of what she is giving up, mixed in with the temptation to grab what happiness she can.
She pushes it all back down, squares her shoulders, and marches into the sunlight.
He looks up, his dark eyes flashing something for a moment. Hope, perhaps, that she has changed her mind. Then he turns away from her. "So. You're going."
She says nothing. _Goodbye_ is too hard a word.
"I could never compete with a dead man," he says, not looking at her.
"Heph, listen to me." Now she's standing only inches away. "I loved you. Maybe I still do." The words are hard to say, their sting of truth. "But I also _know_ you. I know you have only one great love in your life. And that is not me."
She doesn't have to say her brother's name. He knows exactly what she means.
"The two of you will have eternal glory."
"That is ridiculous!" He kicks the bucket of water hard. It flies a few feet, crashes down, and rolls on its side as Ares steps sideways in fright, straining at the rope tying him to the post. Kat watches the hot water seep between the cobblestones. "Why? Why can't it be both? Why must you make everything an impossible choice?"
She lets out a breath, then reaches out to touch his arm. There's a part of her that wants never to let go. And another part of her that knows time will not be good to them. That if she stayed for him, she would resent being trapped, and that entrapment would put reins on her love.
"The truth is, I don't think I ever want to marry."
Heph looks at her with dark eyes rapidly filling with tears. "I love you, Katerina of Erissa. I will always love you."
She strokes his cheek. "You know where you belong. Now it's my turn."
He nods, swallowing hard.
"Thousands of years from now, everyone will remember your name."
There is much more she wants to say—that she will always love him, that she will never forget their time together on Meninx, that in leaving him she is breaking off another part of her already broken heart. But she cannot see him cry. She, too, would cry. And they would comfort each other, with whispers and touches and sighs...and then she might not be able to go.
And so she steps away, back through the door, and into the shadows of the courtyard.
In a rush of feathers, she transforms into a kestrel. Beating her wings, she rises rapidly over the palace complex. The buildings grow smaller, as does the dark-haired young man standing next to a yellow horse, his right hand raised in farewell.
Before she flies west, to the place that calls her, she heads north. For there is something she must see, something to gladden a heart aching with loss.
Once she is there, she waits among the ruins of Troy, flying in circles as shepherds and their flocks perform their slow, ancient dance. Rich foreign visitors pray ostentatiously before the funeral mounds of the heroes. The sick and the lame limp toward the temple of healing high up on the hill, the place where she and Jacob found happiness, for a time. And at night, the tunnels beneath the ruins are silent and empty as the grave.
On the morning of the fourth day, wheeling over the beach, Kat spies Zofia's ship dock. Men lead a brown mare down the ramp, followed by Zofia herself. Within minutes, she is riding like a Fury toward the Temple of Asklepios, its mismatched columns gleaming pink and white and black on the hill. Children are playing on the lawn in front of it, chasing each other in a game of tag. Zofia pulls up some distance away as a little girl stands stock-still, staring openmouthedly at the approaching visitor. Zofia jumps off the horse and runs toward the girl, arms outstretched, calling her name again and again. Roxana runs toward her and throws herself into Zo's embrace, crying out her name. The two whirl around and around in a frenzy of pure joy.
Satisfied, Kat angles her wings and continues on, over the sea and toward the setting sun.
CHAPTER THIRTY-FIVE
DARIUS
"Wine, sire?" Bagoas kneels before him.
Darius nods, pushing the maps and military scrolls off his sandalwood desk to clear a space. The eunuch, now the wealthiest man in the empire, except for Darius himself, pours wine into a cup, sips it, swishes it around in his mouth, and swallows. He nods, fills the royal chalice, and hands it to Darius.
Briefly, the king imagines there might be poison in the royal cup itself. Or in the wine, and Bagoas has already taken an antidote. It has occurred to Darius lately that Bagoas agreed rather too quickly to help him kill Artaxerxes with the poisoned figs. And after that, the remaining Assassins—anyone who might have questioned his loyalty to the throne.
Darius can never forget the last words of Great King Artaxerxes. _You will never be safe._ Was it a curse? A prophecy? The words seem to pump through Darius with every beat of his heart. The birds in the trees seem to sing it. The wind whispers it. Can others hear it? Sometimes he wonders if he is going mad.
He stands and looks out the tower window over the sprawling palace of Persepolis, its famed courtyards and gardens far from the continued battles in the empire's west. At least he is safe from Alexander's armies here. The Persian forces were routed at the Battle of Gordium. The fires Darius set to trap the Greeks proved ineffective when creatures of darkness winged their way through the sky, devouring Persians and Greeks alike, and then the rain came. Since then, Alexander's armies have conquered the western provinces along the Aegean, his success made easier by promising they could keep their local rulers and ways of life. Formerly Greek colonies, they submitted cheerfully, many of them opening their gates as the soldiers flocked to join him.
But Darius has set up a wall of armies in Cappadocia to protect the east. He has withdrawn his forces from Egypt and Palestine to fortify the heart of his empire, which is wide, and rich in people and in resources. The young king's successes in the west are no indication of what will happen if and when he marches farther east. Darius knows he is safe here. He has hired many former Aesarian Lords as his personal bodyguards, the brotherhood dismantled now that the very reason for their existence—to protect the world from the Spirit Eaters—no longer exists.
No, his main concern is not the Greek invasion, but the fact that he has no heir to give stability to his throne. For despite his extensive spy network, he has not been able to learn what happened to his only son, Ochus. He sent spies to search all the mercenary armies in the known world for someone with his son's appearance—broad-shouldered, muscular, with a lion's mane of tawny hair and unusual honey-colored eyes. Perhaps Ochus is dead of a fever or a brawl. He always was prideful to a fault.
"Sire?"
He turns to see Bagoas, his heavily lined dark eyes intense, holding out the royal chalice brimming with wine. Darius removes the tall gem-studded conical crown and sets it on the table. Before becoming Great King, he never understood how heavy it was.
ONE YEAR LATER
CHAPTER THIRTY-SIX
ZOFIA
Babylon: a sprawling complex of towers, domes, and gardens, all undulating in a violet haze through the modest veil covering Zofia's face. It is breathtaking, a place she has often read about and dreamed of before, but in person it is more alive—birds dart through a cerulean sky, and the sounds of chanted prayer fill the warm air.
And there, in the center of it all, stands the royal library.
Zo's heart flutters as she approaches.
She already has so much, she reminds herself. Mandana, now a toddler tripping over her own robes, curious and full of laughter. Then, the miracle of her sister, Roxana, who seemed to come back from the dead, saved by Katerina. And her old nursemaid, too, the original Mandana, plucked from the Sardisian palace when Alexander conquered it, to look after both of them. She knows that she could easily trundle them all back west over the Royal Road and settle down in Alexander's newly conquered territory on the coast. He has promised her whatever riches and property she wants.
And yet...
Zofia has always been determined to create her own fate.
She pushes open the double doors, twice the height of a man, and steps into a large, shadowy hall with a round pool below a circular opening in the roof. Corridors and doors lead off the hall, and beyond another set of open doors is a large colonnaded garden. Scholars in white robes, arms full of scrolls, bustle about.
For a while, she wanders, hesitant to make a disturbance, in awe of the power of knowledge that fills the air. There are uncountable libraries within the library, rooms heaped with scrolls and windows overflung with lush plants and flowers. She moves through the first garden with its splashing fountains and lounging scholars, and past the domed astronomical observatory. For a moment, she almost forgets why she has come, but the parchment folded into her robes rustles gently against her thigh, reminding her softly of its presence. The note she found in Ochus's cloak, hidden in the secret compartment she hadn't known of until Alexander pointed out in Gordium that all of the Assassins have them.
She enters a doorway at the far end of one hall—to the philosophy library, a cool, high-ceilinged room. Men and women reading at the long tables look up at her in surprise.
Light streams in high windows, illuminating an old man painstakingly copying a text.
A young woman reading a scroll absentmindedly drums her hennaed fingers on the table, each tap a subtle drumbeat that momentarily lulls Zo.
In the far corner, a white-robed figure holding scrolls descends a tall ladder. Her heart stops. The wide shoulders. The tawny hair. The indescribable confidence of his every movement, the ease of a trained killer, the elegance of a wild cat.
When he turns, he sees her and drops the scrolls.
His mouth opens and closes. His amber eyes widen, as if to let her in.
"You...came," he says, his voice hoarse.
"It took me a while to find your message," she says, tears already falling down her cheeks.
He comes toward her, reaching out to touch her tentatively, like she might turn to dust and blow away. His hand lands gently against her cheek, and he inhales sharply, as if surprised to find her warm and real and alive.
His arms open, and she falls into them.
TWO YEARS LATER
CHAPTER THIRTY-SEVEN
ALEXANDER
Alex casts a look behind him. His vast army fills the fields as far as he can see. Tens of thousands of men, some mounted, most on foot. Thousands of carts bearing tents and weapons, cook pots, medical supplies, and dismantled catapults. Entire herds of livestock. They are ready to march. For the army is moving eastward, ever eastward, until they surround Persepolis itself and Great King Darius. But he never sets out without Heph. He will wait.
Behind him, men on horses move aside, allowing a rider to pass.
Heph pulls up beside him.
"She's settled, then?"
Heph nods. "I never thought I would be like the Persians, taking my wife with me on campaign."
Alex smiles. "Who knows how many years it will be until we return? You and Sarina need to start working on our next generation of warriors."
Heph grins. "Oh, we're working on it, my liege."
Alex is glad Heph has found happiness with Sarina. After Kat left, two years ago, he seemed so heartbroken that Alex wanted him to have a complete change of scene. And someone had to deal with the Egyptian agitators led by the sultry, dark-skinned priestess, who had kidnapped Arridheus and attempted to assassinate Alexander. And so Heph sailed down the Nile on a diplomatic mission to promise that Alexander would not brutally conquer their land and enslave their people, as the Persians had done. He returned not only with a treaty, but with a bride to seal the alliance.
At first, Alex wondered if Heph had married Sarina to forget Kat. And even though he seems quite happy in his choice, perhaps he did. Sometimes Alex catches a shadow in his eyes. Alex suspects Heph will never quite love his wife as much as he loved Kat, which is exactly as it should be. The love between a man and a woman, Aristotle said, can drive a man mad.
Kat sent word that she is living on the isle of Meninx, also known as the Island of the Lotus-Eaters, off the coast of Carthage. She explores the sea, searching for the inhabitants of Atlantis, Snake Bloods, she says, who long ago intentionally sank with their entire civilization below the waters, where they still live. During those wonderful months of togetherness between the Battle of Gordium and her departure, she once told Alex of her fleeting vision of a city on the bottom of the sea, with many-columned temples and palaces, markets and roads, and people gliding forward, their long hair and full robes trailing behind them in the sea currents.
Since then, Alexander has heard strange tales of a siren off the coast of Lotus Island. Only, this siren doesn't lure men to their deaths, as sirens are wont to do; she swims out to sinking ships and rescues them. The sailors call her Thessalonike and worship her as a goddess of the sea.
Some parts of the vision the Fates showed him in their goblet remain yet to be determined. Will he have the wives and many children he saw in the sparkling water? He does not know. And the long life? It doesn't matter. For, one thing he does know; one thing does matter. He will have the glory he always dreamed of. He will conquer all of Persia. But he will not stop there. He will continue east, to that legendary land where brown men dye their hair green and ride tusked beasts as big as houses. Perhaps he will march to that fabled place far beyond the known world, where worms spin shining threads on trees, and where the earth ends at an endless ocean. And in his path, he will leave justice, thriving trade, and peace. Dozens of new cities called Alexandria will spring up behind him. His name will truly live forever.
And, always at his side, he will have Heph. He looks at him now; they smile at each other and nod. Then he blows the horn, and the army moves forward.
* * * * *
Keep reading for an excerpt from Reign of Serpents by Eleanor Herman.
ACKNOWLEDGMENTS
Alexander the Great didn't conquer the known world by riding east alone on his horse. He was assisted by a devoted, talented group of generals and countless others who strategized, negotiated, and polished weapons to a burnished brightness. Together, they slogged through mud, desert, and a fair share of horse manure into luminous realms of glory.
As with Alexander, so with this fourth and final installment of the Blood of Gods and Royals series. It would not have been possible without the assistance of an extraordinarily capable team: Lexa Hillyer and Kamilla Benko at Glasstown Entertainment; my agent, Stephen Barbara of Inkwell Management, and his incredible team there; my editor, Natashya Wilson at Harlequin Teen, and her excellent team as well, including Gabrielle Vicedomini, Lauren Smulski, Amy Jones, Bryn Collier, Evan Brown, Krista Mitchell, Laura Gianino, Shara Alexander, and Linette Kim. To all of you, it has been great ride. I'm so glad I had you by my side.
AUTHOR'S NOTE
Writing this series has been an epic journey of its own. In doing so, I have traveled to a time and place often more real to me than the here and now, and certainly more interesting! I have dug deep into my soul to channel Alexander's brilliance, Hephaestion's loyalty, Jacob's courage, Katerina's independence, Cynane's resentment, Zofia's determination, and Olympias's endless manipulation. These characters are all pieces of myself and I will miss them.
It has also been a joy to write about the ancient world, blending a great deal of fact with fiction. For instance, the Asklepions of the sort Katerina and Jacob encountered outside Troy were healing temples sacred to Asklepios, the Greek god of medicine. Sacred snakes really did wander the site, communing with patients and priests. Asklepios—who I believe was a real medical practitioner with amazing skill—was known to walk with a staff, and the symbol of a snake twisting around a staff has been used by medical practitioners for centuries, including the American Medical Association.
The epic saga of Troy greatly shaped Alexander's world. Warriors wanted to be like Achilles; women compared their beauty to that of Helen. For centuries, Greeks visited the ruins of Troy on the northwest coast of what is now Turkey to sacrifice to the heroes who died there, including Alexander himself in 334 BC. While we don't know if Achilles, Helen, and the host of other unforgettable characters were real flesh-and-blood people, we do know the site was devastated around 1200 BC, as were most coastal cities in the eastern Mediterranean around that time. The end of this advanced civilization was due, experts now believe, to a combination of climate change (resulting in earthquakes, droughts, tsunamis, floods, and famine), epidemics, large-scale migration as people tried to escape these cataclysms, and invaders taking advantage of them. In the wake of this massive destruction, people moved to the hills and, while focusing on survival, forgot how to read and write, build elaborate palaces and temples, craft gold and silver, and make sophisticated pottery. The _Iliad_ and the _Odyssey_ were, I believe, love songs to a vanished golden age, a mythical explanation of unendurable loss, and a light in four hundred years of darkness.
The story of the Gordian knot is much as I have written it. When the kingdom of Phrygia found themselves without a king, the elders visited an oracle who told them that the next man driving an oxcart to enter the city should be their new ruler. A peasant named Gordios must have been shocked as he drove his oxcart into town and was immediately proclaimed king. To commemorate the historic event, the oxcart was tied to a post with an intricate knot. Another oracle proclaimed that any man who could unravel the knot was destined to become the master of all Asia. When Alexander arrived in Gordium, he tried to untangle it without success. Frustrated, he pulled out his sword and sliced it open. He went on to conquer Asia, thereby fulfilling the prophecy. These days, the term _Gordian knot_ refers to a particularly thorny problem that requires creative thinking to solve, just as Alexander did with the real knot.
The tale of mad honey is one of my favorites from the ancient world. In 67 BC, Roman soldiers in Asia Minor marching against the wily King Mithridates near the Black Sea unexpectedly found luscious, gooey chunks of honeycomb strewed in their path. The hungry men dived into the sticky, sweet stuff, became wildly intoxicated, and were killed by the enemy army lying in wait. Bees make mad honey with nectar from a certain kind of rhododendron flower that contains grayanotoxin, a highly hallucinogenic substance. Even today, hikers and tourists in that part of Turkey are periodically treated for mad honey poisoning.
In the Battle of Chaeronea in 338 BC, King Philip of Macedon led one wing of the army, entrusting Alexander with the other. Philip retreated, but it was a trick to draw the enemy in as Alexander's wing whipped around and attacked from behind. The prince clashed with the Sacred Band of Thebes, one hundred and fifty pairs of gay lovers known as the fiercest warriors on earth, killing them all. When Philip surveyed the battle after his victory, he encountered corpses "heaped one upon another," according to the ancient historian Plutarch. Understanding they were members of the Sacred Band, he wept and said, "Perish any man who suspects that these men either did or suffered anything unseemly." He erected a memorial on the battlefield in the form of a lion as a testament to their courage, which was discovered in the nineteenth century and stands today. Beneath the statue, archeologists found hundreds of skeletons, human ashes, shields, helmets, and spears. After the battle, Philip formed the League of Corinth with his former enemies and prepared to march on Persia, but he was assassinated before he could set sail.
One of my favorite characters is Queen Olympias, Alexander's mother, who was what we today would call "a piece of work." Her marriage to King Philip was quite stormy, especially after he married a girl half Olympias's age, Cleopatra, the niece of Attalus. At the wedding of another Cleopatra, Philip's daughter, Philip was murdered by a former lover, Pausanias, though Olympias was believed to have had a hand in the deed. She returned to her brother's court in Epirus, where she organized the murder of Philip's young widow and their infant son. In proving the old adage that only the good die young, she survived just about everyone mentioned in this story.
The eighty-eight-year-old Great King Artaxerxes III of Persia was reportedly poisoned by his eunuch, Bagoas, and a physician. While details of his death are unknown, I borrowed from the reported poisoning of the seventy-seven-year-old Roman emperor Caesar Augustus in 14 AD. According to one version, his wife Livia painted figs on his favorite fig tree with poison, knowing he never had a taster test them.
Regarding the fate of Katerina—fierce, smart, independent, and a fictional invention of my own—it seemed impossible to me that she would settle down with a man, have kids and a traditional life. There is a legend that Alexander, in his effort to find the Fountain of Youth, found a flask of immortal water he used to wash his sister's hair. Years later, hearing of his death, his sister tried to commit suicide by leaping into the sea. But instead of dying, she became a mermaid who passed judgment on sailors who came her way. To everyone she met, she asked, "Is Alexander the king alive?" If the sailor replied, "He lives and reigns and conquers the world," she would permit the ship to sail on safely. But any less pleasing answer would cause her to become a monster and send the vessel and its entire crew to the bottom of the sea. I borrowed from this story to paint Katerina's fate—it seemed fitting for her memory to take on mythic proportions, just as Alexander's did.
Nor did it seem realistic for Alexander's half sister, Cynane—another of my favorites—to settle down to bake honey cakes for a husband and children. And history agrees with us here. After her disastrous marriage to her cousin Amyntas, she never remarried but remained a warrior until the end. As the ancient Macedonian historian Polyaenus wrote, "Cynane, the daughter of Philip, was famous for her military knowledge: she conducted armies, and in the field charged at the head of them. In an engagement with the Illyrians, she with her own hand slew Caeria, their queen; and with great slaughter defeated the Illyrian army." It seems right, somehow, for her to ride off into the sunset with Alecta and her own battalion of female warriors.
There has been much speculation about the nature of Alexander's relationship with Hephaestion. No ancient sources state that the relationship was sexual, either because it wasn't, or because it was and gay relationships were so commonplace no one cared or thought that was unusual. Whatever the case sexually, Alexander was so close to Hephaestion he considered him to be his other self. When a captured Persian queen mother went up to the two gorgeous young men waiting to receive her and bowed down before Hephaestion, everyone burst out laughing, causing the queen to turn bright red. But Alexander reassured her, "You were not mistaken, Mother; this man, too, is Alexander." When Alexander poured a libation at the tomb of Achilles at Troy, he remarked how fortunate Achilles was to have had such a faithful friend as Patroclus, clearly a reference to Hephaestion.
Whether they were gay or not, both Alexander and Hephaestion married women, Alexander three times to secure political alliances. His first wife, the one he truly loved according to all accounts, was named Roxana, daughter of a Bactrian king. Though Zofia is an invention, I imagined Roxana might have been her own sister, lost and then found again, who, later in life, met and fell in love with Alexander and fulfilled our prophecy of Zofia's blood mingling with that of Alexander, but in a whole new way.
Throughout this book, a major thread—pardon the pun—is that of fate versus free will, a question that has intrigued writers from ancient Greek dramatists to Shakespeare. Many of us ask ourselves, "Is everything in our lives fated and we are just going through the motions of what must be? And if so, what's the point?" I look at it this way: some things are indeed "fated." The bodies we are born with. The parents who welcome us into the world—or not. The people we meet. The situations that appear before us. Free will is what we do with these things. Free will is the essence of what it means to be human, and while genetics and circumstances may set up the playing field, it is our own choices that define who we are and how far we run.
Reign of Serpents
by Eleanor Herman
Chapter One:
Zofia
Zofia, Princess of Sardis, leans forward and digs her heels into the beast's sides. Vata strains, shaking her mane, each beat of her powerful wings launching them higher into the sky. Zo doesn't need to look back to feel the dark energy of the thing—the horrifying, shadowy form hurtling toward her, gaining on her.
A roar splits the morning air.
Vata lets out a terrible whinny and banks hard to the left. Zo's heart plunges into her stomach as she begins to slip, frantically grasping tufts of cottony mane with both hands to steady herself. Far below her, the dry crumpled hills stretch to the horizon.
The Pegasus veers again, her mane whipping Zo's face as the wind stings her eyes and her vision blurs. A shadow falls on her and Vata. Something rakes her back. Sudden warmth floods through her tunic: blood.
She has only a second to register what has happened before white pain explodes across her body. In her agony, she is only vaguely aware that she has stopped holding on to Vata and that she's slipping, falling...
She slams into something brutally hard.
The pain goes numb.
Blackness drowns her.
* * *
Zo wakes in a sweat, flinging her arms and crying out in panic until she realizes that she is inside, lying on a little bed of straw and blankets. Panting heavily, she slowly understands that the flight and the chase were just a nightmare.
The Pegasus is not real—was never real.
Zo calms her breathing, taking in the moss-covered walls, daylight streaming in lazy diagonal shafts through the triangular cave openings. And for the third time that week, Zo forces herself to steady her heartbeat after the dream of flying and of falling.
Both the pain and the dreams are, Zo was told, the result of an avalanche that nearly killed her. Since then, she has lost her sense of what happened. Of what was real. She has images of a destroyed village, full of ash and ravaged bones and bright blood.
Red on Zo's hands, smeared on her thighs.
She tenses on her straw pallet, closing her eyes. No, those memories do not belong together. The blood was not the villagers' blood. It was hers. Her unborn child's. A moan falls from Zo's lips.
"My child?" Cool fingers suddenly trace her cheek, and Zo grabs the wrinkled hand. Blinking, she stares into the cloudy, violet eyes of her savior: Kohinoor the soothsayer.
Gradually, Zo's pulse slows, and her grief subsides. Being around Kohinoor eases her because each time she catches a glimpse of the old woman, she's reminded of miracles.
For it had to have been fate that threw the two of them together, first as captive slaves, then again, when the nearly sightless soothsayer had found Zo, battered and swollen, beneath the stones of the avalanche. Kohinoor had brought her to her home in the Eastern Mountains, allowing her to rest and recover from her wounds. She was there in the night when Zo screamed as her bruised body healed. She was by her side when Zo stood again, taking painful practice steps. And she was there to hold her when Zo had asked the terrible question. Kohinoor was the one who'd gently informed Zo that her unborn baby was no more.
Now Zo leans into the old woman for comfort. The knowledge that she has lost her child still catches her off guard, making her feel as though a giant fist has knocked the wind out of her. The agonizing loss is like another avalanche of rocks crushing her chest, making it nearly impossible to breathe or to think. Her sweet child would never see the world, never inhale fresh air or feel Zo's warm, loving arms. Long ago, in the slave cage, Kohinoor predicted that if Zo ever saw Cosmas again, she would cause his death. And now their child—her permanent link to him—is gone. So much is gone.
Kohinoor helps Zo sit up and hands her a warm mug of tea. Slowly, Zo sips the brew, which tastes like earth and roots and smells like fall leaves. Warmth curls inside her, soothing her and calming her heart.
"Better?" Kohinoor asks.
Zo nods but can't speak. An overwhelming weariness is settling upon her. For even when she sleeps, she does not rest. She battles the dark shadow of despair that threatens to engulf her when she thinks too much about all she has lost on this endless journey: her life as princess of Sardis; her five-year-old sister, Roxana, killed by the slave traders; Cosmas, the man she loves; and their baby. It is too much.
Before she can finish her tea, she lets sleep take her.
* * *
When Zo wakes, Kohinoor is gone. She's not sure how much time has passed, but from the absence of sunlight through the cave's roof, she guesses it is dusk. A tendril of fear rises within her—she hasn't woken up alone before. Kohinoor has always been here. She tells the fear to go away, that Kohinoor will be back soon, either from the fields where she picks herbs or the stream where she traps fish.
From somewhere far away, Zo thinks she hears a drumlike pounding. Sitting up, she frowns, wondering what it is and why she's never heard it before. Though steady, the sound is muted, like the world's heart beating in the center of the earth. Curious, she stands and is grateful that her legs can again bear her weight after a month of healing and training. They no longer ache with movement. In fact, she feels strong. Much stronger than she has been in weeks.
She pours herself some water and drinks as the beat continues. What is it? Could it be the answer she seeks—the truth behind the dreams of flying beasts and monsters? Heart pounding with dread and hope, she feels around on the large table for the fire starter kit as if she were as blind as Kohinoor herself. She strikes iron on flint and soon holds a blazing torch that illuminates the cave. Her pallet rests on one side, Kohinoor's on the other. In between are tables with crockery, jugs, and mortars. Baskets of many different shapes line one wall.
She follows the sound to the back of the cave—nothing there. But through the solid wall of stone, she can still hear the rhythm. Turning her head, Zo presses her ear to the wall, and that's when she sees it. A small, dark opening in the shadowy corner, invisible unless you are looking for it.
"Kohinoor?" Zo calls. "Are you there?"
No response, but the beating is crisper now, louder, a drumbeat calling her to action. Curious, she ducks her head and steps through the opening into the passage. It is so narrow she has to angle her shoulders to get through, and even then she brushes against rock. She follows the passage for several minutes, the torch's light illuminating only a few feet in front of her at a time. Part of her says to go back to the cave, to lie down again before she exhausts herself. But a bigger part tells her to keep going. For the first time since waking in Kohinoor's cave she feels...alive. More awake. More herself. She's tired of days spent sleeping or pacing around the cave and is eager to see something new. To maybe, even, learn more about the Eastern Mountains and their dark secrets.
At last, the passage opens into another cavern, dark except for the torch she holds, and there, in the center, sits Kohinoor. She hunches over a wooden plank, hammering a peg into one end. Relief sighs through Zo, followed quickly by another emotion: disappointment. There are no answers here, no clues to ancient mysteries. Just an old woman building something.
The hammering suddenly stops.
"You're here," Kohinoor says. It isn't a question.
"I am," Zo says. She hesitates before drawing closer, feeling as though she's stumbled upon something private. The old woman has already done so much for her. She doesn't want to be more of a burden.
"I woke and you weren't there," Zo rushes to explain. "And then I heard the hammering and followed it. I'm sorry to disturb your work."
"No harm. Shall we return? I will make you more tea."
"That's all right," Zo said. "Finish what you're doing. What are you making?"
"A table." Kohinoor suddenly slams her hammer into the peg again, and Zo starts as the sound rings around the cave. She is constantly astonished at the soothsayer's ability to see without eyes, and at the strength that allows such a seemingly frail old woman to lift heavy pails of water or this weighty mallet. As the woman returns to her work, Zo raises her torch and sees paintings brushed onto the rough walls.
"What are these?" she mutters to herself, walking over to hold the light closer.
"Paintings from the Hunor," Kohinoor says without turning her head from her woodwork. How can the old woman know what the paintings are without seeing them? Zo shivers.
The Hunor, Zo knows, are an ancient tribe here, in the Eastern Mountains. With her fingertips, she traces a green snake curling around a lotus flower. A few paces away, she sees men with horns dancing around a blazing pyre. Serpents in tall waves racing toward a city of temples and palaces. Three old women weaving on a giant loom atop a hill.
Their colors faded, these images are clearly things of the ancient past, but for some unknown reason, they make Zo's heart pound. It's as though time has instilled a sense of weight to them, a thick patina of importance. Of truth.
The paintings curve around the entire length of the cave wall, and she follows them, her heart hammering in her chest. And then—
Zo gasps. There on the wall, flickering in her torchlight, is a Pegasus, white wings outspread, climbing into the sky. A girl with long dark hair, arms flailing in panic, has just fallen off its back and plummets to the earth. Falling. Falling.
Just like in her dream.
Next to the image of the Pegasus stands a walled city, Persepolis from the looks of it. There, by the gate, is King Artaxerxes's famous Tower of the Sun and Moon, with its great horned battlements, cracked in two, soldiers tumbling out as flames explode in all directions.
Zo can feel her pulse in her throat now. These paintings are ancient, centuries old at least.
But she knows that the tower was constructed within the past couple of years. Are these paintings some sort of prophecy...or warning?
She stares at the flaming tower in Persia's capital. The destruction. The ruin. The tiny figures of people, fleeing. As if in a trance, she reaches the last image and holds the torch close. A winged child, its arms encircling a wax tablet, rises from the earth as darkness descends from above.
She doesn't understand what that last symbol represents, but she senses that it's important—that it shows the culmination of this...this prophecy. Of the fall of Persepolis.
"Kohinoor," she breathes. "What is this?"
The hammering stops. "As I said, pictures from the Hunor."
"Yes, but what do they mean?"
"You know what they mean, child." Kohinoor's voice is but a rasp. "Danger breeds in the heart of Persia."
Zo's blood turns to ice as she remembers the rumors she and Ochus heard at taverns along the Royal Road. Entire villages reduced to ashes. Missing couriers. Empty farmhouses. Vanished horses and oxen. And the village she herself had wandered into the day of her injuries. Doors yanked off hinges. Bloody streaks on the ground. And below a shifting cloud of flies, a heap of bones—human and animal—gouged with deep teeth marks. She had thought, during these weeks of healing, that perhaps that memory, like the Pegasus, had been a dream. A fantasy born of rocks hitting her head.
But if these paintings are true, if they are prophecy...could it be that there was no avalanche? That Kohinoor had found her, unconscious and bloody among rocks, and had assumed she had been caught in a surging tide of stones? That Zo's dream of flying and falling was no dream at all, but a memory? That the Pegasus was real? And the creature that had raked her back with sharp talons... Was it a Spirit Eater?
"But who...what is doing this?" she asks.
The old woman turns sightless eyes to Zo and croons eerily, as if singing a lullaby to a baby, "Spirit Eaters are doing this, girl. The Spirit Eaters' hunger is sharp."
Spirit Eaters. Months ago, on the slave cart, Kohinoor had told Zo it was fated for her blood to mix with that of Prince Alexander of Macedon. The only way to undo the threads of fate that have been woven for you is to find the Spirit Eaters who can negotiate with those goddesses who spin out, weave, and cut the threads of our fate, she had said.
Where do I find these Spirit Eaters? Zo asked.
If they still exist, you will find them in the Eastern Mountains. That is where the Spirit Eaters sprang up from a fissure in the rocks, and there they still live. That is where you must go.
Zo had thought of these magical beings as gods, not monsters, and ridden east with Ochus to find them. Thinking once more of the pile of bones in the abandoned village, she realizes she almost did find them. Or they almost found her.
Mouth dry, Zo licks her lips. "Then we must go and tell the king. We must tell him what is to come before there is more death and loss."
The old woman sets down her hammer and looks up, her smile revealing a few brown teeth.
"Must we? All right, child, let us return to our living quarters to discuss it."
Her calm response unnerves Zo. It's as though she thinks Zo is addled, that she doesn't believe her...and maybe she is right. Maybe there was no Pegasus, no falling from the sky. Zo does remember something about an avalanche, doesn't she? What happened the day Kohinoor found her?
She shakes her head in frustration. There's so much in her mind, and though her thoughts feel sharper than they have in many days, they are still somewhat blunt at the edges, like a dull sword. It's as though she's been living life at the edge of sleep, as though she were downing a sleeping potion instead of water or...instead of tea.
A terrible thought crosses Zo's mind, and once she thinks it, it cannot be unthought.
"Come," the old woman urges. "The passage is over here." Her dry gnarled hand grabs Zo's wrist. The skin is leathery, like that of a crocodile. It's all Zo can do not to pull away in disgust. But Kohinoor rescued her, nursed her back to health. In all likelihood, Zo would have died without her.
So she allows the soothsayer to guide her to the opening in the wall. Just before she ducks into it, Zo notices something glinting in her torchlight: a large cage.
"What's this for?" she asks.
Kohinoor blinks. "I see that soon a dog will come here looking for food, and I will make him my companion. That is where I will keep him until he learns that this is his home."
But the cage is taller than Zo. A dog wouldn't need a cage that high.
"Did you make this yourself?" Zo asks.
The soothsayer laughs, not answering her, and starts down the narrow winding tunnel. As soon as they emerge in the cave below, Kohinoor sets about boiling water in her pot, throwing in leaves and powders, preparing the strengthening tea she has been giving Zo every day to ease her pains. But now the earthy scent makes her stomach roll.
When Kohinoor hands her the clay mug, Zo waves it away. "I...don't want any today."
Kohinoor pushes matted hair out of her face, and her bleary lavender eyes seem to stare at Zo beneath a furrowed brow. "You must drink. For your health." Her rasping voice suddenly seems as strong and deep as a man's.
Zo's discomfort grows. "Very well," she says, then pretends to sip. After a time, the old woman goes back to the jugs on the table, opening them, sniffing the contents, and exploring the inside with her bony fingers.
"You walked very far today, Princess. Are you not tired?" the old woman croaks.
Zo stares into her mug. "Yes," she whispers. She rearranges the straw on her pallet as if to sleep and silently tips the mug into it. A moment later, Kohinoor's clawed hand is there, ready to take the empty vessel from her.
"You rest," Kohinoor says. "I will go and gather more rosemary."
Obediently, Zo lies down and closes her eyes. She can feel the old woman staring at her a long moment, and then she hears her slip out of the cave.
This is the first afternoon Zo hasn't had any tea, and her blood hums with energy. Her thoughts are clear. All sense of lethargy and disorientation is gone. So. Her terrible suspicion is correct; Kohinoor has been drugging her...but for what purpose?
Zo sits up quickly. She doesn't know how much time she has before Kohinoor will be back.
Grabbing the torch, Zo again goes to the back of the cave, into the narrow passage, and back to the painted cavern. She first takes a closer look at the table Kohinoor was making. With her free hand, Zo pulls the wooden planks up, and feels her heart tumble into her stomach.
It's not a table—it's a cradle.
And at the moment, she feels a flutter in her womb. A tiny tremble.
Zo's hand flies to her belly. Her baby... Cosmas's baby.
She's still alive.
Zo almost sways with relief, happiness, and...horror. Not only has Kohinoor been drugging her—which Zo might have dismissed as a well-meaning attempt to prevent her from overexerting herself—but Kohinoor has been lying to her.
A cradle and a cage. One for the infant and one, Zo realizes with rapidly increasing horror, for her. The cage is for Zo.
She lurches back, and as she does, the torch sweeps an orange swath of light across the wall. The paintings are illuminated, and the winged child stands out in sharp relief. In the flickering light, the wings look as if they are beating the air. And as Zo stares at the prophecy, she feels her child kick.
Her heart now a hammer against her chest, her eyes flick to the image beside it: the girl, falling from the Pegasus—her. Her empire's capital burning. And a child, destined to save it from flames.
Her child.
She needs to get out of here, away from destroyed villages and soothsayers, away from iron cages and drugged tea. She must get to Persepolis, to tell the Great King about the missing villages and the warning on the wall.
To find answers.
To save her child.
Copyright © 2017 by Paper Lantern Lit LLC and Eleanor Herman
ISBN-13: 9781488030529
Dawn of Legends
Copyright © 2018 by Glasstown Entertainment LLC and Eleanor Herman
All rights reserved. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this ebook on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereafter invented, without the express written permission of the publisher, Harlequin Enterprises Limited, 22 Adelaide St. West, 40th Floor, Toronto, Ontario M5H 4E3, Canada.
This is a work of fiction. Names, characters, places and incidents are either the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events or locales is entirely coincidental. This edition published by arrangement with Harlequin Books S.A.
® and ™ are trademarks of the publisher. Trademarks indicated with ® are registered in the United States Patent and Trademark Office, the Canadian Intellectual Property Office and in other countries.
www.Harlequin.com
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{
"redpajama_set_name": "RedPajamaBook"
}
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Back in February this year I posted this chart here at TradingView to diagram the structure of the Crude Oil market and its potential upside along with the overhead resistance and the general timeframe to expect based on the chart at the time.
The method I have outlined here is logical and not subjective, but what I do call this is "guessing" because anytime you deal with the future you are dealing with guesses, plain and simple. What I noticed back in January is that crude oil had accumulated for many months from the low in 2016 and formed 13 months of time at one price (the mode) at the $48.25 level. Once the market climbed above or range expanded above that level on a monthly basis, the market was positioned to advance for 13 months with some strong overhead resistance from the breakdown level in 2014 at the $70-$75 level where I have marked "HEAVY RESISTANCE" in the red rectangle .
The advance did end up adhering to the time and to the resistance area . Typically after time runs out for the rally, there is a move BACK TO THE MODE at the $48 level within 13 months. So far it has done that in only 3.
Given that sentiment was so extreme at the high with heavy long positions by traders and short positions by commercial traders, this decline is testing even the most die-hard bull.
If at the end of December the monthly range is less than November, we can look to go long on a break above previous day's highs with stops under the 5-day low.
And join us in the Key Hidden Levels chat room here at TradingView to get additional updates. Or click "agree" and you'll receive updates by email automatically.
@TholeThilis, Sideways chop for a month or two from here. I'm constantly posting daily charts at Key Hidden Levels (chat room here at TradingView) about the technical structure.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Does Liam Smith Have the Goods to Beat Canelo?
By: Brandon Bernica
This week, boxing groaned collectively when Golden Boy Promotions announced Canelo Alvarez's intentions to fights Gennady Golovkin…in 15 months. In the eyes of most fans, this superfight needs about as much marinade as a perfectly cooked Thanksgiving turkey – forget the spices and rubs, you just want to eat it as soon as possible. Even a recent string of thrilling victories could not save Canelo from seething backlash and disappointment.
To make matters worse, Golden Boy revealed Canelo's next opponent to the chagrin of boxing's devoted: Liam Smith of Liverpool. Naturally, an uproar sparked due to Smith's anonymity to casual fans. Overreaction typically ensues when top-tier opponents are neglected for stars like Canelo. My curiosity wonders whether Smith holds up as a quality-enough opponent, even after the hysteria dies down.
At first glance, his unscathed record speaks to consistency and capability. Of course, many of his past fights involved overmatched, homegrown foes, and his resume desperately lacks big names to cement his talent. Many consider him a "paper champion" – one who falls upon a title by the fortune of avoiding the division's best fighters. Labeling someone a fraud for clearing out a novice queue of opponents is unfair; regardless, the jury remains out until England's own passes more staunch tests. At a prime 27 years old, however, you still have to respect his potential.
So, why did Smith fly under the radar with the stealth of a notorious bank robber? That could be due in part to his lineage. He is the 2nd youngest of four brothers, all high-level fighters. His oldest brother, Paul, formerly starred on the boxing reality show "The Contender" and challenged for a world title on numerous occasions. Stephen, the second eldest, defends the WBC Silver Super Middleweight Championship. Perhaps the golden child of the clan, Callum, projects to stardom as an undefeated super middleweight prospect. With the "Fighting Smiths" as siblings, Liam easily finds himself lost in the shuffle. Being constantly contextualized with his brother's accomplishments raises the bar for distinguishing himself among them.
If one saving grace remains in the cards for Liam, it's the most obvious –he can fight. Smith acquits himself very well in the ring. He won't overwhelm you with explosiveness –perhaps to a flawed degree – but he checks out as a well-rounded fighter. Each fight, he aims at pushing the tempo and initiating the action. Whether it's short hooks or a favored uppercut-to-straight right hand, Smith stays on balance. This does leave him susceptible to sustained counterattacks in the pocket, and combined with an absence of head movement, he makes himself a target for anyone tough enough to survive his offensive assault.
Let's take this full-circle and evaluate how he matches up against Canelo. Canelo will thrive on the counterpunching opportunities provided by the hungry Smith, just as he did against James Kirkland and Miguel Cotto. The threat of Smith's wide-ranging arsenal will stick at the forefront of Canelo's mind throughout the night, but that's nothing smarts and timing couldn't overcome. Punching power and experience also tip in the Mexican's favor. What does Smith have that could riddle Canelo enough to capture victory. Intangibles. He's from a boxing family, he's bold, and he's unfamiliar with the losing taste as a professional. With so much pressure on Alvarez to win every fight until the Golovkin showdown, Smith will enjoy the freedom of playing underdog. These intangibles in the hands of a world titlist can be dangerous enough to pull the stunning upset. And with over half of his wins earned by knockout, one punch might be enough to change both of their fates.
I hear you, boxing fans. This is not a PPV worthy fight. This isn't the win that will change Canelo's life. But it isn't reason to fully dismiss Liam Smith. His resolve and skills can invoke some interesting exchanges with Canelo, making for an entertaining affair. As opposed to Canelo's last fight – a spectacular knockout of natural welterweight Amir Khan – Smith will be a sturdy, game foe that is more acclimated to the weight. Just as fans at the NBA Draft jeer every foreign player they've never heard of, boxing fans on this side of the Atlantic have difficulty giving non-American fighters a chance to prove their merit. Yes, it's not Golovkin (unfortunately), but without much to lose, he has a shot to carry the Smith name to new heights.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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package com.damienfremont.blog;
import java.util.Calendar;
import javax.validation.constraints.NotNull;
import javax.validation.constraints.Past;
import javax.validation.constraints.Size;
public class PersonModel {
@NotNull
@Size(min = 1, max = 16)
private String firstName;
@NotNull
@Size(min = 1, max = 16)
private String lastName;
@NotNull
@Past
private Calendar birthDate;
public PersonModel(String firstName, String lastName, Calendar birthDate) {
super();
this.firstName = firstName;
this.lastName = lastName;
this.birthDate = birthDate;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,509
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\section{Introduction and statement of the results.}
Suppose that $N$ is a sufficiently large integer and denote
\[
J(n) = \sum_{p_1 + p_2 = n} \log p_1 \log p_2 .
\]
(From this place the letter $p$, with or without subscripts, is reserved for primes.)
It is expected that if $n$ is a large even integer then $J(n) \sim c_0 \lambda (n) n $, where
\begin{equation} \label{+++1}
\lambda(k) = \prod_{\substack{p \mid k \\ p > 2}} \frac{p-1}{p-2} , \qquad
c_0 = 2 \prod_{p>2} \left( 1 - \frac{1}{(p-1)^2} \right) .
\end{equation}
This conjecture has not been proved so far, but using the Hardy--Littlewood circle method
and Vinogradov's method for estimating exponential sums over primes
(see, for example, Vaughan~\cite{Vaug}, Ch.~2), one can find that
\begin{equation} \label{+++2}
\sum_{\substack{n \le N \\ 2 \mid n}} \left| J(n) - c_0 \lambda (n) n \right|
\ll N^2 \mathcal L^{-A} ,
\end{equation}
where $A>0$ is an arbitrarily large constant and $\mathcal L = \log N$.
Let $r(k)$ be the number of solutions of the equation $x_1^2 + x_2^2 = k$ in integers $x_1, x_2$.
One of the classical problems in prime number theory is the Hardy--Littlewood problem
concerning the representation of large integers as a sum of two squares and a prime.
It was solved by Linnik (see \cite{Linnik}) and related problems have been studied
by Linnik, Hooley and other mathematicians.
For more information we refer the reader to Hooley's book \cite{Hooley},~Ch.5.
In particular, one can show that
\begin{equation} \label{+++3}
\sum_{p \le N} r(p - 1) =
\pi N \mathcal L^{-1}
\prod_{p > 2} \left( 1 + \frac{\chi(p)}{p(p-1)} \right)
+ O \left( N \mathcal L^{-1-\theta_0} \left( \log \mathcal L \right)^5 \right) ,
\end{equation}
where $ \chi(k)$ is the non-principal character modulo 4 and
\begin{equation} \label{+++4}
\theta_0 = \frac{1}{2} - \frac{1}{4} e \log 2 = 0.0029\dots .
\end{equation}
Let $\tau (k)$ be the number of positive divisors of $k$.
Linnik~\cite{Linnik}
(see also Halberstam and Richert~\cite{H-R},~Ch.~3.5.)
solved the Titchmarsh divisor problem and proved
that
\begin{equation} \label{+++5}
\sum_{p \le N} \tau(p - 1) = c_0 N +
O \left( N \mathcal L^{-1} \log \mathcal L \right) , \qquad
c_0 = \prod_p \left( 1 + \frac{1}{p (p-1)}\right) .
\end{equation}
We note that sharper versions of \eqref{+++3} and \eqref{+++5} are known at present
(see Bredihin~\cite{Bredihin},
Bombieri, Friedlander and Iwaniec~\cite{BoFrIw}) and Fouvry~\cite{Fouvry}.
In this paper we state two theorems which are, in some sense, combinations of
\eqref{+++2}, \eqref{+++3} and respectively \eqref{+++2}, \eqref{+++5}.
Denote
\begin{equation} \label{+++5.5}
\mathcal R(n) = \sum_{p_1 + p_2 = n} r(p_1 - 1) \log p_1 \log p_2 .
\end{equation}
After certain formal calculations one may conjecture that for any
sufficiently large even $n$ the quantity
$\mathcal R(n)$ is asymptotically equal to
\begin{equation} \label{+++6}
\mathcal M_{\mathcal R} (n)
= \pi c_0 n
\prod_{p \mid n-1} \left( 1 - \frac{\chi(p)}{p} \right)
\prod_{\substack{p \mid n \\ p > 2 }} \left( 1 + \frac{p + \chi(p) }{p(p-2)} \right)
\prod_{p \nmid n(n-1)} \left( 1 + \frac{2 \chi(p)}{p(p-2)} \right) .
\end{equation}
Our first result is the following:
\begin{theorem} \label{T1}
Suppose that $\theta_0$ is the constant defined by \eqref{+++4}. Then
we have
\begin{equation} \label{+++7}
\sum_{\substack{ n \le N \\ 2 \mid n } }
\left|
\mathcal R(n) - \mathcal M_{\mathcal R} (n)
\right|
\ll N^2 \mathcal L^{- \theta_0} \left( \log \mathcal L \right)^6 .
\end{equation}
\end{theorem}
It is clear that $ n \left( \log \log (10 n) \right)^{-2} \ll \mathcal M_{\mathcal R} (n) \ll n \left( \log \log (10 n) \right)^2$. Also, from \eqref{+++7} it follows that for any positive constant $\theta < \theta_0$
the number of even $n \le N$ for which
$\left| \mathcal R(n) - \mathcal M_{\mathcal R} (n) \right| > N \mathcal L^{-\theta} $
is $ O \left(N \mathcal L^{- (\theta_0 - \theta) } (\log \mathcal L )^6 \right)$.
So, in other words, $\mathcal R(n)$ is close to
$ \mathcal M_{\mathcal R} (n) $ for almost all even $n$.
Theorem~\ref{T1} is related to a recent result of K. Matom\"aki~\cite{Mato}. It is shown in \cite{Mato} that
the number of integers $n \le N$ satisfying $n \equiv 0 \; \text{or} \; 4 \pmod{6} $ and
that cannot be represented as a sum of two primes, one of which of the form
$k^2 + l^2 +1$, is $O \left( N \mathcal L^{-A} \right)$, where $A$ is an arbitrarily large constant.
So Matom\"aki's estimate for the cardinality of this exceptional set is stronger then ours, but her method
does not provide so sharp information about the number of such representations.
Our second result is concerning the quantity
\[
\mathcal T(n)
= \sum_{p_1 + p_2 = n} \tau(p_1 - 1) \log p_1 \log p_2 .
\]
Again, after certain formal calculations, one may conclude that
$\mathcal T(n)$ should be asymptotically equal to
\[
\mathcal M_{\mathcal T}(n) = c_0 n \log n
\prod_{p \mid n-1} \left( 1 - \frac{1}{p} \right)
\prod_{\substack{p \mid n \\ p > 2 }} \left( 1 + \frac{p + 1 }{p(p-2)} \right)
\prod_{p \nmid n(n-1)} \left( 1 + \frac{2}{p(p-2)} \right) .
\]
We can establish:
\begin{theorem} \label{T2}
The following estimate holds
\[
\sum_{\substack{ n \le N \\ 2 \mid n } }
\left|
\mathcal T(n) - \mathcal M_{\mathcal T} (n)
\right|
\ll N^2 \left( \log \mathcal L \right)^3 .
\]
\end{theorem}
We note that
$ n \, \log n \, \left( \log \log (10 n) \right)^{-2} \ll \mathcal M_{\mathcal T} (n) \ll n \, \log n \,
\left( \log \log (10 n) \right)^2 $,
so the quantity $\mathcal T(n)$ is close to $ \mathcal M_{\mathcal T} (n) $
for almost all even $n$.
We prove only Theorem~\ref{T1}. The proof of Theorem~\ref{T2} is similar and simpler.
\section{Some lemmas.}
Suppose that $n \le N$ and let $k$ and $l$ be integers with
$(k,l)=1$ (as usual, $(k,l)$ stands for the greatest common factor
of $k$ and $l$).
Let $\mathcal I $ be the set of all subintervals of the interval $[1, N]$ and let $I \in \mathcal I$.
We denote
\begin{align}
J_{k, l}(n; I)
& =
\sum_{\substack{p_1 + p_2 = n \\ p_1 \equiv l \pmod{k} \\ p_1 \in I}}
\log p_1 \log p_2 ,
\qquad
J_{k, l}(n) = J_{k, l} (n; [1, N]) ;
\label{+++8} \\
\mathfrak S_{k, l}(n)
& =
\begin{cases}
c_0
\lambda(nk)
\qquad
& \text{if} \;\; (k, n-l) = 1 \; \; \text{and} \;\;2 \mid n , \\
0
& \text{otherwise} ;
\end{cases}
\label{+++9} \\
\Phi(n; I)
& =
\sum_{\substack{m_1 + m_2 = n \\ m_1 \in I}} 1 .
\label{+++9.5}
\end{align}
Our first lemma states that the expected formula for $J_{k, l}(n; I) $ is true on average with respect to
$k \le \sqrt{N} \mathcal L^{-B}$ and $n \le N$ and uniformly for $l$ and $I$. More precisely, we have
\begin{lemma} \label{L1}
For any constant $A>0$ there exist $B=B(A) > 0$ such that
\[
\sum_{k \le \sqrt{N} \mathcal L^{-B} } \max_{(l, k)=1} \max_{I \in \mathcal I}
\sum_{ n \le N } \left|
J_{k, l}(n; I) - \frac{\mathfrak S_{k, l}(n)}{\varphi(k)} \Phi(n; I)
\right|
\ll N^2 \mathcal L^{-A} .
\]
\end{lemma}
This lemma is very similar to results of Mikawa~\cite{Mikawa} and Laporta~\cite{Laporta}.
These authors study the equation $p_1 - p_2 = n$ and without the condition $p_1 \in I$.
However inspecting the arguments presented in \cite{Laporta},
the reader will readily see that the proof of Lemma~\ref{L1} can be obtained is the same manner.
The next lemma is an immediate consequence from a classical sieve theory result
(see \cite{H-R},~Ch.~2, Th.~2.4).
\begin{lemma} \label{L2}
Suppose that $h$ is an integer such that $1 \le |h| \le N$. Then the number of
solutions of the equation $p_1 - p_2 = h$ in primes $p_1, p_2 \le N$ is
$O \left( N \mathcal L^{-2} \log \mathcal L \right)$, where the constant in the Landau symbol is absolute.
\end{lemma}
The next two lemmas are due to C.Hooley and play an essential role in the proof of \eqref{+++3},
as well as in the solutions of other related problems.
\begin{lemma} \label{L3}
Suppose that $\omega > 0 $ is a constant and let $F_{\omega}(N)$ be the number of primes $p \le N$ such that
$p-1$ has a divisor lying between $ \sqrt{N} \mathcal L^{-\omega} $ and $ \sqrt{N} \mathcal L^{\omega} $.
Then we have
\[
F_{\omega}(N) \ll N \mathcal L^{-1 - 2 \theta_0} \left( \log \mathcal L \right)^3 ,
\]
where $\theta_0$ is defined by \eqref{+++4} and where
the constant in the Vinogradov symbol depends only on $\omega$.
\end{lemma}
\begin{lemma} \label{L4}
Suppose that $\omega > 0 $ is a constant. Then we have
\[
\sum_{p \le N} \left|
\sum_{\substack{ d \mid p - 1 \\\sqrt{N} \mathcal L^{-\omega} < d < \sqrt{N} \mathcal L^{\omega} } }
\chi(d)
\right|^2
\ll N \mathcal L^{-1} \left( \log \mathcal L \right)^7 ,
\]
where the constant in the Vinogradov symbol depends only on $\omega$.
\end{lemma}
The proofs of very similar results (with $\omega = 48$ and with the condition $d \mid N - p$
rather than $d \mid p-1$) are available in \cite{Hooley},~Ch.5 and the reader will easily see that
the method used there yields also the validity of Lemmas~\ref{L3} and~\ref{L4}.
\section{Proof of Theorem~\ref{T1}.}
\subsection{Beginning.}
Denote by $\mathcal E$ the sum on the left-hand side of \eqref{+++7} and put
\begin{equation} \label{+++10}
D = \sqrt{N} \mathcal L^{-1 - B(1)} ,
\end{equation}
where $B(A)$ is specified in Lemma~\ref{L1}.
Using \eqref{+++5.5} and the well-known identity
$
r(m) = 4 \sum_{d \mid m} \chi(d)
$
we find
\begin{equation} \label{+++11}
\mathcal R(n) = 4
\sum_{p_1 + p_2 = n} \left(
\sum_{d \mid p_1 -1} \chi(d) \right)
\log p_1 \log p_2
= 4 \left( S_1(n) + S_2(n) + S_3(n) \right) ,
\end{equation}
where
\begin{align}
S_1(n)
&=
\sum_{p_1 + p_2 = n} \left(
\sum_{\substack{d \mid p_1 -1 \\ d \le D}} \chi(d) \right)
\log p_1 \log p_2
\label{+++12} \\
S_2(n)
&=
\sum_{p_1 + p_2 = n} \left(
\sum_{\substack{d \mid p_1 -1 \\ D < d < N/D}} \chi(d) \right)
\log p_1 \log p_2
\label{+++13} \\
S_3(n)
&=
\sum_{p_1 + p_2 = n} \left(
\sum_{\substack{d \mid p_1 -1 \\ d \ge N/D}} \chi(d) \right)
\log p_1 \log p_2
\label{+++14}
\end{align}
Therefore from \eqref{+++7} and \eqref{+++11} it follows
\begin{equation} \label{+++15}
\mathcal E \ll
\mathcal E_1 + \mathcal E_2 + \mathcal E_3 ,
\end{equation}
where
\begin{equation} \label{+++16}
\mathcal E_1
=
\sum_{\substack{n \le N \\ 2 \mid n}}
\left|
4 S_1(n) - \mathcal M_{\mathcal R}(n)
\right| ;
\qquad
\mathcal E_j
=
\sum_{\substack{n \le N \\ 2 \mid n }}
\left| S_j(n) \right| , \qquad j = 2,3.
\end{equation}
\subsection{The estimation of $\mathcal E_1$.}
Using \eqref{+++8}, \eqref{+++9.5}, \eqref{+++12} and bearing in mind Lemma~\ref{L1} we find
\[
S_1(n) = \sum_{d \le D} \chi(d) J_{d, 1}(n) = (n-1) S_1^{\prime}(n) + S_1^*(n) ,
\]
where
\begin{align}
S_1^{\prime}(n)
&
= \sum_{d \le D} \chi(d)
\frac{\mathfrak S_{d, 1}(n)}{\varphi(d)} ,
\label{+++17} \\
S_1^*(n)
& =
\sum_{d \le D} \chi(d)
\left( J_{d, 1}(n) - (n-1) \frac{\mathfrak S_{d, 1}(n)}{\varphi(d)}
\right) .
\label{+++17.5}
\end{align}
Hence
\begin{equation} \label{+++18}
\mathcal E_1 \ll \mathcal E_1^{\prime} + \mathcal E_1^* ,
\end{equation}
where
\begin{equation} \label{+++19}
\mathcal E_1^{\prime} = \sum_{\substack{n \le N \\ 2 \mid n}}
\left|
4 (n-1) S_1^{\prime}(n) - \mathcal M_{\mathcal R}(n) \right| , \qquad
\mathcal E_1^* = \sum_{\substack{n \le N \\ 2 \mid n}} \left|
S_1^*(n) \right| .
\end{equation}
By \eqref{+++10}, \eqref{+++17.5}, \eqref{+++19} and Lemma~\ref{L1} it follows that
\begin{equation} \label{+++20}
\mathcal E_1^* \ll N^2 \mathcal L^{-1} .
\end{equation}
Consider $\mathcal E_1^{\prime}$.
From \eqref{+++1}, \eqref{+++9} and \eqref{+++17} we find
\begin{equation} \label{+++21}
S_1^{\prime}(n) = c_0
\sum_{\substack{d \le D \\ (d, n-1)=1}} \frac{\chi(d)}{\varphi(d)}
\lambda (nd)
= c_0 \lambda(n)
\sum_{\substack{d \le D \\ (d, n-1)=1}} f_n(d) ,
\end{equation}
where
\begin{equation} \label{+++22}
f_n(d) = \frac{\chi(d)}{\varphi(d)}
\frac{\lambda (d)}{\lambda((n, d))} .
\end{equation}
Obviously the function $f_n(d)$ is multiplicative with respect to $d$ and
\begin{equation} \label{+++23}
f_n(d) \ll d^{-1} \left( \log \log (10 d) \right)^2
\end{equation}
uniformly with respect to $n$.
To evaluate the sum in right-hand side of \eqref{+++21} we consider the function
\[
F_n(s) = \sum_{\substack{d=1 \\ (d, n-1) = 1 }}^{\infty} f_n(d) d^{-s} .
\]
It is analytic in the half-plane $ Re \, (s) > 0 $ and we may represent it as an Euler product:
\[
F_n(s) = \prod_{p \nmid n-1} T_n(p, s) , \qquad T_n(p, s) = 1 + \sum_{l=1}^{\infty} f_n(p^l) p^{-ls} .
\]
From \eqref{+++1} and \eqref{+++22} we easily find
\[
f_n(p^l) =
\begin{cases}
\chi(p)^l \; p^{1-l} \; (p-1)^{-1}
& \text{if} \; \; p \mid n , \\
\chi(p)^l \; p^{1-l} \; (p-2)^{-1}
& \text{if} \; \; p \nmid n ;
\end{cases}
\]
and respectively
\[
T_n(p, s) =
\left( 1 - \frac{\chi(p)}{ p^{s+1}} \right)^{-1}
T_n^*(p, s) ,
\]
where
\[
T_n^*(p, s) =
\begin{cases}
1 + \chi(p) p^{-s-1} (p-1)^{-1}
& \quad \text{if} \quad p \mid n , \\
1 + 2 \chi(p) p^{-s-1} (p-2)^{-1}
& \quad \text{if} \quad p \nmid n .
\end{cases}
\]
Therefore
\begin{equation} \label{+++24}
F_n(s) = L(s+1, \chi) H_n(s)
\end{equation}
where $L(s, \chi) $ is the Dirichlet $L$-function corresponding to the character $\chi$ and
\begin{equation} \label{+++25}
H_n(s) =
\prod_{p \mid n-1}
\left( 1 - \frac{\chi(p)}{ p^{s+1}} \right)
\;
\prod_{p \mid n}
\left( 1 + \frac{\chi(p)}{ p^{s+1} (p-1)} \right)
\;
\prod_{p \nmid n(n-1)}
\left( 1 + \frac{2 \chi(p)}{ p^{s+1} (p-2)} \right) .
\end{equation}
From \eqref{+++24}, \eqref{+++25} we see that $F_n(s)$ has an analytic continuation to the half-plane
$Re \, (s) > -1$.
It is clear that $H_n(s) \ll n^{\varepsilon}$ for $|Re \, (s)| \ge -1/2$
(here and later $\varepsilon$ is an arbitrarily small positive number).
Also, it is well-known that in the same region
we have $L(s+1, \chi) \ll 1 + |Im \, (s)|^{1/6}$. Hence
\begin{equation} \label{+++26}
F_n(s) \ll N^{\varepsilon} \, T^{1/6} \qquad \text{if} \qquad Re \, (s) \ge -1/2 , \quad | Im \, (s) | \le T
\end{equation}
for any $T > 1 $.
We apply Perron's formula (see, for example \cite{Tenen},~Ch.~II.2) to find
\begin{equation} \label{+++27}
\sum_{\substack{d \le D \\ (d, n-1)=1}} f_n(d) = \frac{1}{2 \pi i}
\int_{\varkappa - i T}^{\varkappa + i T}
F_n(s) \frac{D^s}{s} d s +
O \left( \sum_{d=1}^{\infty}
\frac{D^{\varkappa} \, |f_n(d)|}{ d^{\varkappa} \left(1 + T \left| \log \frac{D}{d} \right| \right)} \right)
\end{equation}
with $\varkappa = 1/10$ and $T = N^{3/4}$.
Using \eqref{+++10} and \eqref{+++23} one can easily verify that
the remainder term in \eqref{+++27} is $O \left( N^{-1/20}\right) $.
To evaluate the integral in \eqref{+++27}
we apply Cauchy's theorem. The residue of the integrand at $ s=0 $ equals
\begin{equation} \label{+++28}
F_n(0) =
\frac{\pi}{4}
\prod_{p \mid n-1} \left( 1 - \frac{\chi(p)}{p} \right)
\prod_{p \mid n} \left( 1 + \frac{\chi(p)}{p(p-1)} \right)
\prod_{p \nmid n(n-1) } \left( 1 + \frac{2 \chi(p)}{p(p-2)} \right) .
\end{equation}
Hence the main term in the right-hand side of \eqref{+++27} is equal to
\begin{equation} \label{+++29}
F_n(0)
+ \frac{1}{2 \pi i}
\left(
\int_{\varkappa - i T}^{-1/2 - i T} + \int_{-1/2 - i T}^{-1/2 + i T} + \int_{-1/2 + i T}^{\varkappa + i T}
\right)
F_n(s) \frac{D^s}{s} d s .
\end{equation}
Using \eqref{+++26} one can easily find that the contribution of the integrals in \eqref{+++29}
is $O \left( N^{-1/20}\right)$.
Therefore
\begin{equation} \label{+++30}
\sum_{\substack{d \le D \\ (d, n-1)=1}} f_n(d) = F_n(0) + O \left( N^{-1/20}\right) .
\end{equation}
From \eqref{+++1}, \eqref{+++6}, \eqref{+++19}, \eqref{+++21}, \eqref{+++28} and \eqref{+++30} it follows that
\[
\mathcal E_1^{\prime} \ll N^2 \mathcal L^{-1} .
\]
Hence, using \eqref{+++18} and \eqref{+++20} we get
\begin{equation} \label{+++31}
\mathcal E_1 \ll N^2 \mathcal L^{-1} .
\end{equation}
\subsection{The estimation of $\mathcal E_2$.}
Clearly, from \eqref{+++16} and Cauchy's inequality it follows that
\begin{equation} \label{+++32}
\mathcal E_2 \ll N^{1/2} \left( \sum_{n \le N} \left| S_2(n) \right|^2 \right)^{1/2}
= N^{1/2} \left( \mathcal E_2^{\prime} \right)^{1/2} ,
\end{equation}
say.
Using \eqref{+++13} we find
\begin{align}
\mathcal E_2^{\prime}
& =
\sum_{n \le N} \sum_{D < d, t < N/D}
\chi(d) \chi(t)
\sum_{\substack{p_1 + p_2 = n \\ p_1 \equiv 1 \pmod{d} }} \log p_1 \log p_2
\sum_{\substack{p_3 + p_4 = n \\ p_3 \equiv 1 \pmod{t} }} \log p_3 \log p_4
\notag \\
& \notag \\
& =
\sum_{p_1 + p_2 = p_3 + p_4 \le N} \log p_1 \log p_2 \log p_3 \log p_4
\sum_{\substack{D < d, t < N/D \\ d \mid p_1 - 1 , \;\; t \mid p_3 - 1}}
\chi(d) \chi(t)
\notag \\
& \notag \\
& \ll
\mathcal L^4 \,
\mathcal E_2^{\prime\prime} + N^{2 + \varepsilon} ,
\label{+++33}
\end{align}
where
\[
\mathcal E_2^{\prime\prime}
=
\sum_{\substack{p_1 + p_2 = p_3 + p_4 \\ p_1, p_2, p_3, p_4 \le N \\ p_1 \not= p_3}}
\left|
\sum_{\substack{D < d < N/D \\ d \mid p_1 - 1 }}
\chi(d)
\right|
\;
\left|
\sum_{\substack{D < t < N/D \\ t \mid p_3 - 1 }}
\chi(t)
\right| .
\]
Denote by $\mathcal F$ the set of primes $p \le N$ such that $p-1$ has a divisor
lying between $D$ and $N/D$.
Using the inequality $uv \le u^2 + v^2$
and taking into account the symmetry with respect to $d$ and $t$
we get
\begin{align}
\mathcal E_2^{\prime\prime}
& \ll
\sum_{\substack{p_1 + p_2 = p_3 + p_4 \\ p_1, p_2, p_4 \le N \\ p_1 \not= p_3 , \; p_3 \in \mathcal F }}
\left|
\sum_{\substack{D < d < N/D \\ d \mid p_1 - 1 }}
\chi(d)
\right|^2
\notag \\
& =
\sum_{p_1 \le N}
\left|
\sum_{\substack{D < d < N/D \\ d \mid p_1 - 1 }}
\chi(d)
\right|^2
\sum_{\substack{ p_3 \in \mathcal F \\ p_3 \not= p_1 }} \;
\sum_{\substack{p_2, p_4 \le N \\ p_4 - p_2 = p_1 - p_3}} 1 .
\label{+++33.5}
\end{align}
Applying Lemmas~\ref{L2} and \ref{L3} we find
\begin{equation} \label{+++33.6}
\sum_{\substack{p_3 \in \mathcal F \\ p_3 \not= p_1 }}
\sum_{\substack{p_2, p_4 \le N \\ p_4 - p_2 = p_1 - p_3}} 1
\ll
N \mathcal L^{-2} ( \log \mathcal L ) \sum_{p \in \mathcal F} 1 \ll
N^2 \mathcal L^{-3- 2 \theta_0} ( \log \mathcal L )^4
\end{equation}
and then using \eqref{+++33.5}, \eqref{+++33.6} and
Lemma~\ref{L4} we get
\begin{equation} \label{+++34}
\mathcal E_2^{\prime\prime}
\ll
N^2
\mathcal L^{-3 - 2 \theta_0} (\log \mathcal L)^4
\sum_{p \le N}
\left|
\sum_{\substack{D < d < N/D \\ d \mid p - 1 }}
\chi(d)
\right|^2
\ll N^3
\mathcal L^{ -4 - 2 \theta_0} (\log \mathcal L )^{11} .
\end{equation}
From \eqref{+++32}, \eqref{+++33} and \eqref{+++34} we conclude that
\begin{equation} \label{+++35}
\mathcal E_2 \ll N^2 \mathcal L^{-\theta_0 } \left( \log \mathcal L \right)^6 .
\end{equation}
\subsection{The estimation of $\mathcal E_3$}
From \eqref{+++14} it follows that
\begin{align}
S_3(n)
& =
\sum_{p_1 + p_2 = n} \log p_1 \log p_2
\sum_{\substack{ m \mid p_1 - 1 \\ \frac{p_1-1}{m} \ge \frac{N}{D} }} \chi \left( \frac{p_1 - 1 }{m} \right)
\notag \\
& =
\sum_{p_1 + p_2 = n} \log p_1 \log p_2
\sum_{j = \pm 1} \chi(j)
\sum_{\substack{m \le \frac{(p_1-1) D}{N} , \; \; 2 \mid m \\ p_1 \equiv 1 + jm \pmod{4m} }} 1 .
\notag
\end{align}
We change the order of summation and use \eqref{+++8} to find
\[
S_3(n) =
\sum_{\substack{m \le D \\ 2 \mid m }} \sum_{j = \pm 1} \chi(j)
J_{4m, 1+jm}(n, I_m) ,
\]
where $I_m$ denotes the interval $\left[1 + m N/D , N \right]$. Having in mind Lemma~\ref{L1}
we write
\begin{equation} \label{+++35.5}
S_3(n) = S_3^{\prime}(n) + S_3^*(n) ,
\end{equation}
where
\begin{align}
S_3^{\prime}(n)
& =
\sum_{\substack{ m \le D \\ 2 \mid m }} \sum_{j = \pm 1} \chi(j) \frac{\mathfrak S_{4m, 1 + jm}(n)}{\varphi (4m)}
\Phi(n, I_m) ,
\notag \\
S_3^*(n)
& =
\sum_{\substack{ m \le D \\ 2 \mid m }} \sum_{j = \pm 1} \chi(j)
\left(
J_{4m, 1+jm}(n, I_m) - \frac{\mathfrak S_{4m, 1 + jm}(n)}{\varphi (4m)} \Phi(n, I_m)
\right) .
\label{+++35.6}
\end{align}
Since $2 \mid n$ it follows from \eqref{+++9} that
\[
\mathfrak S_{4m, 1 + jm}(n) =
\begin{cases}
c_0 \lambda(4mn) & \text{if} \;\; (4m, n-1-jm)=1 , \\
0 & \text{otherwise} .
\end{cases}
\]
However the condition
$(4m, n-1-jm) = 1$ is independent of $j$
(from the set $\{ 1, -1 \} $) and therefore
$\mathfrak S_{4m, 1 + jm}(n) $ is independent of $j$ too. This means that
\[
S_3^{\prime}(n) = 0 .
\]
Hence, using \eqref{+++10}, \eqref{+++16}, \eqref{+++35.5}, \eqref{+++35.6} and Lemma~\ref{L1} we find
\begin{align}
\mathcal E_3
& \ll
\sum_{n \le N} \left| S_3^*(n) \right|
\notag \\
& \notag \\
& \ll
\sum_{\substack{m \le D \\ 2 \mid m }} \sum_{j = \pm 1} \sum_{n \le N}
\left|
J_{4m, 1+jm}(n, I_m) - \frac{\mathfrak S_{4m, 1 + jm}(n)}{\varphi (4m)} \Phi(n, I_m)
\right|
\notag \\
& \notag \\
& \ll
\sum_{k \le 4D} \max_{(l, k)=1} \max_{I \in \mathcal I} \sum_{n \le N}
\left|
J_{k, l}(n, I) - \frac{\mathfrak S_{k, l}(n)}{\varphi (k)} \Phi(n, I)
\right|
\notag \\
& \notag \\
& \ll
N^2 \mathcal L^{-1} .
\label{+++36}
\end{align}
The estimate \eqref{+++7} follows from \eqref{+++15},
\eqref{+++31}, \eqref{+++35} and \eqref{+++36}, so the theorem is proved.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 856
|
\section*{Results}
\begin{figure*}
\includegraphics[width=\textwidth]{PDF/SatEff.pdf}
\caption{\textbf{a},~Photon counting yields conversion response to applied MW field intensity, from thermal noise level ($2.1 {\cdot} 10^3\ \mathrm{phot} / \mathrm{s}$) to conversion saturation limit ($1.5 {\cdot} 10^9\ \mathrm{phot} / \mathrm{s}$). The observed non-thermal added noise is at the level of noise-equivalent temperature $T_{\mathrm{NE}} {=} 53\ \mathrm{K}$ and can be further filtered down to $T_{\mathrm{NE}} {=} 3.8\ \mathrm{K}$. \textbf{b},~Relative conversion efficiency (thermal noise subtracted) with ${>}1$ SNR and ${>}0.5$ efficiency achieved over $57\ \mathrm{dB}$ dynamic range of MW intensity -- from $4.0 {\cdot} 10^{-10}$ to $1.9 {\cdot} 10^{-4}\ \mathrm{W} / \mathrm{m}^2$. The efficiency is normalized to the highest measured value. In both cases we compare the results with a theoretical curve, where the overall efficiency is the only free parameter. Notably, the saturation intensity is well reproduced along with the shape of the curve.}
\label{sateff}
\end{figure*}
\paragraph{Conversion in warm Rydberg vapors}
Atomic vapors constitute a medium adequate for CW conversion . In our demonstration, a cylindrical vapor cell serves as a hot-atom Rydberg converter (Fig. \ref{qura}\textbf{a}). In our experiment, three focused optical fields determine an interaction region, where ground-state atoms are supplied from the remainder of the cell via thermal atomic motion, assuring that no depletion nor other long-term time-dependence occur, thus satisfying CW operational framework. These fields combined with a MW field realize coherent six-wave mixing (Fig. \ref{qura}\textbf{c}) with emission at $776\ \mathrm{nm}$. The conservation of energy and momentum, governing the wave mixing process, assure that temporal and spatial properties of converted photons are preserved. In opposition to previously presented conversion schemes \cite{Han_2018,Vogt_2019,Tu_2022}, this scheme does not require additional MW field, and all the introduced optical fields are spectrally different well enough to be separated with the use of free-space optics, even in a collinear configuration.
Considering field detunings from the energy level structure, we observe that similarly as in \cite{Tu_2022} the best conversion efficiency is achieved for off-resonant realization, as shown in the Fig. \ref{qura}\textbf{d} in relation to the EIT effect. Here the detuning applies to Doppler-broadened level structure, involving a range of atomic velocity classes. We particularly find that the detuning from level $55^2 \mathrm{D}_{5/2}$ plays an important role in increasing the efficiency of conversion, yielding a nearly fivefold improvement in efficiency at $\Delta_{55\mathrm{D}} {=} 20 {\cdot} 2 \pi\ \mathrm{MHz}$ in comparison to the resonant case.
\paragraph{Continuous-wave conversion}
Atomic conversion processes arise from atomic coherences, which in the quantum mechanical approach can be derived from state density matrices $\hat{\rho}$. With the incident weak MW field $E_\mathrm{MW}$ and strong drive fields we obtain an optical coherence $\rho_s$ and thus expect emission of the signal field as $E_s\sim\rho_s=\mathrm{Tr}(|5^2D_{5/2}\rangle\langle5^2P_{3/2}|\hat{\rho})\sim E_\mathrm{MW}$. The time evolution of the atomic state is governed by Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation,
\begin{equation}
\partial_t \hat{\rho} = \frac{1}{i \hbar} [\hat{H}, \hat{\rho}] + \mathcal{L}[\hat{\rho}],
\end{equation}
where by $\hat{H}$ we denote Hamiltonian and by $\mathcal{L}[{\cdot}]$ a superoperator responsible for spontaneous emission and other sources of decoherence. The pulsed approach to conversion assumes tailoring of the Hamiltonian $\hat{H} (t)$ (e.g. by sequentially turning on the driving fields and enabling the conversion for short time only) to maximize the coherence and in consequence the conversion output. This approach, showcased particularly in \cite{Tu_2022}, can be implemented in Doppler-free cold-atom system and leads to high, local conversion efficiency, albeit it is practically limited to operating for as short as $1/1000$ of time sequence, which narrows down the potential applications to the instances where the converter can be triggered to a short signal or where we aim to convert a strong classical field. The antipodal CW approach addresses this gap, enabling uniform, trigger-free conversion of weak or asynchronous signals.
In this approach for conversion, we consider the density matrix in steady state, $\partial_t \hat{\rho} (t) = 0$, similarly to EIT-based Rydberg electrometry systems. We employ the steady-state solution of the GKSL equation as the basis for the parametric numerical simulation utilized as an aid to interpret the experimental results we present. We consider a range of atomic velocity classes contributing to the conversion process and take into consideration the shape of the interaction volume.
\begin{figure*}
\includegraphics[width=\textwidth]{PDF/g2bw.pdf}
\caption{\textbf{a},~Conversion's dependence on MW field detuning (red curve), revealing a bandwidth of $\Gamma_{\mathrm{con}} {=} 16 {\cdot} 2 \pi\ \mathrm{MHz}$ FWHM. Interdependence on the detunings of MW and decoupling fields shows that with commensurate detuning of decoupling field the tunable bandwidth of conversion (dark curve) can be broadened to $59 {\cdot} 2 \pi\ \mathrm{MHz}$ FWHM. This is in agreement with the general theoretical prediction,~\textbf{i}~(numerical simulation). \textbf{b},~Photon autocorrelation (homodyne) measurement yields second order autocorrelation function of thermal photons, confirming the conversion of quantum thermal state. This experimental result agrees perfectly with parameter-free theory prediction based on the conversion band and measured noise properties. \textbf{c},~The decreasing of $g^{(2)} (0)$ with the introduction of coherent state photons rate $\bar{n}_{\mathrm{coh}}$, in agreement with the theory. \textbf{d},~Introducing far detuned ($\Delta \omega {=} 4 \Gamma_{\mathrm{con}} {=} 64 {\cdot} 2 \pi\ \mathrm{MHz}$) coherent MW field induces interference in the autocorrelation function with beat modulation frequency equal to the detuning.}
\label{results}
\end{figure*}
\paragraph{Dynamic range and efficiency}
With the use of a single-photon counter, we measure the converter's response to MW field intensity. As shown in the Fig. \ref{sateff}\textbf{a}, the converted signal ranges from $2.1 {\cdot} 10^3$ to $1.5 {\cdot} 10^9\ \mathrm{phot} / \mathrm{s}$. We identify the upper bound as saturation from energy levels shift, and the lower limit as free-space thermal photons coupling to the converter. We adjust for other sources of noise, directional and polarization coupling to the converter, and the measured conversion band. The electric field spectral density level of observed thermal radiation, $1.59\ \mathrm{nV}\mathrm{cm}^{-1}\mathrm{Hz}^{-1/2}$ (isotropic, both polarizations), agrees very well with the theoretical prediction of $1.64\ \mathrm{nV}\mathrm{cm}^{-1}\mathrm{Hz}^{-1/2}$ (see Methods for derivation). We note, however, that such a remarkable agreement occurs for the employed model that is simpler than the reality of the experiment, i.e. the presence of MW antenna near the converter is not accounted for.
We consider the efficiency of the implemented conversion: Fig. \ref{sateff}\textbf{b} shows relative conversion efficiency ${>}0.5$ and signal-to-noise ratio (SNR) ${>}1$ (in relation to thermal noise) for $57\ \mathrm{dB}$ of MW intensities (from $4.0 {\cdot} 10^{-10}$ to $1.9 {\cdot} 10^{-4}\ \mathrm{W} / \mathrm{m}^2$), which we announce as the converter's dynamic range. The data is presented with normalization to the highest measured value. The converter's response to MW field and conversion efficiency are well-predicted by the numerical simulation, where only the overall efficiency is parametrically matched. As the interaction volume is sub-wavelength to MW field, we choose not to follow the simple approach presented in \cite{Han_2018,Vogt_2019,Tu_2022} to get an absolute conversion efficiency, assuming sub-wavelength effective area. In this case, cavity-assisted conversion \cite{Kumar_2022} remains the only completely reliable method of absolute efficiency estimation.
\paragraph{Conversion band}
With the use of single photon counter we measure the conversion's dependence on MW frequency in the linear response regime (Fig. \ref{results}\textbf{a}), arriving at a bandwidth $\Gamma_{\mathrm{con}} = 16 {\cdot} 2 \pi\ \mathrm{MHz}$ FWHM (full width at half maximum). Further explorations reveal interdependence on MW and decoupling ($1258\ \mathrm{nm}$) fields detunings, that can be utilized as a means to fine-tune the converter to incoming MW field. We show that with such a procedure we are able to widen the tunable conversion bandwidth up to $59 {\cdot} 2 \pi\ \mathrm{MHz}$ FWHM.
\paragraph{Photonic conversion of thermal radiation}
We perform single-photon autocorrelation measurement (Fig. \ref{results}\textbf{b}) on the thermal radiation signal, confirming the Hanbury Brown and Twiss (HBT) effect of thermal photon bunching ($g^{(2)} (0) {>} 1$) for MW photons. The results obtained experimentally agree perfectly with the parameter-free theory drawn from the Wiener-Khinchin theorem,
\begin{equation}
g^{(1)}_{\mathrm{th}} (\tau) = \frac{1}{2 \pi} \int_{- \infty}^{\infty} \left| S(\omega) \right|^2 e^{- i \omega \tau} \mathrm{d} \omega,
\end{equation}
where $|S(\omega)|^2$ is normalized ($\int_{- \infty}^{\infty} \left| S(\omega) \right|^2 \mathrm{d} \omega {=} 2 \pi$) power spectral density, here measured in the Fig. \ref{results}\textbf{a} with the assumption that thermal radiation locally has white noise characteristics. Naturally following, $g^{(2)}_{\mathrm{th}} (\tau){=} 1 {+} \left| g^{(1)}_{\mathrm{th}} (\tau) \right|^2$. We consider measured added non-thermal noise to arrive at the relation applicable to the experiment (see Methods for full consideration). An alternative parametric fitting of exponent function, $g^{(2)} (\tau) {=} 1 {+} \left( g^{(2)} (0) {-} 1 \right) e^{-2 \tau / \tau_0}$, yields $g^{(2)} (0) = 1.868 {\pm} 0.035$ and coherence time $\tau_0 = 26.0 {\pm} 1.5\ \mathrm{ns}$.
Next we introduce a resonant coherent MW field of strength comparable to the thermal field -- in terms of converted photons rates $\bar{n}_{\mathrm{coh}}$ for coherent photons and $\bar{n}_{\mathrm{th}}$ for thermal photons. We estimate the $g^{(2)} (0)$ value for consecutive measurements (Fig. \ref{results}\textbf{c}), in agreement with the theory. Interestingly, the introduction of a stronger but $\Delta \omega$ far-detuned coherent field, while keeping $\bar{n}_{\mathrm{coh}}/\bar{n}_{\mathrm{th}} {=} 1$, leads to a beat modulation of the autocorrelation function (Fig. \ref{results}\textbf{d}). Specifically in this case
\begin{equation}
g^{(2)} (\tau) = 1 + \frac{1}{4} \left( \left| g^{(1)}_{\mathrm{th}} (\tau) + e^{- i \omega \tau} \right|^2 - 1 \right),
\end{equation}
where $g^{(1)}_{\mathrm{th}} (\tau) {=} A(\tau) e^{- i \omega_0 \tau}$ and $\omega_0$ is the mean frequency of the conversion band. The beat modulation is at the frequency of the detuning $\Delta \omega {=} \omega {-} \omega_0$.
\paragraph{Noise figure}
With no external field applied to the converter, we observe photons, which with disabling each field and cross-correlation analysis we identify as consisting of $85\%$ thermal radiation, $13\%$ fluorescence of optical elements induced by $480\ \mathrm{nm}$ laser field and $2\%$ other noise. This proportion results in an overall $5.7{:}1$ signal-to-noise ratio, and as the signal is thermal in this case, we can translate it straightforwardly to a noise-equivalent temperature $T_{\mathrm{NE}} = 53\ \mathrm{K}$. As the added noise is wideband, we utilize additional cavity-assisted filtering of thermal signal, improving the ratio to $77{:}1$, thus $T_{\mathrm{NE}} = 3.8\ \mathrm{K}$. We present these values in the Fig. \ref{sateff}\textbf{a} as a reference to the converter's response characteristics.
\begin{figure}
\includegraphics[width=.48\textwidth]{PDF/55D-54F.pdf}
\caption{
\textbf{a},~Measured conversion in the domain of bare-state detunings from energy levels $55^2\mathrm{D}_{5/2}$ and $54^2\mathrm{F}_{7/2}$ (other detunings being near-zero), unveiling the underlying bright resonances $\ket{+}$ and $\ket{-}$. From this measurement we obtain the optimal level detunings $\Delta_{55\mathrm{D}} {=} 16{\cdot} 2 \pi\ \mathrm{MHz}$, $\Delta_{54\mathrm{F}} {=} 0$. \textbf{b},~The same conversion interdependence predicted in the numerical simulation.}
\label{levmap}
\end{figure}
\paragraph{Observation of bright resonances}
Following the short demonstration of the level-detuned working regime in the Fig. \ref{qura}\textbf{d}, we further explore the conversion's dependence on the detunings from bare atomic levels -- realized with proportional detunings of the fields. We find this domain the most convenient to operate in, as it naturally obeys the conservation of energy in six-wave mixing. We find that the Doppler effect is mostly canceled via the selection of velocity class in the two-photon process, i.e. with probe and coupling fields. In general, with a hot-atom system, the Doppler effect prevents interpretations as simple as in the cold-atom case. However, for our case, we expect that the conversion loop is actually composed of two dressed-state subsystems: the three-level system dressed by coupling and probe fields, and the two-level systems dressed by the decoupling field. Indeed by scanning the $\Delta_{55D}$ level detuning, we observe resonances corresponding to two bright states, that correspond to the simplified and unnormalized eigenstates given by $\ket{B\pm} = \Omega_p^*\ket{5^2S_{1/2}} \pm \sqrt{|\Omega_p|^2{+}|\Omega_c|^2}\ket{5^2P_{3/2}} + \Omega_c\ket{55^2D_{5/2}}$ split by roughly $\Omega_c$ (as seen in the Fig. \ref{levmap}). The dark state $\ket{D}=\Omega_c^*\ket{5^2S_{1/2}}-\Omega_p\ket{55^2D_{5/2}}$ does not take part in the conversion process, but could be accessed in a cavity-enhanced system to yield conversion with lower loss due to 5P state decay.
On the other hand, as we sweep the detuning $\Delta_{54F}$, we observe broadening rather than clear splitting into two dressed states $\ket{\pm}=\ket{54^2F_{7/2}}\pm\ket{5^2D_{5/2}}$, which is due to particular interference of different velocity classes. Via numerical studies we found that this splitting is observed only at much higher decoupling Rabi frequencies in different scans. We also note that the blue-detuned resonance results in a slightly stronger conversion than the red-detuned, which we attribute to the interfering effect of other energy levels (that may take part in wave-mixing process) on the latter.
\section*{Discussion}
The presented proof-of-concept warm-atomic converter exhibits flexibility in terms of high dynamic range and wide bandwidth, which supports its potential for applications. We believe the fields of contemporary astronomy could benefit from conversion-based MW photonic measurements \cite{Riechers_2022,Pankratov_2022,Komatsu_2022,Greaves_2018,Gorski_2021,Kou_2022}, where Rydberg atoms excel in simplicity, adaptability and presented low noise figures. As the conversion scheme utilized here is all-optical, it can be applied in similar systems (i.e. cold trapped atoms and superheterodyne Rydberg electrometry) to avoid the introduction of noise by spectrally close fields.
We highlight many desirable properties of the presented converter. The very ability to perform photon counting of microwave radiation in room temperature along with observation of the HBT interference and coherent-thermal interference is remarkable, as those feats typically require deeply cryogenic conditions \cite{PhysRevLett.107.217401}. The converter presents an extremely large dynamic range and excellent bandwidth, along with many options for both fine and very coarse tuning to different Rydberg levels. The all-optical realization gives further prospects for applications as even strong electromagnetic interference would not damage the device. This could be important for microwave-based communication, where another advantage may come from avoiding shot noise of homodyne detection via photon counting.
There are numerous issues that so far escaped the scope of this work and could be explored in more detail to the benefit of increasing the conversion's range and efficiency. Because thermal photons are spread over the whole conversion band, the limit approached in the Fig. \ref{sateff}\textbf{a} is not fundamental, as efficient narrow spectral filtering or phase-locking measurement would push it down, enabling detection at the single-photon level. The non-thermal noise may be decreased further e.g. by utilizing different optical elements in the setup (specifically addressing the $480\ \mathrm{nm}$ fluorescence) or by exploring non-collinear configuration of the laser fields. Various optimization efforts may also be performed to find working points with greater conversion efficiency, though the operational regime of warm atomic vapors and strong probe field does not allow for simple theoretical predictions. The converter's response to MW pulses is yet to be investigated, as well as pulse-control of laser fields or coherent repumping of atomic population, though the complexity of the setup would then inevitably increase. Additionally, the operation mode of the Rydberg converter in principle can be thoroughly shifted to adapt to emission at the telecom C-band wavelength -- specifically, $4^2\mathrm{D}_{5/2} \rightarrow 5^2\mathrm{P}_{3/2}$ transition occurs at $1530\ \mathrm{nm}$.
The all-optical realization could be retained while designing a setup that exploits the Doppler effect canceling in a wider scope (i.e. by using different transitions), enabling more atoms to take part in the conversion process. On the contrary, a range of atomic velocities may be seen as the means to further widen the tunable conversion band. We believe that with commensurate detunings of all-optical field, the tunable conversion bandwidth can be extended to as much as $600\ \mathrm{MHz}$ (the width of Doppler-broadened probe absorption line), thus with the consideration of the neighboring Rydberg transitions, the converter could efficiently cover the full range of MW frequencies up to $50\ \mathrm{GHz}$, only with the tuning of laser fields. To our knowledge, such an ultrawide adaptable conversion band is possible only with warm Rydberg atomic vapors.
We envisage that further progress will be centered around introducing warm Rydberg atomic vapors to MW cavity systems, as it will enhance the coupling between MW field and atomic interaction volume. Our all-optical scheme is well-suited for that task; in comparison, the six-wave mixing schemes involving two microwave transitions would require a doubly-resonant cavity with close-by but non-degenerate resonances. The design principles for hot-atom MW-cavity systems can be drawn from cavity-enhanced atomic clocks. Even with moderate finesse, we expect the conversion efficiency to be significantly enhanced, likely even leading to radiative cooling of the cavity mode via conversion. At the same time, a lot of potential lies in recent progresses in microfabrication of atomic devices \cite{Kitching_2018}, such as micro- and nanoscale vapor cells \cite{Cutler_2020,Lucivero_2022} and hollow-core photonic bandgap fibers filled with alkali atoms \cite{Peters_2020}. These instruments may enable all-fiber, reproducible applications of the presented converter's model, paving the way to next-generation MW converting sensors.
\section*{Methods}
\paragraph{Density matrix calculation}
As indicated in the main text, we solve the five-level GKSL equation in the steady state, i.e. $\partial_t\hat{\rho}=0$. The Lindbladian is constructed from the Hamiltonian and the jump operators using the QuantumOptics.jl package \cite{kramer2018quantumoptics}. We next solve for the steady-state using standard linear algebra methods finding the zero eigenvalue. We employ jump operators $\hat{J}_n$ for spontaneous decay and for transit-time decay, i.e. atoms exiting the interaction region with ground-state atoms entering instead. Each solution is for a given set of Rabi frequencies and detuning and for a specific velocity class $v$ (along the longitudinal direction). The detunings are modified due to the Doppler effect as $\Delta_n^v {=} \Delta_n {+} k_n v$ with $n {\in} \{p,c,\mathrm{MW},d\}$. We next perform averaging of the resulting state-state density matrix $\hat{\rho}$ over the Doppler velocity profile with weighting function $f(v) {=} \sqrt{m/2\pi k_B T} \exp(-mv^2/k_B T)$ (with $m$ being the $^{85}$Rb atomic mass) and over the Gaussian profile of the beams due to changing Rabi frequencies. Unless noted otherwise, the microwave Rabi frequency is taken well below the saturation point. Finally, we extract the generated signal, always plotted as intesnity, as $|E_s|^2 {\sim} |\mathrm{Tr}(\ket{5^2D_{5/2}}\bra{5^2P_{3/2}}\hat{\rho})|^2$ or the EIT signal as $\mathrm{Im}(\mathrm{Tr}(\ket{5^2S_{1/2}}\bra{5^2P_{3/2}}\hat{\rho}))$
\paragraph{Phase matching}
Efficient conversion requires all atoms interacting with the MW field to emit in phase. This renders the phase-matching condition that determines the conversion efficiency depending on the MW field mode. Thus we introduce the phase-matching factor $\eta_\mathrm{phm}(\theta)$ for a plane-wave MW field $E_\mathrm{MW}(\theta)$ entering the medium at angle $\theta$ to the propagation axis ($z$). The factor is then calculated by projecting the generated electric susceptibility onto the detection mode $u_e$ being a Gaussian beam focused in the center of the cell ($z_0{=}0$) with $w_0{=}100\ \mathrm{\mu m}$. To obtain the projection we assume the spatial dependence of the susceptibility in the form of the product of all interacting fields. This is a simplification compared with the full model, as in general the steady-state density matrix and, in turn, the coherence $\rho_s$ have a more complex dependence on strong drive field amplitudes. Nevertheless, for the purpose of spatial calculation, the simplified approach yields proper results and then the susceptibility takes on a form:
\begin{equation}
\chi_\theta = E_pE_p^*E_cE_d^{*}E_\mathrm{MW}(\theta),
\end{equation}
with the optical fields ($E_p$, $E_c$, $E_d$) taken to be Gaussian beams with the same $z_0$ and $w_0$ as the detection mode $u_s$. Additionally, we account for the relatively strong absorption of the probe ($E_p$) field by multiplying its amplitude by the exponential decay $\exp(-\alpha z)$ with (measured) $\alpha{=}19\ \mathrm{m}^{-1}$. Finally, the coefficient is calculated as the following integral:
\begin{equation}
\eta_\mathrm{phm}(\theta)=\int_{-L/2}^{L/2}\mathrm{d}z \int_0^{2\pi}\mathrm{d}\phi\int_0^{\infty}\rho\mathrm{d}\rho \chi_\theta u_s^{*},
\end{equation}
where $L{=}50\ \mathrm{mm}$ is the length of the glass cell.
\begin{figure}
\includegraphics[width=1\columnwidth]{PDF/qra.pdf}
\caption{Calculated angular dependence of the MW conversion efficiency for two circular polarizations: $\sigma_+$ (antenna polarization) and orthogonal $\sigma_-$. The scale for both plots represents efficiency relative to the maximum of the $\sigma_+$ pattern in $\mathrm{dB}$. The slight drop in efficiency at $0^\circ$ is due to the Gouy phase of optical beams.}
\label{fig:qra}
\end{figure}
\begin{figure*}
\includegraphics[width=\textwidth]{PDF/flat_uklad_methods.pdf}
\caption{
Details of the entire experimental setup.
\textbf{a},~Laser system. A narrowband fiber laser at $1560\ \mathrm{nm}$ serves as a frequency reference for the entire system. The laser is amplified via erbium-doped fiber amplifier (EDFA) and the frequency is doubled using a second-harmonic generation (SHG) process to $780\ \mathrm{nm}$. The $960\ \mathrm{nm}$ external-cavity diode laser (ECDL) is amplified using a tapered amplifier (TA) and frequency-doubled via cavity-enhanced SHG to yield a coupling field at $480\ \mathrm{nm}$. The $960\ \mathrm{nm}$ laser is stabilized to the master laser of $1560\ \mathrm{nm}$ using a common cavity. A similar independent system is introduced for the stabilization of decoupling ($1258\ \mathrm{nm}$) laser, likewise for local oscillator (LO) $776\ \mathrm{nm}$ laser. The second harmonic of the $1560\ \mathrm{nm}$ fiber laser serves as a reference for offset-locking of the probe laser in an optical phase-locked loop (PLL).
\textbf{b},~Microwave generation system. The generated microwaves are attenuated adequately and split into the spectrum analyzer for power and frequency references and to the antenna.
\textbf{c},~Thermal image of the cell and cell holder. The constant temperature of the cell is assured with hot-air heating via hollow channels in the 3D-printer cell holder.
\textbf{d},~The generated signal light exits the converter in a beam with a Gaussian profile shown in the image.
\textbf{e},~Heterodyne measurement with LO yields spectrum of the converted signal: the residual broadening of $\Gamma_{\mathrm{sig}} {=} 86 {\cdot} 2 \pi\ \mathrm{kHz}$ FWHM is due to collective laser-locking phase noise.
\textbf{f},~Scheme of the experimental setup: probe, coupling and decoupling laser beams are combined with dichroic mirrors (DM) into a collinear configuration, and focused inside a rubidium vapor cell of length $50\ \mathrm{mm}$ and diameter $25\ \mathrm{mm}$ with a MW helical antenna pointed there. The circular polarization of lasers is assured with quarter-wave plates (QWP). Probe ($780\ \mathrm{nm}$) signal is registered by an avalanche photodiode (APD), which enables laser calibration by observation of EIT features. Converted $776\ \mathrm{nm}$ signal is spectrally separated and coupled into a single-mode fiber.
\textbf{g},~Different setups are used for detection, with single setup being used at a time. The detection setups include direct photon counting with optional attenuation, cavity-filtered photon counting, photon autocorrelation with two channels simultaneously, and heterodyne detection. For photon counting, a multichannel superconducting nanowire single-photon detector (SNSPD) is used. For heterodyne measurement we combine the signal with LO with a polarization beam splitter (PBS) and split the combined signal $50{:}50$ with a half-wave plate (HWP) and a second PBS. The signal is then registered on a differential photodiode (PD).
}
\label{setup}
\end{figure*}
\paragraph{Thermal modes consideration}
Let us now estimate the effective field root-mean-square amplitude due to thermal black-body radiation in our given conversion band, which takes into account that our converter is both polarization and wavevector sensitive. In order to do so, we follow a standard derivation of Planck's law, however, when considering all electromagnetic modes we account for both the polarization-dependence and phase matching during conversion. The effective mean-square field amplitude at the temperature $T$ may be written as follows:
\begin{align}
\langle E^2_{\mathrm{eff}} \rangle &=\frac{\omega^2\langle \mathcal{E}\rangle}{\pi^2c^3\varepsilon_0} \frac{1}{4\pi}\int_0^{2\pi}\mathrm{d}\phi\int_0^\pi\mathrm{d}\theta\sin(\theta) |\eta(\theta)|^2, \\
\langle \mathcal{E}\rangle &= \frac{\hbar \omega}{e^{\hbar \omega/k_\mathrm{B} T}-1},
\end{align}
where $\omega$ is the frequency of the MW field, $c$ is the speed of light, $\varepsilon_0$ is vacuum permittivity and $k_\mathrm{B}$ is the Boltzmann constant. Importantly, $\eta(\theta)$ is the field conversion efficiency coefficient that arises from projecting the MW field in the plane-wave modes at the given angle $\theta$ onto circular polarization along the conversion axis and multiplying it by the phase matching factor:
\begin{equation}
|\eta(\theta)|^2=\left(\cos\left(\frac{\theta}{2}\right)^4 + \sin\left(\frac{\theta}{2}\right)^4\right)|\eta_\mathrm{phm}(\theta)|^2.
\end{equation}
In fact the efficiency $|\eta(\theta)|^2$ represents the converter reception pattern, which we plot in the Fig. \ref{fig:qra}. From the pattern we see that the contribution to the effective noise-equivalent field at temperature $T$ comes mostly from the phase-matched microwaves coming from an almost right-angled cone with the correct circular polarization.
Finally, we refer the effective field convertible by the atoms $\sqrt{\langle E_\mathrm{eff}^2\rangle}$ to total RMS field fluctuations $\sqrt{\langle E^2\rangle}$, which can be calculated by plugging in $|\eta(\theta)|^2 {=} 2$ (because of two polarizations). In the Fig. \ref{sateff}\textbf{a} we find that the thermal noise level corresponds to an effective field of $\sqrt{\langle E_\mathrm{eff}^2\rangle}=480\ \mathrm{pV cm^{-1} Hz^{-1/2}}$, as calibrated from the A-T splitting at large fields. This is indeed highly consistent with the model prediction of $\sqrt{\langle E_\mathrm{eff}^2\rangle} = 493\ \mathrm{pV cm^{-1} Hz^{-1/2}}$. When referred to the total field fluctuations, we obtain the results quoted in the text, i.e. $\sqrt{\langle E^2\rangle}=1.59\ \mathrm{nVcm^{-1}Hz^{-1/2}}$ measured and $\sqrt{\langle E^2\rangle}=1.64\ \mathrm{nVcm^{-1}Hz^{-1/2}}$ predicted. The agreement is rather remarkable, even tough some effects are not taken into account, such as modification of thermal radiation due to the presence of glass cell, holder, antenna etc. We believe however that this is nevertheless a very strong indication that we observe converted thermal radiation of an expected intensity.
\paragraph{Thermal and coherent states autocorrelation function}
We consider the autocorrelation measurements with three different sources of photons, defined by rates: $\bar{n}_{\mathrm{th}}$ for thermal state photons, $\bar{n}_{\mathrm{coh}}$ for coherent state photons and $\bar{n}_{\mathrm{noise}}$ for photons coming from wideband, non-interfering sources (e.g. various forms of fluorescence). We follow the derivation presented in \cite{Marian_1993} (and later showcased in \cite{Forsch_2019,Kumar_2022}) to arrive with the following general form of second order autocorrelation function
\begin{equation}
g^{(2)} (\tau) = 1 + \frac{ \left| \bar{n}_{\mathrm{th}} g^{(1)}_{\mathrm{th}} (\tau) + \bar{n}_{\mathrm{coh}} e^{- i \omega \tau} \right|^2 - \bar{n}_{\mathrm{coh}}^2}{\left( \bar{n}_{\mathrm{th}} + \bar{n}_{\mathrm{coh}} + \bar{n}_{\mathrm{noise}} \right)^2},
\end{equation}
where $\omega$ is the coherent state frequency. $g^{(1)}_{\mathrm{th}} (\tau)$ can be extracted from power spectral density via the Wiener-Khinchin theorem. In the Fig. \ref{results}\textbf{b},\textbf{c},\textbf{d} $\bar{n}_{\mathrm{noise}} / \bar{n}_{\mathrm{th}} {=} 15/85$, in the Fig. \ref{results}\textbf{d} $\bar{n}_{\mathrm{coh}} / \bar{n}_{\mathrm{th}} {=} 100/85$.
\paragraph{Laser fields parameters}
The laser beams are focused to equal Gaussian waists of $w_0 {=} 100\ \mathrm{\mu m}$ and combined with the use of dichroic mirrors (Fig. \ref{setup}\textbf{f}). The probe beam counterpropagates with respect to the other fields and its transmission through ${}^{85} \mathrm{Rb}$ medium is then utilized as a means for calibration via EIT effects (Fig. \ref{qura}\textbf{d},\textbf{e}). The length of vapor cell is $50\ \mathrm{mm}$. The collinear laser fields configuration is assured with $4 f$ optical systems (not shown on the scheme) with mirrors at focal planes, enabling independent control of every beam's position an propagation angle. The employed laser beam effective peak Rabi frequencies for $780\ \mathrm{nm}$, $480\ \mathrm{nm}$ and $1258\ \mathrm{nm}$ fields are derived from the numerical simulation as $\Omega_p{=}8 {\cdot} 2 \pi\ \mathrm{MHz}$, $\Omega_c{=}22 {\cdot} 2 \pi\ \mathrm{MHz}$ and $\Omega_d{=}17 {\cdot} 2 \pi\ \mathrm{MHz}$ respectively. The optimal dominant detuning from the $55 \mathrm{D}$ level is measured as $\Delta_{55 \mathrm{D}} {=} 16 {\cdot} 2 \pi\ \mathrm{MHz}$ (Fig. \ref{levmap}\textbf{a}) and is used as a working point for measurements, with the other detunings being near-zero. For the Fig. \ref{qura} and \ref{setup}\textbf{e} the microwave Rabi frequency is set to about $\Omega_\mathrm{MW}{=}8 {\cdot} 2 \pi\ \mathrm{MHz}$, while for the Figs. \ref{results} and \ref{levmap}\textbf{a} it is deeply in the unsaturated regime. For the Fig. \ref{levmap}\textbf{a} the decoupling intensity is increased to about $\Omega_d{=}25 {\cdot} 2 \pi\ \mathrm{MHz}$ peak. The calibration of detuning zero-points takes place in the ${}^{85} \mathrm{Rb}$ working temperature to account for pressure shifts of Rydberg energy levels.
\paragraph{Temperature stabilization and microwave shielding}
The measured optimal ${}^{85} \mathrm{Rb}$ working temperature, $T {=} 42 {}^\circ \mathrm{C}$, is ensured via hot air heating, as this method introduces little interference to MW fields propagating through the vapor cell. The air is heated and pumped through a specially designed hollow 3D-printed resin cell holder (Fig. \ref{setup}\textbf{c}), where the heat exchange takes place. The cell is enclosed inside a MW absorbing shield, made from a material (LeaderTech EA-LF500) with ${>}30\ \mathrm{dB}$ loss at $14\ \mathrm{GHz}$, with sub-MW-wavelength apertures for optical beams -- the shield also provides additional thermal isolation, reducing fluctuations of temperature. The temperature of the shield inside, for the reference to black-body radiation, is measured as $26{-}27{}^\circ \mathrm{C}$. The MW helical antenna is placed inside the shield and the collinear propagation is assured with optical fields passing through an aperture at the backplate of the antenna (Fig. \ref{setup}\textbf{f}).
\paragraph{Frequency stabilization and calibration}
The lasers in the experiment are stabilized (Fig. \ref{setup}\textbf{a}) to a narrowband, frequency-doubled fiber laser (NKT Photonics, $1560\ \mathrm{nm}$), which is itself stabilized to a Rb cell via a modulation-transfer lock. The probe laser at $780\ \mathrm{nm}$ is stabilized to the frequency-doubled reference via an optical phase-locked loop (PLL). The $1258\ \mathrm{nm}$ laser (decoupling) the $960\ \mathrm{nm}$ laser (frequency-doubled to yield coupling light at $480\ \mathrm{nm}$, Toptica DL-SHG pro) and the $776\ \mathrm{nm}$ laser acting as LO, are all stabilized to the reference at $1560\ \mathrm{nm}$ using independent cavities via transfer locks. Laser fields are calibrated in low-intensity regimes with the use of EIT effects observed in the probe field transmission, registered on the control avalanche photodiode (Thorlabs APD120A, Fig. \ref{setup}\textbf{f}). MW field, generated via LMX2820 PLL frequency synthesizer, is coupled to the antenna and spectrum analyzer (Agilent N9010A EXA) used for relative reference of field amplitude (Fig. \ref{setup}\textbf{b}). Absolute MW field amplitude is calibrated with a standard measurement of A-T splitting \cite{Sedlacek_2012}. As a method to quantify the converter's ability to distinguish photons incoming from spectrally different sources, we perform a heterodyne measurement, yielding the converter's signal width $\Gamma_{\mathrm{sig}} {=} 86 {\cdot} 2 \pi \mathrm{kHz}$ FWHM (Fig. \ref{setup}\textbf{e}). This value may be interpreted as a measure of collective laser-locking phase noise.
\paragraph{Measurement techniques}
Initially, the $776\ \mathrm{nm}$ signal is reflected from a bandpass filter and then passed through a series of free-space spectral filters (highpass, lowpass, $1.2\ \mathrm{nm}$ spectral width bandpass) and coupled to a fiber. Due to the Gaussian characteristics of the beam (Fig. \ref{setup}\textbf{d}), coupling losses are negligible in this case (${<}25\%$). We apply various detection methods pictured in the Fig. \ref{setup}\textbf{g}. Additional filtering in the measurement of noise-equivalent temperature is performed with an optical cavity ($160\ \mathrm{MHz}$ spectral width, $80$ finesse). The measurement of photon counting is performed with a single-photon detector (ID Quantique ID281 superconducting nanowire single-photon detector) with $85\%$ quantum efficiency, ${<}1\ \mathrm{Hz}$ dark count rate and $35\ \mathrm{ns}$ recovery time. As the detector's reliable response to incoming photons is at the order of ${<}10^7\ \mathrm{phot} / \mathrm{s}$, calibrated neutral density filters are applied to the signal above this value. The autocorrelation measurement is performed on two channels of the single-photon detector after splitting the signal $50{:}50$ with a fiber splitter. The heterodyne measurement is performed using a custom-made differential photodiode with the use of $776\ \mathrm{nm}$ LO laser, $20\ \mathrm{mW}$ power. The image of the beam profile (Fig. \ref{setup}\textbf{d}) is taken on a ${>}50\%$ quantum efficiency CMOS camera (Basler acA2500-14gm). The supplementary measurements of temperature (including Fig. \ref{setup}\textbf{c}) are performed with a calibrated thermal camera (Testo 883).
\section*{Data availability}
Data underlying the results presented in this paper are available in the Ref. \cite{Data23}.
\section*{Author contributions}
S.B. and U.P. built the optical and microwave setups. S.B. took the measurements and analyzed the data assisted by other authors. S.B, M.M. and M.P. prepared the figures and the manuscript. M.M. and M.P. developed the theory assisted by S.B. who facilitated comparison with experiments. M.P. led the project assisted by M.M.
\begin{acknowledgments}
We thank K.~Banaszek for the generous support and W.~Wasilewski for both support and discussions. The "Quantum Optical Technologies" (MAB/2018/4) project is carried out within the International Research Agendas programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund. This research was funded in whole or in part by National Science Centre, Poland grant no. 2021/43/D/ST2/03114.
\end{acknowledgments}
\bibliographystyle{apsrev4-2}
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A love story shot in two continuous 40 minute takes over a sunset and sunrise, Almost in Love is Sam Neave's third feature as writer/director.
A South African writer comes back to New York to deal with life after his wife's suicide.
Six friends reunite for a 30th birthday party but soon realize that they've moved apart in ways they didn't expect.
A short film by Sam Neave that marked his first collaboration with DP Daniel McKeown and composer James Lavino.
A short film about an obsessive piano tuner.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 3,859
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With a rich history, the borough of Turtle Creek, located in Allegheny County, is a great location to live and work. We have a Rupp Fiore location in Turtle Creek located on Osborne Street. If you don't know too much about us, we pride ourselves in being reliable, unbiased, professional, and experienced in providing the best quality insurance for all of our clients. We have a variety of personal and business insurance opportunities and we can give you the most cost-effective and tailored insurance plan for yourself or business.
It can be really hard to go to an insurance company to get advice or a plan because you may feel like your needs are not in their best interest, but at Rupp Fiore, your needs are always our priority. With over 40 regional and national insurance carriers, we will get to know you and build a relationship with you before we offer you the insurance that we feel is best fitting to your needs.
There are so many insurance companies to choose from and that can be overwhelming. You may find yourself asking: What's the best company to work with? Who offers the best price? Who offers the best coverage? The questions go on and on, making the process of getting insurance long and painful. Choosing the right insurance doesn't need to be difficult. It can be simple with us at Rupp Fiore.
If you are looking for insurance in Turtle Creek, Rupp Fiore is the first place you will want to stop. We promise to go above and beyond for all of your insurance needs. Get a quick quote with us today and build a long-lasting relationship with us.
|
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"redpajama_set_name": "RedPajamaC4"
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| 5,914
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Useful information and materials you can use and refer to, during the course of your global talent attraction and retention journey.
We have produced a brochure called "Come join us" which showcases the Gothenburg region as a place to live and work. We have also created videos that both present the region, the benefits of living and working here, and tips and advice. Both the brochure and the videos are available further down on this page.
In addition, you should tip them about www.movetogothenburg.com, where they can find information about the region and its offerings as well as a guide to help them pre- and post-arrival.
A tool created specially to help employers and companies successfully recruit international talent. Here you will find answers to some key questions as well as useful tips and suggestions.
This guide is available in English and Swedish and can be downloaded below.
Find Your Beat in the Gothenburg region, West Sweden!
|
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"redpajama_set_name": "RedPajamaC4"
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| 1,441
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{"url":"http:\/\/mathoverflow.net\/questions\/127866\/weierstrass-factorization-with-l2-estimates?sort=votes","text":"# Weierstrass factorization with $L^2$ estimates?\n\nLet $\\Omega$ be a bounded domain in $\\mathbb{C}$. Let $X$ be a discrete set of points whose boundary is in the boundary of $\\Omega$. Can I find an $L^2$ holomorphic function which vanishes on $X$? Can I solve the problem in weighted $L^2$ spaces?\n\nIf there are counterexamples, are precise conditions on the set $X$ known to ensure the existence of an $L^2$ solution?\n\nI have been learning about Hormander's approach to the $\\bar{\\partial}$-problem, and this seems like a natural question to ask from that perspective, but I have not been able to find any work done on this.\n\n-\n\nYou have to specify what you mean by $L^2$. Is this $L^2$ with respect to Lebesgue measure (area) in $D$? Whatever you mean by $L^2$, the answer is \"no\". The reason is Jensen's formula. It says that a function which has too many zeros must grow fast.\nIf you want to solve it in weighted $L^2$ space, then your weight must be related to the growth rate of the set $X$. If instead you want to fix the weight in $L^2$, the conditions on $X$ will come from the Jensen formula. If you are interested in $L^2$ without weight, look in the books about Bergman space. There you can find the exact conditions on $X$.","date":"2015-03-27 08:11:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.8983062505722046, \"perplexity\": 91.91449727422369}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-14\/segments\/1427131295993.24\/warc\/CC-MAIN-20150323172135-00201-ip-10-168-14-71.ec2.internal.warc.gz\"}"}
| null | null |
Q: Extract data from a MongoDB database using the Mongo Java Driver I'm trying to write a program in Java that will connect to a MongoDB database. I already have the server, port, name, userDB, and username/password of the MongoDB database I'm trying to access.
I wasn't really sure where to start, so after looking around for a bit, I found a Java driver for MongoDB available on GitHub:
https://github.com/mongodb/mongo-java-driver. I downloaded the source code and imported it as a project on Eclipse. The source code for the driver has literally hundreds if not thousands of files. I tried running it as is, and couldn't get the project to compile.
I have no clue where to go from here. There isn't a lot of in-depth instructional material out there for this, and if I took the time to thoroughly inspect all the files in the driver, that could take decades.
A: You are on the right track here. What you need to do instead of downloading the source code is to download the JAR files or alternatively use Maven which is a dependency management tool.
This is what you will need to install the mongodb driver.
After you have successfully added the dependencies to your project. Use this quick start guide which will step you through the basics.
Here is an example of working with maven.
Don't get disheartened by small trials..there's always decades of documentation to read ;)
A: Download the .JAR file from here: https://oss.sonatype.org/content/repositories/releases/org/mongodb/mongodb-driver/3.0.0/
You need mongodb-driver-3.0.0.jar from this list.
Once you have that add it as a library dependency in your Eclipse project and you'll be all set.
|
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\section{Acknowledgments.}
\vskip -0.15in
\textred{We thank Haitham Hassanieh, Vikram Iyer, Vamsi Talla, Justin Chan, Anran Wang, Rajalakshmi Nandakumar, and the anonymous reviewers for their helpful feedback on the paper. This work was funded in part by NSF awards CNS-1812554, CNS-1452494, CNS-1823148, Google Faculty Research Awards and a Sloan Fellowship.}
\section{Applications}
In this section, we
\subsection{IoT Protocol: LoRa}
\subsubsection{Transmitter}
\subsubsection{Receiver}
\subsection{LoRa Spectrum Sensing}
\subsection{Localization}
\section{Research Case Study}
An SDR that is geared for IoT endpoints and can provide I/Q transmission and reception capability opens up opportunities for addressing multiple research questions in IoT networks.
In this section, we focus on the following question: Can a {\it low-power IoT endpoint} device decode multiple concurrent LoRa transmissions at the same time? LoRa supports long range communication for IoT devices and is gaining popularity as a low-power wide area networking (LPWAN) standard. While it supports long ranges, it is also challenging since the range of these devices can be a few kilometers. This increases the number of collisions and probability of concurrent transmissions in a large scale city-scale deployment. While recent work~\cite{lorasigcomm17,netscatter} has explored the feasibility of enabling concurrent LoRa transmissions, they have been targeted for decoding on a gateway USRP device. In fact, most concurrent transmission techniques in our community~\cite{zigzag, anc, lorasigcomm17} have been prototyped on USRPs and it is unclear if a low-power IoT endpoint device can decode concurrent transmissions in real-time within its stringent 1) power constraints, 2) computational constraints and 3) resource constraints. TinySDR enables us to explore such questions and design MAC protocols for decoding concurrent transmissions on IoT endpoints.
{\bf Using orthogonal LoRa codes.} Here we explore a specific way of enabling concurrent transmissions in LoRa: using orthogonal codes. Specifically, to allow multiple LoRa nodes to communicate at the same time, we exploit LoRa's support for orthogonal transmissions~\cite{lora_basics} which can occupy the same frequency channel without interfering with each other. Two chirp symbols are orthogonal when they have a different chirp slope. For a chirp with spreading factor of $SF$ and bandwidth of $BW$, the chirp slope is the bandwidth occupation of the chirp over time: $\frac{BW^{2}}{2^{SF}}$~\cite{netscatter}.
{\bf Decoding concurrent transmissions on \name.} We explore if we can demodulate multiple concurrent LoRa transmissions in real-time on a \name\ platform and evaluate its power consumption, computational resources and delay.
In order to receive concurrent LoRa transmissions, \name\ must be able to demodulate LoRa upchirp symbols with different slopes. So say we have two LoRa transmissions that use different spreading factor and bandwidth configurations: $SF_1, BW_1$ and $SF_2, BW_2$. To decode them concurrently, we implement decoders for each of the chirp configurations in parallel on our FPGA. Specifically, we first generate a corresponding downchirp symbol for each configuration. We then correlate the received signals in the frequency domain with each of the downchirp symbols using time domain multiplication with the received signal. After correlation, we take an FFT of the result and identify the maximum to decode the symbol values. Fig.~\ref{sdsd} shows the decoding pipeline we use.
{\bf Evaluation.} We evaluate three key aspects of our design: 1) the platform's effectiveness in decoding concurrent transmissions across a range of RSSI values, 2) the power consumption at the endpoint device while decoding concurrent transmissions and 3) the computational resources required and the resulting delay in decoding.
We use two SX 1276 LoRa transceivers as our transmitters and set them to transmit continuously at two different settings: they both use a spreading factor of $SF=8$ but have two different bandwidth setups, $BW_1=125 kHz$ and $BW_2=250 kHz$. The transmitters are set to transmit random chirp symbols and are connected to a power combiner and we connect the combined output to the RF input of \name\ through a variable attenuator. The \name\ platform decodes these two concurrent transmissions and computes the chirp symbol error rate for each of these transmissions. We evaluate three main scenarios: 1) when the two transmitters have a similar power level at the receiver, 2) when the first transmitter has a power level XX~dB higher than the second and 3) when the first transmitter has a power level xXX~dB lower than the second.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/8_lora_orthogonal_rx/lora_multiple.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf LoRa Orthogonal Demodulator Evaluation.}}}
\label{fig:lora_multiple_rx}
\vskip -0.15in
\end{figure}
Fig.~\ref{fig:lora_multiple_rx}(a)-(c) plot the measured the chirp symbol error rate vs received power and plot the results in Fig.~\ref{fig:lora_multiple_rx}. The results can be summarized as follows:
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item This plot shows our implementation on \name achieves a sensitivity of -123~dBm for $SF=8$ and $BW=125kHz$. This is expected because XXX explain the. Explain the two extra graphs we are adding
\item Our parallel implementation in Fig.~\ref{sdds} uses only 11\% of the FPGAs resources. Yet it is able to perform the functionality of decoding two concurrent transmissions while also allowing the flexibility for further optimization. Further the delayXXX.
\item Power conusmption XXX and compare it to a single one. XXX
\end{itemize}
Note that Semtech gateway solutions such as SX1308~\cite{sx1308} can receive multiple transmissions like our design. However it is a gateway solution and to the best of our knowledge we are the first to show that concurrent LoRa transmissions can be decoded on a IoT endpoint, which would have been difficult without \name.
\section{Research Study: Concurrent Reception}
An SDR designed for IoT endpoints that can provide I/Q transmission and reception capability opens up opportunities for addressing multiple research questions in IoT networks.
In this section, we focus on the following question: Can a {\it low-power IoT endpoint} device decode multiple concurrent LoRa transmissions at the same time? LoRa supports long range communication for IoT devices and is gaining popularity as a low-power wide area networking (LPWAN) standard. Supporting long ranges introduces new challenges since it increases the probability of collisions in large scale city-wide deployments. While recent works~\cite{lorasigcomm17,netscatter} have explored the feasibility of enabling concurrent LoRa transmissions, they have been designed for decoding on a gateway-style USRP device. In fact, most concurrent transmission techniques in our community~\cite{zigzag, anc, lorasigcomm17} have been prototyped on USRPs and it is unclear if a low-power IoT endpoint device can decode concurrent transmissions in real-time within its stringent power and resource constraints. TinySDR enables us to explore such questions and design MAC protocols for decoding concurrent transmissions on IoT endpoints.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/10_ota_exp/10_ota_time_cdf.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf OTA Programming Time.} We show CDF of OTA programming time for programming LoRa and BLE implementations on \name.}}
\label{fig:ota_exp}
\vskip -0.2in
\end{figure}
{\bf Using orthogonal LoRa codes.} Here we explore a specific way of enabling concurrent transmissions in LoRa: using orthogonal codes. Specifically, to allow multiple LoRa nodes to communicate at the same time, we exploit LoRa's support for orthogonal transmissions~\cite{lora_basics} which can occupy the same frequency channel without interfering with each other. Two chirp symbols are orthogonal when they have a different chirp slope. For a chirp with a spreading factor of $SF$ and bandwidth of $BW$, the chirp slope is given by: $\frac{BW^{2}}{2^{SF}}$~\cite{netscatter}.
{\bf Decoding concurrent transmissions on \name.} In order to receive concurrent LoRa transmissions, \name\ must be able to demodulate LoRa upchirp symbols with different slopes. Suppose we have two LoRa transmissions that use different spreading factor and bandwidth configurations: $SF_1, BW_1$ and $SF_2, BW_2$. To decode them concurrently, we implement decoders similar to Fig.~\ref{fig:lora_rx_block} for each chirp configuration in parallel on our FPGA. Specifically, we first generate a corresponding downchirp symbol for each configuration. We then correlate the received signals with their corresponding downchirp symbols using time domain multiplication. After correlation, we take the appropriate length FFT of the result.
{\bf Evaluation.} We evaluate three key aspects of our design: 1) the platform's effectiveness in decoding concurrent transmissions across a range of RSSI values, 2) the power consumption at the endpoint device while decoding concurrent transmissions and 3) the computational resources required.
We use two SX1276 LoRa transceivers as our transmitters and set them to transmit continuously at two different settings: they both use a spreading factor of $SF=8$ but have two different bandwidth setups, $BW_1=125 kHz$ and $BW_2=250 kHz$. We set the two to send random chirp symbols. The \name platform decodes these two concurrent transmissions and computes the chirp symbol error rate for each transmission. We evaluate two scenarios: 1) when the two transmitters have a similar power level at the receiver, 2) fix the power of one of the transmitters and increase the power of the other one.
Fig.~\ref{fig:lora_multiple_rx_same} shows the results when the two transmissions have similar power at the receiver. We lose around 2~dB and 0.5~dB sensitivity for concurrent demodulation of LoRa configurations with $BW_1=125kHz$ and $BW_2=250kHz$. This is because while in theory the two chirps are orthogonal, in practice, the chirps are created in the digital domain with discrete frequency steps which introduces some non-orthogonality.
Fig.~\ref{fig:lora_multiple_rx_diff} shows the results when the first LoRa transmitter $BW_1=125kHz$ is received near its sensitivity of -123~dBm and and the second LoRa transmitter changes its power. Here, the chirp symbol error rate is affected when the other transmission's power is higher than -116~dBm. When two concurrent transmissions are present, one acts as an interferer when decoding the other. The combined power of noise and the interferer, $P_{I,N}$, determines the error rate. When sweeping the power of interferer, at first the $P_{I,N}$ is dominated by noise and we should not see much effect on error rate. Then at some point the power of them would be equal which results in a 3~dB increase of $P_{I,N}$ and hence 3~dB sensitivity loss and afterwards the the error rate is determined by the interferer power. This demonstrates the need for power control for concurrent transmissions to be received on IoT endpoints.
Our parallel demodulation implementation, uses only 17\% of the FPGAs resources. This concurrent demodulation implementation consumes 207~mW. Note that Semtech gateway solutions such as SX1308~\cite{sx1308} can receive multiple transmissions. But, to the best of our knowledge we are the first to show that concurrent LoRa transmissions can be decoded on a IoT endpoint while meeting its power and computational requirements, which is difficult to do without \name.
\begin{figure}[t]
\begin{subfigure}{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figs/8_lora_orthogonal_rx/lora_multiple.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf Orthogonal Transmissions with Same Received Signal Power.}}}
\label{fig:lora_multiple_rx_same}
\end{subfigure}
\begin{subfigure}{1\linewidth}
\centering
\includegraphics[width=1\linewidth]{figs/8_lora_orthogonal_rx/concurrent_interference.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf Orthogonal Transmissions with Different Received Signal Power}. We set the power of LoRa transmission with $BW_1=125kHz$ to -123~dBm and increase the power of the other one.}}
\label{fig:lora_multiple_rx_diff}
\end{subfigure}
\vskip -0.1in
\caption{\footnotesize{{\bf Orthogonal LoRa Demodulation Evaluation}}}
\label{fig:lora_multiple_rx}
\vskip -0.2in
\end{figure}
\section{System Design}
We design \name platform to fulfill the vision of an SDR based IoT testbed. In particular, \name system has flexible PHY and MAC layer, is capable of updating over the air (OTA) and low power. In this section, we explain the design of the \name system and present design choices and considerations. As shown in fig.~\ref{fig:system}, the system is divided to six different functional blocks: i) SDR, ii) OTA update system, iii) Power Management Unit (PMU), iv) storage devices and v) sensor interface.
\subsection{SDR}
Software-Defined Radio is a programmable PHY layer which converts bits to radio signals and vice versa. At a high level, SDR uses programmable I/Q radio which transforms RF signals to I/Q samples and vice versa. This radio exchanges data with the FPGA for modulation and demodulation. Our SDR is composed of multiple blocks: i) I/Q Radio, ii) Power Amplifier, iii) FPGA and iv) Memory.
\noindent{\bf I/Q Radio.} Current SDR systems use I/Q radio chips that are designed to cover a multi-GHz spectrum and have high ADC/DAC sampling rates to support large bandwidth. For example, AD9361~\cite{ad9361} series which is used in USRP and ADPluto can transmit from xxx MHz to xxx GHz and supports sampling rates as high as tens of MHz. Supporting a wide frequency spectrum with a low receiver noise figure and being to able to operate at tens of megahertz sampling rate, makes these radios power hungry on the order of watts. Moreover, as shown in Table~\ref{table:Radios}, almost all these radio chips costs more than XXX\$. To support IoT testbed vision, we need to use a radio that supports at least 900~MHz and 2.4~GHz ISM bands and is low-power and low-cost to be deployable. To this end, we selected AT86RF215~\cite{at86rf215} radio chip among all the choices. As we can see in Table~\ref{table:Radios}, this chip is the only one supporting both 900~MHz and 2.4~GHz ISM bands, costs less than 10\$ and consumes less than xxxx watts. MAX2831 and SX1257 are the only competitors in terms of cost. MAX2831 supports only 2.4~GHz, consumes twice the current and needs separate ADC and DAC. SX1257 supports only 900~MHz, has less bandwidth and consumes more current in TX mode. AT86RF215 integrates all the necessary blocks including LNA, programmable receive gain and low pass filter and ADC on the RX chain and including DAC to programmable PA with a maximum power of 14~dBm on the TX side. It supports three frequency bands of 389.5-510~MHz, 779-1020~MHz and 2400-2483.5~MHz and has two RF ports - one for sub-GHz and the other for 2.4~GHz. This chip utilizes two PLLs for sub-GHz and 2.4~GHz transceivers. Both PLLs use a shared 26~MHz crystal as reference and they can synthesize any frequencies with 198~Hz step size. Moreover, it has 13 bit ADC and DAC supporting up to 4~Msps of sampling rate for I and Q and hence 4~MHz of bandwidth which is more than the bandwidth of common IoT protocols. The RF front-end has 3-5~dB noise figure which is even lower than the noise figure of the front-end used in Semtech LoRa chipset, SX1276. Receiver/transmitter chain consumes around 28~mA/62mA nearly 5x lower than the current consumption of radio chips used in other SDR platforms~\cite{}. The receiver chain passes signal through an LNA with noise figure of 4.5 and then downconverts the signal using a mixer that is controlled by frequency synthesizer (PLL). In addition, this radio supports various modulations such as MR-FSK, MR-OFDM, MR-O-QPSK and O-QPSK. This allows us to bypass the FPGA entirely and enable greater power savings when we use standard modulations supported by this radio.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/block_diagram/system.eps}
\caption{\footnotesize{\name system block diagram. \textcolor{red}{We will revise this diagram based on new system design explanation. XXX}}}
\label{fig:system}
\end{figure}
\begin{table}[t]
\centering
\caption{\footnotesize{\bf Existing off-the-shelf I/Q radios.}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|C{2.4cm}|C{2.2cm}|c|c|}
\hline
\cellcolor{lightgray}{\bf I/Q Radio} & \cellcolor{lightgray}{\bf Frequency (MHz)} & \cellcolor{lightgray}{\bf Power} & \cellcolor{lightgray}{\bf Cost (\$)}\\
\hline
AD9361~\cite{ad9361} & 70$\sim$6000 & XXX & 282\\
\hline
Lms7002m~\cite{lms7002m} & 10$\sim$3500 & XXX & 110\\
\hline
MAX2831~\cite{max2831} & 2400$\sim$2500 & XXX & 9\\
\hline
SX1257~\cite{sx1257} & 862$\sim$1020 & XXX & 7.5\\
\hline
{\bf AT86RF215~\cite{at86rf215}} & {\bf 389.5$\sim$510 \newline 779$\sim$1020 \newline 2400$\sim$2483} & {\bf XXX} & {\bf 5.5}\\
\hline
\end{tabular}
}
\label{tab:radios}
\end{table}
\noindent{\bf FPGA.} The baseband I/Q signal is exchanged between I/Q radio and FPGA. The radio baseband I/Q interface is Low Voltage Differential Signalling (LVDS).
We use LFE5U-25F~\cite{latticeLFE5U} FPGA from Lattice Semiconductor for baseband processing which supports LVDS and has 24~k logic units which is comparable to other common SDR platforms including XXX, XXX and XXX. We use a small package of this FPGA on our design to decrease the platform physical size. Compared to FPGAs used in other existing platform this FPGA provides higher number of LUTs for the same cost. We use a 32~MHz clock oscillator as the reference clock for the FPGA. When radio operates in transmit mode, we use this clock reference to generate a 64~MHz clock reference for I/Q serial interface. However, when the I/Q operates in receive mode, we turn off this clock oscillator since the I/Q interface provides an clock reference and we use this reference for DSP operations in FPGA.
\noindent{\bf Power amplifier.} Our vision is to design an SDR which can be used for a long-range IoT node implementation. To be able to support long ranges, we need to support high power transmitter. AT86RF215 can transmit maximum power of 14~dBm which is less than the maximum power allowed by FCC in 900~MHz and 2.4~GHz. To this end, we use SE2435L~\cite{SE2435L} 900~MHz module which has 30~dBm PA for sub-GHz RF port. This module has LNA and power amplifier which can be bypassed if needed. This gives us flexibility in different operations. For example, in receive mode we can either pass the incoming signal through the LNA and then connect it to the I/Q radio or completely bypass the LNA and connect the signal directly. The bypass current consumption is around xxx uA and the sleep current of these two power amplifiers is xxx uA. Similarly, in transmit operation we can pass the signal through the PA and amplify the signal or turn off the PA and just pass the signal directly to the antenna. If the required output power is less than 14~dBm, we would bypass the PA to decrease power consumption. We use similar module, SKY66112~\cite{SKY66112}, which can output 27~dBm at 2.4~GHz band with similar capabilities.
\subsection{Over-the-Air (OTA) update system}
Our goal in this paper is to design an IoT platform that is deployable and scalable in a wide area. To achieve this vision, we need to think about how to update these platform's firmware in a large network over time without the need of a wired infrastructure. In this section, we explain our OTA update system where we can update each \name node wirelessly.
\noindent{\bf OTA wireless protocol.}
Supporting wide area requires a long-range wireless protocol for the OTA. We used LoRa protocol for two reasons. First, LoRa gives high sensitivity and in turn kilometer ranges and also support a wide range of data rates from from XXX to XXX.
Moreover, LoRa is becoming more and more wide-spread specially in the US. Table~\ref{tab:OTA_protocols} compares different long-range protocols and their corresponding data-rates. We use SX1276 Semtech chipset~\cite{sx1276} to implement LoRa protocol. As shown in Fig.~\ref{fig:OTA_XXX}, MCU receives the baseband bits from the SX1276 which has compressed firmware bitstream for both FPGA and MCU. Then, it stores this data in the external flash memory.XXXXX
\noindent{\bf Flash Memory.} We store the firmware update bitstream in an external flash memory. FPGA programming bitstream for our FPGA on \name has a size of 572~KB and MCU is XXX~KB. In our platform, we use 2~MB flash memory to be able to store multiple bitstreams for FPGA and MCU at the same time.
\subsection{Controller}
Controller is the central brain of the system and is implemented on a low-power MCU. The MCU needs to have low sleep current, able to support multiple control interfaces, requires enough memory resources to support IoT MAC protocols and also able to run decompression algorithm. We used MSP432P401R~\cite{msp432P401R} a 32-Bit Cortex M4F MCU which has less than 1~uA sleep current, includes 64~KB SRAM and 256~KB flash memory. MCU controls I/Q radio parameters, backbone radio parameters, power amplifiers, RF switches, all the regulators and FPGA programming. By controlling the regulators, it does power-gating of different voltage domains and hence puts components supported by these domain to sleep or wakes them up. The sleep process which is controlled by the MCU is the following, it sets the timer, puts all the regulators to shut-down mode and then it goes to low power mode consuming only 550~nA.
\subsection{Power Management Unit}
In this section, we present the design of power management unit working with 4V xxxx battery with the goal of decreasing power consumption. Specifically, to be able to support long battery life we need to be able to duty-cycle and power-gate the the system. In addition, different components in the system have different supply voltage requirements. To minimize the power, we need to provide components with the lowest voltage possible. This means that to minimize power we need to support similar voltage components with the voltage domain supported by the same voltage regulator. Additionally, voltage of some of the components needs to be changed over time to get the desired performances like power efficiency. So, ideally we would want have separate controllable voltage regulator for each major power consuming component in the system. However, having many different voltage regulators and being able to control them increases the number of devices and in turn \name price significantly. Moreover, control signals routing and having different power planes on the PCB to support these voltage domains makes the PCB design exponentially complicated. Therefore, trade-off exists between the number of voltage domains and price and complexity of the design. Table~\ref{tab:voltages} shows the supply voltages needed for each component and the voltage domain supporting it.
\begin{table}[t]
\centering
\caption{\footnotesize{\bf Voltage domains in \name.}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|C{2.4cm}|C{3cm}|c|}
\hline
\cellcolor{lightgray}{\bf Component} & \cellcolor{lightgray}{\bf Voltage (V)} & \cellcolor{lightgray}{\bf Voltage Domain}\\
\hline
MCU & 1.8 & V1\\
\hline
FPGA & 1.1, 1.8, 2.5, 1.8< V5 <3.6 & V2, V3, V4, V5\\
\hline
I/Q Radio & 1.8< V5 <3.6 & V5\\
\hline
Backbone Radio & 1.8< V5 <3.6 & V5\\
\hline
sub-GHz PA & 3.5 & V6\\
\hline
2.4~GHz PA & 1.8, 3.0 & V3, V7\\
\hline
FLASH Memory & 1.8 & V3\\
\hline
Micro SD Memory& 3.0 & V7\\
\hline
\end{tabular}
}
\vskip -0.2 in
\label{tab:voltages}
\end{table}
\noindent{\bf Voltage domain V1.} We use linear regulator for V1 which supports MCU. Since MCU is the central controller, it needs to be powered at all times. Therefore, it is powered by a separate voltage domain. Moreover, to decrease the sleep current we need to use a voltage regulator that has low quiescent current. Although switching voltage regulator has higher voltage conversion efficiency when it is active, its high quiescent current prevents it from being used for V1. Considering a wake-up power consumption of 100~mW XXX and duty-cycles as high as 1000x, the sleep power consumption should be much lower than $\frac{100mW}{1000} = 100uW$ or $\frac{100uW}{5} = 20uW$. Having a battery voltage of 4V results in less than 5uA sleep current. We use XXX which is a linear regulator and has only XXX uA quiescent current.
\noindent{\bf Voltage domain V2.} This voltage domain supports FPGA core voltage which needs constant voltage of 1.1~V. Since the voltage difference between the battery voltage and this domain's voltage is high, we need to use switching voltage regulator to achieve high efficiency. Since we are using SRAM-based FPGA, it has high core and I/O leakage current. Therefore, in order to decrease sleep current, this voltage domain needs to be shut-down during sleep. This means that the regulator needs to have low shut-down current of less than 1~uA. We use TPS62240~\cite{TPS62240} to support this domain since it has high voltage conversion efficiency and low shut-down current.
\noindent{\bf Voltage domain V3, V4 and V7.} We need 1.8~V voltage supply for I/O banks and programming of the FPGA, digital supply of the 2.4~GHz PA and Flash memory and is provided by V3 voltage domain. In addition, FPGA I/O banks need a supply voltage of 2.5~V other than their main I/O supply voltage. V4 provides this supply voltage. Voltage regulators for these domains should have low shut-down current during sleep and high efficiency. We use 3~V voltage supply to power microSD card and analog supply of the 2.4~GHz PA. The voltage regulator for this domain needs to support currents as high as 120~mA for the 2.4~GHz PA in addition to having low shut-down current and high efficiency. Similar to V2, we use three TPS62240 switching voltage regulators for V3, V4 and V7 because of its low shut-down current, high efficiency and being able to provide as high as 300~mA current.
\noindent{\bf Voltage domain V5.} V5 is a shared voltage domain for I/Q radio, backbone LoRa radio and FPGA I/O bank connected to LVDS interface of the I/Q radio. This voltage domain is initially set to 1.8V to decrease the power consumption. If the output of the I/Q radio needs to increase, we would need to increase this voltage. Therefore, we need controllable voltage regulator for this voltage domain besides having high efficiency and low shut-down current. To this end, we use Semtech SC195ULTRT~\cite{SC195ULTRT} which provides a range from 1.8~V to 3.6~V.
\noindent{\bf Voltage domain V6.} This voltage domain supplies the 900~MHz PA. This PA consumes as high as 550~mA current when transmitting 30~dBm. Therefore, the voltage regulator for this domain needs to be able to supply this current while having high efficiency and low shut-down current. We use TPS62080~\cite{TPS62080} which is able to provide 1.2A while satisfying the other two requirements. Note that, since this PA can consumes as much as 2~W, we carefully design ground and power plane on PCB to support enough current and also avoid heating issues on the \name board.
\section{\name\ Platform}
Our IoT platform has low-power, supports a few MHz of bandwidth, achieves microsecond latency and includes over-the-air update of the wireless protocol it can run on its FPGA. Here, we first describe the different components in our hardware. We then present the interfaces between the different hardware components. Finally, we describe the over-the-air update protocol including the compression and decompression algorithm as well as over-the-air FPGA programming.
\subsection{Hardware Design}
Building a platform that satisfies the above requirements needs a number of careful design considerations. Specifically, we seek to minimize power consumption and cost while still offering the flexibility of an SDR which can read and process raw samples. Here, we outline the system architecture along with a detailed analysis of our design decisions.
\subsubsection{Designing the Software Radio}
The core block on our platform is the software-defined radio, a programmable PHY layer that processes and converts bits to radio signals and vice versa. We begin by explaining our choices for the primary components of an SDR which are a radio chip that provides an interface for sending and receiving raw samples of an RF signal as well as an FPGA that can process these signals in real time. We then discuss the supporting peripherals for these devices such as a power amplifier (PA) to boost the output of the radio chip and non-volatile memory for the FPGA to read and write data from.
{\bf Choosing a Radio Chip.}
We begin by choosing a radio chip as its specs define the requirements for the FPGA and other blocks. Our primary requirement is that the chip supports reading and writing raw complex I/Q samples of the RF signal. Current SDR systems use I/Q radio chips that are designed to cover a multi-GHz spectrum and have high ADC/DAC sampling rates to support large bandwidth. For example, AD93XX~\cite{XXX} series which is used in USRP and ADPluto can transmit from xxx MHz to xxx GHz and supports sampling rates as high as tens of MHz. Each of these spec, wide bandwidth, low noise figure, and high sampling rate represent fundamental trade offs of power for performance, and therefore consume watts of power. Moreover, almost all these radio chips costs more than \$XX.
We instead take a different approach when designing \name: identify the minimum required specs and find a radio that supports them. Specifically, an IoT platform must be able to operate in at least the 900~MHz and 2.4~GHz ISM bands, have at least XX~MHz of bandwidth, while otherwise minimizing power and ideally costing less than \$10. We analyze all of the commercially available radio chips that provide these capabilities and list them in Table~\ref{table:Radios}. As shown, only the AT86RF215 supports all of our requirements. In addition to lower cost and support for both frequency bands, it also consumes less power than the MAX2831 and the SX1257. Moreover, the AT86RF215 integrates all the necessary blocks including LNA, programmable receive gain and low pass filter, ADC on the RX chain, as well as a DAC and programmable PA with a maximum power of 14~dBm on the TX side. This chip supports three frequency bands of 389.5-510~MHz, 779-1020~MHz and 2400-2483.5~MHz which includes major ISM bands and provides 4~MHz of bandwidth which meets our IoT protocol requirements.
Next, we evaluate its noise figure to determine whether it compromises performance to achieve these specs. For comparison, the RF front-end has 3-5~dB noise figure which is even better than the noise figure of the front-end used in Semtech SX1276 LoRa chipset, suggesting it should be able to achieve long range performance. At the same time it consumes 5x less power than the radios used on other SDRs~\cite{XXX} and has built in support for standard modulations such as MR-FSK, MR-OFDM, MR-O-QPSK and O-QPSK that can save FPGA resources or power by bypassing the FPGA entirely.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/block_diagram/system.eps}
\caption{\footnotesize{\name system block diagram.}}
\label{fig:system}
\end{figure}
\begin{table}[t]
\centering
\caption{\footnotesize{\bf Existing off-the-shelf I/Q radios.}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|C{2.4cm}|C{2.2cm}|c|c|}
\hline
\cellcolor{lightgray}{\bf I/Q Radio} & \cellcolor{lightgray}{\bf Frequency (MHz)} & \cellcolor{lightgray}{\bf Power} & \cellcolor{lightgray}{\bf Cost (\$)}\\
\hline
AD9361~\cite{ad9361} & 70$\sim$6000 & XXX & 282\\
\hline
LMS7002M~\cite{lms7002m} & 10$\sim$3500 & XXX & 110\\
\hline
MAX2831~\cite{max2831} & 2400$\sim$2500 & XXX & 9\\
\hline
SX1257~\cite{sx1257} & 862$\sim$1020 & XXX & 7.5\\
\hline
{\bf AT86RF215~\cite{at86rf215}} & {\bf 389.5$\sim$510 \newline 779$\sim$1020 \newline 2400$\sim$2483} & {\bf XXX} & {\bf 5.5}\\
\hline
\end{tabular}
}
\vskip -0.15in
\label{tab:radios}
\end{table}
{\bf Picking an FPGA.} Now that we have chosen a radio chip, the next step in our design process is to find an FPGA that can interface with it. Aside from minimizing power, we would also like to maintain a small form factor, and so we also consider the physical package size of the FPGA. We use LFE5U-25F~\cite{latticeLFE5U} FPGA from Lattice Semiconductor for baseband processing which has 24~k logic units. This chip provides a greater number of LUTs than other SDR platforms such as the XXX and XXX but at lower cost. We also choose a small package of this FPGA to decrease the platform's size.
{\bf Adding a power amplifier.} AT86RF215 only supports a maximum transmit power of 14~dBm that is traditionally used by IoT radios but which is less than the 30~dBm allowed by the FCC for transmissions in ISM bands. To provide flexibility, we add a PA to maximize range within FCC constraints. Considering the high cost and power requirements of wide-band PAs that could operate at both 900~MHz and 2.4~GHz we instead select two different chips: the SE2435L~\cite{SE2435L} for 900~MHz and SKY66112~\cite{SKY66112} for 2.4~GHz. Our 900~MHz PA supports up to 30~dBm output power, and the 2.4~GHz PA can output up to 27~dBm. Both chips also include an LNA for receive mode and include a built in circuit to bypassing either of these components for maximum flexibility and power savings. For example, in receive mode we can either pass the incoming signal through the LNA and then connect it to the I/Q radio or completely bypass the LNA and connect the signal directly. The bypass current consumption is a negligible xxx uA and the sleep current of both two power amplifiers is xxx uA. Similarly, in transmit operation we can pass the signal through the PA and amplify the signal or turn off the PA and just pass the signal directly to the antenna if we need to transmit less than 14~dBm.
\subsubsection{Designing hardware for Over-the-Air (OTA) update system.}
Designing a wireless IoT platform that is useful for research and performing experiments presents a practical issue: how can we send firmware updates to these nodes with new IoT protocols to run? While all of the discussion above enables a small, low power, low cost SDR for easy deployment, FPGAs and microcontrollers typically require a wired interface for reprogramming. To make a large scale network practical we develop an OTA update system to reconfigure and program \name nodes wirelessly.
{\bf OTA wireless chipset.} A key question when designing an OTA update system is, what wireless protocol should be used? Considering our goal is to support wide area networking, we focus on protocols designed for long range operation. We analyze all of the available long range protocols and select LoRa for our OTA system for a number of reasons. First, LoRa receivers have a high sensitivity which enables kilometer ranges. LoRa also support a wide range of data rates from XXX to XXX which allows us to trade off rate for range depending on the deployment scenario. Moreover, LoRa is becoming more and more wide-spread in the US. We use the SX1276 Semtech chipset~\cite{sx1276} which is available for only \$XX, minimizing the cost overhead.
{\bf Flash Memory.} While our FPGA achieves good performance at low power, size and cost, it is SRAM based and does not include onboard non-volatile memory for storing programming data. We instead store the firmware bitstream on a separate flash memory chip. The FPGA programming bitstream is 572~KB and the MCU programs require up to XXX~KB. When choosing a flash chip we select the XXX which has 2~MB of memory. Although this is far more than the minimum size required, it allows for \name to store multiple bitstreams and MCU programs to quickly switch between stored protocols or configurations without re-sending the programming data over the air.
\subsubsection{Picking the microcontroller}
So far we explained how to assemble a highly flexible and low power SDR design with wireless update capabilities. To control all of these individual chips and toggle all of these power saving options, we use a low-power microcontroller (MCU).
What are the requirements for our MCU? In addition to having a low sleep current it must be able to support multiple control interfaces, requires enough memory resources to support IoT MAC protocols and also able to run a decompression algorithm for our OTA system. We select the MSP432P401R~\cite{msp432P401R} a 32-Bit Cortex M4F MCU which meets all of our requirements with its less than 1~uA sleep current, and 64~KB of onboard SRAM and 256~KB of onboard flash memory. In addition to controlling the I/Q radio parameters and backbone radio parameters, and reprogramming of the FPGA, the MCU performs the important function of power management. It is responsible or toggling on and off the power amplifiers, as well as performing power-gating by turning ON and OFF different voltage regulators to turn various components in our system ON or OFF.
\subsubsection{Designing the Power Management Unit}
Next, we present the design of our power management unit which seeks to maximize the system lifetime when running off of a 3.7~V Lithium battery. To enable long battery lifetimes we need to be able to duty-cycle our system and toggle each of the above blocks ON and OFF when they are not used. Further, different components have different supply voltage requirements and we wish to provide each one with the lowest voltage possible to minimize power usage.
Ideally we would want separate controllable voltage regulator for each major power consuming component in the system. However, having many different voltage regulators and being able to control all of them significantly increases the complexity and number of components which in turn increases price. Moreover, it complicates the PCB design requiring many control signals to be routed and a multitude of power planes. Therefore, there exists a trade-off between the granularity of power control and the price/complexity of a design. To better understand the needs of our system, we outline the supply voltages needed for each component and the voltage domain supporting it in Table~\ref{tab:voltages}. We describe each of these voltage domains and how we group components to balance power optimization with complexity below.
\begin{table}[t]
\centering
\caption{\footnotesize{\bf Voltage domains in \name.}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|c|c|c|}
\hline
Component & Voltage [V] & Voltage Domain\\
\hline
MCU & 1.8V & V1\\
\hline
FPGA & 1.1, 1.8, 2.5, Vlvds & V2, V3, V4, V5\\
\hline
I/Q Radio & 1.8< V5 <3.6 & V5\\
\hline
Backbone Radio & 1.8< V5 <3.6 & V5\\
\hline
sub-GHz PA & 3.5V & V6\\
\hline
2.4~GHz PA & 1.8, 3.0 & V3, V7\\
\hline
FLASH Memory & 1.8 & V3\\
\hline
Micro SD Memory& 3.0 & V7\\
\hline
\end{tabular}
}
\vskip -0.2 in
\label{tab:voltages}
\end{table}
{\bf Voltage domain V1 (MCU).} Since MCU is the central controller which implements power management, it needs to be powered at all times and therefore has its own voltage domain. Moreover, to decrease the sleep current we need to use a voltage regulator with a low quiescent current. Although switching voltage regulator designs have higher conversion efficiency when active, they also have high quiescent currents. So we instead use a linear regulator. When choosing these components we want to make sure this regulator and the microcontroller sleep power do not dominate the sleep power of the whole systemt. Since our system consumes around 100~mW when active and duty-cycling of IoT devices can be as much as 1000x, our sleep power consumption should be much lower than $\frac{100mW}{1000} = 100uW$; we target $20uW$. Having a battery voltage of 3.7 V results in around 5uA sleep current. We therefore choose the XXX which is a linear regulator with a quiescent current of XXX uA.
{\bf Voltage domain V2.} This voltage domain supports the FPGA core voltage which needs to be a constant 1.1~V. Since the voltage difference between the battery voltage and this domain's voltage is high and the FPGA can be power gated when not in use, we use a switching voltage regulator to achieve high efficiency. Because our FPGA is SRAM-based, it has a relatively high core and I/O leakage current, therefore wasting power even in sleep mode. In order to decrease the overall sleep power of our system, we shut down this voltage domain during sleep. This means that the regulator needs to have low shut-down current of less than 1~uA. We use TPS62240~\cite{TPS62240} here since it has high voltage conversion efficiency and low shut-down current.
{\bf Voltage domains V3, V4 and V7.} We need a 1.8~V supply for the I/O banks and programming of the FPGA, the digital supply of the 2.4~GHz PA and flash memory which are all provided by V3 voltage domain. In addition, the FPGA I/O banks need a supply voltage of 2.5~V given by V4 in addition to V3. Considering these components will all be turned off when not operating the voltage regulators for these domains should have low shut-down current during sleep and high efficiency. Additionally, we need a 3~V voltage supply provided by V7 to power the microSD card and analog supply of the 2.4~GHz PA. The voltage regulator for V7 has the additional requirement of supporting currents as high as 120~mA when the 2.4~GHz PA is active. Similar to V2, we use three TPS62240 switching voltage regulators which can be set to output each of the required voltages for V3, V4 and V7 because of its low shut-down current, high efficiency and ability to provide up to 300~mA of current.
{\bf Voltage domain V5.} V5 is a shared voltage domain for I/Q radio, backbone LoRa radio and FPGA I/O bank connected to the I/Q radio. This voltage domain is initially set to 1.8V to minimize power consumption, however components such as the radio chips can require higher voltage to achieve maximum power output. Therefore, in addition to high efficiency and low shut-down current as with the others, this domain should be programmable as well. To this end, we use Semtech SC195ULTRT~\cite{SC195ULTRT} which provides an adjustable output that can be set from 1.8~V to 3.6~V.
{\bf Voltage domain V6.} This voltage domain supplies the 900~MHz PA. This PA consumes as much as 550~mA current when transmitting at 30~dBm. We use TPS62080~\cite{TPS62080} which is able to provide 1.2A. Note that, we carefully design the ground and power plane on PCB to support enough current while minimizing heat dissipation issues. We also note that in normal operations the PA is turned off since IoT protocols do not such high transmit powers.
\section{\Name\ Platform}
We first describe our design choices for the different components of our hardware shown in Fig.~\ref{fig:system} and explain the interfaces between them. Next we present the power management module which enables our ultra-low-power sleep mode. Finally, we describe our over-the-air update protocol including decompression algorithms and over-the-air reprogramming.
\vspace{-0.2 in}
\subsection{Hardware Design}
\label{sec:hardware}
We seek to minimize power consumption and cost while offering the flexibility of an SDR to process raw samples.
\subsubsection{Designing the Software Radio}
The core block on our platform is the software-defined radio, a programmable PHY layer that processes and converts bits to radio signals and vice versa. We begin by explaining our choices for the primary components of an SDR which are a radio chip that provides an interface for sending and receiving raw samples of an RF signal as well as an FPGA that can process these signals in real time. We then discuss the supporting peripherals for these devices such as a power amplifier (PA) to boost the output of the radio chip and non-volatile memory for the FPGA to read and write data from.
{\bf Choosing a radio chip.} We begin by choosing a radio chip as its specs define the requirements for the FPGA and other blocks. Our primary requirement is that the chip supports reading and writing raw complex I/Q samples of the RF signal. As shown in Table~\ref{tab:radios}, current SDR systems use I/Q radio chips that are designed to cover a multi-GHz spectrum and have high ADC/DAC sampling rates to support large bandwidth. For example, AD936x~\cite{ad9361} series which is used in USRP and ADPluto can transmit from 325~MHz to 3.8~GHz and supports sampling rates as high as tens of MHz. Each of these specs such as wide bandwidth, low noise, and high sampling rate represent fundamental trade offs of power for performance, and therefore these chips consume watts of power. Moreover, some of these radio chips costs more than \$100.
We instead take a different approach: identify the minimum required specs and find a radio that supports them. Specifically, an IoT platform must be able to operate in at least the 900~MHz and 2.4~GHz ISM bands, have 4~MHz of bandwidth, while otherwise minimizing power and ideally costing less than \$10. We analyze all of the commercially available radio chips that provide baseband I/Q samples and list them in Table~\ref{tab:radios}, where only the AT86RF215 supports all of our requirements. In addition to lower cost and support for both frequency bands, it also consumes less power than the MAX2831 and the SX1257. Moreover, the AT86RF215 integrates all the necessary blocks including an LNA, programmable receive gain, {automatic gain control (AGC)} and low pass filter, ADC on the RX chain, as well as a DAC and programmable PA with a maximum power of 14~dBm on the TX side. In terms of noise, the RF front-end has a 3-5~dB noise figure which is even better than the noise figure of the front-end used in Semtech SX1276 LoRa chipset, suggesting it should be able to achieve long range performance. It consumes 5x less power than the radios used on other SDRs as shown in Fig.~\ref{fig:platform_power} and has built in support for common modulations such as MR-FSK, MR-OFDM, MR-O-QPSK and O-QPSK that can save FPGA resources or power by bypassing the FPGA entirely.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\linewidth]{figs/tinysdr_diagram.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf \Name System Block Diagram.} A complete system diagram showing all of the components of \name. This includes the software radio consisting of the radio, amplifiers, and FPGA, OTA programmer which uses a LoRa radio and flash memory to store programs, and a power managment system with the flexibility to turn off power consuming components. Each of these subsystems are controlled in software running on the MCU.}}
\vskip -0.23in
\label{fig:system}
\end{figure*}
\begin{table}[t]
\centering
\caption{\footnotesize{\bf Existing Off-the-Shelf I/Q Radio Modules.}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|C{2cm}|C{2.2cm}|c|c|}
\hline
\cellcolor{lightgray}{\bf I/Q Radio} & \cellcolor{lightgray}{\bf Frequency (MHz)} & \cellcolor{lightgray}{\bf RX Power (mW)} & \cellcolor{lightgray}{\bf Cost}\\
\hline
AD9361~\cite{ad9361} & 70$\sim$6000 & 262 & \$282\\
\hline
AD9363~\cite{ad9363} & 325$\sim$3800 & 262 & \$123\\
\hline
AD9364~\cite{ad9364} & 70$\sim$6000 & 262 & \$210\\
\hline
LMS7002M~\cite{lms7002m} & 10$\sim$3500 & 378 & \$110\\
\hline
MAX2831~\cite{max2831} & 2400$\sim$2500 & 276 & \$9\\
\hline
SX1257~\cite{sx1257} & 862$\sim$1020 & 54 & \$7.5\\
\hline
{\bf AT86RF215~\cite{at86rf215}} & {\bf 389.5$\sim$510 \newline 779$\sim$1020 \newline 2400$\sim$2483} & {\bf 50} & {\bf \$5.5}\\
\hline
\end{tabular}
}
\vskip -0.3in
\label{tab:radios}
\end{table}
{\bf Picking an FPGA.} Now that we have chosen a radio chip, the next step in our design process is to find an FPGA that can interface with it. Aside from minimizing power {and cost}, we would also like to maintain a small form factor {and short wake-up time}. {Although flash-based FPGAs are capable of fast wake-ups, they are more expensive compared to SRAM-based FPGAs with the same number of logic elements.} We use LFE5U-25F~\cite{latticeLFE5U} FPGA from Lattice Semiconductor for baseband processing which is an SRAM-based and has 24~k logic units. This chip provides a greater number of look up tables (LUTs) than the FPGAs on the PlutoSDR and LimeSDR mini, and at lower cost.
{\bf Adding a power amplifier (PA).} AT86RF215 only supports a maximum transmit power of 14~dBm which is traditionally used by IoT radios but is less than the 30~dBm maximum allowed by the FCC. To provide flexibility, we add optional PAs. Given the high cost and power requirements of wide-band PAs that could operate at both 900~MHz and 2.4~GHz we instead select two different chips: the SE2435L~\cite{SE2435L} for 900~MHz and SKY66112~\cite{SKY66112} for 2.4~GHz. Our 900~MHz PA supports up to 30~dBm output power, and the 2.4~GHz PA can output up to 27~dBm. Both chips also include an LNA for receive mode and a built in circuit to bypass either of these components for power savings. In receive mode, we can either pass the incoming signal through the LNA and then connect it to the radio or completely bypass the LNA and connect the signal directly. The maximum bypass current is 280 uA and the sleep current of both power amplifiers is only 1 uA. In transmit operation we can pass the signal through the PA and amplify the signal or turn off the PA and pass the signal directly to the antenna for transmit power $<$ 14~dBm.
{\bf Picking the microcontroller.} We use a microcontroller to control all the individual chips and toggle all of these power saving options. In addition to having a low sleep current it must be able to support multiple control interfaces, have enough memory resources to support IoT MAC protocols and also be able to run a decompression algorithm for our OTA system. We select the MSP432P401R~\cite{msp432P401R} a 32-Bit Cortex M4F MCU which meets all of our requirements with less than 1~uA sleep current, has 64~KB of onboard SRAM and 256~KB of onboard flash memory. In addition to controlling the I/Q and backbone radio parameters, and reprogramming of the FPGA, the MCU performs the important function of power management. It is responsible for toggling ON and OFF the power amplifiers, as well as performing power-gating by turning ON and OFF different voltage regulators in~\xref{sec:powermanagement}.
\subsubsection{Designing OTA Update Hardware}
While the above discussion enables a small, low power, low cost SDR for easy deployment, FPGAs and microcontrollers typically require a wired interface for reprogramming. Here we present the hardware for the OTA update system to reconfigure and program \name nodes wirelessly.
{\bf OTA wireless chipset.} A key question when designing an OTA update system is, what wireless protocol should be used? To support wide area networking, we focus on protocols designed for long range operation. We analyze all of the available long range protocols and select LoRa for our OTA system for a number of reasons. First, LoRa receivers have a high sensitivity which enables kilometer ranges. LoRa also support a wide range of data rates from 11~bps to 37~kbps which allows us to trade off rate for range depending on the deployment scenario. Moreover, LoRa is becoming more and more wide-spread in the US. We use the SX1276 Semtech chipset~\cite{sx1276} which is available for \$4.5, minimizing cost.
{\bf Flash Memory.} Our FPGA is SRAM based and does not include on-chip non-volatile memory for storing programming data. We instead store the firmware bitstream on a separate flash memory chip. The FPGA programming bitstream is 579~KB and the MCU programs require a maximum of 256~KB. We chose the MX25R6435F flash chip with 8~MB memory. Although this is far more than the size required, it allows \name to store multiple FPGA bitstreams and MCU programs to quickly switch between stored protocols without having to re-send the programming data over the air.
\section{Conclusion and Discussion}
This paper presents the first SDR platform specifically tailored to the needs of IoT endpoints that can be used for large scale IoT network deployments. The goal of \name is to provide a platform that can catalyze research in IoT networks.
{\bf Research on PHY/MAC protocols.} \Name presents an opportunity for researchers to avoid the time consuming endeavor of building their own custom IoT evaluation platforms and instead focus on PHY/MAC protocol innovations for the unique challenges of IoT networks. These include the fact that protocols trade-off data rate for range resulting in long packets during which channels may change significantly. What is the trade-off between packet length and overall throughput? Are there benefits of rate adaptation? We could also explore other questions like whether we can develop systems that support concurrent transmissions or creating multi-hop IoT PHY/MAC innovations, which have not been explored well given the lack of a flexible platform.
{\bf Research on IoT localization.} \Name could also be used to build localization systems as it gives access to the raw signals and therefore phase information across 2.4~GHz and 900~MHz bands, which forms the basis for many localization algorithms. In addition to exploring novel techniques such as distributed localization solutions where we could combine the phase information across a distributed set of sensors to create a large MIMO sensing system, \name presents an opportunity to evaluate and verify existing techniques on a common platform and understand how they would perform within the power and computation constraints of an IoT device.
{\bf Machine learning on IoT devices.} The FPGA on \name opens up exciting opportunities for exploring machine learning algorithms on-board. This would allow researchers to explore trade-offs between the power overhead of running an on-board classifier versus repeatedly turning on the radio to send data to the cloud. This could also enable use of high bandwidth sensors such as cameras and microphones in which the time and therefore power bottleneck may be communication rather than sensing, as well as the ability to immediately act on sensor data during network outages for example to sound a local alert or alarm.
In addition to IoT research opportunities, below we outline ways in which we can improve our platform.
{\bf Improving OTA programming.}
Currently, \name uses a simple MAC protocol for programming with a focus on using minimal system resources to allow for other custom software. However sequentially updating each node, could present significant overhead for systems with >100 nodes. Potential research directions involve exploring modified MAC protocols to reduce this time by exploiting the wireless broadcast nature, as well as a distributed update system in which nodes could help propagate update packets through the network.
{\bf Programming interface.}
TinySDR currently requires users to write Verilog or VHDL to program the FPGA and C code for programming the microcontroller. Future versions can incorporate a pipeline to use high level synthesis tools or integration with GNUradio for easy prototyping.
\section{Conclusion and Research Opportunities}
This paper presents the first SDR platform specifically tailored to the needs of IoT endpoints that can be used for large scale IoT network deployments. The goal of \name is to provide a platform that can catalyze research in IoT networks.
{\bf Research on PHY/MAC protocols.} \Name presents an opportunity for researchers to avoid the time consuming endeavor of building their own custom hardware and instead focus on PHY/MAC protocol innovations across the stack: What is the trade-off between packet length and overall throughput? Are there benefits of rate adaptation? What about concurrent transmissions from IoT devices? One can also create multi-hop IoT PHY/MAC innovations, which have not been explored well given the lack of a flexible platform.
{\bf Research on IoT localization.} \Name could also be used to build localization systems as it gives access to I/Q signals and therefore phase across 2.4~GHz and 900~MHz bands, which forms the basis for many localization algorithms\cite{sensys18}. One can also explore distributed localization solutions that combine the phase information across a distributed set of sensors to create a large MIMO sensing system.
{\bf Machine learning on IoT devices.} The FPGA on \name opens up exciting opportunities~\cite{deeplearning} for exploring machine learning algorithms on-board. This would allow researchers to explore trade-offs between the power overhead of running an on-board classifier versus sending data to the cloud. This could also enable use of high bandwidth sensors such as cameras and microphones where the power bottleneck may be communication rather than sensing.
{\bf Low power backscatter readers.} Recent work on ambient backscatter~\cite{abc,wifibackscatter,nsdi16,interscatter,fmbackscatter} aims to achieve ultra-low power communication for IoT devices. Many of these proposals require either a single-tone generator~\cite{nsdi16} or a custom receiver to decode the backscatter transmissions~\cite{3dprinted,3dprinted2,nsdi2018camera}. TinySDR can be used as a building block to achieve a battery-operated backscatter signal generation and receiver.
{\bf Better programming interface and protocols.} In addition to IoT research opportunities, we can also improve our platform in multiple ways. TinySDR currently requires users to write Verilog or VHDL to program the FPGA and C code for programming the microcontroller. Future versions can incorporate a pipeline to use high level synthesis tools or integrate with GNUradio for easy prototyping. Further, \name uses a simple MAC protocol for programming with a focus on using minimal system resources to allow for other custom software; however we could explore modified MAC protocols that simultaneously broadcast the updates across the network to reduce programming time.
\section{Benchmarks and Evaluation}\label{eval}
In this section, we characterize \name platform while it is running different operations.
\begin{figure}[t]
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=1\linewidth]{figs/block_diagram/rx_block.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf I/Q Receiver.}}}
\label{fig:rx_block}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=1\linewidth]{figs/block_diagram/tx_block.eps}
\caption{\footnotesize{{\bf Single Tone Transmitter.}}}
\label{fig:tx_block}
\end{subfigure}
\vskip -0.15in
\caption{\footnotesize{{\bf FPGA Block Diagrams.} XXX}}
\label{fig:fpga_blocks}
\end{figure}
\subsection{Single Tone Transmitter}
One of the basic operations of an SDR platform is transmitting a single tone wave. We program the radio front-end to transmit I/Q samples. Then, we use FPGA architecture, shown in Fig.~\ref{fig:tx_block} block diagram, to generate single tone wave and interface with radio front-end. This block diagram includes the basic architecture for any transmitting operation on \name platform. This block diagram includes a PLL module, clock divider, modulator, IQ Serializer and two LVDS modules. We use PLL to convert 50~MHz clock oscillator to 64~MHz clock that is required to interface with radio front-end serial interface. We divide this 64~MHz clock to 4~MHz clock that is the maximum sampling rate of the radio. Then we use this clock frequency to generate I/Q samples in signal modulator which in this example is a single tone modulator. We design an I/Q serializer module to convert parallel samples to serial. Then we connect serial output alongside the 64~MHz serial clock to LVDS modules to interface with radio font-end.
We set our radio to operate at following settings; Sampling rate of 4~MHz, power amplifier (PA) DC voltage of $2.4V$. We sweep the power output of the radio by changing PA index power and record the total power consumption of our platform. Fig.~\ref{fig:bench_tx_power} shows the power consumption of \name operating at 900~MHz and 2.4~GHz ISM bands. This plot shows XXXX.
In addition, we evaluate the area size of the single tone transmitter design on FPGA to show XXX.
\subsection{I/Q Receiver}
Next, we evaluate our platform in receive mode. To do this, we design a simple I/Q receiver on FPGA which receives I/Q samples from radio front-end and stores them in FPGA memory. Fig.~\ref{fig:rx_block} shows the FPGA block diagram of I/Q receiver. We design an I/Q deserializers that receives serial data from LVDS modules and converts to parallel I/Q samples. Then, deserializer module interface with SRAM memory to store I/Q samples. This is a universal design in our platform which provides I/Q samples to other FPGA blocks such as LoRa demodulator.
\subsection{Clock Synchronization}
\subsection{Frequency Hopping}
\section{Evaluation}
\subsection{LoRa}
To show the capability of the sub-GHz radio front-end in \name, we implemented LoRa modulator and demodulator on FPGA on our platform. Here, we explain the block diagram for modulator and demodulator that we implement on FPGA.
\noindent {\bf LoRa Modulator.} Fig.~\ref{fig:lora_tx_block} shows the block diagram of our LoRa modulator. We implement this modulator chain in Verilog HDL programming. Modulator starts with determining LoRa configurations such as $Spreading Factor$ and $Bandwidth$ and symbols values in Packet Generator unit. Then, the Chirp Generator unit uses these information to generate each chirp symbol. It uses a phase accumulator along with two lookup tables for $Sin$ and $Cosin$ functions to generate I/Q samples. We feed I/Q samples to IQ Serializer to generate serial stream information compatible with radio front-end interface. Finally, serial I/Q samples along with TX clock go to two LVDS interface that is connected to radio front-end.
\noindent {\bf LoRa Demodulator.} Fig.~\ref{fig:lora_rx_block} shows the block diagram of our LoRa demodulator. Demodulator interfaces with radio fornt-end through two LVDS connection; one for I/Q serial data and the other for synchronization clock. We implement an IQ Deserializer to convert serial I/Q data to parallel I/Q for further signal processing. Then, we run these data through a low-pass FIR filter. We use a memory in FPGA to buffer raw samples between the front-end and our demodulator. We use the Chirp Generator that we implemen for LoRa modulator to generate a baseline upchirp/downchirp symbol and then we multiply that with the received chirp symbol using Chirp Multiplier unit. Then, we use FFT core implementation provide by Lattice to perform FFT operation on multiplied signal and then detect symbol value and symbol type (upchirp or downchirp) using our Symbol Detector implementation.
\subsection{Zigbee De/Modulator}
\subsection{Power Consumption}
We begin by evaluating the power consumption of \name to demonstrate it is both lower power than any SDR platform on the market and more importantly that its power requirements are within an acceptable range for use as an IoT endpoint. We measure the power consumption for different functions such as running a transmitter or receiver for various protocols by running each function xx times and measuring the required current and voltage using an xxx digital multimeter. We show the results in Table~\ref{tab:power}. To give some comparison, the XXX LoRa radio consumes XX~mW in receive mode making \name only xxx more power expensive than these custom solutions.
While Table~\ref{power} shows the power consumption of the receiver, the transmitter power is highly dependent on the output level of the radio and presents a trade-off versus range: high transmit power enables longer range but requires additional power. To understand this relationship we use an xxx spectrum analyzer to measure the output power of the radio for different gain settings while it is transmitting a single tone signal and again use an xxx mulitmeter to measure the DC power required. Fig~\ref{fig:bench_tx_power} shows the results. We note that the sharp increase in power above xx~dBm indicates the maximum output power of the XXX radio front end and transmitting at higher power requires turning on the external power amplifier (PA).
\begin{table}[]
\centering
\begin{tabular}{|c|c|}
\hline
Mode & Power (mW) \\
\hline
LoRa TX & xxx\\
\hline
LoRa RX & xxx\\
\hline
Zigbee TX & xxx\\
\hline
Zigbee RX & xxx\\
\hline
OTA programming & xxx \\
\hline
OTA sleep & xxx \\
\hline
All components in sleep mode & xxx \\
\hline
\end{tabular}
\caption{\footnotesize{\bf Power consumption of our SDR platform while performing different operations.}}
\label{tab:power}
\end{table}
\subsection{Duty-cycling}
Considering many IoT nodes often perform short, simple tasks they are typically duty cycled to allow for further power savings. In \name, the microcontroller has the functionality to disable power consuming components such as the radio and FPGA and enter a low power sleep mode. Table~\ref{tab:wakeup} shows both the time required to wake up from sleep mode as well as the average power consumed by the device running at an xx\% duty cycle.
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|}
\hline
Mode & Wake-up time (ms) & Average power (mW) \\
\hline
FPGA TX & xxx & xxx\\
\hline
MCU TX & xxx & xxx\\
\hline
Radio TX & xxx & xxx\\
\hline
\textbf{Total TX} & \textbf{xxx} & \textbf{xxx}\\
\hline
FPGA RX & xxx & xxx\\
\hline
MCU RX & xxx & xxx\\
\hline
Radio RX & xxx & xxx\\
\hline
\textbf{Total RX} & \textbf{xxx} & \textbf{xxx}\\
\hline
\end{tabular}
\caption{\footnotesize{\bf Wake-up times for each component of our SDR platform.}}
\label{tab:wakeup}
\end{table}
To further demonstrate the capabilities of \name, we also evaluate it in a typical IoT use case. Specifically, we measure the power consumption of our platform as it wakes up from sleep mode, measures a sensor value, and transmits a packet. Fig.~\ref{fig:wakeup_curve} shows the results for this process, and the average power is xx. This means that \name can achieve a battery life of xx using an xx battery.
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{figs/1_lora_tx_rssi/lora_tx_per.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa Modulator Evaluation.} We evaluate our LoRa modulator in comparision with Semtech SX1276 LoRa chip.}}
\label{fig:bench_lora_tx_per}
\vskip -0.15in
\end{figure}
\subsection{Protocols}
To evaluate the use of \name as an IoT endpoint, we implement the LoRa and Zigbee protocols as described in section~\xref{xxx}. Table~\ref{tab:protocol} shows the size for implementing each on the FPGA to show it has sufficient resources to support encoding and decoding packets and leave space for additional custom operations.
\begin{table}[]
\centering
\begin{tabular}{|c|c|c|}
\hline
Protocol & \# of LUT4 & FPGA Utilization\\
\hline
Single Tone Modulator & 58 & 0.24\%\\
\hline
LoRa Modulator & 1964 & 8\%\\
\hline
LoRa Demodulator & xxx & xxx\\
\hline
Zigbee Modulator & xxx & xxx\\
\hline
Zigbee Demodulator & xxx & xxx\\
\hline
\end{tabular}
\caption{\footnotesize{\bf FPGA resources required to implement different protocols.}}
\label{tab:protocols}
\end{table}
Table~\ref{tab:protocols} shows the resources required to implement each protocol, next we evaluate the performance of our implementations. First, we use our LoRa modulator to generate packets with payload size three bytes with an SF of 9 and bandwidth of 500~kHz which we transmit at xx~dBm. We connect the output of \name through a rotary attenuator (xxx) to a Semtech SX1276 LoRa receiver~\cite{sx1257} which we use to measure the packet error rate versus transmitted power and plot the results in Fig.~\ref{fig:bench_lora_tx_per}. We compare our implementation to transmissions from an SX1276 LoRa chip and find that we can achieve a comparable sensitivity of -123~dBm which is the LoRa sensitivity with $SF=9$ and $BW=500kHz$.
Next we evaluate our LoRa demodulator. We connect the output of the Semtech SX1276 LoRa transmitter with an output of -11.5~dBm to \name through a rotary attenuator. We record the received RF signal in the FPGA memory and run it through our demodulator to compute a chirp symbol error rate. We use the attenuator to vary the power and plot the results in Fig.~\ref{fig:bench_lora_rx_per}. Our LoRa demodulator can demodulate chirp symbols up to -123~dBm which is LoRa protocol sensitivity at $SF=8$ and $BW=250kHz$.
\subsection{Over-the-air Update}
In order to understand how our SDR system can perform in a real world deployments that may require updating code to run different experiments we evaluate our over the air update system. An effective OTA update system should both minimize use of system resources such as power as well as network downtime. As explained above, the components of our OTA update system consumes a peak power of xx for xx during the update and only xx in sleep mode. By scaling this to once per hour the average power would only be xx, while scaling it to once per day would require as little as xx.
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{figs/2_lora_rx_rssi/lora_rx_chirp.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa Demodulator Evaluation.} We evaluate our LoRa demodulator by demodulating chirp symbols at different signal powers.}}
\label{fig:bench_lora_rx_per}
\vskip -0.15in
\end{figure}
We also evaluate the time required for updating a node. Considering the actual time to reprogram the FPGA and microcontroller is fixed, the downtime for programming a node depends only on the throughput which varies with SNR. While compression can reduce the transmission time, it also requires additional time to uncompress the data. To see the impact on a real deployment, we evaluate the total downtime when sending both the raw programming data and the same data compressed using xxx. We set up a LoRa transmitter connected to an xxx antenna transmitting at xxx~dBm and measure the time required to send the programming data at different distances.
\begin{table}[]
\centering
\begin{tabular}{|c|c|c|}
\cline{1-3}
\multicolumn{2}{|c|}{Components} & Price (\$) \\\hline
\multirow{2}{*}{DSP} & FPGA & 8.69 \\\cline{2-3}
& Oscillator & 0.9 \\\hline
\multirow{2}{*}{IQ Front-End} & Radio & 5.08 \\\cline{2-3}
& Crystal & 0.53 \\\cline{2-3}
& 2.4~GHz Balun & 0.36 \\\cline{2-3}
& Sub-GHz Balun & 0.3 \\\hline
\multirow{2}{*}{Backbone} & Radio & 4.5 \\\cline{2-3}
& Crystal & 0.4 \\\cline{2-3}
& Flash Memory & 1.6 \\\hline
\multirow{2}{*}{MAC} & MCU & 3.89 \\\cline{2-3}
& Crystals & 0.68 \\\hline
\multirow{2}{*}{RF} & Switch & 3.14 \\\cline{2-3}
& Sub-GHz PA & 1.54 \\\cline{2-3}
& 2.4~GHz PA & 1.72 \\\hline
\multirow{1}{*}{Power Management} & Regulators & 3.3 \\\hline
\multirow{1}{*}{Supporting Components} & -- & 4.5 \\\hline
\multirow{2}{*}{Production} & Fabrication & 4 \\\cline{2-3}
& Assembly & 10 \\\hline
\multirow{1}{*}{\bf Total} & -- & {\bf 55.13} \\\hline
\end{tabular}
\caption{\footnotesize{\bf Cost breakdown of all the components in \name for 1000 board production.}}
\label{tab:cost}
\end{table}
\subsection{Latency}
\subsection{Cost}
\section{Evaluation}
\subsection{Single-Tone Modulator}
To evaluate \name platform in basic operations at different frequency bands, we implement a single-tone modulator on FPGA. This modulator generates I/Q samples for a continues single-tone transmission and interfaces with LVDS serial interface of I/Q radio front-end. The transmitter power is highly dependent on the output level of the radio and presents a trade-off versus range: high transmit power enables longer range but requires additional power. To understand this relationship we use an xxx spectrum analyzer to measure the output power of the radio for different gain settings while it is transmitting a single tone signal and use an xxx mulitmeter to measure the DC power required. Fig~\ref{fig:bench_tx_power} shows the results. We note that the sharp increase in power above xx~dBm indicates the maximum output power of the XXX radio front end and transmitting at higher power requires turning on the external power amplifier (PA).
\subsection{LoRa}
To demonstrate the capability of \name at sub-GHz frequency band, we implemented LoRa modulator and demodulator by putting AT86RF215 in 900~MHz I/Q and programming the FPGA by LoRa baseband. LoRa uses Chirp Spread Scpectrum (CSS) modulation. In CSS, data is modulated using linearly increasing frequency signals or upchirp symbol. Each upchirp symbol has two main features; Spreading Factor (SF) and Bandwidth (BW). Spreading factor determines the length of each upchirp sysmbol and also is equivalent to number of bits that each upchirp symbol reperesents~\cite{netscatter,lorabackscatter}. LoRa uses SF values from 6 to 12. BW determines the total frequency spectrum that each upchirp symbol utilizes. Each upchirp symbol covers the whole BW frequency and the starting point of the symbol in frequency domain determines the symbol value~\cite{lora_basics}. In this section, we first explain our LoRa modulator/demodulator implementation and then we evaluate our implementation.
\noindent {\bf LoRa Modulator.} Fig.~\ref{fig:lora_tx_block} shows the block diagram of our LoRa modulator. We implement this modulator chain in Verilog HDL programming. Modulator starts with determining LoRa configurations such as SF and BW and symbols values in {Packet Generator} module. Then, the {Chirp Generator} module uses these information to generate each chirp symbol. It uses a phase accumulator along with two lookup tables for $Sin$ and $Cosin$ functions to generate I/Q samples. We feed I/Q samples to {I/Q Serializer} to serialize I/Q samples and generate serial information compatible with radio front-end interface. Finally, we feed serial I/Q samples along with TX clock to two LVDS interfaces that are connected to radio front-end.
\noindent {\bf LoRa Demodulator.} Fig.~\ref{fig:lora_rx_block} shows the block diagram of our LoRa demodulator. This demodulator interfaces with radio fornt-end using two LVDS connections; one for I/Q serial data (RXD) and the other for synchronization clock (RXCLK). We implement an {I/Q Deserializer} module to convert serial I/Q data to parallel I/Q for further signal processing. Then, we run these data through a low-pass FIR filter. We use a memory in FPGA to buffer raw samples between the front-end and our demodulator. We use the Chirp Generator that we implemented for LoRa modulator to generate a baseline upchirp/downchirp symbol and then we multiply that with the received chirp symbol using Chirp Multiplier unit. Then, we use FFT core implementation provide by Lattice to perform FFT operation on multiplied signal and then detect symbol value and symbol type (upchirp or downchirp) using our Symbol Detector implementation.
To show the performance of our LoRa implementation, first we evaluate our LoRa modulator. To do this, we use our LoRa modulator to generate packets with three bytes payload with $SF=9$ and $BW=500~kHz$ which we transmit at xx~dBm. We connect the output of \name through a rotary attenuator to a Semtech SX1276 LoRa transceiver~\cite{sx1257} which we use to measure the packet error rate versus transmitted power and plot the results in Fig.~\ref{fig:bench_lora_tx_per}. We compare our LoRa modulator implementation to transmissions from an SX1276 LoRa chip and find that we can achieve a comparable sensitivity of -123~dBm which is the LoRa sensitivity with $SF=9$ and $BW=500kHz$ configuration.
Next we evaluate our LoRa demodulator. We connect the output of the Semtech SX1276 LoRa transmitter with an output of -11.5~dBm to \name through a rotary attenuator. We record the received RF signal in the FPGA memory and run it through our demodulator to compute a chirp symbol error rate. We use the attenuator to vary the power and plot the results in Fig.~\ref{fig:bench_lora_rx_per}. Our LoRa demodulator can demodulate chirp symbols up to -123~dBm which is LoRa protocol sensitivity at $SF=8$ and $BW=250kHz$.
Next we evaluate the resource utilization of our LoRa implementation on FPGA. Table~\ref{tab:protocol} shows the size for implementing modulator and demodulator on FPGA. This table shows that our FPGA has sufficient resources to support different modules for LoRa protocol while it has sufficient resources for additional custom operations.
\subsection{Zigbee}
\subsection{Sleep Mode}
Since \name implementation includes multiple components, we define sleep mode operation in terms of the state mode for each component. Therefore, we evaluate power consumption, latency and XXX in following sections based on this definition.
In sleep mode, the MCU controlls every component to reduce the power consumption. For I/Q transceiver we use sleep mode that is defined on the transceiver itself and consumes only 30~nA current. Similarly, we put LoRa radio in sleep mode which consumes 200~nA current. To reduce the static power consumption of the FPGA, we shutdown the FPGA by disabling voltage regulators that provide power to each FPGA IO bank and core voltage. Similarly, we turn off the voltage regulators that provide voltage for sub-GHz PA and 2.4~GHz PA in sleep mode. Finally, we put the MCU in sleep mode XXX and setup timer for wake up to reduce even more power.
\begin{table}[]
\centering
\begin{tabular}{|c|c|}
\hline
Mode & Power (mW) \\
\hline
LoRa TX & xxx\\
\hline
LoRa RX & xxx\\
\hline
Zigbee TX & xxx\\
\hline
Zigbee RX & xxx\\
\hline
OTA programming & xxx \\
\hline
OTA sleep & xxx \\
\hline
All components in sleep mode & xxx \\
\hline
\end{tabular}
\caption{\footnotesize{\bf Power consumption of our SDR platform while performing different operations.}}
\label{tab:power}
\end{table}
\subsection{Duty-cycling}
Many IoT nodes often perform short, simple tasks they are typically duty cycled to allow for further power savings. In \name, the microcontroller has the functionality to disable power consuming components such as the radio, power amplifiers and FPGA and enter a low power sleep mode. Table~\ref{tab:wakeup} shows both the time required to wake up from sleep mode as well as the average power consumed by each component running at an xx\% duty cycle. At this table shows, FPGA boot up from shutdown state has the dominant time in our platform. The total time for \name to be completely functional in RX/TX modes is XXX which is the sum of wake-up time for FPGA and MCU. This is because we perform I/Q radio setup in parallel to the FPGA boot up.
\begin{table}[H]
\centering
\caption{\footnotesize{\bf Wake-up times for each component of \name.}}
\vskip -0.15 in
\begin{tabular}{|c|c|c|}
\hline
Mode & Wake-up time (ms) & Average power (mW) \\
\hline
FPGA & 35 & xxx\\
\hline
MCU & 0.17 & xxx\\
\hline
I/Q Radio & 4.9 & xxx\\
\hline
{\bf Total} & 35.17 & xxx\\
\hline
\end{tabular}
\label{tab:wakeup}
\end{table}
To further demonstrate the capabilities of \name, we also evaluate it in a typical IoT use case. Specifically, we measure the power consumption of our platform as it wakes up from sleep mode, measures a sensor value, and transmits a packet. Fig.~\ref{fig:wakeup_curve} shows the results for this process, and the average power is xx. This means that \name can achieve a battery life of xx using an xx battery.
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{figs/1_lora_tx_rssi/lora_tx_per.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa Modulator Evaluation.} We evaluate our LoRa modulator in comparision with Semtech SX1276 LoRa chip.}}
\label{fig:bench_lora_tx_per}
\vskip -0.15in
\end{figure}
\begin{table}[]
\centering
\caption{\footnotesize{\bf FPGA resources required to implement different protocols.}}
\vskip -0.15 in
\begin{tabular}{|c|c|c|}
\hline
Protocol & \# of LUT4 & FPGA Utilization\\
\hline
Single Tone Modulator & 58 & 0.24\%\\
\hline
LoRa Modulator & 1964 & 8\%\\
\hline
LoRa Demodulator & xxx & xxx\\
\hline
Zigbee Modulator & xxx & xxx\\
\hline
Zigbee Demodulator & xxx & xxx\\
\hline
\end{tabular}
\label{tab:protocols}
\end{table}
\subsection{Over-the-air Update}
In order to understand how our SDR system can perform in a real world deployments that may require updating code to run different experiments we evaluate our over the air update system. An effective OTA update system should both minimize use of system resources such as power as well as network downtime. As explained above, the components of our OTA update system consumes a peak power of xx for xx during the update and only xx in sleep mode. By scaling this to once per hour the average power would only be xx, while scaling it to once per day would require as little as xx.
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{figs/2_lora_rx_rssi/lora_rx_chirp.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa Demodulator Evaluation.} We evaluate our LoRa demodulator by demodulating chirp symbols at different signal powers.}}
\label{fig:bench_lora_rx_per}
\vskip -0.15in
\end{figure}
We also evaluate the time required for updating a node. Considering the actual time to reprogram the FPGA and microcontroller is fixed, the downtime for programming a node depends only on the throughput which varies with SNR. While compression can reduce the transmission time, it also requires additional time to uncompress the data. To see the impact on a real deployment, we evaluate the total downtime when sending both the raw programming data and the same data compressed using xxx. We set up a LoRa transmitter connected to an xxx antenna transmitting at xxx~dBm and measure the time required to send the programming data at different distances.
\begin{table}[]
\centering
\begin{tabular}{|c|c|c|}
\cline{1-3}
\multicolumn{2}{|c|}{Components} & Price (\$) \\\hline
\multirow{2}{*}{DSP} & FPGA & 8.69 \\\cline{2-3}
& Oscillator & 0.9 \\\hline
\multirow{2}{*}{IQ Front-End} & Radio & 5.08 \\\cline{2-3}
& Crystal & 0.53 \\\cline{2-3}
& 2.4~GHz Balun & 0.36 \\\cline{2-3}
& Sub-GHz Balun & 0.3 \\\hline
\multirow{2}{*}{Backbone} & Radio & 4.5 \\\cline{2-3}
& Crystal & 0.4 \\\cline{2-3}
& Flash Memory & 1.6 \\\hline
\multirow{2}{*}{MAC} & MCU & 3.89 \\\cline{2-3}
& Crystals & 0.68 \\\hline
\multirow{2}{*}{RF} & Switch & 3.14 \\\cline{2-3}
& Sub-GHz PA & 1.54 \\\cline{2-3}
& 2.4~GHz PA & 1.72 \\\hline
\multirow{1}{*}{Power Management} & Regulators & 3.3 \\\hline
\multirow{1}{*}{Supporting Components} & -- & 4.5 \\\hline
\multirow{2}{*}{Production} & Fabrication & 3 \\\cline{2-3}
& Assembly & 10 \\\hline
\multirow{1}{*}{\bf Total} & -- & {\bf 54.13} \\\hline
\end{tabular}
\caption{\footnotesize{\bf Cost breakdown of all the components in \name for 1000 board production.}}
\label{tab:cost}
\end{table}
\subsection{Cost}
\section{Evaluation}
In this section, we evaluate the performance of \name to show how that it meets our design requirements for prototyping IoT nodes. We begin with benchmarks and specs detailing radio performance, power consumption
\subsection{Single-Tone Modulator}
To evaluate \name platform in basic operations at different frequency bands, we implement a single-tone modulator on FPGA. This modulator generates I/Q samples for a continues single-tone transmission and interfaces with LVDS serial interface of I/Q radio front-end. The transmitter power is highly dependent on the output level of the radio and presents a trade-off versus range: high transmit power enables longer range but requires additional power. To understand this relationship we use an xxx spectrum analyzer to measure the output power of the radio for different gain settings while it is transmitting a single tone signal and use an xxx mulitmeter to measure the DC power required. Fig~\ref{fig:bench_tx_power} shows the results. We note that the sharp increase in power above xx~dBm indicates the maximum output power of the XXX radio front end and transmitting at higher power requires turning on the external power amplifier (PA).
\subsection{LoRa Evaluation}
To show the performance of our LoRa implementation, first we evaluate our LoRa modulator. To do this, we use our LoRa modulator to generate packets with three bytes payload with $SF=9$ and $BW=500~kHz$ which we transmit at xx~dBm. We connect the output of \name through a rotary attenuator to a Semtech SX1276 LoRa transceiver~\cite{sx1257} which we use to measure the packet error rate versus transmitted power and plot the results in Fig.~\ref{fig:bench_lora_tx_per}. We compare our LoRa modulator implementation to transmissions from an SX1276 LoRa chip and find that we can achieve a comparable sensitivity of -123~dBm which is the LoRa sensitivity with $SF=9$ and $BW=500kHz$ configuration.
Next we evaluate our LoRa demodulator. We connect the output of the Semtech SX1276 LoRa transmitter with an output of -11.5~dBm to \name through a rotary attenuator. We record the received RF signal in the FPGA memory and run it through our demodulator to compute a chirp symbol error rate. We use the attenuator to vary the power and plot the results in Fig.~\ref{fig:bench_lora_rx_per}. Our LoRa demodulator can demodulate chirp symbols up to -123~dBm which is LoRa protocol sensitivity at $SF=8$ and $BW=250kHz$.
Next we evaluate the resource utilization of our LoRa implementation on FPGA. Table~\ref{tab:protocol} shows the size for implementing modulator and demodulator on FPGA. This table shows that our FPGA has sufficient resources to support different modules for LoRa protocol while it has sufficient resources for additional custom operations.
\subsection{BLE Evaluation}
\subsection{Sleep Mode}
Since \name implementation includes multiple components, we define sleep mode operation in terms of the state mode for each component. Therefore, we evaluate power consumption, latency and XXX in following sections based on this definition.
In sleep mode, the MCU controls every component to reduce the power consumption. For I/Q transceiver we use sleep mode that is defined on the transceiver itself and consumes only 30~nA current. Similarly, we put LoRa radio in sleep mode which consumes 200~nA current. To reduce the static power consumption of the FPGA, we shutdown the FPGA by disabling voltage regulators that provide power to each FPGA IO bank and core voltage. Similarly, we turn off the voltage regulators that provide voltage for sub-GHz PA and 2.4~GHz PA in sleep mode. Finally, we put the MCU in sleep mode XXX and setup timer for wake up to reduce even more power.
\begin{table}[]
\centering
\begin{tabular}{|c|c|}
\hline
Mode & Power (mW) \\
\hline
LoRa TX & xxx\\
\hline
LoRa RX & xxx\\
\hline
Zigbee TX & xxx\\
\hline
Zigbee RX & xxx\\
\hline
OTA programming & xxx \\
\hline
OTA sleep & xxx \\
\hline
All components in sleep mode & xxx \\
\hline
\end{tabular}
\caption{\footnotesize{\bf Power consumption of our SDR platform while performing different operations.}}
\label{tab:power}
\end{table}
\subsection{Duty-cycling}
Many IoT nodes often perform short, simple tasks they are typically duty cycled to allow for further power savings. In \name, the microcontroller has the functionality to disable power consuming components such as the radio, power amplifiers and FPGA and enter a low power sleep mode. Table~\ref{tab:wakeup} shows both the time required to wake up from sleep mode as well as the average power consumed by each component running at an xx\% duty cycle. At this table shows, FPGA boot up from shutdown state has the dominant time in our platform. The total time for \name to be completely functional in RX/TX modes is XXX which is the sum of wake-up time for FPGA and MCU. This is because we perform I/Q radio setup in parallel to the FPGA boot up.
\begin{table}[H]
\centering
\caption{\footnotesize{\bf Wake-up times for each component of \name.}}
\vskip -0.15 in
\begin{tabular}{|c|c|c|}
\hline
Mode & Wake-up time (ms) & Average power (mW) \\
\hline
FPGA & 35 & xxx\\
\hline
MCU & 0.17 & xxx\\
\hline
I/Q Radio & 4.9 & xxx\\
\hline
{\bf Total} & 35.17 & xxx\\
\hline
\end{tabular}
\label{tab:wakeup}
\end{table}
To further demonstrate the capabilities of \name, we also evaluate it in a typical IoT use case. Specifically, we measure the power consumption of our platform as it wakes up from sleep mode, measures a sensor value, and transmits a packet. Fig.~\ref{fig:wakeup_curve} shows the results for this process, and the average power is xx. This means that \name can achieve a battery life of xx using an xx battery.
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{figs/1_lora_tx_rssi/lora_tx_per.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa Modulator Evaluation.} We evaluate our LoRa modulator in comparison with Semtech LoRa chip.}}
\label{fig:bench_lora_tx_per}
\vskip -0.15in
\end{figure}
\begin{table}[]
\centering
\caption{\footnotesize{{\bf FPGA utilization for LoRa protocol.}}}
\vskip -0.15 in
\begin{tabular}{|c|c|c|}
\hline
SF & LoRa TX (LUT4) & LoRa RX (LUT4)\\
\hline
6 & 976 (4\%) & 2656 (10\%)\\
\hline
7 & 976 (4\%) & 2670 (10\%)\\
\hline
8 & 976 (4\%) & 2700 (11\%)\\
\hline
9 & 976 (4\%) & 2742 (11\%)\\
\hline
10 & 976 (4\%) & 2786 (11\%)\\
\hline
11 & 976 (4\%) & 2794 (11\%)\\
\hline
12 & 976 (4\%) & 2818 (11\%)\\
\hline
\end{tabular}
\label{tab:lora_utilization}
\end{table}
\begin{table}[]
\centering
\caption{\footnotesize{{\bf Programming data compression.}}}
\vskip -0.15 in
\begin{tabular}{|C{2.6cm}|C{2.2cm}|C{2.2cm}|}
\hline \rowcolor{lightgray}
{\bf Protocol Implementation} & {\bf FPGA Compression Ratio} & {\bf MCU Compression Ratio}\\
\hline
LoRa RX and TX & XXX\% & XX\%\\
\hline
BLE Beacon & XXX\% & XX\%\\
\hline
\end{tabular}
\label{tab:ota_compression}
\end{table}
\subsection{Over-the-air Update}
In order to understand how our SDR system can perform in a real world deployments that may require updating code to run different experiments we evaluate our over the air update system. An effective OTA update system should both minimize use of system resources such as power as well as network downtime. As explained above, the components of our OTA update system consumes a peak power of xx for xx during the update and only xx in sleep mode. By scaling this to once per hour the average power would only be xx, while scaling it to once per day would require as little as xx.
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{figs/2_lora_rx_rssi/lora_rx_chirp.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa Demodulator Evaluation.} We evaluate our LoRa demodulator by demodulating chirp symbols at different signal received strength.}}
\label{fig:bench_lora_rx_per}
\vskip -0.15in
\end{figure}
We also evaluate the time required for updating a node. Considering the actual time to reprogram the FPGA and microcontroller is fixed, the downtime for programming a node depends only on the throughput which varies with SNR. While compression can reduce the transmission time, it also requires additional time to uncompress the data. To see the impact on a real deployment, we evaluate the total downtime when sending both the raw programming data and the same data compressed using xxx. We set up a LoRa transmitter connected to an xxx antenna transmitting at xxx~dBm and measure the time required to send the programming data at different distances.
\begin{table}[]
\centering
\begin{tabular}{|c|c|c|}
\cline{1-3}
\multicolumn{2}{|c|}{Components} & Price \\\hline
\multirow{2}{*}{DSP} & FPGA & \$8.69 \\\cline{2-3}
& Oscillator & \$0.9 \\\hline
\multirow{2}{*}{IQ Front-End} & Radio & \$5.08 \\\cline{2-3}
& Crystal & \$0.53 \\\cline{2-3}
& 2.4~GHz Balun & \$0.36 \\\cline{2-3}
& Sub-GHz Balun & \$0.3 \\\hline
\multirow{2}{*}{Backbone} & Radio & \$4.5 \\\cline{2-3}
& Crystal & \$0.4 \\\cline{2-3}
& Flash Memory & \$1.6 \\\hline
\multirow{2}{*}{MAC} & MCU & \$3.89 \\\cline{2-3}
& Crystals & \$0.68 \\\hline
\multirow{2}{*}{RF} & Switch & \$3.14 \\\cline{2-3}
& Sub-GHz PA & \$1.54 \\\cline{2-3}
& 2.4~GHz PA & \$1.72 \\\hline
\multirow{1}{*}{Power Management} & Regulators & \$3.3 \\\hline
\multirow{1}{*}{Supporting Components} & -- & \$4.5 \\\hline
\multirow{2}{*}{Production} & Fabrication & \$3 \\\cline{2-3}
& Assembly & \$10 \\\hline
\multirow{1}{*}{\bf Total} & -- & {\bf \$54.13} \\\hline
\end{tabular}
\caption{\footnotesize{\bf Cost breakdown of all the components in \name for 1000 board production.}}
\label{tab:cost}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{figs/8_lora_orthogonal_rx/lora_multiple.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf Orthogonal Demodulator Evaluation.} XXX}}
\label{fig:lora_multiple_rx}
\vskip -0.15in
\end{figure}
\subsection{Cost}
\section{Evaluation}
We evaluate \name's performance to show that it meets our design requirements for prototyping IoT nodes. We begin with benchmarks and specs detailing radio performance and power consumption followed by cost estimates. We then present the results with our two case studies. Following this, we evaluate the performance of our over the air update system.
\subsection{Benchmarks and Specifications}
{\bf Radio performance.} As a first test of \name's capabilities, we implement a single-tone modulator on the FPGA that generates the appropriate I/Q samples and streams them over LVDS to the radio. We connect the output to an MDO4104b-6~\cite{mdo4104b} spectrum analyzer and observe a single tone, shown in Fig.~\ref{sdds}, with no unexpected harmonics introduced by the modulator and compliant with the specs in the radio datasheet.
While the receiver's power consumption is dependent on the specific decoder and protocol being used, the transmitter's power consumption
is largely dominated by the radio frontend. To evaluate this, we measure the transmitter's DC power consumption which is highly dependent on the RF output power of the radio and presents a trade-off versus range: high RF transmit power enables longer range but requires additional DC power. To understand this relationship we again use the spectrum analyzer to measure the output power of the radio for different gain settings while it is transmitting a single tone signal and use a Fluke 287 mulitmeter to measure the DC power required. Fig~\ref{fig:bench_tx_power} shows the power conusmption drawn by our SDR platform, for both 900~MHz and 2.4~GHz operation. As expected the power consumption increases with the radio output power. The power consumption at a radio output power of 0 and 12~dBm at 900~MHz was 231 and 275~mW. This is 16 and 15 times better than the USRP E310. Further, in comparison with a non-SDR IoT LoRa chipset such as SX1276, we consume only 5 times more power, while providing full SDR capabilities.
\begin{table}[t]
\centering
\caption{\footnotesize{\bf Wake-up times for each component of \name.}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|c|c|c|}
\hline \rowcolor{lightgray}
{\bf Operation} & {\bf Duration (ms)}\\
\hline
Sleep to Radio Operation & 21\\
\hline
Radio Setup & XXX\\
\hline
TX to RX & 0.045\\
\hline
RX to TX & 0.011\\
\hline
Frequency Switch & 0.220\\
\hline
\end{tabular}
}
\label{tab:wakeup}
\end{table}
{\bf Duty-cycling.} Many IoT and sensor nodes often perform short, simple tasks allowing them to be heavily duty cycled which is the main way they can achieve battery lifetimes of years. We design \name with this critical need in mind such that the microcontroller can toggle on and off power consuming components such as the radio, power amplifiers and FPGA and enter a low power sleep mode.
We do this by first turning off the the I/Q transceiver and LoRa radios by shutting down their voltage regulator. To reduce the static power consumption of the FPGA, we shut it down by disabling the voltage regulators that provide power to each IO bank and core voltage. Similarly, we turn off the voltage regulators that provide power for the PAs. Finally, we put the MCU in sleep mode LPM3 running only a wakeup timer. As shown in Table.~\ref{tab:protocol_pow}, the measured total system sleep power in this mode was around 30~uW.
The low sleep power allows for significant power savings, but also introduces delay in terms of wake time. Table~\ref{tab:wakeup} shows the time required to wake up from sleep mode to having an active radio (Tx/Rx). Because we can perform the I/Q radio setup in parallel with booting the FPGA the total wakeup time for RX and TX is 21~ms. The I/Q radio setup takes XXX and so the wakeup time is dominated by the time to boot up the FPGA which itself takes 21~ms. We compare this to a SmartSense Temperature sensor~\cite{smartthings_temp} and find that \name has only a 4x longer wakeup time even though it requires programming unlike commercial products that use a single protocol radio. Additionally many IoT devices operate at low duty cycles waiting in sleep mode for seconds or more making \name's wakeup latency insignificant.
{\bf Switching delays.} We also measure the switching delays for different operations on the radio. Specifically, as shown in Table.~\ref{tab:wakeup}, it takes 45~$\mu$s and 11~$\mu$s to switch from TX to RX mode and RX to TX mode respectively. As we see later, this is sufficient to meet the timing requirements of IoT packet acknowledgements and MAC protocols. Further, the delay for switching between different frequencies is only 220~us. To measure these numbers, we switch between 2.426~GHz and 2.480~GHz on the radio. This switching delay is again sufficient to meet the requirements of frequency hopping during Bluetooth advertising.
{\bf Cost.} In addition to the technical specs of \name, we analyze the cost which is an important practical consideration for real world deployment at scale. Table~\ref{tab:cost} shows a detailed breakdown of cost including each component as well as PCB fabrication and assembly, where the overall cost is <\$55.
\begin{table}[t]
\centering
\caption{\footnotesize{\bf \Name cost breakdown for 1000 board production.}}
\vskip -0.15in
\footnotesize{
\begin{tabular}{|c|c|C{1.5cm}|}
\cline{1-3}\rowcolor{lightgray}
\multicolumn{2}{|c|}{\bf Components} & {\bf Price} \\\hline
\multirow{DSP} & FPGA & \$8.69 \\\cline{2-3}
& Oscillator & \$0.9 \\\hline
\multirow{IQ Front-End} & Radio & \$5.08 \\\cline{2-3}
& Crystal & \$0.53 \\\cline{2-3}
& 2.4~GHz Balun & \$0.36 \\\cline{2-3}
& Sub-GHz Balun & \$0.3 \\\hline
\multirow{Backbone} & Radio & \$4.5 \\\cline{2-3}
& Crystal & \$0.4 \\\cline{2-3}
& Flash Memory & \$1.6 \\\hline
\multirow{MAC} & MCU & \$3.89 \\\cline{2-3}
& Crystals & \$0.68 \\\hline
\multirow{RF} & Switch & \$3.14 \\\cline{2-3}
& Sub-GHz PA & \$1.54 \\\cline{2-3}
& 2.4~GHz PA & \$1.72 \\\hline
\multirow{Power Management} & Regulators & \$3.7 \\\hline
\multirow{Supporting Components} & -- & \$4.5 \\\hline
\multirow{Production} & Fabrication & \$3 \\\cline{2-3}
& Assembly & \$10 \\\hline
\multirow{\bf Total} & -- & {\bf \$54.53} \\\hline
\end{tabular}
}
\label{tab:cost}
\end{table}
\subsection{Evaluating Protocols}
To better evaluate the performance of \name, we evaluate our two protocol implementations.
{\bf LoRa using \name.}
We evaluate various different components of \name\ using LoRa as a case study.
{\it LoRa modulator.}
To evaluate this, we use our LoRa modulator to generate packets with three byte payloads using a spreading factor of $SF=9$ and bandwidths of 500~kHz and 250~kHz which we transmit at -13~dBm. We receive the output of \name on a Semtech SX1276 LoRa transceiver~\cite{sx1257} which we use to measure the packet error rate versus received RSSI and plot the results in Fig.~\ref{fig:bench_lora_tx_per}. We compare our LoRa modulator implementation to standard transmissions from an SX1276 LoRa transceiver. The plots show that we can achieve a comparable sensitivity of -123~dBm which is the LoRa sensitivity with $SF=9$ and $BW=500kHz$ configuration. This is true for all the three spreading factors, which shows that our low-power SDR can meet the sensitivity requirement of LPWAN IoT protocols.
{\it Lora demodulator.} Next we evaluate our LoRa demodulator on \name. To test this, we use transmissions from a Semtech SX1276 LoRa transmitter with an output of -11.5~dBm. We use the \name\ platform to receive these transmissions. The LoRa transmitter transmits packets with two configureations using a spreading factor of 8 and bandwidths of 250~kHz and 125~kHz. We record the received RF signal in the FPGA memory and run it through our demodulator to compute a chirp symbol error rate. We plot the results in Fig.~\ref{fig:bench_lora_rx_per} as a function of the RSSI values for the LoRa transmissions. Our LoRa demodulator can demodulate chirp symbols up to -123~dBm which is LoRa protocol sensitivity at $SF=8$ and $BW=250kHz$.
{\it Resource allocation.} Next we evaluate the resource utilization of our LoRa implementation on FPGA. Table.~\ref{tab:lora_utilization} shows the size for implemented modulator and demodulator on FPGA performing using different SFs. Our LoRa modulator supports all LoRa configurations with different SF with no additional cost. However, in LoRa demodulator, we need FFT blocks with different sizes to support different SF configurations. This table shows that our FPGA has sufficient resources to support different modules for LoRa protocol while it has sufficient resources for additional custom operations.
{\it LoRa MAC.} ??? There should be something to evaluate it? XXXX Delay for ACK? or scheduling? XXXX CDF?
{\it Power consumption.} XXXWhat is the system power consumption of transmitting a Lora packet? What about receiving? XXXX What are the individual power consumption?XXX
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/1_lora_tx_rssi/lora_tx_per.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa modulator evaluation.} We evaluate our LoRa modulator in comparison with Semtech LoRa chip.}}
\label{fig:bench_lora_tx_per}
\vskip -0.15in
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/2_lora_rx_rssi/lora_rx_chirp.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa demodulator evaluation.} We evaluate our LoRa demodulator by demodulating chirp symbols at different signal received strength.}}
\label{fig:bench_lora_rx_per}
\vskip -0.15in
\end{figure}
{\bf BLE using \name.} Next, we evaluate \name\ using BLE beacons as a case study. First, we measure the impact of our BLE beacons transmitted from \name\ using the TI CC2650~\cite{cc2650} BLE chip as a receiver. We do this by configuring \name to transmit BLE beacons at a rate of 1 packet per second. We transmit 100 packets at and set the CC2650 BLE chip to report bit error rate (BER). Fig~\ref{fig:bench_ble_tx_ber} shows the BER as a function of the received RSSI as reported by the CC2650 BLE chip. The plot shows that we achieve a sensitivity of -94~dBm. This is within 2~dB of the CC2650 BLE chipset's sensitivity defined by a BER threshold of $10^-3$.
Next we evaluate the latency of our BLE implementation as BLE beacons are typically transmitted in sequence by hopping between three different advertising channels. We measure the minimum time \name takes to switch between these frequencies by connecting its output to a 2.4~GHz envelope detector and using an MDO4104B-6 oscilloscope to measure the time delay between transmissions. Fig.~\ref{fig:ble_timing} plots a sequence of three BLE beacons transmitted on the advertising channels and show our system can transmit packets in 220~us. Fig.~\ref{SDSD} shows a CDF of this timing value for the BLE beacons transmitted by \name\ and the CCS2650 over 130 packets. The plots show that the variance for \name\ is small and is comparable to the CC2650's delay between BLE beacons of xxx~us.
Finally, our implementation of BLE beacons only requires 3\% of our FPGA resources leaving plenty of space for custom processing.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/11_ble_tx_rssi/ble_tx_ber.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf BLE evaluation.} We evaluate our BLE packet generator at different power levels.}}
\label{fig:bench_ble_tx_ber}
\vskip -0.15in
\end{figure}
\begin{table}[h]
\begin{minipage}[t]{0.45\linewidth}
\caption{\footnotesize{{\bf FPGA utilization for LoRa protocol.}}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|c|C{1.2cm}|C{1.4cm}|}
\hline \rowcolor{lightgray}
{\bf SF} & {\bf LoRa TX (LUT)} & {\bf LoRa RX (LUT)}\\
\hline
6 & 976 (4\%) & 2656 (10\%)\\
\hline
7 & 976 (4\%) & 2670 (10\%)\\
\hline
8 & 976 (4\%) & 2700 (11\%)\\
\hline
9 & 976 (4\%) & 2742 (11\%)\\
\hline
10 & 976 (4\%) & 2786 (11\%)\\
\hline
11 & 976 (4\%) & 2794 (11\%)\\
\hline
12 & 976 (4\%) & 2818 (11\%)\\
\hline
\end{tabular}
}
\label{tab:lora_utilization}
\end{minipage}
\hspace{0.2in}
\begin{minipage}[t]{0.45\linewidth}
\caption{\footnotesize{\bf Power consumption of different protocols.}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|C{1.5cm}|C{1.4cm}|}
\hline \rowcolor{lightgray}
{\bf Protocol} & {\bf Power (mW)}\\
\hline
LoRa RX & 182\\
\hline
LoRa TX & 283\\
\hline
BLE Beacon & 259\\
\hline
\end{tabular}
}
\label{tab:protocol_pow}
\end{minipage}
\end{table}
\subsection{Over-the-Air Programming}
In order to understand how our SDR system can perform in a real world deployments that may require updating code to run different experiments we evaluate our over the air programming system. An effective OTA programming system should both minimize use of system resources such as power as well as network downtime. As explained above, the components of our OTA programming system consumes a peak power of xx for xx during the update and only xx in sleep mode. By scaling this to once per hour the average power would only be xx, while scaling it to once per day would require as little as xx.
We also evaluate the time required for updating a node. Considering the actual time to reprogram the FPGA and microcontroller is fixed, the downtime for programming a node depends only on the throughput which varies with SNR. To see the impact on a real deployment, we evaluate the time required to programm a \tinysdr node at 10 locations around a university campus. We set up a LoRa transmitter connected to an patch antenna transmitting the update data at 14~dBm according to our protocol and measure the time it takes to program a \tinysdr node at each location. Fig.~\ref{fig:ota_exp} plots a CDF of the results, showing updates take an average of xx.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/10_ota_exp/10_ota_time_cdf.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf OTA Programming Time.} We show CDF of OTA programming time for programming LoRa and BLE implementations on \name.}}
\label{fig:ota_exp}
\vskip -0.2in
\end{figure}
\section{Evaluation}
We evaluate \name's performance to show that it meets our design requirements for prototyping IoT nodes. We begin with benchmarks and specs detailing radio performance and power consumption followed by cost estimates. We then present the results with our two case studies. Following this, we evaluate the performance of our over the air update system.
\subsection{Benchmarks and Specifications}
{\bf Platform transmit performance.} As a first test of \name's capabilities, we implement a single-tone modulator on the FPGA that generates the appropriate I/Q samples and streams them over LVDS to the radio. We connect the output to an MDO4104b-6~\cite{mdo4104b} spectrum analyzer and observe a single tone, shown in Fig.~\ref{sdds}, with no unexpected harmonics introduced by the modulator and compliant with the specs in the radio datasheet.
While the receiver's power consumption is dependent on the specific decoder and protocol being used, the transmitter's power consumption
is largely dominated by the radio frontend. To evaluate this, we measure the transmitter's power consumption which is highly dependent on the RF output power of the radio and presents a trade-off versus range: high RF transmit power enables longer range but requires additional power. To understand this relationship we again use the spectrum analyzer to measure the output power of the radio for different gain settings while it is transmitting a single tone signal and use a Fluke 287 mulitmeter to measure the required power. Fig~\ref{fig:bench_tx_power} shows the power consumption drawn by our SDR platform, for both 900~MHz and 2.4~GHz operation. As expected the power consumption increases with the radio output power. The total power consumption at a radio output power of 0 and 14~dBm at 900~MHz was 231 and 283~mW. This is 16 and 15 times better than the USRP E310. Further, in comparison with a non-SDR IoT LoRa chipset such as SX1276, we consume only 3 times more power, while providing full SDR capabilities.
\begin{table}[t]
\centering
\caption{\footnotesize{\bf Wake-up times for each component of \name.}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|c|c|c|}
\hline \rowcolor{lightgray}
{\bf Operation} & {\bf Duration (ms)}\\
\hline
Sleep to Radio Operation & 22\\
\hline
Radio Setup & 1.2\\
\hline
TX to RX & 0.045\\
\hline
RX to TX & 0.011\\
\hline
Frequency Switch & 0.220\\
\hline
\end{tabular}
}
\label{tab:wakeup}
\end{table}
{\bf Duty-cycling.} Many IoT and sensor nodes often perform short, simple tasks allowing them to be heavily duty cycled which is the main way they can achieve battery lifetimes of years. We design \name with this critical need in mind such that the microcontroller can toggle on and off power consuming components such as the radio, power amplifiers and FPGA and enter a low power sleep mode.
We do this by first turning off the the I/Q transceiver and LoRa radios by shutting down their voltage regulator. To reduce the static power consumption of the FPGA, we shut it down by disabling the voltage regulators that provide power to each IO bank and core voltage. Similarly, we turn off the voltage regulators that provide power for the PAs. Finally, we put the MCU in sleep mode LPM3 running only a wakeup timer. As shown in Table.~\ref{tab:protocol_pow}, the measured total system sleep power in this mode was 30~uW.
The low sleep power allows for significant power savings, but also introduces delay in terms of wake time. Table~\ref{tab:wakeup} shows the time required to wake up from sleep mode to having an active radio (Tx/Rx). Because we can perform the I/Q radio setup in parallel with booting the FPGA the total wakeup time for RX and TX is 22~ms. The I/Q radio setup takes 1.2~ms and so the wakeup time is dominated by the time to boot up the FPGA which itself takes 22~ms. We compare this to a SmartSense Temperature sensor~\cite{smartthings_temp} and find that \name has only a 4x longer wakeup time even though it requires programming unlike commercial products that use a single protocol radio. Additionally many IoT devices operate at low duty cycles waiting in sleep mode for seconds or more making \name's wakeup latency insignificant.
{\bf Switching delays.} We also measure the switching delays for different operations on the radio. Specifically, as shown in Table.~\ref{tab:wakeup}, it takes 45~$\mu$s and 11~$\mu$s to switch from TX to RX mode and RX to TX mode respectively. As we see later, this is sufficient to meet the timing requirements of IoT packet acknowledgements and MAC protocols. Further, the delay for switching between different frequencies is only 220~us. To measure these numbers, we switch between 2.426~GHz and 2.480~GHz on the radio. This switching delay is again sufficient to meet the requirements of frequency hopping during Bluetooth advertising.
{\bf Cost.} In addition to the technical specs of \name, we analyze the cost which is an important practical consideration for real world deployment at scale. Table~\ref{tab:cost} shows a detailed breakdown of cost including each component as well as PCB fabrication and assembly, where the overall cost is <\$55.
\begin{table}[t]
\centering
\caption{\footnotesize{\bf \Name cost breakdown for 1000 board production.}}
\vskip -0.15in
\footnotesize{
\begin{tabular}{|c|c|C{1.5cm}|}
\cline{1-3}\rowcolor{lightgray}
\multicolumn{2}{|c|}{\bf Components} & {\bf Price} \\\hline
\multirow{DSP} & FPGA & \$8.69 \\\cline{2-3}
& Oscillator & \$0.9 \\\hline
\multirow{IQ Front-End} & Radio & \$5.08 \\\cline{2-3}
& Crystal & \$0.53 \\\cline{2-3}
& 2.4~GHz Balun & \$0.36 \\\cline{2-3}
& Sub-GHz Balun & \$0.3 \\\hline
\multirow{Backbone} & Radio & \$4.5 \\\cline{2-3}
& Crystal & \$0.4 \\\cline{2-3}
& Flash Memory & \$1.6 \\\hline
\multirow{MAC} & MCU & \$3.89 \\\cline{2-3}
& Crystals & \$0.68 \\\hline
\multirow{RF} & Switch & \$3.14 \\\cline{2-3}
& Sub-GHz PA & \$1.54 \\\cline{2-3}
& 2.4~GHz PA & \$1.72 \\\hline
\multirow{Power Management} & Regulators & \$3.7 \\\hline
\multirow{Supporting Components} & -- & \$4.5 \\\hline
\multirow{Production} & Fabrication & \$3 \\\cline{2-3}
& Assembly & \$10 \\\hline
\multirow{\bf Total} & -- & {\bf \$54.53} \\\hline
\end{tabular}
}
\label{tab:cost}
\end{table}
\subsection{Evaluating Protocols}
To better evaluate the performance of \name, we evaluate our two protocol implementations.
{\bf LoRa using \name.}
We evaluate various different components of \name\ using LoRa as a case study.
{\it LoRa modulator.}
To evaluate this, we use our LoRa modulator to generate packets with three byte payloads using a spreading factor of $SF=9$ and bandwidths of 500~kHz and 250~kHz which we transmit at -13~dBm. We receive the output of \name on a Semtech SX1276 LoRa transceiver~\cite{sx1257} which we use to measure the packet error rate versus received RSSI and plot the results in Fig.~\ref{fig:bench_lora_tx_per}. We compare our LoRa modulator implementation to standard transmissions from an SX1276 LoRa transceiver. The plots show that we can achieve a comparable sensitivity of -123~dBm which is the LoRa sensitivity with $SF=9$ and $BW=500kHz$ configuration. This is true for all the three spreading factors, which shows that our low-power SDR can meet the sensitivity requirement of LPWAN IoT protocols.
{\it Lora demodulator.} Next we evaluate our LoRa demodulator on \name. To test this, we use transmissions from a Semtech SX1276 LoRa transmitter with an output of -11.5~dBm. We use the \name\ platform to receive these transmissions. The LoRa transmitter transmits packets with two configureations using a spreading factor of 8 and bandwidths of 250~kHz and 125~kHz. We record the received RF signal in the FPGA memory and run it through our demodulator to compute a chirp symbol error rate. We plot the results in Fig.~\ref{fig:bench_lora_rx_per} as a function of the RSSI values for the LoRa transmissions. Our LoRa demodulator can demodulate chirp symbols up to -123~dBm which is LoRa protocol sensitivity at $SF=8$ and $BW=250kHz$.
{\it Resource allocation.} Next we evaluate the resource utilization of our LoRa implementation on FPGA. Table.~\ref{tab:lora_utilization} shows the size for implemented modulator and demodulator on FPGA performing using different SFs. Our LoRa modulator supports all LoRa configurations with different SF with no additional cost. However, in LoRa demodulator, we need FFT blocks with different sizes to support different SF configurations. This table shows that our FPGA has sufficient resources to support different modules for LoRa protocol while it has sufficient resources for additional custom operations.
{\it LoRa MAC.} ??? There should be something to evaluate it? XXXX Delay for ACK? or scheduling? XXXX CDF?
{\it Power consumption.} XXXWhat is the system power consumption of transmitting a Lora packet? What about receiving? XXXX What are the individual power consumption?XXX
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/1_lora_tx_rssi/lora_tx_per.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa modulator evaluation.} We evaluate our LoRa modulator in comparison with Semtech LoRa chip.}}
\label{fig:bench_lora_tx_per}
\vskip -0.15in
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/2_lora_rx_rssi/lora_rx_chirp.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa demodulator evaluation.} We evaluate our LoRa demodulator by demodulating chirp symbols at different signal received strength.}}
\label{fig:bench_lora_rx_per}
\vskip -0.15in
\end{figure}
{\bf BLE using \name.} Next, we evaluate \name\ using BLE beacons as a case study. First, we measure the impact of our BLE beacons transmitted from \name\ using the TI CC2650~\cite{cc2650} BLE chip as a receiver. We do this by configuring \name to transmit BLE beacons at a rate of 1 packet per second. We transmit 100 packets at and set the CC2650 BLE chip to report bit error rate (BER). Fig~\ref{fig:bench_ble_tx_ber} shows the BER as a function of the received RSSI as reported by the CC2650 BLE chip. The plot shows that we achieve a sensitivity of -94~dBm. This is within 2~dB of the CC2650 BLE chipset's sensitivity defined by a BER threshold of $10^-3$.
Next we evaluate the latency of our BLE implementation as BLE beacons are typically transmitted in sequence by hopping between three different advertising channels. We measure the minimum time \name takes to switch between these frequencies by connecting its output to a 2.4~GHz envelope detector and using an MDO4104B-6 oscilloscope to measure the time delay between transmissions. Fig.~\ref{fig:ble_timing} plots a sequence of three BLE beacons transmitted on the advertising channels and show our system can transmit packets in 220~us. Fig.~\ref{SDSD} shows a CDF of this timing value for the BLE beacons transmitted by \name\ and an iPhone 6s over 130 packets. The plots show that the variance for \name\ is small and is less than the iPhone's delay between BLE beacons of 350~us.
Finally, our implementation of BLE beacons only requires 3\% of our FPGA resources leaving plenty of space for custom processing.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/11_ble_tx_rssi/ble_tx_ber.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf BLE evaluation.} We evaluate our BLE packet generator at different power levels.}}
\label{fig:bench_ble_tx_ber}
\vskip -0.15in
\end{figure}
\begin{table}[h]
\begin{minipage}[t]{0.45\linewidth}
\caption{\footnotesize{{\bf FPGA utilization for LoRa protocol.}}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|c|C{1.2cm}|C{1.4cm}|}
\hline \rowcolor{lightgray}
{\bf SF} & {\bf LoRa TX (LUT)} & {\bf LoRa RX (LUT)}\\
\hline
6 & 976 (4\%) & 2656 (10\%)\\
\hline
7 & 976 (4\%) & 2670 (10\%)\\
\hline
8 & 976 (4\%) & 2700 (11\%)\\
\hline
9 & 976 (4\%) & 2742 (11\%)\\
\hline
10 & 976 (4\%) & 2786 (11\%)\\
\hline
11 & 976 (4\%) & 2794 (11\%)\\
\hline
12 & 976 (4\%) & 2818 (11\%)\\
\hline
\end{tabular}
}
\label{tab:lora_utilization}
\end{minipage}
\hspace{0.2in}
\begin{minipage}[t]{0.45\linewidth}
\caption{\footnotesize{\bf Power consumption of different protocols.}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|C{1.5cm}|C{1.4cm}|}
\hline \rowcolor{lightgray}
{\bf Protocol} & {\bf Power (mW)}\\
\hline
LoRa RX & 182\\
\hline
LoRa TX & 283\\
\hline
BLE Beacon & 259\\
\hline
\end{tabular}
}
\label{tab:protocol_pow}
\end{minipage}
\end{table}
\subsection{Over-the-Air Programming}
In order to understand how our SDR system can perform in a real world deployments that may require updating code to run different experiments we evaluate our over the air programming system. An effective OTA programming system should both minimize use of system resources such as power as well as network downtime. As explained above, the components of our OTA programming system consumes a peak power of xx for xx during the update and only xx in sleep mode. By scaling this to once per hour the average power would only be xx, while scaling it to once per day would require as little as xx.
We also evaluate the time required for updating a node. Considering the time to reprogram the FPGA and microcontroller from flash is fixed, the downtime for programming a node depends only on the throughput which varies with SNR. To see the impact on a real deployment, we evaluate the time required to program a \tinysdr node at 10 locations around a university campus. We set up a LoRa transmitter connected to an patch antenna transmitting the update data at 14~dBm according to our protocol and measure the time it takes to program a \tinysdr node at each location. Fig.~\ref{fig:ota_exp} plots a CDF of the results, showing updates take an average of xx.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/10_ota_exp/10_ota_time_cdf.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf OTA Programming Time.} We show CDF of OTA programming time for programming LoRa and BLE implementations on \name.}}
\label{fig:ota_exp}
\vskip -0.2in
\end{figure}
\section{Evaluation}
We deploy a testbed of 20 \name devices across our institution's campus as shown in Fig.~\ref{fig:ota_map}; we removed the details of the map for anonymity.
To see if \name meets the requirements for IoT endpoint devices, we characterize its power, computational resource usage, delays and cost when operating in different modes and running different protocols.
\subsection{Benchmarks and Specifications}
{\bf Sleep mode power.} Many IoT and sensor nodes often perform short, simple tasks allowing them to be heavily duty cycled which is the main way they can achieve battery lifetimes of years. We design \name with this critical need in mind such that the microcontroller can toggle on and off power consuming components such as the radio, power amplifiers and FPGA and enter a low power sleep mode.
We do this by first turning off the the I/Q transceiver and LoRa radios. To reduce the static power consumption of the FPGA, we shut it down by disabling the voltage regulators that provide power to each I/O bank and core voltage. Similarly, we also turn off the PAs. Finally, we put the MCU in sleep mode LPM3 running only a wakeup timer. The measured total system sleep power in this mode was 30~uW.
The low sleep power allows for significant power savings, but also introduces delay in terms of wake time. Table~\ref{tab:wakeup} shows the time required to wake up from sleep mode to having an active radio (Tx/Rx). Because we can perform the I/Q radio setup in parallel with booting the FPGA the total wakeup time for RX and TX is 22~ms. The I/Q radio setup takes 1.2~ms and so the wakeup time is dominated by the time to boot up the FPGA which itself takes 22~ms. We compare this to a SmartSense Temperature sensor~\cite{smartthings_temp} and find that \name has only a 4x longer wakeup time even though it requires programming unlike commercial products that use a single protocol radio. Additionally many IoT devices operate at low duty cycles waiting in sleep mode for seconds or more making \name's wakeup latency insignificant.
{\bf Switching delays.} We also measure the switching delays for different operations on the I/Q radio as is an important parameter for meeting MAC and protocol timing requirements. Table.~\ref{tab:wakeup} shows that it takes 45~$\mu$s and 11~$\mu$s to switch from TX to RX mode and RX to TX mode respectively. As we see later, this is sufficient to meet the timing requirements of IoT packet ACKs and MAC protocols. Further, the delay for switching between different frequencies is only 220~us. To measure this number, we switch between 2.402~GHz, 2.426~GHz and 2.480~GHz frequencies on the radio. This switching delay is again sufficient to meet the requirements of frequency hopping during Bluetooth advertising.
\begin{table}[t]
\centering
\caption{\footnotesize{\bf \Name Cost Breakdown for 1000 Units}}
\vskip -0.15in
\footnotesize{
\begin{tabular}{|c|c|C{1.5cm}|}
\cline{1-3}\rowcolor{lightgray}
\multicolumn{2}{|c|}{\bf Components} & {\bf Price} \\\hline
\multirow{DSP} & FPGA & \$8.69 \\\cline{2-3}
& Oscillator & \$0.9 \\\hline
\multirow{IQ Front-End} & Radio & \$5.08 \\\cline{2-3}
& Crystal & \$0.53 \\\cline{2-3}
& 2.4~GHz Balun & \$0.36 \\\cline{2-3}
& Sub-GHz Balun & \$0.3 \\\hline
\multirow{Backbone} & Radio & \$4.5 \\\cline{2-3}
& Crystal & \$0.4 \\\cline{2-3}
& Flash Memory & \$1.6 \\\hline
\multirow{MAC} & MCU & \$3.89 \\\cline{2-3}
& Crystals & \$0.68 \\\hline
\multirow{RF} & Switch & \$3.14 \\\cline{2-3}
& Sub-GHz PA & \$1.54 \\\cline{2-3}
& 2.4~GHz PA & \$1.72 \\\hline
\multirow{Power Management} & Regulators & \$3.7 \\\hline
\multirow{Supporting Components} & -- & \$4.5 \\\hline
\multirow{Production} & Fabrication~\cite{pcbminions} & \$3 \\\cline{2-3}
& Assembly~\cite{pcbminions} & \$10 \\\hline
\multirow{\bf Total} & -- & {\bf \$54.53} \\\hline
\end{tabular}
}
\vskip -0.2in
\label{tab:cost}
\end{table}
{\bf Transmitter performance.} First, we implement a single-tone modulator on the FPGA that generates the appropriate I/Q samples and streams them over LVDS to the radio. We connect the output to an MDO4104b-6~\cite{mdo4104b} spectrum analyzer and observe a single tone, shown in Fig.~\ref{fig:spectrum}, with no unexpected harmonics introduced by the modulator.
While the receiver's power consumption is dependent on the specific decoder and protocol being used, the transmitter's power consumption
is largely dominated by the radio frontend. To evaluate this, we measure the transmitter's power consumption which is highly dependent on the RF output power of the radio. To understand this relationship we again use the spectrum analyzer to measure the output power of the radio for different gain settings while it is transmitting a single tone signal and use a Fluke 287 multimeter to measure the required power. Fig~\ref{fig:bench_tx_power} shows the power consumption drawn by our SDR platform, for both 900~MHz and 2.4~GHz operation. As expected the power consumption increases with the radio output power. {The total power consumption of \name platform including I/Q radio, FPGA, MCU and regulators at radio output power of 0 and 14~dBm at 900~MHz is 231 and 283~mW. This is 16 and 15 times better than the USRP E310.}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/5_bench_power/bench_tx_pow.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf Single-Tone Transmitter Power Consumption.} We show the total power consumption of \name including I/Q radio, FPGA, MCU and regulators at different transmitter output power. This is 15-16 times lower power consumption than the USRP E310 embedded SDR.}}
\label{fig:bench_tx_power}
\vskip -0.2in
\end{figure}
{\bf Cost.} We also analyze the cost which is an important practical consideration for real world deployment at scale. Table~\ref{tab:cost} shows a detailed breakdown of cost including each component as well as PCB fabrication and assembly based on quotes for 1000 units~\cite{XXX}, where the overall cost is around \$55.
\subsection{Evaluating the Case Studies}
{\bf LoRa using \name.}
We evaluate various different components of \name\ using LoRa as a case study.
{\it LoRa modulator.}
To evaluate this, we use our LoRa modulator to generate packets with three byte payloads using a spreading factor of $SF=8$ and bandwidths of 250~kHz and 125~kHz which we transmit at -13~dBm. We receive the output of \name on a Semtech SX1276 LoRa transceiver~\cite{sx1257} which we use to measure the packet error rate versus RSSI and plot the results in Fig.~\ref{fig:bench_lora_tx_per}. We compare our LoRa modulator to transmissions from an SX1276 LoRa transceiver. The plots show that we can achieve a comparable sensitivity of -126~dBm which is the LoRa sensitivity for $SF=8$ and $BW=125kHz$ configuration. This is true for both configurations, which shows that our low-power SDR can meet the sensitivity requirement of LPWAN IoT protocols.
{\it Lora demodulator.} Next we evaluate our LoRa demodulator on \name. To test this, we use transmissions from a Semtech SX1276 LoRa transceiver and use the \name\ platform to receive these transmissions. The LoRa transceiver transmits packets with two configurations using a spreading factor of 8 and bandwidths of 250~kHz and 125~kHz. We record the received RF signal in the FPGA memory and run it through our demodulator to compute a chirp symbol error rate. \textcolor{red}{Note that the Semtech LoRa transceiver does not give access to symbol error rate but since we have access to I/Q samples, we can compute it on our platform.} We plot the results in Fig.~\ref{fig:bench_lora_rx_per} as a function of the LoRa RSSI values. Our LoRa demodulator can demodulate chirp symbols up to -126~dBm which is LoRa protocol sensitivity at $SF=8$ and $BW=125kHz$. Both the LoRa modulator and demodulator run in real-time.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/1_lora_tx_rssi/lora_tx_per.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf LoRa Modulator Evaluation.} We evaluate our LoRa modulator in comparison with Semtech LoRa chip.}}
\label{fig:bench_lora_tx_per}
\vskip -0.15in
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/2_lora_rx_rssi/lora_rx_chirp.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf LoRa Demodulator Evaluation.} We evaluate our LoRa demodulator by demodulating chirp symbols at different RSSI.}}
\label{fig:bench_lora_rx_per}
\vskip -0.2in
\end{figure}
\begin{table}[t]
\centering
\caption{\footnotesize{{\bf FPGA Utilization for LoRa Protocol}}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|C{1cm}|C{2.5cm}|C{2.5cm}|}
\hline \rowcolor{lightgray}
{\bf SF} & {\bf LoRa TX (LUT)} & {\bf LoRa RX (LUT)}\\
\hline
6 & 976 (4\%) & 2656 (10\%)\\
\hline
7 & 976 (4\%) & 2670 (10\%)\\
\hline
8 & 976 (4\%) & 2700 (11\%)\\
\hline
9 & 976 (4\%) & 2742 (11\%)\\
\hline
10 & 976 (4\%) & 2786 (11\%)\\
\hline
11 & 976 (4\%) & 2794 (11\%)\\
\hline
12 & 976 (4\%) & 2818 (11\%)\\
\hline
\end{tabular}
}
\label{tab:lora_utilization}
\vskip -0.2in
\end{table}
{\it Resource allocation.} Next, we evaluate the resource utilization of our LoRa PHY implementation on FPGA. Table.~\ref{tab:lora_utilization} shows the size for implemented modulator and demodulator on FPGA performing using different SFs. Our LoRa modulator supports all LoRa configurations with different SF with no additional cost. However, in LoRa demodulator, we need FFT blocks with different sizes to support different SF configurations. This table shows that our FPGA has sufficient resources to support different modules for LoRa protocol while it has sufficient resources for additional custom operations.
{\it LoRa MAC.} We implement the LoRa MAC based on TTN's Arduino libraries~\cite{ttnarduino}. TTN protocol together with control for I/Q radio, backbone radio, FPGA and PMU and decompression algorithm for OTA take 18\% of MCU resources. Also, as shown in Table~\ref{tab:wakeup}, our timings are well within the requirements for LoRaWAN specifications~\cite{loraalliance} by LoRa Alliance.
We also measure the power consumption of our platform for LoRa packet transmission and reception. LoRa packet transmission with $SF=9$ and $BW=500~kHz$ and radio output power of 14~dBm consumes a total power of 287~mW from which 179~mW is for the radio and the rest is for the FPGA and MCU. LoRa packet reception consumes 186~mW with radio taking 59~mW.
{\bf BLE using \name.} Next, we evaluate \name\ using BLE beacons as a case study. First, we measure the impact of our BLE beacons transmitted from \name\ using the TI CC2650~\cite{cc2650} BLE chip as a receiver. We do this by configuring \name to transmit BLE beacons at a rate of 1 packet per second. We transmit 100 packets and set the CC2650 BLE chip to report bit error rate (BER). Fig~\ref{fig:bench_ble_tx_ber} shows the BER as a function of the received RSSI as reported by the CC2650 BLE chip. The plot shows that we achieve a sensitivity of -94~dBm. This is within 2~dB of the CC2650 BLE chipset's sensitivity, defined by a BER threshold of $10^{-3}$.
Next we evaluate the latency of our BLE implementation as BLE beacons are typically transmitted in sequence by hopping between three different advertising channels. We measure the minimum time \name takes to switch between these frequencies by connecting its output to a 2.4~GHz envelope detector and using an MDO4104B-6 oscilloscope to measure the time delay between transmissions. Fig.~\ref{fig:ble_timing} plots a sequence of three BLE beacons in time-domain transmitted on the advertising channels and shows that our system can transmit packets with 220~us delay between advertising beacons. The corresponding result when a iPhone 8 transmits beacons is 350~us. Finally, our implementation of BLE beacons only requires 3\% of the FPGA resources on the platform \textcolor{red}{ and can run for over 2 years on a 1000~mAh battery when transmitting once per second}
\subsection{Over-the-Air Programming}
In order to understand how our SDR system can perform in a real world deployments that may require updating firmware to run different experiments we evaluate our over the air programming system. An effective OTA programming system should both minimize use of system resources such as power as well as network downtime. Considering the time to reprogram the FPGA and microcontroller from flash is fixed, the downtime for programming a node depends only on the throughput which varies with SNR. To see the impact on a real deployment, we evaluate the time required to program a \tinysdr node at 20 locations around a university campus. We set up LoRa transceiver with $SF=8$, $BW=500~kHz$ and $CodingRate=6$ configuration connected to an patch antenna transmitting at 14~dBm as an AP and measure the time it takes to program a \tinysdr node at each location according to our protocol. We send FPGA compressed programming bitstream for LoRa (BLE) with 95~kB (37~kB) programming data size. Fig.~\ref{fig:ota_exp} plots a CDF of the results, showing OTA programming of the FPGA with LoRa (BLE) bitstream takes an average of 150s (59s). Our OTA programming system components, backbone radio and MCU, consumes an average energy of 6144~mJ (2342~mJ) with 14~dBm output power for \name backbone radio for LoRa (BLE). Having a 1000~mAh LiPo battery, we can OTA program \name 2100 (5600) times for similar FPGA compressed bitstream file sizes. Assuming OTA programming of once per day, the average power consumption would be 71~uW and 27~uW respectively.
\section{Evaluation}
We deploy a testbed of 20 \name devices across our institution's campus as shown in Fig.~\ref{fig:ota_map} (details removed for anonymity).
To see if \name meets the requirements for IoT endpoint devices, we characterize its power, computational resource usage, delays and cost when operating in different modes and running different protocols.
\vspace{-0.1 in}
\subsection{Benchmarks and Specifications}
{\bf Sleep mode power.} Many IoT nodes perform short, simple tasks allowing them to be heavily duty cycled which allows them to achieve battery lifetimes of years. We design \name with this critical need in mind such that the MCU can actively toggle on and off power consuming components such as the radio, PAs, and FPGA to enter a low power sleep mode.
We do this by first turning off the the I/Q transceiver and LoRa radios. To reduce the static power consumption of the FPGA, we shut it down by disabling the voltage regulators that provide power to its I/O banks and core voltage. Similarly, we also turn off the PAs. Finally, we put the MCU in sleep mode LPM3 running only a wakeup timer. The measured total system sleep power in this mode was 30~uW.
The low sleep power allows for significant power savings, but also introduces latency. Table~\ref{tab:wakeup} shows the time required to wake up from sleep mode until the radio is active. Because we can perform the I/Q radio setup in parallel with booting the FPGA, the total wakeup time for RX and TX is 22~ms. The I/Q radio setup takes 1.2~ms, so the wakeup time is dominated by booting up the FPGA which itself takes 22~ms. We compare this to a SmartSense Temperature sensor~\cite{smartthings_temp} and find that \name has only a 4x longer wakeup time even though it requires programming unlike commercial products that use a custom single protocol radio. Additionally many IoT devices operate at low duty cycles waiting in sleep mode for seconds or more making \name's wakeup latency insignificant.
{\bf Switching delays.} We also measure the switching delays for different operations on the I/Q radio as this is an important parameter for meeting MAC and protocol timing requirements. Table~\ref{tab:wakeup} shows that it takes 45~$\mu$s and 11~$\mu$s to switch from TX to RX mode and RX to TX mode respectively. As we see later, this is sufficient to meet the timing requirements of IoT packet ACKs and MAC protocols. Further, the delay for switching between different frequencies is only 220~us. To measure this number, we switch between 2.402~GHz, 2.426~GHz and 2.480~GHz. This switching delay is again sufficient to meet the requirements of frequency hopping during Bluetooth advertising.
\begin{table}[t]
\centering
\caption{\footnotesize{\bf \Name Cost Breakdown for 1000 Units.}}
\vskip -0.15in
\footnotesize{
\begin{tabular}{|C{3cm}|C{2cm}|C{1.5cm}|}
\hline
\cline{1-3}\rowcolor{lightgray}
\multicolumn{2}{|c|}{\bf Components} & {\bf Price} \\\hline
\multirow{2}{*}{DSP} & FPGA & \$8.69 \\\cline{2-3}
& Oscillator & \$0.9 \\\hline
\multirow{4}{*}{IQ Front-End} & Radio & \$5.08 \\\cline{2-3}
& Crystal & \$0.53 \\\cline{2-3}
& 2.4~GHz Balun & \$0.36 \\\cline{2-3}
& Sub-GHz Balun & \$0.3 \\\hline
\multirow{3}{*}{Backbone} & Radio & \$4.5 \\\cline{2-3}
& Crystal & \$0.4 \\\cline{2-3}
& Flash Memory & \$1.6 \\\hline
\multirow{2}{*}{MAC} & MCU & \$3.89 \\\cline{2-3}
& Crystals & \$0.68 \\\hline
\multirow{3}{*}{RF} & Switch & \$3.14 \\\cline{2-3}
& Sub-GHz PA & \$1.54 \\\cline{2-3}
& 2.4~GHz PA & \$1.72 \\\hline
\multirow{1}{*}{Power Management} & Regulators & \$3.7 \\\hline
\multirow{1}{*}{Supporting Components} & -- & \$4.5 \\\hline
\multirow{2}{*}{Production} & Fabrication~\cite{pcbminions} & \$3 \\\cline{2-3}
& Assembly~\cite{pcbminions} & \$10 \\\hline
\multirow{1}{*}{\bf Total} & -- & {\bf \$54.53} \\\hline
\end{tabular}
}
\vskip -0.2in
\label{tab:cost}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/5_bench_power/bench_tx_pow.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf Single-Tone Transmitter Power Consumption.} We show the total power consumption of \name including I/Q radio, FPGA, MCU and regulators at different transmitter output power. This is 15-16 times lower power consumption than the USRP E310 embedded SDR.}}
\label{fig:bench_tx_power}
\vskip -0.22in
\end{figure}
{\bf Transmitter performance.} First, we implement a single-tone modulator on the FPGA that generates the appropriate I/Q samples and streams them over LVDS to the radio. We connect the output to an MDO4104b-6~\cite{mdo4104b} spectrum analyzer and observe a single tone, shown in Fig.~\ref{fig:spectrum}, with no unexpected harmonics introduced by the modulator.
Next we measure the end-to-end DC power consumption of our system including the I/Q radio, FPGA, MCU and regulators to see how it scales with RF output power. We vary our radio output power while transmitting a single tone and use a Fluke 287 multimeter to measure its DC power draw. Fig.~\ref{fig:bench_tx_power} shows the power consumption of \name for 900~MHz and 2.4~GHz operation. Interestingly, we observe the DC power is constant at low RF power but increases as expected beyond some RF power level. \Name consumes 231~mW when transmitting at 0~dBm, and for comparison the end-to-end power consumption of the USRP E310 is 16x higher under the same conditions. Similarly \name consumes 283~mW at its 14~dBm setting while the USRP E310 is 15x higher.
{\bf Cost.} We also analyze the cost which is an important practical consideration for real world deployment at scale. Table~\ref{tab:cost} shows a detailed breakdown of cost including each component as well as PCB fabrication and assembly based on quotes for 1000 units~\cite{pcbminions}, where the overall cost is around \$55.
\subsection{Evaluating the Case Studies}
{\bf LoRa using \name.}
We evaluate various different components of \name\ using LoRa as a case study.
{\it LoRa modulator.}
To evaluate this, we use our LoRa modulator to generate packets with three byte payloads using a spreading factor of $SF=8$ and bandwidths of 250~kHz and 125~kHz which we transmit at -13~dBm. We receive the output of \name on a Semtech SX1276 LoRa transceiver~\cite{sx1257} which we use to measure the packet error rate (PER) versus RSSI and plot the results in Fig.~\ref{fig:bench_lora_tx_per}. We compare our LoRa modulator to transmissions from an SX1276 LoRa transceiver. The plots show that we can achieve a comparable sensitivity of -126~dBm which is the LoRa sensitivity for $SF=8$ and $BW=125kHz$ configuration. This is true for both configurations, which shows that our low-power SDR can meet the sensitivity requirement of LPWAN IoT protocols.
{\it LoRa demodulator.} Next we evaluate our LoRa demodulator on \name. To test this, we use transmissions from a Semtech SX1276 LoRa transceiver and use \name\ to receive these transmissions. The LoRa transceiver transmits packets with two configurations using a spreading factor of 8 and bandwidths of 250~kHz and 125~kHz. We record the received RF signals in the FPGA memory and run them through our demodulator to compute a chirp symbol error rate. {Note that the Semtech LoRa transceiver does not give access to symbol error rate but since we have access to I/Q samples, we can compute it on our platform.} We plot the results in Fig.~\ref{fig:bench_lora_rx_per} as a function of the LoRa RSSI values. Our LoRa demodulator can demodulate chirp symbols down to -126~dBm which is LoRa protocol sensitivity at $SF=8$ and $BW=125kHz$. Both the LoRa modulator and demodulator run in real-time.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/1_lora_tx_rssi/lora_tx_per.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf LoRa Modulator Evaluation.} We evaluate our LoRa modulator in comparison with Semtech LoRa chip.}}
\label{fig:bench_lora_tx_per}
\vskip -0.15in
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/2_lora_rx_rssi/lora_rx_chirp.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf LoRa Demodulator Evaluation.} We evaluate our LoRa demodulator by demodulating chirp symbols at different RSSI.}}
\label{fig:bench_lora_rx_per}
\vskip -0.2in
\end{figure}
\begin{table}[t]
\centering
\caption{\footnotesize{{\bf FPGA Utilization for LoRa Protocol.}}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|C{1cm}|C{2.5cm}|C{2.5cm}|}
\hline \rowcolor{lightgray}
{\bf SF} & {\bf LoRa TX (LUT)} & {\bf LoRa RX (LUT)}\\
\hline
6 & 976 (4\%) & 2656 (10\%)\\
\hline
7 & 976 (4\%) & 2670 (10\%)\\
\hline
8 & 976 (4\%) & 2700 (11\%)\\
\hline
9 & 976 (4\%) & 2742 (11\%)\\
\hline
10 & 976 (4\%) & 2786 (11\%)\\
\hline
11 & 976 (4\%) & 2794 (11\%)\\
\hline
12 & 976 (4\%) & 2818 (11\%)\\
\hline
\end{tabular}
}
\label{tab:lora_utilization}
\vskip -0.2in
\end{table}
{\it Resource allocation.} Next, we evaluate the resource utilization of our LoRa PHY implementation on the FPGA. Table~\ref{tab:lora_utilization} shows the size for implementing the modulator and demodulator on our FPGA performing using different SFs. Our LoRa modulator supports all LoRa configurations with different SF with no additional cost. However, in the LoRa demodulator, we need FFT blocks with different sizes to support different SF configurations. This table shows that our FPGA has sufficient resources to support multiple configurations of LoRa and still leave space for other custom operations.
{\it LoRa MAC.} We implement the LoRa MAC based on TTN's Arduino libraries~\cite{ttnarduino}. TTN protocol together with control for the I/Q radio, backbone radio, FPGA, PMU and decompression algorithm for OTA take only 18\% of MCU resources. Also, as shown in Table~\ref{tab:wakeup}, our timings are well within the requirements for LoRaWAN specifications~\cite{loraalliance}.
We also measure the power consumption of our platform for LoRa packet transmission and reception. LoRa packet transmission with $SF=9$ and $BW=500~kHz$ and radio output power of 14~dBm consumes a total power of 287~mW from which 179~mW is for the radio and the rest is for the FPGA and MCU. LoRa packet reception consumes 186~mW with radio taking 59~mW.
{\bf BLE using \name.} Next, we evaluate \name\ using BLE beacons as a case study. First, we measure the impact of our BLE beacons transmitted from \name\ using the TI CC2650~\cite{cc2650} BLE chip as a receiver. We do this by configuring \name to transmit BLE beacons at a rate of 1 packet per second. We transmit 100 packets and set the CC2650 BLE chip to report bit error rate (BER). Fig.~\ref{fig:bench_ble_tx_ber} shows the BER as a function of the received RSSI as reported by the CC2650 BLE chip. The plot shows that we achieve a sensitivity of -94~dBm. This is within 2~dB of the CC2650 BLE chipset's sensitivity, defined by a BER threshold of $10^{-3}$.
Next we evaluate the latency of our BLE implementation as BLE beacons are typically transmitted in sequence by hopping between three different advertising channels. We measure the minimum time \name takes to switch between these frequencies by connecting its output to a 2.4~GHz envelope detector and using an MDO4104B-6 oscilloscope to measure the time delay between transmissions. Fig.~\ref{fig:ble_timing} plots the envelope of three BLE beacons in the time-domain transmitted on the different advertising channels and shows that our system can transmit packets with as little as 220~us delay between beacons. The corresponding result when a iPhone 8 transmits beacons is 350~us. Finally, generating BLE beacons requires only 3\% of the FPGA resources on the \name and it could run for over 2 years on a 1000~mAh battery when transmitting once per second.
\subsection{Over-the-Air Programming}
An effective OTA programming system should both minimize use of system resources such as power as well as network downtime. Considering the time to reprogram the FPGA and microcontroller from flash is fixed, the downtime for programming a node depends on the amount of data sent and the throughput which varies with SNR.
Raw programming files for our FPGA are 579~kB, however we compress our data using miniLZO. While the exact compression ratio depends on FPGA utilization, our LoRa program compresses to 99~kB and BLE to 40~kB. Our microcontroller programs for both LoRa and BLE are approximately 78~kB and are both compressed to 24~kB. When dividing the files into packets, we would ideally minimize the preamble length and maximize packet length to reduce overhead, however long packets with short preambles lead to higher PER. We choose a preamble of 8 chirps and packets of 60~B which we find balances the trade-off of protocol overhead versus range in our experiments.
To see the impact on a real deployment, we evaluate the time required to program \name nodes in our 20 device testbed shown in Fig.~\ref{fig:ota_map}. We set up a LoRa transceiver configured with $SF=8$, $BW=500~kHz$ and $CodingRate=6$ connected to a patch antenna transmitting at 14~dBm as an AP and measure the time it takes to program the \name devices at each location, according to our protocol. We transmit the compressed FPGA and MCU programming data for LoRa and BLE and plot the results as a CDF in Fig.~\ref{fig:ota_exp}. The plots show that the LoRa FPGA requires an average programming time of 150~s while BLE, FPGA, and MCU require 59~s and 39~s respectively due to their smaller file size. { Decompressing these received files only takes a maximum of 450~ms.}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/11_ble_tx_rssi/ble_tx_ber.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf BLE evaluation.} BLE beacons at different power levels.}}
\label{fig:bench_ble_tx_ber}
\vskip -0.2in
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/12_ble_timing/12_ble_timing.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf BLE Beacons Signal.} We show BLE beacon transmissions on three advertising channels from \name using an envelope detector.}}
\label{fig:ble_timing}
\vskip -0.2in
\end{figure}
Our OTA programming system components, backbone radio and MCU, consume an average energy of 6144~mJ for receiving a LoRa FPGA update and 2342~mJ for a BLE FPGA update when using 14~dBm output power. Using a 1000~mAh LiPo battery, we could OTA program each \name node with LoRa 2100 times and BLE 5600 times. Assuming OTA programming of once per day, the average power consumption would be 71~uW and 27~uW respectively for LoRa and BLE.
\section{Hardware Implementations}
Next we describe the implementation details of our hardware prototypes such as the PCB design and clocking. We build two prototypes: \name V1, in which we implement the core software radio components and \name V2 in which we design the final cost and form factor optimized version including the OTA update system. We explain the details of each below.
\subsection{\name V1}
Before designing the full system we begin by prototyping the core SDR modules including the FPGA, radio, MCU, and SD card to validate our key performance specs such as sensitivity and ability to process samples in real time. \name V1 is shown in Fig.~\ref{fig:tinysdrv1}. The design consists of a four layer FR-4 PCB and uses the M2GL010-FGG484~\cite{igloo2} IGLOO2 FPGA from Microsemi. The IGLOO FPGA has a large internal SRAM memory which allows for easy debugging by simply recording I/Q samples in memory which can be saved to a computer using a JTAG debugger. We verify the performance of our LVDS interface and I/Q radio as well as as the ability to write data to the SD card. Additionally this platform allows us to test the interface between the radio and MSP430FR5969~\cite{msp430} and evaluate the characteristics of the radio output signal. While the IGLOO2 allows for easy prototyping, its high price ($\approx \$40$), limited number of logic units, and large form factor make it impractical for a final design.
\subsection{\name V2}
After verifying the core components on \name V1, we developed \name V2 which fully optimizes the size, power and form factor to achieve our original design specs. We implement \name V2 on a 6 layer FR-4 PCB with dimension of $3cm\times 5cm$ as shown in Fig.~\ref{fig:sdr_hardware}. In this version, we implement the power management system described above with multiple controllable voltage domains. We use MSP432P401R~\cite{msp432P401R} microcontroller which is in the same family as the MCU described above but provides more FLASH and SRAM memory necessary for data compression in our OTA system. In addition, it support multiple SPI interfaces which we use to interface with FPGA, I/Q radio and backbone radio.
In this design we also exchange the large, expensive IGLOO2 for the smaller form factor LFE5U-25F-BG256~\cite{latticeLFE5U} FPGA from Lattice Semiconductor and has almost 2x more logic units. We connect the FPGA with I/Q radio using LVDS, and these connections require careful design. Each differential signal has two trace lines on PCB which must be the same length to have similar timing characteristics. We use tools in design software to implement differential lines with same length. In addition, LVDS requires 100~$\ohm$ differential impedance between two lines. To do this, we simulate differential impedance between LVDS lines based on material properties, differential line width and separation and PCB layers and use the results to set the appropriate dimensions.
In addition to the PCB design considerations we also have to coordinate the various clocks and timing on each device. Both of the radios require crystals which they use as a reference for their onboard PLLs which synthesize the high frequency RF signals. While the FPGA has a dedicated 32~MHz reference oscillator and onboard PLL it uses for generating the 64~MHz clock when transmitting samples over LVDS, and it maintains synchronization when reading by instead using a clock reference output by the radio. The MCU also has both a high frequency mode which can run up to 48~MHz to perform computation intensive tasks like compression and an ultra-low power watchdog timer when in sleep mode.
In addition, we add SX1276 radio for backbone support along with the supporting hardware as well as the various voltage regulators discussed previously to minimize power. \name V2 also exposes a number of external interfaces to the MCU such as SPI, I2C and ADC pins which can be used to connect a variety of analog or digital sensors to create a complete IoT sensor node.
\section{Hardware Implementations}
\section{Hardware Implementation}
In this section we go over the implementation of \name hardware and present design choices.
Fig.~\ref{fig:blcok} shows the block diagram of \name system. We choose various components of our implementation based on this block diagram to fulfill our goals for \name that mentioned in XXX.
We use a low-power low-cost RF front-end, AT86RF215~\cite{ref:AT86RF215} from Microchip as the I/Q transceiver for our design. This radio can operate in 400~MHz, 900~MHz and 2.4~GHz frequency bands which satisfy our goal of supporting different ISM bands. It provides all the necessary blocks of the transceiver chain from the antenna to the digital baseband data including LNA/PA, downconversion/upconversion mixers, band-pass/low-pass filters, ADC/DAC, synthesizer and voltage-controlled oscillator. To exchange digitized I/Q baseband data between radio and baseband processing unit, it uses 64~MHz double data-rate low voltage differential signaling (LVDS). The maximum sampling frequency for the I/Q data is 4~MHz which means that it can support modulations with maximum of 2~MHz bandwidth. This chip is capable of 13~bits ADC/DAC which is even better than the analog front-end (AD93XX~\cite{xxxx}) used by USRP~\cite{xxxx}. To receive I/Q samples from this radio, we need to interface with an FPGA with LVDS capability.
We use LFE5U-25F~\cite{} FPGA from Lattice Semiconductor for baseband processing. This FPGA has 24k logic units, it supports multiple I/O voltages for different interfaces and also it has LVDS capability. We use a small form-factor of this FPGA on our design. Hoever, the logic unit size is more than most of the existing SDR platforms including XXX, XXX and XXX. We use an external FLASH memory with XX~Mb size which can store multiple boots of this FPGA at the same time. We connect FPGA to flash with a Quad SPI interface to enable dynamic programmable capability for the FPGA.
To enable over-the-air updates, we use a Semtech SX1276~\cite{sx1276} LoRa radio as the backbone radio. SX1276 can operate at different LoRa configurations, it provides long-range coverage and it is power efficient. This chip supports frequency range of 137~MHz to 1020~MHz with bitrate range of 0.018~kbps to 37.5~kbps.
In \name platform, we control every components with a low-power microcontroller. We use MSP432P401R~\cite{} 32-Bit Cortex M4F microcontroller to controll two radios, communicate with baseband processor and power management. This microcontroller has 64~kB RAM along with 256~kB FLASH memory for programming. This microcontroller provides huge flexibility for performing compression algorithms, complex MAC layer implementation IoT protocols and sensor reading applications.
Finally, to implement the RF section we use two RF paths, one for sub-GHz frequency bands and the other for 2.4~GHz frequency band. We use 2.4~GHz power amplifier ,SE2431L~\cite{}, and 900~MHz band power amplifier, SE2435L~\cite{}, to increase the maximum output power of our radios. Since both I/Q radio and backbone radio share the sub-GHz RF path, we use an ADG904~\cite{} RF switch to select between I/Q radio, RX and TX mode of LoRa radio.
We integrate all components on a 6-layers FR-4 PCB with the dimension of $30mm\times50mm$ as shown in Fig.~\ref{fig:board}. Our board provides different sensor interfaces including SPI and I2C as external GPIOs for sensor reading applications.
\begin{figure}[t]
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=1\linewidth]{figs/block_diagram/lora_tx_block.eps}
\vskip -0.1in
\caption{\footnotesize{XXX}}
\label{fig:lora_tx_block}
\end{subfigure}
\vskip -0.15in
\caption{\footnotesize{LoRa FPGA Block Diagrams. XXX}}
\label{fig:lora_blocks}
\end{figure}
\section{Hardware Implementation}
In this section we go over the implementation of \name hardware and present design choices. The hardware is implemented based on the system design presented in \xref{sec:system} in fig.~\ref{fig:system} and components are chosen to meet our goal of designing a system that is suitable for IoT testbed implementation.
\subsection{MAC Layer}
We use MSP432P401R~\cite{} a 32-Bit Cortex M4F MCU with floating point unit to control two I/Q and backbone transceiver, communicate with baseband processor and managing power on the platform. This MCU has 64~kB RAM along with 256~kB FLASH memory for programming and provides huge flexibility for performing compression algorithms, complex MAC layer implementation of IoT protocols and sensor reading applications. It supports up to eight SPI, four UART and four I2C interfaces. It can operate at low frequencies, such as 128~kHz, as well as high frequencies up to 48~MHz. It has two 32-bits timers and more importantly it supports different low-power modes. As an example, it supports LPM4.5 mode which only consumes 25~nA current. These features make this MCU a suitable choice for \name to operate at low-power and also perfomr accurate duty-cycling.
\subsection{PHY Layer}
We use a low-power low-cost RF front-end, AT86RF215~\cite{ref:AT86RF215} from Microchip. We choose this chip as it is a highly integrated solution designed for operation in all of our target ISM bands (400~MHz, 900~MHz and 2.4~GHz ISM) and most importantly is one of the few commercial chips that also provides an I/Q output to allow for custom protocols and modulation. It provides all the blocks of the transceiver chain required between the antenna and digital baseband data including LNA/PA, downconversion/upconversion mixers, band-pass/low-pass filters, ADC/DAC, frequency synthesizer and voltage-controlled oscillator. This chip provides two RF ports for sub-Ghz and 2.4~GHz frequencies. Each RF port is access with balanced differential signal pairs. We use 0896BM15E0025E~\cite{} balun for sub-GHz RF port and 2450FB15A050E~\cite{} balun for 2.4~GHz RF port. The receiver chain passes signal through an LNA and then downconverts the signal using a mixer that is controlled by frequency synthesizer (PLL). This chip utilizes two PLLs for sub-GHz and 2.4~GHz transceivers. Both PLLs use a 26~MHz crystal as reference and they can synthesize any frequencies with the step of around 198~Hz. Then signal goes through an analog bandpass filter and then sampled by a 13-bits Sigma-Delta analog to digital convertor (ADC). On the transmitter side, the digital controller feeds I/Q samples to digital front-end. Then, the signal is sampled by a 13-bits digital to analog converter (DAC). This signal goes to a analog lowpass filter and upcoverts by the mixer which uses the same PLL. Finally, signal is amplified with internal power amplifier of this chip. Each RF port has one internal power amplifier (PA) that can output signal up to 14~dBm. This optimal power, however, is achieved when the RF port is matched to 50~$\ohm$ differential.
\textcolor{red}{Add noise figure, mention it provides two RF outputs (separate paths for sub GHz and 2.4 GHz, add specs for each internal block like LNA noise, PA efficiency, xx order filter, frequency synthesizer specs/step sizes. Mention what kind of architecture it is (e.g. superheterodyne, direct conversion etc.)What kind of clock input does it require?}
To exchange digitized I/Q baseband data between radio and baseband processing unit, it uses 64~MHz double data-rate low voltage differential signaling (LVDS).
\textcolor{red}{What is LVDS? Knowing nothing about the protocol I'm assuming it's low power and differential to reject common mode but how does it work? Add 1-2 sentences about it, if there's anything special you have to do for people who haven't heard of it.}
The maximum sampling frequency for the I/Q data is 4~MHz which means that it can support modulations with maximum of 2~MHz bandwidth. This chip is capable of 13~bits ADC/DAC which is even better than the analog front-end (AD93XX~\cite{xxxx}) used by USRP~\cite{xxxx}. The current consumption of this chip is several tens of milli-ampere which is around an order of magnitude\textcolor{red}{lower than} the AD93XX series used in common SDR platfroms like USRP. Note that AD93XX series have higher sampling rate and frequency spectrum coverage which are not required considering our target application of IoT node implementation. Moreover, AD93XX series' price is at least 100\$ or even several hundreds of dollar depending on the version compared to our I/Q transceiver which is about 5\$. In conclusion, AT86RF215 has unique properties which are suitable for IoT node implementation.
\textcolor{red}{Expand this: Additionally the rf front end chip can be set to implement different basic modulations (FSK OFDM etc). This allows us to bypass the FPGA entirely and enable greater power savings when implementing standard modulations while developing a MAC protocol.}
We use LFE5U-25F~\cite{} FPGA from Lattice Semiconductor for baseband processing which talks to the I/Q transceiver using LVDS interface. This FPGA has 24k logic units which is comparable to other common SDR platforms including XXX, XXX and XXX. We use a small form-factor of this FPGA on our design to decrease the platform physical size. Compared to FPGAs used in other existing platform this FPGA provides higher number of LUTs for the same cost. Since this is an SRAM-based FPGA, our design requires an external FLASH memory with XX~Mb size capable of storing multiple boots of this FPGA is included in the design. The FPGA talks to the flash with Quad SPI interface.
\textcolor{red}{What happens when this goes into sleep mode and wakes up? Explain details of reprogramming (e.g. on wake up it reads from xxx memory address in flash for bitstream). How big is the bitstream to program the whole thing? Does this vary by design? You mention you can have multiple boots in this flash, can you use the MCU to select which one to load. One reason I can think of is mabye you want to do an experiment comparing two protocols one after another? Mention if this FPGA has IP cores for encryption algorithms or machine learning as well, just to add other features. Also what is the cost of the flash, does flash + fpga make the chip more expensive than others? For clocking does it have a PLL, does it get a reference from an external clock/crystal?}
Finally, to implement the RF section we use two RF paths, one for sub-GHz frequency bands and the other for 2.4~GHz frequency band. We use 2.4~GHz power amplifier ,SE2431L~\cite{}, and 900~MHz band power amplifier, SE2435L~\cite{}, to increase the maximum output power of our radios. Since both I/Q radio and backbone radio share the sub-GHz RF path, we use an ADG904~\cite{} RF switch to select between I/Q radio, RX and TX mode of LoRa radio.\textcolor{red}{Talk about bypass option, explain the switch network, how this handles all TX/RX switching.}
\subsection{Over-the-Air Update}
We employed LoRa protocol at 900~MHz for backbone communication. LoRa is a propriety protocol by Semtech which uses Chirp Spread Spectrum to increase the sensitivity and achieve long-ranges. In \name, we used SX1276 from Semtech that is designed for sub-GHz. Depending on the range, the update rate can be higher than xxx~kbps which is enough considering the compressed programming data of FPGA and MCU. For example with 30~dBm output power, $SF = xxx$, $BW = xxx$ and a 2~dBi antenna, we can achieve xxx~kbps at xxxx~km.
Over-the-air updates is enabled using
Semtech SX1276~\cite{sx1276} as backbone radio talking to the access point by LoRa protocol\textcolor{red}{Why this particular Semtech chip? Explain for people who don't know LoRa is proprietary so it has to be Semtech, but also within their line of chips why this one? What kind of range can you get for different rates/SF with the PA (and mention that this would be when transmitting at xx power with a 2 dBi antenna)? Put some numbers}. Depending on the range, the update rate can be between 0.018~kbps to 37.5~kbps which is enough considering the compressed programming data of FPGA and MCU. In section~\xref{}, these rates and also compressed programming data volume is evaluated for different cases to measure the update duration.
\subsection{Power Management Unit}
\textcolor{red}{Add a paragraph about power, what voltages regulators etc. Can you use this to power an external sensor? What voltages do you have? What components can you turn off, what is always on?}
We integrate all components on a 6-layers FR-4 PCB with the dimension of $30mm\times50mm$ as shown in Fig.~\ref{fig:board}. Our board provides different sensor interfaces including SPI and I2C as external GPIOs for sensor reading applications.
\begin{figure}[t]
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=1\linewidth]{figs/block_diagram/lora_tx_block.eps}
\vskip -0.1in
\caption{\footnotesize{LoRa Modulator}}
\label{fig:lora_tx_block}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=1\linewidth]{figs/block_diagram/lora_rx_block.eps}
\caption{\footnotesize{LoRa Demodulator}}
\label{fig:lora_rx_block}
\end{subfigure}
\vskip -0.15in
\caption{\footnotesize{LoRa implementation FPGA block diagrams.}}
\label{fig:lora_blocks}
\end{figure}
\section{Hardware Implementation}
In this section we go over the implementation of \name hardware and present design choices. The hardware is implemented based on the system design presented in \xref{sec:system} in Fig.~\ref{fig:system} and components are chosen to meet our goal of designing a system that is suitable for IoT testbed implementation.
\subsection{MAC Layer}
We use MSP432P401R~\cite{} a 32-Bit Cortex M4F MCU with floating point unit to control two I/Q and backbone transceiver, communicate with baseband processor and managing power on the platform. This MCU has 64~kB RAM along with 256~kB FLASH memory for programming and provides huge flexibility for performing compression algorithms, complex MAC layer implementation of IoT protocols and sensor reading applications. It supports up to eight SPI, four UART and four I2C interfaces. It can operate at low frequencies, such as 128~kHz, as well as high frequencies up to 48~MHz. It has two 32-bits timers and more importantly it supports different low-power modes. As an example, it supports LPM4.5 mode which only consumes 25~nA current. These features make this MCU a suitable choice for \name to operate at low-power and also perfomr accurate duty-cycling.
\subsection{PHY Layer}
We use a low-power low-cost RF front-end, AT86RF215~\cite{ref:AT86RF215} from Microchip. We choose this chip as it is a highly integrated solution designed for operation in all of our target ISM bands (400~MHz, 900~MHz and 2.4~GHz ISM) and most importantly is one of the few commercial chips that also provides an I/Q output to allow for custom protocols and modulation. It provides all the blocks of the transceiver chain required between the antenna and digital baseband data including LNA/PA, downconversion/upconversion mixers, band-pass/low-pass filters, ADC/DAC, frequency synthesizer and voltage-controlled oscillator. This chip provides two RF ports for sub-Ghz and 2.4~GHz frequencies. Each RF port is access with balanced differential signal pairs. We use 0896BM15E0025E~\cite{} balun for sub-GHz RF port and 2450FB15A050E~\cite{} balun for 2.4~GHz RF port. The receiver chain passes signal through an LNA with noise figure of 4.5 and then downconverts the signal using a mixer that is controlled by frequency synthesizer (PLL). This chip utilizes two PLLs for sub-GHz and 2.4~GHz transceivers. Both PLLs use a shared 26~MHz crystal as reference and they can synthesize any frequencies with 198~Hz step size. Then signal goes through an analog bandpass filter and then sampled by a 13-bits Sigma-Delta analog to digital converter (ADC). On the transmitter side, the digital controller feeds I/Q samples to digital front-end. Then, the signal is sampled by a 13-bits digital to analog converter (DAC). This signal goes to a analog lowpass filter and upconverts by the mixer which uses the same PLL. Finally, signal is amplified with internal power amplifier. Each RF port has one internal power amplifier (PA) that can output signal up to 14~dBm. This optimal power, however, is achieved when the RF port is matched to 50~$\ohm$ differential.
\textcolor{red}{Add noise figure, mention it provides two RF outputs (separate paths for sub GHz and 2.4 GHz, add specs for each internal block like LNA noise, PA efficiency, xx order filter, frequency synthesizer specs/step sizes. Mention what kind of architecture it is (e.g. superheterodyne, direct conversion etc.)What kind of clock input does it require?}
To exchange digitized I/Q baseband data between radio and baseband processing unit, it uses 64~MHz double data-rate low voltage differential signaling (LVDS). LVDS is a differential serial communication protocol which operates at low power and can run at very high speeds. This protocol transmits information as the difference between the voltages on a pair of wires. The two wire voltages are compared at the receiver. To achieve the best connection between two components with LVDS interface, the differential impedance between each pair should be 100~$\ohm$ which we consider when we design \name PCB.
The maximum sampling frequency for the I/Q data is 4~MHz which means that it can support modulations with maximum of 2~MHz bandwidth. This chip is capable of 13~bits ADC/DAC which is even better than the analog front-end (AD93XX~\cite{xxxx}) used by USRP~\cite{xxxx}. \textcolor{red}{The current consumption of this chip is several tens of milli-ampere which is around an order of magnitude lower than the AD93XX series used in common SDR platfroms like USRP}. Note that AD93XX series have higher sampling rate and frequency spectrum coverage which are not required considering our target application of IoT node implementation. Moreover, AD93XX series' price is at least 100\$ or even several hundreds of dollar depending on the version compared to our I/Q transceiver which is about 5\$. In addition, our I/Q transceiver support various modulations such as MR-FSK, MR-OFDM, MR-O-QPSK and O-QPSK. This allows us to bypass the FPGA entirely and enable greater power savings when we use standard modulations supported by our transceiver. In conclusion, AT86RF215 has unique properties which are suitable for IoT node implementation.
We use LFE5U-25F~\cite{} FPGA from Lattice Semiconductor for baseband processing which talks to the I/Q transceiver using LVDS interface. This FPGA has 24k logic units which is comparable to other common SDR platforms including XXX, XXX and XXX. We use a small form-factor of this FPGA on our design to decrease the platform physical size. Compared to FPGAs used in other existing platform this FPGA provides higher number of LUTs for the same cost. Since this is an SRAM-based FPGA, our design requires an external FLASH memory with XX~Mb size capable of storing multiple boots of this FPGA which is included in the design. However, the price of the FLASH memory is not significant compare to the total price of the platform based on Table~\ref{tab:cost}. We use a 32~MHz clock oscillator as the reference clock for the FPGA. When I/Q radio operates in transmit mode, we use this clock reference to generate a 64~MHz clock reference for I/Q serial interface. However, when the I/Q operates in receive mode, we turn off this clock oscillator since the I/Q interface provides an clock reference and we use this reference for DSP operations in FPGA.
Note: Explain the programming chain in detail.
\textcolor{red}{What happens when this goes into sleep mode and wakes up? Explain details of reprogramming (e.g. on wake up it reads from xxx memory address in flash for bitstream). How big is the bitstream to program the whole thing? Does this vary by design? You mention you can have multiple boots in this flash, can you use the MCU to select which one to load. One reason I can think of is mabye you want to do an experiment comparing two protocols one after another? Mention if this FPGA has IP cores for encryption algorithms or machine learning as well, just to add other features. Also what is the cost of the flash, does flash + fpga make the chip more expensive than others? For clocking does it have a PLL, does it get a reference from an external clock/crystal?}
Finally, to implement the RF section we use two RF paths, one for sub-GHz port and the other for 2.4~GHz port. Since the I/Q transceiver can only go up to 14~dBm output power~\cite{}, we use a power amplifier for each port. Since both I/Q radio and backbone radio share the sub-GHz RF port, we use an ADG904~\cite{} RF switch to select between different modes. There are three 900~MHz RF paths that share sub-GHz RF port including I/Q Transceiver RX/TX, backbone radio RX and backbone radio TX. We connect all these three paths to the RF switch and the output of the switch goes to sub-GHz PA. We use SE2435L~\cite{} PA for sub-GHz RF port. This PA has bypass capabilities, LNA and power amplifier. This means gives us flexibility in different operations. For example, in receive mode we can either pass the incoming signal through the LNA and then connect it to the transceiver of completely bypass the LNA and connect the signal directly. Similarly, in transmit operation we can pass the signal through the PA and amplify the signal or turn off the PA and just pass the signal directly to the antenna. We use similar PA, SE2431L~\cite{}, for 2.4~GHz RF port with identical capabilities. Since the 2.4~GHz RF port is only used by I/Q transceiver, it does not require an RF switch and we connect the PA directly to the I/Q transceiver. Using PA with bypass and sleep mode options gives us great capability to reduce power consumption in the case of lower power transmissions.
\subsection{Over-the-Air Update}
We employed LoRa protocol at 900~MHz for backbone communication. LoRa is a propriety protocol by Semtech which uses Chirp Spread Spectrum (CSS) to increase the sensitivity and achieve long-ranges. In \name, we used SX1276 from Semtech that is designed for sub-GHz. Depending on the range, the update rate can be higher than xxx~kbps which is enough considering the compressed programming data of FPGA and MCU. For example with 30~dBm output power, $SF = xxx$, $BW = xxx$ and a 2~dBi antenna, we can achieve xxx~kbps at xxxx~km. In section~\xref{}, these rates and also compressed programming data volume is evaluated for different cases to measure the update duration.
\noindent{\bf Compression.} We compress the over-the-air updates before transmitting them to our testbed nodes. We use miniLZO compression algorithm presented in~\cite{compressionWebsite}. miniLZO is a very lightweight subset of the Lempel–Ziv–Oberhumer (LZO) compression algorithm. Current implementation of miniLZO requires memory allocation with the size of the uncompressed file. Since we have limited SRAM memory available on the MCU, we first divide the original FPGA programming file into smaller block size. Then we perform compression algorithm on each block separately and then we transmit these files to the \name end-node. On the \name platform, we use the same algorithm on MCU to decompresss each file and write it to the flash memory.
\subsection{Power Management Unit}
We implement power management unit with the purpose of having maximum control over all components' power supplies. Table~\ref{tab:voltages} shows various voltage requirements for each component in \name. To power the FPGA, we use three switching regulators to provide 1.1V, 1.8V and 2.5V. We use switching regulators since they have better efficiency and XXX compare to XXX. MCU requires 1.8V power supply which is in common with FPGA. Since we want to completely power of the FPGA, we cannot share V2 voltage domain with MCU. We use an XXX LDO to provide V4 voltage domain for MCU. The reason is XXX. We use a shared voltage domain for I/Q transceiver and backbone radio which we provide the power with a programmable switch regulator, Semtech SC195ULTRT~\cite{}. Using this voltage regulator, we are able to adjust the radio voltage based on the radio mode and maximum power output. Therefore, we can optimize power consumption of the radio in different modes. We power the sub-GHz PA with another switching regulator that provides V6 voltage domain. Similarly, we use a separate voltage regulator for analog voltage power supply of the 2.4~GHz PA and we share its digital power supply with V2 voltage domain. Finally, we share power supplies for FLASH memory and Micro SD memory with other power domain since their functionalities has overlap with other components and they are not power hungry components.
We integrate all components on a 6-layers FR-4 PCB with the dimension of $30mm\times50mm$ as shown in Fig.~\ref{fig:board}. Our board provides different sensor interfaces including SPI and I2C as external GPIOs for sensor reading applications.
\begin{table}[H]
\centering
\caption{\footnotesize{\bf Voltage domains in \name.}}
\vskip -0.15 in
\begin{tabular}{|c|c|c|}
\hline
Component & Voltage & Voltage Domain\\
\hline
FPGA & 1.1V, 1.8V, 2.5V & V1, V2, V3\\
\hline
MCU & 1.8V & V4\\
\hline
I/Q Transceiver & 1.8V< V5 <3.6V & V5\\
\hline
Backbone Radio & 1.8V< V5 <3.6V & V5\\
\hline
sub-GHz PA & 4.0V & V6\\
\hline
2.4~GHz PA & 1.8V, 3.0V & V2, V7\\
\hline
FLASH Memory & 1.8V & V2\\
\hline
Micro SD Memory& 3.0V & V7\\
\hline
\end{tabular}
\label{tab:voltages}
\end{table}
\subsection{Interfacing Between Blocks}
\label{sec:interface}
\subsubsection{Reading and Writing I/Q Samples}
The AT86RF215 radio chipset samples baseband signals at 4~MHz with a 13~bit resolution for both I and Q. Operating at the full rate therefore requires an interface which can support a throughput of over 100~Mbps without consuming a large amount of power to meet our design objectives. To do this we use low-voltage differential signaling (LVDS)~\cite{lvdsTechnology} which is a high-speed digital interface that reduces power by using lower voltage signals but maintains good SNR by sending data over two differential lines to reduce common mode noise.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{./figs/block_diagram/iq_structure.eps}
\vskip -0.15 in
\caption{\footnotesize{{\bf I/Q Word Structure Used in I/Q Interface.}}}
\label{fig:iq_structure}
\vskip -0.2 in
\end{figure}
{\bf Receiving serial I/Q data.}
Our system communicates over LVDS to the FPGA in serial mode to transfer I/Q data with a physical interface consisting of 4 I/O lines, pairs of which are used to send data and clock signals. The radio outputs 32-bit serial data words at 4~Mwords/s using the format in Fig.~\ref{fig:iq_structure}. Each data word starts with the $I\_{SYNC}$ pattern which indicates the start of the $I$ sample which we use for synchronization. Next, it has 13 bits of $I\_{Data}$ followed by a control bit. The same format follows for $Q$, beginning with a synchronization pattern $Q\_{SYNC}$ and then 13 bits for $Q\_{Data}$ and the final control bit. The required 128~Mbps data rate is achieved using a 64~MHz clock provided by the radio operating at double data rate by sampling at both the rising and falling edges of the clock. We implement an I/Q deserializer on the FPGA to read the data which samples the input at both the rising and falling edges of the clock, uses the $I\_{SYNC}$ and $Q\_{SYNC}$ to detect the beginning of the data fields and loads the $I$ and $Q$ values into 13~bit registers for parallel processing.
{\bf Transmitting I/Q samples.} In TX mode we need to do the opposite of the above sequence to convert from the parallel representation on the FPGA to a serialized LVDS stream. To do this, we use the FPGA's onboard PLL to generate the 64~MHz clock signal. Next to create our double data rate output signal that varies on both the positive and negative edges of this clock signal using a dual-edge D flip-flop design~\cite{hildebrandt2011pseudo} resulting in the desired 128~Mbps data rate. We use this to generate the same I/Q word structure described above.
\subsubsection{Memory Interfaces}
After reading the raw data from the LVDS lines using the I/Q deserializer described above, we store the samples into a FIFO buffer implemented using the FPGA's embedded SRAM. We implement a simple memory controller to write data to the FIFO which generates the memory control signals and writes a full data word on each cycle. The embedded memory can run at rates significantly greater than 4~MHz meaning it is not a limiting factor for real time processing. The SRAM can buffer up to 126~kB. The data stored in the FIFO can then be sent to signal processing blocks to implement filters, cryptographic functions, etc. or to non-volatile flash memory. For flash memory, we use microSD cards which support two modes: native SD mode and standard SPI mode. In native SD mode, we use 4 parallel data lines to read/write data to/from the microSD card. This mode supports a higher data rate compared to the SPI mode which only supports 1 bit serial interface. However, we implement SPI mode since it supports the 104~Mbps data rate which we need to write data in real time. This allows us to re-use the same, simpler SPI block for multiple functions and save resources on the FPGA.
\subsubsection{RF, Control and Sensor Interfaces}
The AT86RF215 provides differential RF signals for both 900~MHz and 2.4~GHz and has an integrated TX/RX switch for both. At 2.4~GHz, the differential signal is transformed to a single-ended output using the 2450FB15A050E~\cite{24balun} balun and fed to the SKY66112~\cite{SKY66112} front-end with the bypassable LNA and PA. Finally, after passing through a matching network, the 2.4~GHz signal is connected to an SMA output.
On the 900~MHz side, the differential output of the AT86RF215 is connected to 0896BM15E0025E~\cite{900balun} to convert it to a single-ended output. This must be shared between the backbone radio's two separate RF paths for transmit and receive and AT86RF215's 900~MHz single-ended signal. We choose between them using a ADG904~\cite{adg904} SP4T RF switch. The single port side is connected to the SE2435L~\cite{SE2435L} 900~MHz front-end which is similar to the 2.4~GHz front-end. The MCU communicates with the I/Q radio, backbone radio, FPGA and Flash memory through SPI which it uses to send commands for changing the frequency, selecting the outputs, etc. It also has control signals for FPGA programming, 900~MHz and 2.4~GHz front-end modules, RF switch and voltage regulators for active power control. {The I2C and SPI serial interfaces and analog to digital converter (ADC) inputs of the MCU are broken out on \name board to support both digital and analog sensors.}
\subsection{Power Management Unit}
\label{sec:powermanagement}
Next, we present the design of our power management unit which seeks to maximize the system lifetime when running off of a 3.7~V Lithium battery. To enable long battery lifetimes we need to be able to duty-cycle our system and allow the MCU to toggle each of the above blocks ON and OFF when they are not in use. Further, different components have different supply voltage requirements and we wish to provide each one with the lowest voltage possible to minimize power usage.
Ideally we would want separate controllable voltage regulators for each component in the system. However, having many different regulators with individual controls significantly increases the complexity, number of components, and price. Moreover, it complicates the PCB design by requiring many control signals and a multitude of power planes. Therefore, there exists a trade-off between the granularity of power control and the price/complexity of a design. We outline the supply voltages needed for each component and the power domain supporting it in Table~\ref{tab:voltages}. Below, we show how we group components to balance power and complexity.
\begin{table}[t]
\centering
\caption{\footnotesize{\bf Power Domains in \Name.}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|c|c|c|}
\hline \rowcolor{lightgray}
{\bf Component} & {\bf Voltage [V]} & {\bf Power Domain}\\
\hline
MCU & 1.8V & V1\\
\hline
FPGA & 1.1, 1.8, 2.5, Vlvds & V2, V3, V4, V5\\
\hline
I/Q Radio & 1.8< V5 <3.6 & V5\\
\hline
Backbone Radio & 1.8< V5 <3.6 & V5\\
\hline
sub-GHz PA & 3.5V & V6\\
\hline
2.4~GHz PA & 1.8, 3.0 & V3, V7\\
\hline
FLASH Memory & 1.8 & V3\\
\hline
Micro SD Memory& 3.0 & V7\\
\hline
\end{tabular}
}
\vskip -0.23 in
\label{tab:voltages}
\end{table}
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Power domain V1 (MCU).} Since the MCU is the central controller which implements power management, it needs to be powered at all times and therefore has its own power domain. To minimize its sleep current we need to use a voltage regulator with a low quiescent current. Although switching voltage regulators have higher conversion efficiency when active, they also have high quiescent currents so we instead select the TPS78218 linear regulator.
\item {\bf Power domains V2, V3, V4, V6 and V7.} These power domains provide power to blocks such as the FPGA, memory blocks, and PAs. Since these components can all be turned off when not operating, the voltage regulators for these domains should have low shut-down current during sleep and high efficiency when active. We therefore choose the TPS62240 which has a shutdown current of only 0.1~uA. It is highly efficient and is rated to support the current draw required by all components except the 900~MHz PA. To support this PA at its maximum output power we use the TPS62080 switching regulator which supports the required current.
\item {\bf Power domain V5.} V5 is a shared power domain for I/Q radio, backbone LoRa radio and FPGA I/O bank. This power domain is initially set to 1.8V to minimize power consumption, however components such as the radio chips can require higher voltage to achieve maximum output power. Therefore, in addition to high efficiency and low shut-down current like the others, this domain should be programmable. To do this, we use Semtech SC195ULTRT~\cite{SC195ULTRT} which provides an adjustable output that can be set from 1.8~V to 3.6~V.
\end{itemize}
\subsection{Over-the-Air Programming protocol}
\label{sec:ota}
{\bf OTA AP and MAC protocol.} To update a network of \name devices, we use an AP with a LoRa radio to communicate with each device sequentially. In order to propagate updates throughout a testbed or to specific \name nodes, we design a MAC layer for the LoRa PHY.
We pre-program a timer on the MCU to periodically turn off the FPGA and switch from IQ radio mode to the backbone radio to listen for new firmware updates. If there is an update, the AP sends a programming request as a LoRa packet with specific device IDs indicating the nodes to be programmed along with the time they should wake up to receive the update. Upon processing this packet and detecting its ID, the \name node switches into update mode and sends a ready message to the AP at the scheduled time. Then, the AP transmits the firmware update as a series of LoRa packets with sequence numbers. Upon receiving each packet, the \name node checks the sequence number and CRC. For a correct packet it writes the data to its flash memory and transmits an ACK to indicate correct reception. In the case of failure no ACK is sent and the AP re-transmits the corrupted packet after a timeout. After sending all the firmware data, the AP sends a final packet indicating the end of firmware update which tells the \name node to reprogram itself and switch back to normal operation.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{./figs/block_diagram/lora_packet.eps}
\vskip -0.1 in
\caption{\footnotesize{{\bf LoRa Packet Structure.}}}
\vskip -0.25in
\label{fig:lora_packet}
\end{figure}
{\bf Compressing and decompressing the bitstream.} Our system compresses data to reduce update times, however this compression must be compatible with the resources available on \name. We choose the miniLZO compression algorithm~\cite{miniLZO}, which is a lightweight subset of the Lempel–Ziv–Oberhumer (LZO) algorithm. Our implementation of miniLZO only requires a memory allocation equal to the size of the uncompressed data. We perform compression on the AP. The compression ratio of bitstream file varies based on the content of the bitstream, and in the worst case the compressed file could have almost the same size of the original file. This would require a maximum memory allocation of 579~kB which we cannot afford on a low-cost MCU. Instead, we first divide the original update file into blocks of 30~kB that will fit in the MCU memory. Then we compress each block separately and transmit them to the \name node one by one. Considering the LoRa radio takes more power than the MCU, we immediately write the data to our dedicated programming flash memory using an SPI interface.
\begin{figure}[t]
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=1\linewidth]{figs/block_diagram/lora_tx_block.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa Modulator}}}
\label{fig:lora_tx_block}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=1\linewidth]{figs/block_diagram/lora_rx_block.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa Demodulator}}}
\label{fig:lora_rx_block}
\end{subfigure}
\vskip -0.1in
\caption{\footnotesize{{\bf LoRa Implementation Block Diagrams.}}}
\label{fig:lora_blocks}
\vskip -0.2in
\end{figure}
After receiving all the data we turn off the LoRa radio and decompress data. First, we allocate memory on the MCU's SRAM equal to the block size and load a block of data from flash. Next, we perform decompression and write the data in the allocated SRAM memory. Finally, we write the decompressed data back to the flash beginning at the corresponding address of the programming boot file. We repeat these steps until we decompress the full firmware update.
{\bf Over-the-air FPGA programming.} After storing uncompressed programming data in flash memory, we program the FPGA. We use the MCU to set the FPGA into programming mode. When the FPGA switches to programming mode, it automatically reads { its firmware} directly from the flash memory using a 62~MHz quad SPI interface and programs itself. Reading from flash using quad SPI achieves programming times of 22~ms which is similar to FPGAs with embedded flash memory and results in minimal system down time. After programming is complete, it resumes operation and begins running the new firmware.
\section{System Implementation}
In this section, we explain the operation of each of the above blocks in detail. To do this we walk through the technical challenges of reading the digitized signal into the FPGA at the full sampling rate of the radio output and into internal memory followed by saving them to an SD card. Next we give an end to end description of our over-the-air update system including the access point communication protocol, compression and reprogramming sequence.
\subsection{Reading and Writing I/Q Samples}
The AT86RF215 radio chipset samples baseband signals at a rate of 4~MHz with a 13~bit resolution for both I and Q. Operating at the full rate therefore requires an interface which can support a throughput of over 100~Mbps without consuming a large amount of power to meet our design objectives. To do this we use a standard called low-voltage differential signaling (LVDS)~\cite{XXX} which is provided by AT86RF215. LVDS is a high-speed digital interface that reduces power by using lower voltage signals but maintains good SNR by sending data over two differential lines to reduce common mode noise.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{./figs/block_diagram/iq_structure.eps}
\vskip -0.15 in
\caption{\footnotesize{I/Q word structure used in I/Q interface.}}
\label{fig:iq_structure}
\vskip -0.2 in
\end{figure}
\noindent{\bf Receiving serial I/Q data.}
Our system communicates by LVDS to the FPGA in synchronous serial mode to transfer I/Q data with a physical interface consisting of 2 pairs of differential I/O lines, one pair for data and one for clock. The radio outputs 32-bit serial data words at 4~Mwords/s using the format in Fig.~\ref{fig:iq_structure}. Each data word starts with the $I\_{SYNC}$ pattern which indicates the start of the $I$ sample which we use for synchronization. Next, it has 13 bits $I\_{Data}$ followed by a control bit. The same format follows for $Q$, beginning with a synchronization pattern $Q\_{SYNC}$ and then 13 bits for $Q\_{Data}$ and the final control bit. The required 128~Mbps data rate is achieved using a 64~MHz double data rate clock provided by the radio, meaning that the data should be sampled at both the rising and falling edges of the clock signal. We implement an I/Q deserializer on the FPGA module to read the data. The deserializer samples the data signal at both the rising and falling edges of the clock, uses the $I\_{SYNC}$ and $Q\_{SYNC}$ to detect the beginning of the data fields and loads the $I$ and $Q$ values into 13~b registers to allow for parallel processing.
\noindent{\bf Transmitting I/Q samples.} In TX mode we need to do the opposite of the above sequence to convert from the parallel representation on the FPGA to a serialized LVDS stream. To do this, we use the FPGA's onboard PLL to generate the 64~MHz clock signal. Next to create our double data rate output signal that varies on both positive and negative edges of this clock signal, we use a dual-edge D flip-flop design~\cite{hildebrandt2011pseudo} resulting in the desired 128~Mbps data rate. We use this to generate the same I/Q word structure described above.
\subsection{RF Interfaces}
Our goal in designing \name is supporting 900~MHz and 2.4~GHz ISM bands. AT86RF215 provides differential RF signal for both 900~MHz and 2.4~GHz and has integrated TX/RX switch for both paths. On the 2.4~GHz side, the differential signal is transformed to single-ended using 2450FB15A050E~\cite{24balun} balun and then fed to SKY66112~\cite{SKY66112} front-end which has LNA and PA. The LNA and PA of SKY66112 can be bypassed if needed. Finally, after passing through matching network, the 2.4~GHz signal is connected to SMA. On the 900~MHz side, the differential signal of the AT86RF215 is connected to 0896BM15E0025E~\cite{900balun} to get single-ended signal.
There are three signal lines that need to share the 900~MHz path to the antenna. Backbone radio's two separate RF paths for transmit and receive and AT86RF215's 900~MHz single-ended signal. We choose between these three using ADG904~\cite{adg904} SP4T RF switch. The single port side of the switch is connected to the SE2435L~\cite{SE2435L} 900~MHz front-end which similar to the 2.4~GHz front-end has PA and LNA that can be bypassed.
\subsection{Control and Sensor Interfaces}
As mentioned previously, MCU is the central controller and controls all the components of the system. It talks to I/Q radio, backbone radio, FPGA and Flash memory through SPI. Moreover, it has control signals for FPGA programming, 900~MHz and 2.4~GHz front-end modules, RF switch and voltage regulators. To enable testing of different application scenarios, \name is utilized with common sensor interfaces like ADC, I2C and SPI. These interfaces are handled by the MCU.
\subsection{Memory Interfaces}
After reading the raw data from the LVDS lines using the I/Q deserializer described above, the next step is to store the samples into a FIFO buffer implemented using the FPGA's embedded SRAM to prepare them for processing by FPGA. We implement a simple memory controller to write data to the FIFO as shown in Fig.~\ref{fig:memory_block} which generates the memory control signals and writes a full data word on each cycle. The embedded memory can run at rates significantly greater than 4~MHz meaning it is not a limiting factor for real time processing. The SRAM alone can buffer up to 126 kB, and additional embedded memory options such as DRAM could allow for larger buffer sizes as well.
The data stored in the FIFO can then be sent to signal processing blocks to implement filters, cryptographic functions, etc. or to non-volatile flash memory. For flash we use microSD cards which support two modes: native SD mode and standard SPI mode. In native SD mode, we can use up to 4 bit parallel data lines to read/write data to/from the microSD card. This mode supports a higher data rate compare to the SPI mode which only supports 1 bit serial interface. However, we implement SPI mode since it supports the data rate of XX~Mbps which our system requires to write data in real time and has a simpler implementation. This allows us to re-use the same, simpler SPI block for multiple functions and save resources on the FPGA.
\subsection{Over-the-Air Update}
In this section, we explain the operation of \name's over-the-air system. This system requires addressing multiple challenges including a protocol for interacting with the centralized access point to ensure each node receives the full bit stream, as well as minimizing transmission length to reduce network downtime. First, we explain how we prepare updates on the access point and handle transmissions to each node. Then, we explain the details of storing the update data and reprogramming the system.
\noindent{\bf OTA AP and MAC Protocol.} To control a network of multiple \name devices in a testbed, we use a single centralized access point with a LoRa radio to communicate with each node. In order to propagate updates throughout a testbed network or to specific \name nodes, we design a MAC layer system over the LoRa physical layer protocol. In order to avoid the chance of corrupted firmware due to packet loss during the update, and considering each update takes only XXX~s as shown in Fig.~\ref{xxx}, our access point updates each \name node individually.
First, the AP sends an update request as a LoRa packet with a specific device ID indicating the node to be updated in the payload. Upon processing this packet and detecting its ID, the \name node switches into update mode and sends a ready message to the access point. Then, the AP transmits the firmware update as a series of LoRa packets with sequence numbers. Upon receiving each packet, the \name node checks the sequence number and CRC. For a correct packet it writes the data to its flash memory and transmits an ACK to indicate correct reception. In the case of failure, the AP retransmits the corrupted packet. After the AP has sent all the firmware data, it sends a final packet indicating the end of firmware update, and the \name node switch its state to normal operation upon receiving this.
\noindent{\bf Compressing and decompressing the bitstream.} Our system uses compression to minimize the amount of data sent in each update, however this compression must be compatible with the resources available on \name. We choose the miniLZO compression algorithm~\cite{miniLZO}, which is a lightweight subset of Lempel–Ziv–Oberhumer (LZO) compression algorithm. Our current implementation of miniLZO decompression algorithm only requires a memory allocation equal to the size of the uncompressed data. We perform compression on the AP. The compression ratio of bitstream file varies based on the content of the bitstream. In worst case scenario, the compressed file has exactly the same size of the original file which is the lowest compression ratio. In this case, we need memory allocation on the MCU on \name with the size of original bitstream file. This is around XXX~kB SRAM memory which we cannot afford on a low-cost MCU. To address this, we first divide the original update file into blocks of 20~kB that will fit in the MCU memory. Then we compress each block separately and transmit them to the \name node one by one. Considering the LoRa radio takes more power than the MCU, we immediately write the data to our dedicated programming flash memory using an SPI interface.
After receiving all the data we turn off the LoRa radio and decompress the data. First, we allocate memory on SRAM on the MCU equal to the block size and load a block of data from flash. Next, we perform decompression and write the data in the allocated SRAM memory. Finally, we write the decompressed data back to the flash beginning at the corresponding address of the programming boot file. We repeat these steps until we decompress the full firmware update.
\noindent{\bf Programming.} After storing uncompressed programming data in flash memory, we perform FPGA programming. We use the MCU to set the FPGA into programming mode. When FPGA switches to programming mode, it automatically reads programming data directly from the flash memory using a quad SPI interface and programs itself. Reading from flash using quad SPI, which can run up to 62~Mbps, achieves programming times of xx~ms which is similar to FPGAs with embedded flash memory and results in minimal system down time. After programming is complete, it resumes operation and begins running the new firmware.
\section{Introduction}
\section{Introduction}
Recent years have seen significant work on wireless communication protocols for IoT endpoint devices. There is currently no de-facto communication standard for these devices as protocols such as LoRa, Sigfox and NB-IoT have only been introduced in the past few years; this has led to a number of works in this space such as xxx, xxx, xxx, xxx.
Despite the interest in developing solutions for IoT endpoints, the research community is limited by the lack of tools to develop and evaluate these protocols. While commercial chipsets implement a single IoT protocol, they aim to provide a turnkey solution for products and offer little to no flexibility for modifying or improving any layer of the network stack and eliminate any possibility of clean slate designs. Moreover, some implementations such as LoRa are proprietary and even details of the standard are not publicly available. Considering the rapid growth of IoT technologies, using one or a combination of inflexible, single protocol chipsets to develop dedicated hardware for individual projects is not sustainable for research community.
In contrast software defined radios (SDRs) solve this problem by offering access to the raw radio signals. SDR platforms such as the USRP have become an essential tool in the wireless research community enabling diverse works ranging from WiFi and LTE access points to backscatter and localization techniques; however we observe that a focus on achieving high bandwidth and wide spectrum coverage leads SDR platforms to be bulky, expensive, and power hungry. While these high performance specs are ideal for development of access points, existing SDR platforms do not meet the following three constraints unique to \textit{IoT endpoints}:
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Computation.}
An IoT endpoint should be standalone and include onboard computation to run its basic functions such as periodically waking up from sleep mode to sample a sensor and process data, transmit a message, and go to back to sleep mode. The need for many SDR platforms to be tethered via USB or Ethernet to a laptop is acceptable for use as a gateway or access point but makes them infeasible for use as IoT nodes.
\item {\bf Power consumption.}
IoT endpoints are often deployed in locations without dedicated power infrastructure and must be battery powered. Unlike plugged in access points, low power consumption is a paramount concern and existing SDR platforms often consume xx~W.
\item {\bf Cost and scalability.}
Many real world IoT deployments require a large number of nodes, and studying the complex networking challenges requires a scalable testbed. This adds the additional constraint of requiring individual nodes to be low cost.
\end{itemize}
In this work we present \name: an SDR platform specifically tailored to the needs of IoT endpoint research. We exploit recent trends that have produced small, low power radios, microcontrollers, and FPGAs to develop our SDR platform. Specifically \name has the following capabilities:
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\it Raw samples.} Our platform outputs raw I/Q samples which enable phase based applications like localization and can be processed in real time or stored to an onboard SD card for offline processing.
\item {\it Low power.} The whole platform consumes xx~mW in transmit mode and only xx~mW in receive mode.
\item {\it Small form factor.} The total size of our platform is xx $\times$ xx cm.
\item {\it Frequency bands.} Our platform covers both the 900~MHz and 2.4~GHz ISM bands allowing for use with a variety of IoT protocols.
\item {\it Transmit/Receive capability.} Despite the small size and low power, it can transmit up to xx~dBm and has a receiver noise figure of only xx~dB.
\item {\it Sensitivity.} This platform can achieve the sensitivity of commercial LoRa chipsets allowing for long range operation.
\item {\it Bandwidth.} The xx~MHz bandwidth is sufficient for a variety of IoT protocols.
\item {\it Computation/programmability.} An included FPGA and microcontroller allow for standalone operation and flexibility to implement custom algorithms onboard.
\item {\it Sensor inputs} The microntroller allows for analog and digital sensor inputs, enabling a complete plug and play IoT sensor node solution.
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/3_platform_power/platform_pow.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf Radio Power Consumption.} We show radio power consumption for each platform. We normalized TX power consumption for output equals to 14~dBm.\newline
**RTL-SDR does not support transmitter mode.}}
\label{fig:platform_power}
\vskip -0.15in
\end{figure}
To demonstrate the functionality of our platform we further present an application scenario: distributed localization and mapping of sensor nodes. In We use our platform to propose the first solution that allows a network of IoT nodes to \textit{localize each other} without the need for a central gateway or access point.
[TODO insert description of algorithm]
We summarize our contributions below:
\begin{itemize}
\item We develop the first SDR platform tailored to the needs of IoT endpoints. Our platform is capable of implementing numerous IoT protocols, gives access to raw samples, and achieves low power cost and size needed for large scale deployments.
\item We extensively characterize and evaluate our platform to show its capabilities including implementations of LoRa and XX as well as sensitivity and synchronization experiments.
\item We present a novel distributed localization system capable of localizing IoT nodes without the need for a centralized access point.
\end{itemize}
\section{Introduction}
The grand vision for Internet of Things (IoT) technologies is to extend internet connectivity to everyday physical objects across homes, vehicles, hospitals, farms, factories and more. The diversity of needs in each of these different sectors has led to significant work on wireless communication protocols for IoT endpoint devices in recent years; however there is currently no de-facto communication standard as protocols like LoRa, Sigfox, NB-IoT and others compete for market share.
This leads to a host of problems for researchers, consumers, and chipset manufacturers alike. On the one hand, chipset manufacturers are incentivized to produce proprietary protocols for a specific industry offering little to no flexibility for modifying or improving \textcolor{red}{any layer of the network stack -> LoRa MAC can be changed} or interacting with other technologies; however customers must then bet on the long term success of a particular IoT technology, or attempt to interconnect them. \textcolor{red}{ Similarly, while this presents an opportunity for researchers to develop novel solutions the lack of flexibility stifles innovation.} Additionally, while recent works in the networking community have sought to solve problems in the IoT space~\cite{xxx,xxx,xxx}, their scope and potential for commercial adoption is limited by the lack of a large scale testbed for real-world deployments.
Similar to how PlanetLab~\cite{planetlab} presented a novel tool for developing disruptive technologies for the Internet, a large scale flexible testbed of IoT nodes would open the door for researchers to address the \textcolor{red}{ outstanding challenges} outlined above. In this paper we take the first step towards building such a testbed by developing \name, a platform for prototyping IoT endpoint nodes.
In order to allow for flexibility, \name uses a software defined radio (SDR) which offers access to the raw radio signals. Existing SDR platforms such as the USRP have become an essential tool in the wireless research community enabling diverse works ranging from WiFi and LTE access points to backscatter and localization techniques; however we observe that a focus on achieving high bandwidth and wide spectrum coverage leads existing SDR platforms to be bulky, expensive, and power hungry. While these high performance specs are ideal for development of access points, existing SDR platforms do not meet the following three constraints unique to \textit{IoT endpoints}:
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/3_platform_power/platform_pow.eps}
\vskip -0.1in
\caption{\footnotesize{Radio power consumption for each platform. We normalized TX power consumption for output power of 14~dBm.\newline
**GalioT does not support transmitter mode.}}
\label{fig:platform_power}
\vskip -0.2in
\end{figure}
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Computation.}
An IoT endpoint should be standalone and include onboard computation to run its basic functions such as periodically waking up from sleep mode to sample a sensor and process data, transmit a message, and go to back to sleep mode. The need for many SDR platforms to be tethered via USB or Ethernet to a laptop is acceptable for use as a gateway or access point but makes them infeasible for use as IoT nodes.
\item {\bf Power consumption.}
IoT endpoints are often deployed in locations without dedicated power infrastructure and must be battery powered. Unlike plugged in access points, low power consumption is a paramount concern and existing SDR platforms often consume xx~W.
\item {\bf Cost and scalability.}
Solving real world problems for IoT deployments requires a large number of nodes, and studying the complex networking challenges requires a scalable testbed. This adds the additional constraint of requiring individual nodes to be low cost.
\end{itemize}
In contrast \name is an SDR platform specifically tailored to the needs of IoT endpoint research. We exploit recent trends that have produced small, low power radios, microcontrollers, and FPGAs to develop our SDR platform. Specifically \name has the following capabilities:
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\it Raw samples.} Our platform outputs raw I/Q samples which enable phase based applications like localization and can be processed in real time or stored to an onboard SD card for offline processing.
\item {\it Low power.} The whole platform consumes xx~mW in transmit mode and only xx~mW in receive mode.
\item {\it Small form factor.} The total size of our platform is xx $\times$ xx cm.
\item {\it Frequency bands.} Our platform covers both the 900~MHz and 2.4~GHz ISM bands allowing for use with a variety of IoT protocols.
\item {\it Transmit/Receive capability.} Despite the small size and low power, it can transmit up to xx~dBm and has a receiver noise figure of only xx~dB.
\item {\it Sensitivity.} This platform can achieve the sensitivity of commercial LoRa chipsets allowing for long range operation.
\item {\it Bandwidth.} The xx~MHz bandwidth is sufficient for a variety of IoT protocols.
\item {\it Computation/programmability.} An included FPGA and microcontroller allow for standalone operation and flexibility to implement custom algorithms onboard.
\item {\it Sensor inputs} The microntroller allows for analog and digital sensor inputs, enabling a complete plug and play IoT sensor node solution.
\end{itemize}
We summarize our contributions below:
\begin{itemize}
\item We develop the first SDR platform tailored to the needs of IoT endpoints. Our platform is capable of implementing numerous IoT protocols, gives access to raw samples, and achieves low power cost and size needed for large scale deployments.
\item We extensively characterize and evaluate our platform to show it is capable of implementing existing protocols such as LoRa and XX to as a starting point for researchers to develop custom solutions.
\end{itemize}
\section{Introduction}
The grand vision for Internet of Things (IoT) technologies is to extend internet connectivity to everyday physical objects across homes, vehicles, hospitals, farms, factories and more. The diversity of needs in each of these different sectors has led to significant work on wireless communication protocols for IoT endpoint devices in recent years; however there is currently no de-facto communication standard as protocols like LoRa, Sigfox, NB-IoT and others compete for market share.
This leads to a host of problems for researchers, consumers, and chipset manufacturers alike. On the one hand, chipset manufacturers are incentivized to produce proprietary protocols for a specific industry offering little to no flexibility for modifying or improving PHY or MAC layer or interacting with other technologies; however customers must then bet on the long term success of a particular IoT technology, or attempt to interconnect them. \textcolor{red}{ Similarly, while this presents an opportunity for researchers to develop novel solutions the lack of flexibility stifles innovation.} Additionally, while recent works in the networking community have sought to solve problems in the IoT space~\cite{xxx,xxx,xxx}, their scope and potential for commercial adoption is limited by the lack of a large scale testbed for real-world deployments.
Similar to how PlanetLab~\cite{planetlab} presented a novel tool for developing disruptive technologies for the Internet, a large scale flexible testbed of IoT nodes would open the door for researchers to address the outstanding challenges outlined above. As shown in fig.~\ref{fig:vision}, our vision is to build a testbed to enable large-scale innovation in IoT domain. This testbed would have hundreds and thousands IoT endpoints distributed around a large area talking to each other or an access point. These endpoints should be flexible to be able to be programmed with different PHY and MAC layer codes so that users can test their customized protocols. In this paper we take the first step towards building such a testbed by developing \name, a platform for prototyping IoT nodes.
In order to allow for flexibility, \name uses a software defined radio (SDR) architecture which offers access to the raw radio signals. Existing SDR platforms such as the USRP have become an essential tool in the wireless research community enabling diverse works ranging from WiFi and LTE access points to backscatter and localization techniques; however we observe that a focus on achieving high bandwidth and wide spectrum coverage leads these platforms to be bulky, expensive, and power hungry. While these high performance specs are suitable for development of access points, existing SDR platforms do not meet the following constraints unique to IoT node development platform:
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/map.eps}
\vskip -0.1in
\caption{\footnotesize{Our vision is to build an IoT testbed including hundreds of IoT endpoints spanning across a large area.}}
\label{fig:vision}
\vskip -0.2in
\end{figure}
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Cost and scalability.}
Solving real world problems for IoT deployments requires a large number of nodes, and studying the complex networking challenges requires a scalable testbed. This imposes the constraint of requiring individual nodes to be low cost.
\item {\bf Standalone operation.}
An IoT endpoint should be standalone and include onboard computation to run its basic functions such as periodically waking up from sleep mode to sample a sensor and process data, transmit a message, and go back to sleep mode. Although, existing SDR platform's FPGA are capable of reprogramming, it is not practical and efficient to develop such functions on them and in most cases they are tethered via USB or Ethernet to a laptop which is acceptable for use as a gateway or access point but makes them infeasible for use as IoT nodes.
\item {\bf Over-the-air (OTA) update.}
In the described testbed, there are hundreds and thousands of flexible IoT nodes which are deployed across a large area. Existing SDR platforms are lack of wireless backbone support to update PHY and MAC layer implementations on the IoT endpoint.
\item {\bf Power consumption.}
IoT endpoints are often deployed in locations without dedicated power infrastructure and must be battery powered. Unlike plugged in access points, low power consumption is a paramount concern and existing SDR platforms often consume several watts of power. Moreover, they are not capable of being duty-cycled which results in significantly short battery life times.
\item {\bf \textcolor{red}{Duty-cycling.}}
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/3_platform_power/platform_pow.eps}
\vskip -0.1in
\caption{\footnotesize{Radio power consumption for each platform. We normalized TX power consumption for output power of 14~dBm.\newline
**GalioT does not support transmitter mode.}}
\label{fig:platform_power}
\vskip -0.2in
\end{figure}
In contrast \name is an SDR platform specifically tailored to the needs of IoT endpoint research. We exploit recent trends that have produced small, low power radios, microcontrollers, and FPGAs to develop our SDR platform. Specifically \name has the following capabilities and characteristics:
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\it Over-the-air update.} We enable over-the-air update functionality on \name by using an extra backbone radio with LoRa protocol.
\item {\it Granular power management.} We design power distribution in \name to have full control over all power sources and optimize power consumption in each operation mode.
\item {\it Raw samples.} Our platform outputs raw I/Q samples which enable phase based applications like localization and can be processed in real time or stored to an onboard mico SD card for offline processing.
\item {\it Low power.} The whole platform consumes xx~mW in transmit mode and only xx~mW in receive mode.
\item {\it Small form factor.} The total size of our platform is $3\times5~cm$.
\item {\it Frequency bands.} Our platform covers both the 900~MHz and 2.4~GHz ISM bands allowing for use with a variety of IoT protocols.
\item {\it Transmit/Receive capability.} Despite the small size and low power, it can transmit up to xx~dBm and has a receiver noise figure of only 5~dB.
\item {\it Sensitivity.} This platform can achieve the sensitivity of commercial LoRa chipsets allowing for long range operation.
\item {\it Bandwidth.} The 4~MHz bandwidth is sufficient for a variety of IoT protocols.
\item {\it Computation/programmability.} An included FPGA and microcontroller allow for standalone operation and flexibility to implement custom algorithms onboard.
\item {\it Sensor inputs.} The microntroller allows for analog and digital sensor inputs, enabling a complete plug and play IoT sensor node solution.
\end{itemize}
We summarize our contributions below:
\begin{itemize}
\item We develop the first SDR platform tailored to implement the IoT testbed. Our platform is low cost, capable of implementing numerous IoT protocols both in 900~MHz and 2.4~GHz ISM bands, have large computation resources to do various on-board processing, modular power management and can be duty-cycled and more importantly its codes can be updated through a remote access point.
\item We extensively characterize and evaluate our platform. To show the flexibility of \name, two IoT protocols in two frequency bands of 900~MHz and 2.4~GHz is implemented. The power consumption is measured in multiple implementation scenarios. Over-the-air update is characterized for multiple different programming data and different ranges of an access point and platform and update times are measured. Moreover, we show cost breakdown for building 1000 \name.
\end{itemize}
\section{Introduction}
Recent years have seen development on wireless protocols for Internet of Things (IoT) devices. In addition to longtime standards such as Bluetooth and Zigbee, a number of other protocols such as LoRa, Sigfox, NB-IoT, LTE-M, and 802.11h have been developed and due to the lack of a de-facto standard this space remains an active area of development for both industry and academia. The rapid advances in this space however presents practical challenge for researchers: each of these protocols requires a dedicated radio chipset to evaluate, and these proprietary solutions often leave little room for protocol modification. The academic community is therefore severely handicapped by the lack of a \textit{flexible} platform as even a complex multi-radio prototype cannot adapt to evaluate new protocols or even customize existing solutions. The current ecosystem therefore discourages researchers from investigating the important questions that arise when scaling up IoT networks, and more importantly taking a systematic approach to develop new protocols from the ground up.
Ideally, we would like a large scale network testbed with the flexibility to run \textit{any} IoT protocol and push changes using simple over-the-air software updates. Fig~\ref{fig:vision} illustrates this vision of a large area network testbed with hundreds of endpoints across a city. This would allow for side-by-side performance comparisons to investigate the trade offs of existing standards and showcase the advantages of an entirely new custom solution. Moreover, to make such a system representative of real-world deployments, we would like for the individual network nodes to model the constraints of IoT endpoints. Specifically, these devices should be able to run on a battery, have appropriate power controls and options to duty cycle transmissions, and interfaces to connect sensors like real IoT nodes. The ability to run these endpoints on batteries would also allow for flexibility of deployment in spaces without dedicated power access, or even on mobile devices.
Realizing this vision however is limited by the hardware platforms that exist today for prototyping IoT systems. Specifically, such a system requires a software defined radio (SDR) to maximize flexibility for implementing different IoT protocols, but there is currently no SDR platform that meets all of our requirements. If we look at commercially available SDR systems, we notice that they are expensive, consume large amounts of power, require wired infrastructure and often a dedicated computer, and a wired backbone for updates. Said differently, devices on the market focus on achieving high performance in terms of bandwidth, etc. because they are tailored to the needs of gateway style devices and there is currently no SDR platform that serves the unique needs of IoT endpoints.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{./figs/foo.eps}
\vskip -0.2 in
\caption{\footnotesize{{\bf \name Hardware Design.}}}
\label{fig:sdr_hardware}
\vskip -0.2 in
\end{figure}
Driven by a need for such a platform in our own research, we build \name.
\name is the first SDR platform specifically tailored to the needs of IoT endpoint research. Our SDR platform provides an entirely standalone solution that incorporates a radio front end, FPGA and microcontroller for custom processing, wireless update capabilities, a microSD card interface for storage, and a highly granular power managment options to enable battery powered operation. It is capable of transmitting and receiving in both the 900~MHz and 2.4~GHz ISM bands, supports 4~MHz of bandwidth, and can achieve the sensitivity of commercial solutions such as LoRa chipsets. Additionally it includes a variety of analog and digital I/O options for connecting sensors and measures only $3\times5~cm$.
We exploit recent trends that have produced small, low power radios, microcontrollers, and FPGAs to develop our SDR platform, however building this platform required a number of careful design decisions and multiple engineering challenges:
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Hardware architecture.} Achieving a small form-factor, low power design requires a minimalist design approach to identify only the key system requirements. We analyze the priorities for IoT endpoints and then design and build a complete system to meet these needs.
\item {\bf OTA update system.} Enabling a truly scalable system requires a wireless updates, however also introduces the challenges of how to communicate these robustly to each device in the network while minimizing downtime. We address this by using a dedicated LoRa backbone subsystem complete with a MAC protocol and compression.
\item {\bf Power management.} Fully optimizing power consumption requires shutting down parts of the system when not in use. This presents a design trade off however between the complexity of being able to toggle each power source on and off and the cost of additional circuitry to do so. We address this challenge by grouping components into a small number of controllable power domains.
\end{itemize}
Our final prototype of \name measures $3\times5~cm$, and costs \$xx to produce at volumes of xxx or more, and we perform detailed measurements of power consumption showing it consumes only XX~uW in its lowest power sleep mode. Additionally, we measure the performance of our over-the-air update system for and show it requires only xx~ms of network downtime at ranges as far as xx~m.
We further evaluate the performance of our \name by presenting case studies of two common protocols: LoRa and Bluetooth. We show that we can implement a LoRa modulator and demodulator using only xx\% of the FPGA resources and can achieve a sensitvity of -xxx~dBm which is the same as the SXxx LoRa chipset. We then evaluate the power consumption of generating LoRa transmissions versus output power and show it is as low as xxx for 14~dBm output. Additionally we demonstrate a MAC protocol on our MCU which is compatible with the existing Things Network and show \name can meet the timing requirements of xxx.
Similarly, we demonstrate 2.4 GHz transmissions by generating BLE advertisement packets. We implement the full baseband packet generation on the FPGA using only xx\% of its resources with a power consumption of xx~mW when transmitting at 0~dBm. We demonstrate a delay between sending packets on different channels of only xx and show our implementation achieves a sensitivity of xx which is comparable to the commercial BLE chips.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/3_platform_power/platform_pow.eps}
\vskip -0.1in
\caption{\footnotesize{Radio power consumption for each platform. We normalized TX power consumption for output power of 14~dBm.\newline
**GalioT does not support transmitter mode.}}
\label{fig:platform_power}
\vskip -0.2in
\end{figure}
We summarize our contributions below:
\begin{itemize}
\item We develop the first SDR platform tailored for the needs of an IoT testbed. Our platform is low cost, capable of implementing numerous IoT protocols both in 900~MHz and 2.4~GHz ISM bands, has an computation resources to do on-board processing, and highly granular power management for duty-cycled operation.
\item We develop a complete over the air update system its codes can be updated through a remote access point.
\item We extensively characterize and evaluate our platform. To show the flexibility of \name, including case studies of two IoT protocols at 900~MHz and 2.4~GHz. We provide detailed measurements of the required FPGA resources and power needed for each. We also characterize our over-the-air update system for different programming data and measure ranges of an access point and platform and update times are measured.
\end{itemize}
\section{Introduction}
Recent years have seen development of numerous wireless protocols for Internet of Things (IoT) devices. In addition to longtime standards such as Bluetooth and Zigbee, a number of new protocols including LoRa, Sigfox, NB-IoT and LTE-M have been developed that achieve long ranges of more than a few kilometers. Due to the lack of a de-facto standard, this space remains an active area of research for both industry and academia. The rapid advances in this space however present practical challenges for researchers: each of these protocols requires a dedicated radio chipset to evaluate, and these proprietary solutions often leave little room for protocol modification. The academic community is therefore {severely handicapped} by the lack of a \textit{flexible} platform, as even a complex multi-radio prototype cannot adapt to evaluate new protocols or even customize existing solutions. The current ecosystem therefore discourages researchers from investigating the important questions that arise when scaling up IoT networks, and more importantly taking a systematic approach to developing new protocols from the ground up.
\begin{figure}[t]
\vskip -0.3in
\centering
\includegraphics[width=0.865\linewidth]{./figs/tinysdr_arrows.eps}
\vskip -0.1 in
\caption{\footnotesize{{\bf \Name\ Hardware Platform.} It has two antenna ports for running IoT PHY and MAC protocols at 2.4~GHz and 900~MHz. {This image is the actual size of the platform on printed paper.}}}
\label{fig:sdr_hardware}
\vskip -0.2 in
\end{figure}
Ideally, we would like a large scale IoT network testbed with the flexibility to run \textit{any} IoT protocol at the PHY and MAC layers. Further, since many of these IoT testbeds can span hundreds of endpoints across a large campus or even a city, we need the ability to push changes to the PHY and MAC layers, using simple over-the-air software updates. This would allow for performance comparisons on a single testbed to investigate the trade-offs between existing standards as well as showcase the advantages of an entirely new custom protocol. Moreover, to make such a system representative of real-world deployments, individual network nodes should model the constraints of IoT endpoints. Specifically, these devices should have appropriate power controls and options to duty cycle transmissions, have an ultra-low power sleep mode and also have interfaces to connect sensors. Finally, the ability to run these endpoints on batteries would also allow for flexibility of deployment in spaces without dedicated power access, or even in mobile scenarios.
\begin{table*}[t!]
\centering
\footnotesize{
\begin{tabular}{|c|C{1.3cm}|c|c|c|C{1.1cm}|C{0.7cm}|c|C{1.1cm}|}
\hline
\rowcolor{lightgray}
{\bf Platform} & {\bf Sleep Power} & {\bf Standalone} & {\bf OTA} & {\bf Cost} & {\bf Max BW (MHz)} & {\bf ADC (bits)} & {\bf Frequency Spectrum (MHz)}& {\bf Size (cm)}\\
{USRP E310 \cite{usrpe310, ad9361}} & 2820~mW & \cmark & \xmark & \$3000 & 30.72 & 12 & 70$\sim$6000 & 6.8$\times$13.3\\
\hline
{USRP B200mini \cite{usrpb200mini, ad9364}} & N/A & \xmark & \xmark & \$733 & 30.72 & 12 & 70$\sim$6000 & 5$\times$8.3\\
\hline
{bladeRF 2.0 \cite{bladerf2, ad9361}} & 717~mW & \cmark & \xmark & \$720 & 30.72 & 12 & 47$\sim$6000 & 6.3$\times$12.7\\
\hline
{LimeSDR Mini \cite{limesdr, limesdrmini, lms7002m}} & N/A & \xmark & \xmark & \$159 & 30.72 & 12 & 10$\sim$3500 & 3.1$\times$6.9\\
\hline
{PlutoSDR \cite{ad9363}} & N/A & \xmark & \xmark & \$149 & 20 & 12 & 325$\sim$3800 & 7.9$\times$11.7\\
\hline
{$\mu$SDR \cite{max2831, ad9228, max5189}} & 320~mW & \cmark & \xmark & \$150 & 40 & 8 & 2400$\sim$2500 & 7$\times$14.5\\
\hline
{GalioT \cite{galiot, rtl2832}} & 350~mW & \cmark & \xmark & \$60 & 14.4 & 8 & 0.5$\sim$1766 & 2.5$\times$7\\
\hline
{\bf{\Name}} & {\bf 0.03~mW} & {\bf \cmark} & {\bf \cmark} & {\bf \$55} & {\bf 4} & {\bf 13} & {\bf 389.5$\sim$510, 779$\sim$1020, 2400$\sim$2483} & {\bf 3$\times$5}\\
\hline
\end{tabular}
\vskip -0.1in
\caption{\footnotesize{\bf Comparison Between Different SDR Platforms.} Costs are based on sale prices for commercial products without a public bill of materials (BOM) and published BOM prices for research prototypes. OTA refers to over-the-air programming capabilities.}
\label{tab:platforms}
\vskip -0.2in
}
\end{table*}
Realizing this vision however is challenging with existing software defined radio (SDR) platforms. Specifically, we require an SDR for the flexibility of implementing different PHY protocols; but there is currently no SDR platform that meets the requirements of IoT endpoints (see Table~\ref{tab:platforms}). Existing SDR systems consume large amounts of power for transmitting data, do not support ultra-low power sleep modes, require wired infrastructure and often a dedicated computer and furthermore, are expensive. More importantly, none of the existing SDR platforms support over-the-air programming to update PHY or MAC protocols. Finally, IoT devices prioritize power consumption and communication range and hence use limited radio bandwidth --- LoRa, Sigfox, NB-IoT, LTE-M, Bluetooth and ZigBee use only 500~kHz, 200~Hz, 180~kHz, 1.4~MHz, 2~MHz and 2~MHz respectively. In contrast, existing SDR platforms focus on achieving high performance in terms of bandwidth because {\it they are tailored to the needs of gateway devices and not for IoT endpoint devices.}
Driven by a need for such a platform in our own research, we design \name as shown in Fig.~\ref{fig:sdr_hardware}, the first SDR platform tailored to the needs of IoT endpoints. TinySDR provides an entirely standalone solution that incorporates a radio front-end, FPGA and microcontroller for custom processing, over-the-air FPGA and microcontroller programming capabilities, a microSD card interface for storage, ultra-low power sleep modes and highly granular power management options to enable battery-powered operation. It is capable of transmitting and receiving in both the 900~MHz and 2.4~GHz ISM bands, supports 4~MHz of bandwidth which is sufficient for most IoT protocols including Bluetooth, Zigbee, LoRa, Sigfox, NB-IoT and LTE-M, and can achieve the high sensitivities of commercial solutions such as LoRa chip~\cite{sx1276}. Additionally it includes multiple analog and digital I/O options for connecting sensors.
Designing such a SDR platform required addressing multiple systems, architecture, power and engineering challenges:
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Low-power hardware architecture.}
Achieving a small form-factor, low-power SDR requires a minimalist design approach that can satisfy the real-time needs of IoT protocols and ensure flexibility at the PHY and MAC layers. To do this, we exploit recent advances in small, low-power microcontrollers, FPGAs and flash memory to pick the right components for our platform (see~\xref{sec:hardware}). We use a low-power FPGA to run the PHY layer while the microcontroller runs the MAC protocols as well as handles the I/O operations between the FPGA, radio, memory and sensor interfaces (see~\xref{sec:interface}).
\item {\bf Efficient power management.} Achieving highly granular power management needed for battery-powered operation and enabling ultra-low power sleep modes requires shutting down parts of SDR when not in use. This is important for IoT endpoints that {perform} duty-cycle operations and require an ultra-low power sleep mode to achieve a long battery life. This presents a design tradeoff between the complexity of toggling the power of each hardware component ON and OFF, and the cost of additional circuitry to do so. We address this challenge in~\xref{sec:powermanagement} and achieve sleep power as low as 30~$\mu$W.
\item {\bf Over-the-air SDR programming.} Enabling a truly scalable system requires the ability to update the PHY and MAC layers on the platform, over-the-air, in a testbed deployment. This however also introduces the challenge of over-the-air FPGA and microcontroller programming as well as communicating these updates robustly to each device in the network while minimizing power consumption and network utilization. We use a dedicated wireless backbone subsystem complete with a MAC protocol and its own flash memory to program both the microcontroller and FPGA. Additionally we leverage compression and low-power decompression algorithms to minimize network downtime during the updates (see~\xref{sec:ota})
\end{itemize}
Fig.~\ref{fig:platform_power} shows the power consumption of the radio module in \name\ compared to existing SDR platforms. We evaluate \name's performance by presenting case studies of two common protocols: LoRa and BLE beacons, and also evaluate \name\ in a campus-testbed of 20 devices.
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item LoRa modulation and demodulation use 4\% and 11\% of the FPGA resources respectively and achieve a sensitivity of -126~dBm for 3.12~kbps, which is similar to an SX1276~\cite{sx1276} LoRa chip with the same configuration. Further, the FPGA supports real-time modulation and demodulation of all LoRa spreading factors from 6 to 12. A LoRa MAC implementation on our MCU is compatible with the {\it The Things Network}.
\item \Name supports 2.4 GHz BLE beacon transmissions. The full baseband packet generation on the FPGA uses 3\% of its resources. The platform can perform frequency hopping with a delay of 220~us and achieves a sensitivity of -94~dBm which is comparable to the commercial BLE chipsets~\cite{cc2650}.
\end{itemize}
{Finally, we present a case study of how the unique capabilities of \name could be used to answer new research questions. Recent work has explored techniques to enable concurrent transmissions in LoRa networks~\cite{lorasigcomm17, netscatter}; however these solutions were prototyped on USRPs and it is unclear if IoT endpoints can decode concurrent transmissions in real-time within their power and resource constraints. We implement a custom decoder on \name to demonstrate for the first time that IoT endpoints \textit{can} receive concurrent transmissions.}
{\bf Contributions. }To summarize, we design the first SDR platform tailored to the needs of IoT endpoint devices. By making careful design and architectural choices, our platform achieves low power, supports IoT protocols at both 900~MHz and 2.4~GHz and has computation resources to do on-board processing. We present a highly granular power management scheme that enables duty-cycled operation and 10,000x lower power sleep modes. {We also develop the first over-the-air SDR programming capability to support PHY and MAC updates in a wireless testbed.} We characterize and evaluate our platform with case studies of LoRa and BLE beacons. Finally, we present a research exploration of concurrently receiving multiple LoRa transmissions on our SDR platform.
{\bf Platform availability.} We will release the \name platform, along with its hardware schematics and software, for others to use and contribute to, before the conference.
\section*{Availability}
{
\bibliographystyle{abbrv}
\section{Platform}
To motivate the need for an SDR platform tailored to prototyping IoT endpoints, we begin by outlining the unique challenges such as the power, size, and cost requirements of these devices as well as what capabilities this platform needs in terms of computation, frequency and bandwidth. Next, we present a detailed analysis of the capabilities of existing SDR platforms and show that there is currently no solution to support the growing research interests in this area. We will then outline the capabilities of our device.
\subsection{Requirements for IoT endpoints}
We outline the unique needs of IoT endpoints to understand what a useful prototyping platform requires.
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Flexibility.} Unlike technologies such as Wi-Fi, there is no de-facto communication standard for IoT; in contrast, IoT protocols themselves are in active development which requires a platform that is flexible enough to implement, modify, and improve them. A common SDR platform that gives access to the raw signals allows for physical layer work on the protocols. Additionally, an SDR implementation of proprietary protocols such as LoRa would allow researchers to modify and improve the standard as well as enable higher layer work such as experimenting with MAC protocols.
\item{\bf Frequency and Bandwidth}
A radio for IoT must operate at least at both 900~MHz and 2.4~GHz as IoT devices typically operate in these ISM bands. A single SDR platform that can operate at both frequencies would allow for side-by-side comparison of different protocols with a common radio front-end. Additionally, IoT sensor nodes produce relatively small amounts of data. Therefore unlike Wi-Fi which requires 20~MHz of bandwidth, IoT protocols such as LoRa and Sigfox are designed to use less than 500~kHz bandwidth and require lower sampling rates of maximum 1~MHz.
\item {\bf Scalibility.}
IoT nodes are often designed for large scale deployments on farms, in warehouses, across cities, and in many other environments without reliable power infrastructure. Therefore the ability to operate the platform on a battery is imperative and adds the additional constraint of low power consumption. A useful research platform should be able to operate for more than a year on a single charge when it is operating at 0.1 percent duty cycle with a 1000~mAh Lithium-Ion battery. Moreover, large batteries are expensive which raises the additional constraint of cost; an SDR platform should minimize cost with a target price of approximately \$xx per node in order to make larger scale deployments viable. Considering IoT nodes are operated as standalone devices, an SDR platform must also include all of the necessary computation and memory within this low power, low cost package. In addition, an IoT endpoint suitable for an IoT testbed requires wireless backbone support so it can perform PHY and MAC layer update remotely.
\end{itemize}
\subsection{Existing SDR Platforms}
In order to understand why existing SDR platforms do not meet the above requirements we analyze their capabilities.
\vskip 0.05in\noindent{\bf Commercial SDR platforms.} Ettus Research (National Instruments) is one of the world's leaders in SDR platform technologies. They provide a variety of platforms such as USRP N210~\cite{}, Embedded USRP~\cite{} and USRP networked series~\cite{}. Many USRP models use peripheral buses such as USB or Ethernet to connect to a computer for controlling its function and recording data. The Embedded USRP series integrates a Dual Core ARM A9 processor and runs embedded Linux to allow for standalone applications but is priced at \$3000 per unit. Lime microsystems provides SDR platforms such as LimeSDR~\cite{} and LimeSDR Mini~\cite{}, which have open source design and use their custom transceiver IC. Nuand is another commercial SDR provider which recently released the bladeRF 2.0 micro SDR platform to provide a lower cost alternative to USRPs. More recently, Analog Devices has released their education focused PlutoSDR~\cite{} platform which uses AD9363~\cite{} radio front-end.
\vskip 0.05in\noindent{\bf Research SDR platforms.}
Over the past decade there has been multiple works on designing SDR platforms. Tick~\cite{} achieves good programmability, high throughput and low latency by software-hardware co-design solutions. However, it uses a high power radio on the order of several watts~\cite{AD9371} and the whole design costs more than \$3000. SORA~\cite{sora} is a research platform that uses PCI-Express bus to provide sub 1~$\mu$s latency. However, it requires host PC for backbone processing and hence it ends up with a large form-factor. $\mu$SDR solves the latency challenge by using an advanced microcontroller bus architecture (AMBA) along with a Cortex-M3 ARM processor~\cite{musdr} and flash-based FPGA to process data. However, it is power-hungry for an IoT network deployment and relatively expensive. SODA~\cite{} proposed a fully programmable SDR architecture IC simulation which meets the requirements of high-end wireless protocols and consist of four SIMD cores. However, it has a slightly customized VLIW+SIMD ISA which needs users to hand-tune their program. KUAR~\cite{} is a research SDR platform that has a large form-factor design and requires PCI-Express connection to a host an embedded PC-like environment to operate. GalioI~\cite{galiot} presents an SDR platform including a RTL2832U radio~\cite{rtl2832} connected to a Raspberry pi computer. This platform can only receive signal and the receiver RF front-end is power-hungry as shown in Fig.~\ref{fig:platform_power}.
\vskip 0.05in\noindent{\bf Scalibility comparison.}
These commercial SDR platforms are designed to operate over frequencies up to 6~GHz and support bandwidths of 60~MHz, which requires high sampling rate ADCs and DACs and more importantly high digital baseband throughput to the processing unit and a computationally extensive processing unit. Moreover, the radio front-ends must also maintain good performance over this large frequency spectrum and bandwidth. Achieving good wide-band performance makes the fundamental trade-off of burning more power, making these platforms consume a few watts. Fig.~\ref{fig:platform_power} shows the power consumption of only the radio module in existing SDR platforms. Additionally, Table~\ref{tab:platforms} shows a comparison of other factors affecting deployment including XXX, XXX, and cost. This table shows that no existing solution meets all of the deployability concerns of IoT endpoints.
\subsection{Our Capabilites}
Our platform achieves the requirements of IoT endpoints by providing all of the capabilities shown in Table~\ref{xxx}. Additionally it consumes only xx~mW of power.
We achieve this low power consumption using the following key insight: IoT protocols only require a few MHz of bandwidth and do not require microsecond latency therefore leaving an over-designed gap between the current SDR designs and what is required for IoT communication. By designing specifically for IoT solutions \name achieves low-power and last XXX years using only a XXX battery.
\section{SDR Requirements for IoT devices}
To motivate the need for XXXX XXXX XXXXXXXX XXXXX XXXXX XXXXX XXXXxx xx We outline the unique needs of IoT endpoints to understand what a useful prototyping platform requires.
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Flexibility.} Unlike technologies such as Wi-Fi, there is no de-facto communication standard for IoT; in contrast, IoT protocols themselves are in active development which requires a platform that is flexible enough to implement, modify, and improve them. A common SDR platform that gives access to the raw signals allows for physical layer work on the protocols. Additionally, an SDR implementation of proprietary protocols such as LoRa would allow researchers to modify and improve the standard as well as enable higher layer work such as experimenting with MAC protocols.
\item{\bf Frequency and Bandwidth}
A radio for IoT must operate at least at both 900~MHz and 2.4~GHz as IoT devices typically operate in these ISM bands. A single SDR platform that can operate at both frequencies would allow for side-by-side comparison of different protocols with a common radio front-end. Additionally, IoT sensor nodes produce relatively small amounts of data. Therefore unlike Wi-Fi which requires 20~MHz of bandwidth, IoT protocols such as LoRa and Sigfox are designed to use less than 500~kHz bandwidth and require lower sampling rates of maximum 1~MHz.
\item {\bf Scalibility.}
IoT nodes are often designed for large scale deployments on farms, in warehouses, across cities, and in many other environments without reliable power infrastructure. Therefore the ability to operate the platform on a battery is imperative and adds the additional constraint of low power consumption. A useful research platform should be able to operate for more than a year on a single charge when it is operating at 0.1 percent duty cycle with a 1000~mAh Lithium-Ion battery. Moreover, large batteries are expensive which raises the additional constraint of cost; an SDR platform should minimize cost with a target price of approximately \$xx per node in order to make larger scale deployments viable. Considering IoT nodes are operated as standalone devices, an SDR platform must also include all of the necessary computation and memory within this low power, low cost package. In addition, an IoT endpoint suitable for an IoT testbed requires wireless backbone support so it can perform PHY and MAC layer update remotely.
\end{itemize}
\subsection{Existing SDR Platforms}
In order to understand why existing SDR platforms do not meet the above requirements we analyze their capabilities.
\vskip 0.05in\noindent{\bf Commercial SDR platforms.} Ettus Research (National Instruments) is one of the world's leaders in SDR platform technologies. They provide a variety of platforms such as USRP N210~\cite{}, Embedded USRP~\cite{} and USRP networked series~\cite{}. Many USRP models use peripheral buses such as USB or Ethernet to connect to a computer for controlling its function and recording data. The Embedded USRP series integrates a Dual Core ARM A9 processor and runs embedded Linux to allow for standalone applications but is priced at \$3000 per unit. Lime microsystems provides SDR platforms such as LimeSDR~\cite{} and LimeSDR Mini~\cite{}, which have open source design and use their custom transceiver IC. Nuand is another commercial SDR provider which recently released the bladeRF 2.0 micro SDR platform to provide a lower cost alternative to USRPs. More recently, Analog Devices has released their education focused PlutoSDR~\cite{} platform which uses AD9363~\cite{} radio front-end.
\vskip 0.05in\noindent{\bf Research SDR platforms.}
Over the past decade there has been multiple works on designing SDR platforms. Tick~\cite{} achieves good programmability, high throughput and low latency by software-hardware co-design solutions. However, it uses a high power radio on the order of several watts~\cite{AD9371} and the whole design costs more than \$3000. SORA~\cite{sora} is a research platform that uses PCI-Express bus to provide sub 1~$\mu$s latency. However, it requires host PC for backbone processing and hence it ends up with a large form-factor. $\mu$SDR solves the latency challenge by using an advanced microcontroller bus architecture (AMBA) along with a Cortex-M3 ARM processor~\cite{musdr} and flash-based FPGA to process data. However, it is power-hungry for an IoT network deployment and relatively expensive. SODA~\cite{} proposed a fully programmable SDR architecture IC simulation which meets the requirements of high-end wireless protocols and consist of four SIMD cores. However, it has a slightly customized VLIW+SIMD ISA which needs users to hand-tune their program. KUAR~\cite{} is a research SDR platform that has a large form-factor design and requires PCI-Express connection to a host an embedded PC-like environment to operate. GalioI~\cite{galiot} presents an SDR platform including a RTL2832U radio~\cite{rtl2832} connected to a Raspberry pi computer. This platform can only receive signal and the receiver RF front-end is power-hungry as shown in Fig.~\ref{fig:platform_power}.
\vskip 0.05in\noindent{\bf Scalibility comparison.}
These commercial SDR platforms are designed to operate over frequencies up to 6~GHz and support bandwidths of 60~MHz, which requires high sampling rate ADCs and DACs and more importantly high digital baseband throughput to the processing unit and a computationally extensive processing unit. Moreover, the radio front-ends must also maintain good performance over this large frequency spectrum and bandwidth. Achieving good wide-band performance makes the fundamental trade-off of burning more power, making these platforms consume a few watts. Fig.~\ref{fig:platform_power} shows the power consumption of only the radio module in existing SDR platforms. Additionally, Table~\ref{tab:platforms} shows a comparison of other factors affecting deployment including XXX, XXX, and cost. This table shows that no existing solution meets all of the deployability concerns of IoT endpoints.
\section{SDR Requirements for IoT Nodes}
To motivate the need for \name and inform our design decisions we begin by identifying the key requirements for an IoT endpoint platform that can be used to build the testbed vision described above. These include operation in the 900 MHz and 2.4~GHz bands, low power operation which requires the ability to transition to an ultra-low power sleep mode, standalone operation which requires an onboard control unit to duty cycle the radio, an OTA programming interface to manage large scale deployments, and low cost per node. While there are a number of commercially available SDRs~\cite{usrp} on the market and numerous SDR research prototypes~\cite{dinesh,sora,swarun,warp}, all of them have been designed to implement gateway devices. We analyze the shortcomings of each of these platforms in the context of IoT endpoint requirements below and show that \name is the only one that can meet these needs.
\subsection{Low power operation and sleep mode}
How do SDR platforms compare in terms of power? Fig.~\ref{fig:platform_power} compares the power consumption of the radio module alone in existing SDR platforms considering each one has different peripherals. Interestingly we find that most SDR platforms consume 200-300~mW in receive mode, but almost 10x when transmitting. While this may be acceptable for a gateway typical IoT endpoints do the opposite and are required to transmit data like sensor information. Moreover, real IoT nodes spend a very short time transmit before quickly transitioning to ultra-low power sleep modes. Although IoT radios often consume milliwatts of power, and the key to achieving long battery lifetimes is exploiting the microwatt power sleep modes. Table~\ref{tab:platforms} shows that none of the other platforms can benefit from duty cycling as they consume more power in sleep mode than \name does when transmitting. In contrast \name's microwatt power consumption in sleep mode enables the benefits of duty cycling and allows for dramatic power savings.
\subsection{Standalone operation and cost}
We first observe that half of these platforms do not allow for standalone operation, meaning they cannot be used in a testbed deployment without an external computer. Among the ones that do, Embedded USRP and bladeRF cost \$700 or more per unit making large scale deployments exorbitantly expensive. GalioI~\cite{galiot} presents an SDR platform using the low cost RTL2832U radio~\cite{rtl2832} connected to a Raspberry Pi computer which allows for standalone operation, however it does not support the 2.4~GHz band. Moreover, this platform is \textit{receiver only} and cannot be used to prototype a typical IoT node that transmits data. This leaves \name as the only low cost platform that can transmit and receive in both ISM bands without being tethered to an external computer.
\subsection{OTA programming}
As shown in Table~\ref{tab:platforms}, all of the existing SDR platforms rely on wired interfaces for programming. This means that even if one of these systems were connected to a battery, running an experiment would require either tethering each one to a wired network or individually programming each one. An OTA programming system is crucial to realizing the goal of a large scale wide area testbed as it leaves researchers to decide between limiting themselves to deployment scenarios with wired infrastructure which are not representative of most real IoT use cases or traveling over \textit{kilometer} distances to update individual nodes for \textit{each} experiment or minor modification to a protocol which is unmanageable.
\section{SDR Requirements for IoT Nodes}
To motivate the need for \name and inform our design decisions, we begin by identifying the key requirements for an IoT endpoint. These include 1) operation in the 900 MHz and 2.4~GHz bands, 2) low power operation which requires the ability to transition to ultra-low power sleep mode, 3) standalone operation which requires an on-board control unit to duty cycle the radio, 4) over-the-air programming capabilities for large scale IoT testbeds, 5) low cost per node, and 6) at least 2~MHz bandwidth to support IoT protocols including LoRa, SIGFOX, LTE-M, NB-IoT, ZigBee and Bluetooth. While there are a number of commercially available SDRs such as the USRP, BladeRF, PlutoSDR, and LimeSDR~\cite{usrpe310,bladerf2,usrpX300,limesdrmini,plutoSDR} on the market and SDR research prototypes such as WARP, Argos, SORA, SODA, KUAR, Tick, $\mu$SDR, OpenMili, and GalioT~\cite{amiri2007warp,khattab2008warp,anand2010warplab,tan2011sora,guddeti2019sweepsense,galiot,musdr,kuo2012compact,dutta2010putting,zhang2016openmili,soda,wu2017tick, minden2007kuar,sutton2010iris,ng2010airblue,argo1,argo2}, all of them are designed as {\it gateway devices} and do not satisfy many of the above constraints. Here, we analyze the shortcomings of these platforms in the context of these requirements.
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Low power operation and sleep mode.} Fig.~\ref{fig:platform_power} compares the power consumption of the radio module {\it alone} in existing SDR platforms, since each one has different peripherals. We find that most SDR platforms consume 200-300~mW in receive mode, but a lot more power when transmitting. While this may be acceptable for a gateway devices that are more often receiving, typical IoT endpoints do the opposite and are required to transmit data like sensor information. Moreover, real IoT nodes spend a very short time transmitting before transitioning to ultra-low power sleep modes. Although IoT radios often consume tens to hundreds of milliwatts of power, the key to achieving long battery lifetimes is exploiting their microwatt power sleep modes. Table~\ref{tab:platforms} shows that none of the other platforms can benefit from duty cycling as they consume more power in sleep mode than \name does when transmitting; \name's microwatt power consumption in sleep mode enables dramatic power savings with duty cycling.
\item{\bf Standalone operation and cost.} We observe that some of these platforms do not allow for standalone operation, i.e., they cannot be used in a testbed deployment without an external computer. Among the ones that do, the Embedded USRP and bladeRF cost \$700 or more per unit making large scale deployments expensive. $\mu$SDR allows for standalone operation but only operates at 2.4~GHz and cannot support protocols like LoRa. GalioT~\cite{galiot} uses the low cost RTL2832U radio~\cite{rtl2832} connected to a Raspberry Pi computer which allows for standalone operation, however it does not support 2.4~GHz band. Moreover, this platform is \textit{receiver only} and cannot be used to prototype a typical IoT node that transmits data.
\begin{figure}[t]
\centering
\includegraphics[width=1.03\linewidth]{figs/3_platform_power/platform_pow.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf Radio Module Power Consumption for Each Platform.} The TX output power of each radio module is shown on top of it.}}
\label{fig:platform_power}
\vskip -0.2in
\end{figure}
\item{\bf Over-the-air (OTA) programming.} As shown in Table~\ref{tab:platforms}, all existing SDR platforms rely on wired interfaces for programming. This means that even if one of these systems were connected to a battery, running an experiment would require either tethering each one to a wired network or individually programming them. An OTA programming system is crucial to realizing the goal of a large scale wide area testbed as without it, researchers have to decide between limiting themselves to deployment scenarios with wired infrastructure that are not representative of real IoT use cases or traveling over \textit{kilometer} distances to update individual nodes for \textit{each} minor protocol modification, which would be unmanageable at scale.
\squishend
\section{Implementing Protocols}
\subsection{LoRa}
To demonstrate the capability of \name at sub-GHz frequencies, we implement a LoRa modulator and demodulator. We choose LoRa as it is gaining popularity for IoT solutions due to its long range capabilities described above. Considering LoRa is a proprietary standard, we begin by describing the basics of its modulation and packet structure followed by the implementation details of our modulator and demodulator.
\subsubsection{LoRa Protocol Primer}
\vskip -0.1 in
LoRa achieves long ranges by using Chirp Spread Scpectrum (CSS) modulation. In CSS, data is modulated using linearly increasing frequency signals or upchirp symbol. Each upchirp symbol has two main features; Spreading Factor (SF) and Bandwidth (BW). Spreading factor determines determines the number of bits that each upchirp symbol represents~\cite{netscatter,lorabackscatter} and $\frac{1}{BW}$ shows the length of each upchirp symbol in time. The SF and BW settings therefore allows users to trade data rate for range. LoRa uses SF values from 6 to 12. The BW determines the total frequency spectrum that each upchirp symbol utilizes which results in a data rate of $\frac{2^{SF}}{BW}\times CR$. The final CR term represents any coding of the data in the payload itself. Each upchirp symbol covers the whole BW frequency. The starting point of the symbol in frequency domain which is the cyclic shift of the upchirp symbol determines the chirp symbol value~\cite{lora_basics}.
Fig.~\ref{fig:lora_packet} shows the LoRa packet structure which begins with a preamble which consists of 10 zero symbols (upchirps with zero cyclic-shift). This is followed by the Sync field with two upchirp data symbols that can indicate information about the network or configuration. Next, a sequence of 2.25 downchirp symbols (chirp symbol with linearly decreasing frequency) indicate the beginning of the payload. The payload then consists of a sequence of upchirp symbols which encode a header, payload and CRC.
\subsubsection{LoRa modulator}
\vskip -0.1 in
Fig.~\ref{fig:lora_tx_block} shows the block diagram of our LoRa modulator. Because CSS is not a pre-programmed modulation option on the AT86RF215, we use our FPGA to implement a LoRa modulator in Verilog and stream data to the radio in I/Q mode. The modulator begins with the {Packet Generator} module reading the data symbols to be sent either from FPGA memory for transmitting fixed packets or from the MCU, as well as LoRa configuration parameters such as SF, and BW. This module then determines all of the necessary parameters for generating a chirp such as its length, chirp slope, and step resolution of the linear frequency sweep. Next, the {Packet Generator} sends these parameters along with the symbol values to the {Chirp Generator} module, which generates the complex values for each chirp symbol in the packet. This block uses a phase accumulator along with two lookup tables for $Sin$ and $Cosin$ functions to generate I/Q samples for a linear chirp as demonstrated in~\cite{lorabackscatter}. We then feed the complex samples into {I/Q Serializer} explained above to stream them over the LVDS interface to the I/Q radio.
\subsubsection{LoRa demodulator}
Fig.~\ref{fig:lora_rx_block} shows the block diagram of our LoRa demodulator. This demodulator begins by reading data from the I/Q radio into the {I/Q Deserializer} module on the FPGA which converts the serial I/Q stream to parallel I/Q for further signal processing. Next, we run these data through an XXX tap XXX low-pass FIR filter to xxxxxx. We store the filtered samples in a buffer implemented using the FPGAs onboard memory. To decode the data, we use the {Chirp Generator} mpodule from the {LoRa modulator} described above to generate a baseline upchirp/downchirp symbol, and then we multiply that with the received chirp symbol using our {Chirp Multiplier} unit which gives the coding gain of CSS. The output of the multiplication then goes to an XXX-point FFT block implemented using a standard IP core from Lattice. Finally the {Symbol Detector} scans the output of the FFT for peaks and records the frequency of the peak to determine the symbol value. To detect chirp type (upchirp/downchirp), we multiply each chirp symbol with both upchirp and downchirp and then compare the amplitude of their FFT peak. The higher peak in FFT result shows higher correlation which we use to detect the chirp type.
\subsubsection{LoRa MAC Layer}
To demonstrate that our LoRa implementation can be used in a real network, we need a MAC protocol to manage transmissions between devices in addition to the physical layer described above. We show that \name is compatible with the existing LoRa based {The Things Network} (TTN)~\cite{thingsnetwork} by implementing its MAC protocol. We adopt the design from TTN's open source Arduino libraries~\cite{ttnarduino} and implement it on \name's MCU. TTN uses two methods for device association; Over-the-air activation (OTAA) and activation by personalization (ABP). OTAA is a secure way to connect with TTN in which nodes perform a join-procedure with the network. During this process a dynamic device address is assigned to the node and security keys are negotiated. ABP allows for a simplified approach in which the device address as well as the security keys are hard coded on the device allowing it to skip the join procedure. Our implementation supports both OTAA and ABP methods.
\subsection{Bluetooth Low Energy}
In addition to the sub-GHz radio our AT86RF215 includes a 2.4~GHz radio as well. To demonstrate \name's 2.4~GHz capabilities we implement Bluetooth beacons which are commonly used for IoT devices. We begin with a brief primer on the BLE physical layer followed by details about our implementation.
\subsubsection{BLE Advertising Primer}
\vskip -0.1 in
Bluetooth Low Energy includes a variety of different modes and packet types, we specifically implement non-connectable BLE advertisements (ADV\_NON\_CONN\_IND) which are broadcast packets used for beacons. These packets allow a low power device to simply boardcast its data to any listening receiver within range without the power overhead of exchanging packets needed to negotiate a connection. These packets themselves have a bit rate of 1~Mbps in Bluetooth 4.0 or up to 2~Mbps in Bluetooth 5.0 and are generated using GFSK with a modulation index of 0.45-0.55. The GFSK modulation is simply binary frequency shift keying (BFSK) with the addition of a Gaussian filter to the square wave pulses to be transmitted which reduces the spectral width. While BLE divides the 2.4~GHz band into a total of 40 channels each spaced 2~MHz apart, only 3 of these throughout the band are set aside for advertisements. A typical BLE advertisement is transmitted on each of the three advertising channels one after another to mitigate the chance of collisions as this protocol trades off a more sophisticated MAC protocol for power savings.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{./figs/block_diagram/ble_tx_block.eps}
\vskip -0.15 in
\caption{BLE implementation on FPGA.}
\label{fig:ble_tx_block}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{./figs/block_diagram/ble_packet.eps}
\vskip -0.15 in
\caption{\footnotesize{BLE packet structure.}}
\label{fig:ble_packet}
\vskip -0.2 in
\end{figure}
\subsubsection{Generating a BLE packet}
The first step to transmitting a BLE advertisement is to generate the bitstream for a valid BLE packet. The structure of an advertising packet is shown in Fig.~\ref{fig:ble_packet}. Advertisements range from 6-37 octets, beginning with a fixed preamble of 0xAA followed by the access address field which is also set to a fixed sequence 0x8E89BED6 which indicates to a receiver the packet is a non-connectable advertisement. These fields are followed by the packet data unit (PDU) which begins with a 2 byte length field indicating the size of the PDU followed by the data segment which includes an advertisement address which can indicate specific device or device type followed by the packet data such as a 128 bit UUID. In addition to this packet structure, BLE requires a 3 byte CRC as well as whitening of the PDU to create a valid packet which we explain below.
\noindent {\bf CRC Generation.} The Bluetooth CRC is a 3 byte sequence generated using the 24-bit linear feedback shift register (LFSR) with the polynomial $x^{24} + x^{10} + x^{9} + x^{6} + x^{4} + x^{3} + x + 1$. The LFSR is set to a starting state of 0x555555 and each byte of the PDU beginning with the length field is input to the LFSR LSB first. The final state of the LFSR after all of the data bits pass through then becomes the CRC. We implement this in xxx on the xxx.
\noindent {\bf Data Whitening.} Data whitening is performed to eliminate long strings of zeros or ones within a packet. This is also done using a 7-bit LFSR with polynomial $x^7 + x^4 + 1$. The LFSR is initialized with the lower 7 bits of the channel number the packet will be transmitted on, and each byte is input LSB first. We implement this in xxx on the xxx.
\noindent {\bf Transmitting a packet.} Now that we have the bitstream, we need to generate the I/Q samples to stream to the radio. To do this we begin by taking the bitstream and upsampling it to XX~MHz to generate a square wave corresponding to the segments of high and low frequencies for BLE's FSK modulation. Before actually modulating the data however, we first smooth the waveform using a Gaussian filter which eliminates the sharp transitions between bits and suppresses unwanted sidelobes. Finally, to generate the BLE waveform we use a modified version of our phase accumulator block described above that creates frequency modulated signals for LoRa. This block us the integrates the frequency to get the phase which we feed to a sine and cosine function to get the I and Q signals. These samples are then sent to the I/Q serializer and streamed to the radio.
\section{Protocol Case Studies}
To demonstrate the use of \name at 900~MHz and 2.4~GHz, we explore two case studies: LoRa and Bluetooth beacons.
\subsection{LoRa Protocol with \name}
We choose LoRa as it is gaining popularity for IoT solutions due to its long range capabilities. Since LoRa is a proprietary standard, we begin by describing the basics of its modulation and packet structure followed by the implementation details of our modulator, demodulator and MAC protocol.
{\bf LoRa Protocol Primer.}
LoRa achieves long ranges by using Chirp Spread Scpectrum (CSS) modulation. In CSS, data is modulated using linearly increasing frequency signals or upchirp symbol. Each upchirp symbol has two main features; Spreading Factor (SF) and Bandwidth (BW). SF determines the number of bits of each upchirp symbol~\cite{lorasigcomm17,netscatter,lorabackscatter} and BW is the difference between upper and lower frequency of the chirp which together with SF determines the length of each upchirp symbol. The SF and BW settings trade data rate for range. Data is modulated by $2^{SF}$ cyclic-shifts of an upchirp symbol. The starting point of the symbol in frequency domain which is the cyclic shift of the upchirp symbol determines the chirp symbol value~\cite{lora_basics}. LoRa uses SF values from 6 to 12 and BW values from 7.8125~KHz to 500~KHz which achieves PHY-layer rate of $\frac{BW}{2^{SF}}\times SF$.
Fig.~\ref{fig:lora_packet} shows the LoRa packet structure which begins with a preamble with 10 zero symbols (upchirps with zero cyclic-shift). This is followed by the Sync field with two upchirp symbols that indicate information about the network or configuration. Next, a sequence of 2.25 downchirp symbols (chirp symbol with linearly decreasing frequency) indicate the beginning of the payload. The payload then consists of a sequence of upchirp symbols which encode a header, payload and CRC.
{\bf LoRa Modulator.}
Fig.~\ref{fig:lora_tx_block} shows the block diagram of our LoRa modulator. Because CSS is not a pre-programmed modulation option on the AT86RF215, we use our FPGA to implement a LoRa modulator in Verilog and stream data to the radio in I/Q mode. The modulator begins with the {\it Packet Generator} module reading the data symbols to be sent either from FPGA memory for transmitting fixed packets or from the MCU, as well as LoRa configuration parameters such as SF, coding and BW. This module then determines all of the necessary parameters for generating a chirp such as its length, chirp slope, and step resolution of the linear frequency sweep. Next, the {\it Packet Generator} sends these parameters along with the symbol values to the {\it Chirp Generator} module, which generates the complex values for each chirp symbol in the packet. This block uses a phase accumulator along with two lookup tables for $Sin$ and $Cosin$ function to generate I/Q samples for a linear chirp similar to~\cite{lorabackscatter}. We then feed the complex samples into {I/Q Serializer} to stream them over the LVDS interface to the I/Q radio.
{\bf LoRa Demodulator. }
Fig.~\ref{fig:lora_rx_block} shows the block diagram of our LoRa demodulator. It begins by reading data from the I/Q radio into the {\it I/Q Deserializer} module on the FPGA which converts the serial I/Q stream to parallel I/Q for further signal processing. Next, we run these data through a 14 tap FIR low-pass filter to suppress high frequency noise and interference. We store the filtered samples in a buffer implemented using the FPGA memory blocks. To decode the data, we use the {\it Chirp Generator} module from the {\it LoRa Modulator} described above to generate a baseline upchirp/downchirp symbol, and then we multiply that with the received chirp symbol using our {\it Chirp Multiplier} unit which gives the coding gain of CSS. The output of the multiplication then goes to an FFT block implemented using a standard IP core from Lattice. Finally the {\it Symbol Detector} scans the output of the FFT for peaks and records the frequency of the peak to determine the symbol value. To detect chirp type (upchirp/downchirp), we multiply each chirp symbol with both upchirp and downchirp and then compare the amplitude of their FFT peak. The higher peak in FFT result shows higher correlation which we use to detect the chirp type.
{\bf LoRa MAC Layer.}
To demonstrate that LoRa implementation on \name is compatible with existing LoRa networks such as {The Things Network} (TTN)~\cite{thingsnetwork}, we adopt LoRa MAC design from TTN's Arduino libraries~\cite{ttnarduino} and implement it on \name's MCU. TTN uses two methods for device association; Over-the-air activation (OTAA) and activation by personalization (ABP). {In OTAA, each node performs a join-procedure with the network, during which a dynamic device address is assigned with the device. However, in ABP we can hard-code the device address in the device which makes it simpler since the node skips the join procedure. Our implementation supports both OTAA and ABP methods.}
\subsection{BLE Beacons with \name}
To demonstrate \name's 2.4~GHz capabilities we implement Bluetooth beacons which are commonly used for IoT devices.
{\bf BLE Beacon Primer.}
BLE includes a variety of different modes and packet types, we specifically implement non-connectable BLE advertisements (ADV\_NON\_CONN\_IND) which are broadcast packets used for beacons. These packets allow a low power device to broadcast its data to any listening receiver within range without the power overhead of exchanging packets needed to negotiate a connection. These packets themselves have a bit rate of 1~Mbps in Bluetooth 4.0 or up to 2~Mbps in Bluetooth 5.0 and are generated using GFSK with a modulation index of 0.45-0.55. The GFSK modulation is simply binary frequency shift keying (BFSK) with the addition of a Gaussian filter to the square wave pulses to be transmitted which reduces the spectral width.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{./figs/block_diagram/ble_tx_block.eps}
\vskip -0.15 in
\caption{\footnotesize{{\bf BLE implementation block diagram.}}}
\label{fig:ble_tx_block}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{./figs/block_diagram/ble_packet.eps}
\vskip -0.15 in
\caption{\footnotesize{{\bf BLE packet structure.}}}
\label{fig:ble_packet}
\vskip -0.2 in
\end{figure}
{\bf Generating a BLE Packet.}
The structure of an BLE beacon is shown in Fig.~\ref{fig:ble_packet}. Advertisements range from 6-37 octets, beginning with a fixed preamble of 0xAA followed by the access address field which is also set to a fixed sequence 0x8E89BED6 which indicates to a receiver the packet is a non-connectable advertisement. These fields are followed by the packet data unit (PDU) which begins with a 2 byte length field indicating the size of the PDU followed by the data segment which includes an advertisement address which can indicate specific device or device type followed by the packet data such as a 128 bit UUID. In addition to this packet structure, BLE requires a 3 byte CRC as well as whitening of the PDU to create a valid packet.
{\bf CRC Generation.} The Bluetooth CRC is a 3 byte sequence generated using the 24-bit linear feedback shift register (LFSR) with the polynomial $x^{24} + x^{10} + x^{9} + x^{6} + x^{4} + x^{3} + x + 1$. The LFSR is set to a starting state of 0x555555 and each byte of the PDU beginning with the length field is input to the LFSR LSB first. The final state of the LFSR after all of the data bits pass through then becomes the CRC. We implement this block in Verilog on the FPGA.
{\bf Data Whitening.} Data whitening is performed to eliminate long strings of zeros or ones within a packet. This is also done using a 7-bit LFSR with polynomial $x^7 + x^4 + 1$. The LFSR is initialized with the lower 7 bits of the channel number the packet will be transmitted on, and each byte is input LSB first. We implement this block in Verilog on the FPGA.
{\bf Transmitting a packet.} From this bitstream, we need to generate the I/Q samples to feed to the I/Q radio. As shown in Fig.~\ref{fig:ble_tx_block}, first, we filter the generated bitstream by a Gaussian filter. This gives us the frequency which we then integrate it to get the phase. We feed the phase to sine and cosine function to get I and Q samples, which are passed to I/Q serializer to send them to I/Q radio through LVDS interface.
{\bf MAC protocol.} While BLE divides the 2.4~GHz band into a total of 40 channels each spaced 2~MHz apart, a typical BLE beacon is transmitted on each of the three advertising channels in a sequential order separated by a few hundred microseconds, without carrier sense. This sequence of three beacons are re-transmitted every advertising interval which is a minimum of 20~ms~\cite{ble_spec}.
\section{Case Studies: LoRa and BLE Beacons}
\subsection{LoRa Protocol with \name}
We choose LoRa as it is gaining popularity for IoT solutions due to its long range capabilities. Since LoRa is a proprietary standard, we begin by describing the basics of its modulation and packet structure followed by the implementation details of our modulator, demodulator and MAC protocol.
{\bf LoRa Protocol Primer.}
LoRa achieves long ranges by using Chirp Spread Spectrum (CSS) modulation. In CSS, data is modulated using linearly increasing frequency upchirp symbol. Each upchirp symbol has two main features: Spreading Factor (SF) and Bandwidth (BW). SF determines the number of bits in each upchirp symbol~\cite{lorasigcomm17,netscatter,lorabackscatter} and BW is the difference between upper and lower frequency of the chirp which together with SF determines the length of an upchirp symbol. SF and BW trade data rate for range. Data is modulated by $2^{SF}$ cyclic-shifts of an upchirp symbol. The starting point of the symbol in frequency domain, which is the cyclic shift of the upchirp symbol, determines its value~\cite{lora_basics}. LoRa uses SF values from 6 to 12 and BW values from 7.8125~KHz to 500~KHz to achieve PHY-layer rates of $\frac{BW}{2^{SF}}\times SF$.
Fig.~\ref{fig:lora_packet} shows the LoRa packet structure which begins with a preamble of 10 zero symbols (upchirps with zero cyclic-shift). This is followed by the Sync field with two upchirp symbols. Next, a sequence of 2.25 downchirp symbols (chirp symbol with linearly decreasing frequency) indicate the beginning of the payload. The payload then consists of a sequence of upchirp symbols which encode a header, payload and CRC.
{\bf LoRa Modulator.}
Fig.~\ref{fig:lora_tx_block} shows the block diagram of our LoRa modulator. We use our FPGA to implement a LoRa modulator in Verilog and stream data to AT86RF215 in I/Q mode. The modulator begins with the {\it Packet Generator} module which reads data either from FPGA memory for transmitting fixed packets or from the MCU, as well as LoRa configuration parameters such as SF, coding and BW. This module determines each symbol value and its corresponding cyclic-shift. Next, the {\it Packet Generator} sends these parameters along with the symbol values to the {\it Chirp Generator} module, which generates the I/Q samples of each chirp symbol in the packet using a squared phase accumulator and two lookup tables for $Sin$ and $Cos$ function~\cite{lorabackscatter}. We then feed these I/Q samples into {I/Q Serializer} to stream them over the LVDS interface to the I/Q radio. We generate 64~MHz transmission clock using internal PLL of the FPGA.
\begin{figure}[t!]
\centering
\includegraphics[width=0.75\linewidth]{figs/10_ota_exp/map.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf Evaluation Testbed Map.}}}
\label{fig:ota_map}
\vskip -0.2in
\end{figure}
{\bf LoRa Demodulator. }
Fig.~\ref{fig:lora_rx_block} shows the block diagram of our LoRa demodulator. It begins by reading data from the I/Q radio into the {\it I/Q Deserializer} module on the FPGA which converts the serial I/Q stream to parallel I/Q for further signal processing. Next, we run the data through a 14 tap FIR low-pass filter to suppress high frequency noise and interference. We store the filtered samples in a buffer implemented using the FPGA's memory blocks. To decode the data, we use the {\it Chirp Generator} module from the {\it LoRa Modulator} described above to generate a baseline upchirp/downchirp symbol, and then we multiply that with the received chirp symbol using our {\it Complex Multiplier} unit. The output of the multiplication then goes to an FFT block implemented using a standard IP core from Lattice. Finally the {\it Symbol Detector} scans the output of the FFT for peaks and records the frequency of the peak to determine the symbol value. To detect chirp type (upchirp/downchirp), we multiply each chirp symbol with both an upchirp and downchirp and then compare the amplitudes of their FFT peaks. The higher peak in the FFT shows higher correlation which indicates the chirp type.
{\bf LoRa MAC Layer.}
To demonstrate that our LoRa implementation on \name is compatible with existing LoRa networks such as the LoRa Alliance's~\cite{loraalliance} {The Things Network} (TTN)~\cite{thingsnetwork}, we adopt their LoRa MAC design from TTN's Arduino libraries~\cite{ttnarduino} and implement it on \name's MCU. TTN uses two methods for device association; Over-the-air activation (OTAA) and activation by personalization (ABP). {In OTAA, each node performs a join-procedure during which a dynamic device address is assigned to a node. However, in ABP we can hard-code the device address in the device which makes it simpler since the node skips the join procedure. Our platform can support both OTAA and ABP methods.}
\subsection{BLE Beacons with \name}
To demonstrate \name's 2.4~GHz capabilities we implement Bluetooth beacons which are commonly used by IoT devices.
{\bf BLE Beacon Primer.}
We implement non-connectable BLE advertisements (ADV\_NON\_CONN\_IND) which are broadcast packets used for beacons. These packets allow a low power device to broadcast its data to any listening receiver within range without the power overhead of exchanging packets to setup a connection. These packets have a bit rate of 1~Mbps in Bluetooth 4.0 or up to 2~Mbps in Bluetooth 5.0 and are generated using GFSK with a modulation index of 0.45-0.55. The GFSK modulation is binary frequency shift keying (BFSK) with the addition of a Gaussian filter to the square wave pulses to reduce the spectral width.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/15_spectrum/15_spectrum_plot.eps}
\vskip -0.15in
\caption{\footnotesize{{\bf \Name Single-Tone Frequency Spectrum.}}}
\label{fig:spectrum}
\vskip -0.11in
\end{figure}
\begin{table}[t!]
\centering
\caption{\footnotesize{\bf Different Operation Timing for \Name.}}
\vskip -0.15 in
\footnotesize{
\begin{tabular}{|C{4cm}|C{2cm}|}
\hline \rowcolor{lightgray}
{\bf Operation} & {\bf Duration (ms)}\\
\hline
Sleep to Radio Operation & 22\\
\hline
Radio Setup & 1.2\\
\hline
TX to RX & 0.045\\
\hline
RX to TX & 0.011\\
\hline
Frequency Switch & 0.220\\
\hline
\end{tabular}
}
\vskip -0.2in
\label{tab:wakeup}
\end{table}
{\bf Generating a BLE Packet.}
Bluetooth advertisements consist of 6-37 octets, beginning with fixed preamble and access address fields indicating the packet type set to 0xAA and 0x8E89BED6 respectively. This is followed by the packet data unit (PDU) beginning with a 2 byte length field and followed by a manufacturer specific advertisement address and data. The final 3 bytes of the packet consist of a CRC generated using a 24-bit linear feedback shift register (LFSR) with the polynomial $x^{24} + x^{10} + x^{9} + x^{6} + x^{4} + x^{3} + x + 1$. The LFSR is set to a starting state of 0x555555 and the PDU is input LSB first. The final LFSR state after inputting the PDU becomes the CRC. Data whitening is then performed over the PDU and CRC fields to eliminate long strings of zeros or ones within a packet. This is also done using a 7-bit LFSR with polynomial $x^7 + x^4 + 1$. The LFSR is initialized with the lower 7 bits of the channel number the packet will be transmitted on, and each byte is input LSB first. We implement both these blocks in Verilog on the FPGA.
{\bf Packet Transmission and MAC Protocol.} From this bitstream, we need to generate the I/Q samples to feed to the I/Q radio. First, we upsample and apply a Gaussian filter to the bitstream. This gives us the desired changes in frequency which we integrate to get the phase. We then feed the phase to sine and cosine functions to get the final I and Q samples, which are passed to I/Q serializer and sent to the I/Q radio. BLE divides the 2.4~GHz band into channels, each spaced 2~MHz apart, but BLE beacons are only transmitted on three advertising channels without carrier sense, typically in sequential order separated by a few hundred microseconds. This sequence is re-transmitted every advertising interval~\cite{ble_spec}.
\section{The Case for \name and Related Work}
To demonstrate that \name is the best platform for network research community to deploy large-scale IoT networks, we first compare it with existing SDR platforms in terms of XXX, XXX and XXX. We then discuss prior work on SDR platforms and compare \name with existing solutions.
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Power consumption.} Conventional SDR platforms are designed to operate at high frequency bandwidth which result in using high sampling rate ADCs and DACs. Existing platforms such as Embedded USRP E310~\cite{XXX}, bladeRF~\cite{XXX} and LimeSDR~\cite{XXX} operate up to 60~MHz bandwidth ans support wide-band frequency spectrum. Therefore, they require to use radios with high sampling rate front-end and also optimize radio noise figure over large frequency spectrum. In contrast, IoT protocols such as LoRa, Sigfox use less than 500~kHz bandwidth and operate at low sampling rate. Fig.~\ref{fig:platform_power} shows the power consumption of radio module in conventional SDR platforms. This plot shows that existing SDR platforms use power-hungry RF front-ends which consumes in the order of few watts of power in transmit mode and few hundred milliwatts of power in receive mode. High power consuming radios are not suitable for a battery-powered SDR platform which is the our target for large-scale research deployments. This requirements lead us to redesigning existing SDR solutions and optimizing them for IoT protocols. We design \name that can operate at low-power and last XXX years using only a XXX battery.
\item {\bf Latency.} Existing conventional SDR platforms focused on providing a low-latency design. SORA~\cite{sora} uses PCI-Express bus to provide sub 1~$mu$s latency. USRP uses peripheral buses such as USB or Ethernet to connect to a computer with operating system for its controlling signal. Embedded USRP E310 benefits from a Dual Core ARM A9 processor for processing high data-rate information. $\mu$SDR solves the latency challenge by using an advanced microcontroller bus architecture (AMBA) along with a Cortex-M3 ARM processor. These efforts for designing low-latency platform is only because of the need for processing high rate information and supporting delay-bounded protocols such as Wi-Fi~\cite{}. However, IoT protocols have higher flexibility in design. For example, LoRa MAC protocols requires to support receive windows for downlink message in XXX and XXX seconds after uplink message~\cite{}. This requirement, however, is achievable with low-computation processing units which result in much simpler design compare to conventional platforms.
\item {\bf Deployability.} Recent work in IoT networks~\cite{choir, netscatter} are required to deploy at large-scale to show the feasibility of new inventions in IoT network domain. However, building large-scale IoT networks with existing SDR platforms is very expensive since they are bulky in design, hardware is expensive in nature and most importantly they require power and communication backbone infrastructure. With the advent of new IoT protocols~\cite{}, the demand for a universal inexpensive platform with processing capabilities for IoT protocols increases. Table~\ref{tab:platforms} shows existing platforms and compares them in terms of size, cost and power requirement. We define large-scale deployability of a platform in terms of form-factor size, cost and infrastructure requirements. We design a cheap battery-powered SDR platform which does not require any infrastructure to operate and is optimized for IoT network research.
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/1_platform_power/platform_pow.eps}
\vskip -0.1in
\caption{\footnotesize{{\bf Radio Power Consumption.} We show radio power consumption for each platform. We normalized TX power consumption for output equals to 14~dBm.\newline
**RTL-SDR does not support transmitter mode.}}
\label{fig:platform_power}
\vskip -0.15in
\end{figure}
In summary, \name provides the smallest SDR solution which enables large-scale deployments for network research community at very low cost. XXX
Related work in SDR platform domain falls into two categories which we explain here.
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item{\bf Commercial SDR platforms.} Ettus research is one of the world's leaders in SDR platform technologies. They provide a variety of platforms such as USRP N210~\cite{}, Embedded USRP~\cite{} and USRP networked series~\cite{}. Lime microsystem provides SDR platforms such as LimeSDR~\cite{} and LimeSDR Mini~\cite{}. Nuand is another commercial SDR provider which recently realsead bladeRF 2.0 micro SDR platform. More recently, Analog Devices has released PlutoSDR~\cite{} platform which uses AD9363~\cite{} radio front-end. Commercial SDR platforms are designed to support large spectrum frequency and provide flexibility for different applications. However, they are bulky in design, expensive and power-hungry in nature. Here, we target to build a platform for IoT network research community which provides flexibility in design while can it can be battery-powered.
\item{\bf Research SDR platforms.} Over the past decade there has been multiple work on designing SDR platforms. SODA~\cite{} proposed a fully programmable SDR architecture IC simulation which meets the requirements of high-end wireless protocols and consist of four SIMD cores. KUAR~\cite{} and SORA~\cite{} are research SDR platforms that have large form-factor design and required PCI-Express connection to a host PC to operate. $\mu$SDR designed an integrated SDR that uses flash-based FPGA and AMR Cortex-M3 to process data. However, this design is power-hungry for an IoT network deployment and relatively expensive. GalioI~\cite{galiot} presents an SDR platform including a RTL2832U radio~\cite{rtl2832} connected to a Raspberry pi computer. This platform can only receive signal and the receiver RF front-end is power-hungry as shown in Fig.~\ref{fig:platform_power}. XXXX
\end{itemize}
{\footnotesize{
\centering
\vskip 0.2in
\begin{tabular}{|C{1.4cm}|C{0.8cm}|C{0.8cm}|C{1cm}|C{1cm}|C{1cm}|}
\hline \rowcolor{black}
\centering
{\bf\color{white}Platform}
& {\bf\color{white}Size (inches)}
& {\bf\color{white}Cost}+
& {\bf\color{white}Standalone}
& {\bf\color{white}Battery-Powered}
& {\bf\color{white}Deployability}
\\
\hline \rowcolor{lightgray}
{USRP E310\cite{usrpe310, ad9361digikey}} & 5.23 $\times$ 2.68 & \$3057 & \cmark & \xmark & \xmark\\
\hline \rowcolor{lightgray}
{USRP B200mini\cite{usrpb200mini, ad9364digikey}} & 3.26 $\times$ 2.0 & \$733 & \cmark & \xmark& \xmark\\
\hline \rowcolor{lightgray}
{bladeRF 2.0\cite{bladerf2, ad9361digikey}} & 5.0 $\times$ 2.5 & \$720 & \cmark & \xmark& \xmark\\
\hline \rowcolor{lightgray}
{LimeSDR Mini \cite{limesdr, limesdrmini, lms7002m}} & 3.9 $\times$ 2.36 & \$159 & \cmark & \xmark & \xmark\\
\hline \rowcolor{lightgray}
{PlutoSDR\cite{XXX, XXX}} & XXX $\times$ XXX & \$149 & \cmark & \xmark & \xmark\\
\hline \rowcolor{lightgray}
{$\mu$SDR\cite{XXX, XXX}} & XXX $\times$ XXX & \$150 & \cmark & \xmark & \xmark\\
\hline \rowcolor{lightgray}
{GalioT\cite{galiot, rtl2832}} & 2.75 $\times$ 1.0 & \$60 & \cmark & \xmark & \xmark\\
\hline \rowcolor{lightgray}
{\bf{\name}} & XX $\times$ XXX & \$XX & \cmark & \cmark & \cmark\\
\hline \rowcolor{lightgray}
{\bf{\nameMini}} & XX $\times$ XXX & \$XX & \cmark & \cmark & \cmark\\
\hline
\end{tabular}
\captionof{table}{\footnotesize{\bf SDR Platforms.}}
\label{tab:platforms}
}}
\section{The Case for \name and Related Work}
To demonstrate that \name is the best fit for network research community to deploy large-scale IoT networks, we first need to understand what are the requirements in IoT communication protocol implementation and deployments and what are the performance parameters of currently available SDR designs in terms of these requirements. The determining performance metrics for SDR platforms used in IoT network research are power consumption, latency and deployability (xxxx why these three?). Next, we explain how previous SDR platforms perform with these metrics and argue that current they are not the optimal choice for supporting IoT wireless technologies' growth.
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Latency.}
Many existing SDR platforms focused on designing a low-latency design. These efforts are only because of the need for processing high rate of information and supporting delay-bounded protocols such as Wi-Fi~\cite{}. However, IoT protocols have relaxed requirements in terms of latency and throughput. For example, LoRa MAC protocols requires to support receive windows for downlink message in XXX and XXX seconds after uplink message~\cite{}. \textcolor{red}{This requirement, however, is achievable with low-computation processing units which result in much simpler design compare to conventional platforms. seems weird}
\item {\bf Power consumption.} Existing SDR platforms can operate at wide frequency spectrum and support up to 60~MHz bandwidth which results in using high sampling rate ADCs and DACs and more importantly high digital baseband throughput to the processing unit and computationally extensive processing unit. Moreover, they require to use radio front-ends with high sampling rate and noise performance over large frequency spectrum and bandwidth. As a result, their whole system, radio front-end and processing unit, consumes several watts of power. Fig.~\ref{fig:platform_power} shows the power consumption of only radio module in existing SDR platforms. In contrast, IoT protocols such as LoRa, Sigfox use less than 500~kHz bandwidth and require low sampling rate of maximum 1~MHz. This means that there is an \textcolor{red}{over-designed} gap between the current SDR designs and what is required for IoT communication protocol implementation. Fig.~\ref{fig:platform_power} shows that these platforms use power-hungry RF front-ends which consumes in the order of few watts of power in transmit mode and few hundred milliwatts of power in receive mode. High power consuming radios are not suitable for a battery-powered SDR platform which is the our target for large-scale research deployments. These requirements lead us to rethink existing SDR solutions and optimize them for IoT protocols. We design and build \name that is low-power and last XXX years using only a XXX battery.
\item {\bf Deployability.} IoT communication network protocols are needed to be implemented at large-scale to be fully evaluated and show their potential~\cite{choir, netscatter}. However, building large-scale IoT networks with existing SDR platforms involves multiple challenges because of their large form-factor, expensive hardware, high power and processing backbone infrastructure. With the advent of new IoT protocols~\cite{}, the demand for a universal inexpensive platform with processing capabilities supporting IoT protocols increases. Table~\ref{tab:platforms} shows existing platforms and compares them in terms of size, cost and power requirement. We define large-scale deployability of a platform in terms of form-factor size, cost and infrastructure requirements. As shown in the Table~\ref{tab:platforms}, none of the previous platforms satisfy the requirements of deployability while \name is cheap, small form-factor, battery-powered and without any need for backbone infrastructure which makes it deployable and optimal fit for IoT network research.
\end{itemize}
In summary, \name provides the smallest SDR solution which enables large-scale deployments for network research community at very low cost. XXX
Related work in SDR platform domain falls into two categories which we explain here.
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item{\bf Commercial SDR platforms.} Ettus research is one of the world's leaders in SDR platform technologies. They provide a variety of platforms such as USRP N210~\cite{}, Embedded USRP~\cite{} and USRP networked series~\cite{}. USRP uses peripheral buses such as USB or Ethernet to connect to a computer with operating system for its controlling signal. Embedded USRP E310 benefits from a Dual Core ARM A9 processor for processing high data-rate information. Lime microsystems provides SDR platforms such as LimeSDR~\cite{} and LimeSDR Mini~\cite{}. Nuand is another commercial SDR provider which recently realsead bladeRF 2.0 micro SDR platform. More recently, Analog Devices has released PlutoSDR~\cite{} platform which uses AD9363~\cite{} radio front-end. Commercial SDR platforms are designed to support large spectrum frequency and provide flexibility for different applications. However, they are bulky in design, expensive and power-hungry in nature. Here, we target to build a platform for IoT network research community which provides flexibility in design while can it can be battery-powered.
\item{\bf Research SDR platforms.} Over the past decade there has been multiple works on designing SDR platforms. Tick~\cite{} achieves good programmability, high throughput and low latency by software-hardware co-design solutions. However, it uses a high power radio on the order of several watts~\cite{AD9371} and the whole design costs more than 3000 dollars. SORA~\cite{sora} is research platform that uses PCI-Express bus to provide sub 1~$\mu$s latency. However, it requires host PC for backbone processing and hence large form-factor. $\mu$SDR solves the latency challenge by using an advanced microcontroller bus architecture (AMBA) along with a Cortex-M3 ARM processor~\cite{musdr} and flash-based FPGA to process data. However, it is power-hungry for an IoT network deployment and relatively expensive. SODA~\cite{} proposed a fully programmable SDR architecture IC simulation which meets the requirements of high-end wireless protocols and consist of four SIMD cores. However, it has a slightly customized VLIW+SIMD ISA which needs users to hand-tune their program. KUAR~\cite{} is a research SDR platform that have large form-factor design and required PCI-Express connection to a host an embedded PC-like environment to operate. GalioI~\cite{galiot} presents an SDR platform including a RTL2832U radio~\cite{rtl2832} connected to a Raspberry pi computer. This platform can only receive signal and the receiver RF front-end is power-hungry as shown in Fig.~\ref{fig:platform_power}. XXXX
\end{itemize}
\begin{table}[htb
\caption{Global caption}
\begin{subtable}[t]{.5\textwidth}
\caption{This is a very very very long Caption}
\raggedright
\begin{tabular}{@{\extracolsep{1pt}}lcccc}
\hline\hline \\
Statistic & \multicolumn{1}{c}{Mean} & \multicolumn{1}{c}{St. Dev.} & \multicolumn{1}{c}{Min} & \multicolumn{1}{c}{Max} \\
\hline \\
A & 1.000 & 0.000 & 1 & 1 \\
B & 0.968 & 0.76 & 0.138 & 1.000 \\
C & 0.969 & 0.009 & 0.917 & 1.000 \\
D & 0.966 & 0.070 & 0.495 & 0.83 \\
\hline \\
\end{tabular}
\end{subtable}%
\begin{subtable}[t]{.5\textwidth}
\raggedleft
\caption{This is a very very very long Caption}
\begin{tabular}{@{\extracolsep{1pt}}lcccc}
\hline\hline \\
Statistic & \multicolumn{1}{c}{Mean} & \multicolumn{1}{c}{St. Dev.} & \multicolumn{1}{c}{Min} & \multicolumn{1}{c}{Max} \\
\hline \\
A & 1.000 & 0.000 & 1 & 1 \\
B & 0.968 & 0.76 & 0.138 & 1.000 \\
C & 0.969 & 0.009 & 0.917 & 1.000 \\
D & 0.966 & 0.070 & 0.495 & 0.83 \\
\hline \\
\end{tabular}
\end{subtable}
\end{table}
\subsection{Network Deployments}\label{sec:deploy}
We evaluate three key network parameters:
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\it Network PHY bit-rate.} This is the bit rate achieved across all the end-devices during the payload part of the packet.
\item {\it Link-layer data rate.} This is the data rate achieved in the network which is defined as the data rate for sending useful payload bits, after considering overheads including the AP's query message and the preamble of the packet transmission.
\item {\it Network latency.} This is the latency to successfully get the payload bits from all the end-devices in the network.
\end{itemize}
\subsubsection{Backscatter deployment}\label{sec:back_deploy}
We set each \tagname bit-rate to 1~kbps, $BW_{agg}=500~KHz$, $SF=9$ and a payload size of five bytes. With the above settings and considering the $\Delta FFT_{bin}$ results shown in Fig.~\ref{fig:fft_var}, the number of empty cyclic shifts is one corresponding to $SKIP=2$ in our design. This means that our network supports {256} \tagnames talking concurrently. We deploy these backscatter end-devices across the floor of an office building with more than eight rooms.\footnote{{Note that RFID EPC-Gen2 readers' receiver decoding sensitivity is -85~dBm~\cite{rfidReader} for battery-powered backscatter tags. With BW=500~kHz and SF=9, the sensitivity for CSS modulation is -123~dBm. This 38~dB gain
allows CSS-based backscatter systems to support long ranges and deployments that are not possible with EPC-Gen2.}}
\begin{figure*}[t]
\begin{subfigure}{0.32\linewidth}
\centering
\includegraphics[width=1.05\linewidth]{./figs/12_active_dataRate_latency/phy_rate_bar.eps}
\label{fig:active_phy}
\vskip -0.25in
\caption{Network Physical Rate}
\end{subfigure}
\begin{subfigure}{0.32\linewidth}
\centering
\includegraphics[width=1.05\linewidth]{./figs/12_active_dataRate_latency/link_rate_bar.eps}
\label{fig:active_link}
\vskip -0.25in
\caption{Link-layer Data Rate}
\end{subfigure}
\begin{subfigure}{0.32\linewidth}
\centering
\includegraphics[width=1.05\linewidth]{./figs/12_active_dataRate_latency/latency_bar.eps}
\label{fig:activ_latency}
\vskip -0.25in
\caption{Latency}
\end{subfigure}
\vskip -0.15in
\caption{{\bf Wide-Area Radio Deployment.} We evaluate network physical rate, link-layer data rate and latency for \nameDesign and compare it with Choir.}
\label{fig:active_link_rate}
\vskip -0.15in
\end{figure*}
We compare three schemes: i) LoRa backscatter~\cite{lorabackscatter}, ii) LoRa backscatter with rate adaptation and iii) \nameDesign. Note that the authors of~\cite{lorabackscatter} did not publicly release the code and so, we replicate the implementation adding the missing details and using $BW=500~KHz$ and $SF=9$. We also note that ~\cite{lorabackscatter} is not designed to work with more than one user. Here, we use TDMA with scheduling when there are more users where the AP queries each end-devices. While LoRa backscatter does not support rate adaptation, we wanted to compare with an ideal approach that maximizes the bit-rate of each end-device by picking the optimal SF and BW. To do so, we measure RSSI from each of the backscatter \tagnames and compute the bit rate using the SNR table in~\cite{lora1276}.
Fig.~\ref{fig:passive_link_rate} shows the results for our backscatter network deployment. The plots show that with 256 backscatter end-devices, \nameDesign increases the PHY bit-rate by 6.8x and 26.2x over LoRa backscatter with and without rate adaptation. The gains are lower with the ideal rate adaptation since it could pick the maximum LoRa bit rate of 32~Kbps. We also note that \nameDesign sets the bit rate of each end-device to 1~kbps and hence has better gains as the number of end-devices increase. Further, the gains at the link-layer are however higher for \nameDesign at 20x and 98.5x respectively. This is because, in \nameDesign, the added overhead of sending AP's query and \tagnames' preambles happen once and at the same time for all \tagnames. But the other schemes need to do TDMA which means that sending query and preamble will not happen concurrently for all \tagnames.
Finally, \nameDesign has a latency reduction of 21.8x and 106x over the other approaches.
\subsubsection{Wide-Area Radio Deployment}\label{sec:wi}
Unlike backscatter devices which can be deployed within a single building, the logistics of deploying 256 radio end-devices across the campus is prohibitive in terms of {man-power as well as the time that it takes}. Instead we emulate a 256 node network across the campus by moving 10 radio \tagnames across 256 different locations across the campus. Note that the actual technical feasibility of \nameDesign at scale has already been demonstrated in the backscatter experiments, so these set of experiments are for completeness.
Specifically, to emulate a network of {256} \tagnames, we do 26 different deployments across campus and record the results using USRP. Each deployment involves all ten radio \tagnames. Moreover, the spatial locations and cyclic shifts assigned to the \tagnames in each deployment is different from other deployments. Note that, we assign \tagnames based on ~\xref{sec:nearfar} at different power levels and they use the power adjustment to stay within that power level after deployment. We then time-synchronize and combine the time-domain recordings across all these 26 deployments. We input this addition result to the \nameDesign decoder and decode all the concurrent 256 \tagnames. A question the reader might have is: If we have a network involving {256} \tagnames, would the result of the USRP recording be the same as our combined deployment results? For a linear system, adding two inputs will result in an output which is equal to adding their corresponding outputs. Since the end-devices are far from the AP and a have weak signal strength, the analog front-end of the USRP is far from being saturated and works in linear region. This means that, the addition of the USRP recordings of different deployments is functionally similar as a recording of deployment which involves all the \tagnames.
We compare \nameDesign with Choir~\cite{lorasigcomm17} which enables up to ten concurrent LoRa transmissions. To support more than 10 transmissions in Choir, we again use TDMA where the AP sends a query message to different sets of ten transmitters. Fig.~\ref{fig:active_link_rate} shows the results for our deployment. Note that each node in Choir transmits an average of 9~kbps and since it can support up to ten concurrent transmissions, our gains are more prominent with a large number of \tagnames. Further, when the number of \tagnames is less than 90, Choir can outperform \nameDesign since the latter requires a large number of users to fully utilizes the network capacity. With 256 \tagnames, the PHY and link-layer gains over Choir (LoRa TDMA with rate adaptation) for \nameDesign was 2.5x (4.1x) and 4.8x (27.8x) respectively. Also, with corresponding latency reduction is around 5.4x (30.1x) which is significant in practical networks. Note that, for clarity of comparison between \nameDesign and Choir, Fig.~\ref{fig:active_link_rate} does not include the results of the LoRa TDMA.
\section{System Design}
\begin{figure}[t]
\centering
\includegraphics[width=0.2\linewidth]{./figs/sdr_hardware.png}
\caption{\footnotesize{{\bf \name Hardware Design.}}}
\label{fig:sdr_hardware}
\end{figure}
\textcolor{red}{write it based on the Mobicom paper}
In this section, we go over the PHY and MAC layer design of the \name system and present design choices and considerations.
\subsection{PHY Layer}
In this section, we go over the different parts of the \name system and present design choices and considerations. \name platform is composed of analog front-end, FPGA, MCU, sensor interface and microSD card interface. Fig.~\ref{fig:block_diagram} shows the block diagram of the \name platform.
In the receive chain, analog front-end receives RF signal and provides I/Q signal to the FPGA for base-band processing. The transmit chain works in the opposite direction meaning that the I/Q signal will be upconverted to RF frequency to send it over the antenna. FPGA does baseband modulation and demodulation. MCU provides sensor interface, controls the analog front-end and power management unit (PMU). We provision microSD card to record raw I/Q samples and packet information for research projects that need data collection.
\subsection{Analog Front-End}
The RF front-end is based on AT86RF215~\cite{ref:AT86RF215} from Microchip which can operate in 400~MHz, 900~MHz and 2.4~GHz frequency bands which makes this design suitable for these ISM bands xxxx and is capable of exchanging digitized raw I/Q samples with the baseband processing unit. This chip provides all the necessary blocks of the transceiver chain from the antenna to the digital baseband data including LNA/PA, downconversion/upconversion mixers, band-pass/low-pass filters, ADC/DAC, synthesizer and voltage-controlled oscillator. To exchange digitized I/Q baseband data between the baseband processing block, it uses 64~MHz double data-rate low voltage differential signaling (LVDS). The maximum sampling frequency for the I/Q data is 4~MHz which means that it can support modulations with maximum of 2~MHz bandwidth. Note that, most IoT communication protocols have low bandwidth xxxx (e.g. LoRa, SigFox \cite{xxxx}). Moreover, the digital analog baseband front-end, ADC/DAC, has 13 bits which is even better than the analog front-end (AD93XX~\cite{xxxx}) used by USRP \cite{xxxx}. In addition, the low bandwidth and sampling rate of this chip makes it low power so that it can be run out of a battery. Fig. \ref{fig:} shows the power consumption comparison of this transceiver compared to other transceivers that are used in similar products. As mentioned in ~\xref{sec:case}, our goal is to enable inexpensive SDR platform. Our other consideration in choosing this chip is its very low price of less than 5 dollars. In summary, AT86RF215 satisfies all the requirements of the wide-spread inexpensive SDR for IoT applications.
- explain the LVDS part
- explain different options for control
\subsection{Processing Unit}
The processing is composed of FPGA and MCU and is responsible for communication protocol implementation including modulation and demodulation and MAC layer. FPGA is responsible for doing modulationa and demodulation and MCU for MAC layer implementation. We selected FPGA by analyzing different IoT communication protocols' receiver algorithm, FPGA computational resource needed to enable these algorithms and also computational resource needed to do localization xxxx. Moreover, it needs to support LVDS interface to be able to talk to the transceiver chip. We have used M2GL010T from IGLOO 2 series of Microsemi for our first prototype. We chose MSP430FR5969 from TI MSP430 series for MCU because of its power performance.
\subsection{Control Unit}
As shown in Fig. \ref{xx}, the other block that we have used in our design is control unit which is implemented on a low power MSP430~\cite{xxxx} MCU. This unit serves multiple purposes. First, it talks to the radio to adjust its settings including synthesizer frequency, analog front-end gain, TX transmit power,sampling rate, filtering parameters and etc. depending on the application and communication protocol. Moreover, it talks to the power management unit to set voltage supply of different blocks and optimize them to minimize power consumption. In addition, it provides different communication interfaces (SPI, I2C and ...) to talk to different sensors that may need to be added.
\subsection{MicroSD Card}
The purpose of this unit is to enable recording high amounts of data (e.g. raw I/Q data or baseband bits) for later post-processing. MicroSD cards provide two modes of communication to the host: 1- xxxx 2- SPI based communication. The SPI-based communication is much simpler than the first option and can provide around xxxx~Mbps. To fast prototype our design, we have implemented the SPI-based communication between the FPGA and the MicroSD card.
\section{System Design}
\begin{figure}[t]
\centering
\includegraphics[width=0.2\linewidth]{./figs/foo.eps}
\caption{\footnotesize{{\bf \name Hardware Design.}}}
\label{fig:sdr_hardware}
\end{figure}
In this section, first, we go over the PHY and MAC layer design and implementations of the \name system and present design choices and considerations. Next, we explain the microSD card interface and sensor interface that has been added to enable the system to act as a sensor node and increase its functionality. Fig.~\ref{fig:block_diagram} shows the block diagram of the \name platform.
\subsection{PHY Layer}
\name platform's PHY layer is composed of analog front-end and FPGA. In the receive chain, analog front-end receives RF signal and provides I/Q signal to the FPGA for base-band processing, modulation and demodulation. The transmit chain works in the opposite direction meaning that the I/Q signal will be upconverted to RF frequency to send it over the antenna.
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Analog Front-End.}
The RF front-end is based on AT86RF215~\cite{ref:AT86RF215} from Microchip which can operate in 400~MHz, 900~MHz and 2.4~GHz frequency bands which makes this design suitable for these ISM bands xxxx and is capable of exchanging digitized raw I/Q samples with the baseband processing unit. This chip provides all the necessary blocks of the transceiver chain from the antenna to the digital baseband data including LNA/PA, downconversion/upconversion mixers, band-pass/low-pass filters, ADC/DAC, synthesizer and voltage-controlled oscillator. To exchange digitized I/Q baseband data between the baseband processing block, it uses 64~MHz double data-rate low voltage differential signaling (LVDS). The maximum sampling frequency for the I/Q data is 4~MHz which means that it can support modulations with maximum of 2~MHz bandwidth. Note that, most IoT communication protocols have low bandwidth of 500~KHz (e.g. LoRa, SigFox \cite{xxxx}). Moreover, the digital analog baseband front-end, ADC/DAC, has 13 bits which is even better than the analog front-end (AD93XX~\cite{xxxx}) used by USRP \cite{xxxx}. \textcolor{red}{the low bandwidth and sampling rate xxxx, mention it or not?}. Fig. \ref{fig:} compares power consumption this transceiver with other transceivers that are used in existing products. As can be seen, power consumption is more than \textcolor{red}{5 xxx} times less than the closest SDR platform. \textcolor{red}{Moreover, the price of this transceiver chip is less than 5 dollars which makes this design very low cost.} In summary, AT86RF215 satisfies all the requirements of the wide-spread cheap SDR for IoT applications.
\item {\bf Baseband-Processing Unit.}
The processing is composed of FPGA and MCU and is responsible for communication protocol implementation including modulation and demodulation and MAC layer. FPGA is responsible for doing modulationa and demodulation and MCU for MAC layer implementation. We selected FPGA by analyzing different IoT communication protocols' receiver algorithm, FPGA computational resource needed to enable these algorithms and also computational resource needed to do localization xxxx. Moreover, it needs to support LVDS interface to be able to talk to the transceiver chip. We have used M2GL010T from IGLOO 2 series of Microsemi for our first prototype. We chose MSP430FR5969 from TI MSP430 series for MCU because of its power performance.
\end{itemize}
MCU provides sensor interface, controls the analog front-end and power management unit (PMU). We provision microSD card to record raw I/Q samples and packet information for research projects that need data collection.
\subsection{MAC Layer}
As shown in Fig. \ref{xx}, the other block that we have used in our design is control unit which is implemented on a low power MSP430~\cite{xxxx} MCU. This unit serves multiple purposes. First, it talks to the radio to adjust its settings including synthesizer frequency, analog front-end gain, TX transmit power,sampling rate, filtering parameters and etc. depending on the application and communication protocol. Moreover, it talks to the power management unit to set voltage supply of different blocks and optimize them to minimize power consumption. In addition, it provides different communication interfaces (SPI, I2C and ...) to talk to different sensors that may need to be added.
\subsection{MicroSD Card}
The purpose of this unit is to enable recording high amounts of data (e.g. raw I/Q data or baseband bits) for later post-processing. MicroSD cards provide two modes of communication to the host: 1- xxxx 2- SPI based communication. The SPI-based communication is much simpler than the first option and can provide around xxxx~Mbps. To fast prototype our design, we have implemented the SPI-based communication between the FPGA and the MicroSD card.
\section{System Design}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{./figs/foo.eps}
\caption{\footnotesize{{\bf \name Hardware Design.}}}
\label{fig:sdr_hardware}
\end{figure}
In this section, first, we go over the PHY and MAC layer design and implementations of the \name system and present design choices and considerations. Next, we explain the microSD card interface and sensor interface that has been added to enable the system to act as a sensor node and increase its functionality. Fig.~\ref{fig:block_diagram} shows the block diagram of the \name platform.
\subsection{PHY Layer}
\name platform's PHY layer is composed of analog front-end and FPGA. In the receive chain, analog front-end receives RF signal and provides I/Q signal to the FPGA for base-band processing, modulation and demodulation. The transmit chain works in the opposite direction meaning that the I/Q signal will be upconverted to RF frequency to send it over the antenna.
\begin{itemize}[itemsep=1pt,parsep=2pt,topsep=3pt,partopsep=0pt,leftmargin=0em, itemindent=1em,labelwidth=1em,labelsep=0.5em]
\item {\bf Analog Front-End.}
The RF front-end is based on AT86RF215~\cite{ref:AT86RF215} from Microchip which can operate in 400~MHz, 900~MHz and 2.4~GHz frequency bands which makes this design suitable for these ISM bands xxxx and is capable of exchanging digitized raw I/Q samples with the baseband processing unit. This chip provides all the necessary blocks of the transceiver chain from the antenna to the digital baseband data including LNA/PA, downconversion/upconversion mixers, band-pass/low-pass filters, ADC/DAC, synthesizer and voltage-controlled oscillator. To exchange digitized I/Q baseband data between the baseband processing block, it uses 64~MHz double data-rate low voltage differential signaling (LVDS). The maximum sampling frequency for the I/Q data is 4~MHz which means that it can support modulations with maximum of 2~MHz bandwidth. Note that, most IoT communication protocols have low bandwidth of 500~KHz (e.g. LoRa, SigFox \cite{xxxx}). Moreover, the digital analog baseband front-end, ADC/DAC, has 13 bits which is even better than the analog front-end (AD93XX~\cite{xxxx}) used by USRP \cite{xxxx}. \textcolor{red}{the low bandwidth and sampling rate xxxx, mention it or not?}. Fig. \ref{fig:} compares power consumption this transceiver with other transceivers that are used in existing products. As can be seen, power consumption is more than \textcolor{red}{5 xxx} times less than the closest SDR platform. \textcolor{red}{Moreover, the price of this transceiver chip is less than 5 dollars which makes this design very low cost.} In summary, AT86RF215 satisfies all the requirements of the wide-spread cheap SDR for IoT applications.
\item {\bf Baseband-Processing Unit.}
The processing is composed of FPGA and MCU and is responsible for communication protocol implementation including modulation and demodulation and MAC layer. FPGA is responsible for doing modulationa and demodulation and MCU for MAC layer implementation. We selected FPGA by analyzing different IoT communication protocols' receiver algorithm, FPGA computational resource needed to enable these algorithms and also computational resource needed to do localization xxxx. Moreover, it needs to support LVDS interface to be able to talk to the transceiver chip. We have used M2GL010T from IGLOO 2 series of Microsemi for our first prototype. We chose MSP430FR5969 from TI MSP430 series for MCU because of its power performance.
\end{itemize}
MCU provides sensor interface, controls the analog front-end and power management unit (PMU). We provision microSD card to record raw I/Q samples and packet information for research projects that need data collection.
\subsection{MAC Layer}
As shown in Fig. \ref{xx}, the other block that we have used in our design is control unit which is implemented on a low power MSP430~\cite{xxxx} MCU. This unit serves multiple purposes. First, it talks to the radio to adjust its settings including synthesizer frequency, analog front-end gain, TX transmit power,sampling rate, filtering parameters and etc. depending on the application and communication protocol. Moreover, it talks to the power management unit to set voltage supply of different blocks and optimize them to minimize power consumption. In addition, it provides different communication interfaces (SPI, I2C and ...) to talk to different sensors that may need to be added.
\subsection{MicroSD Card}
The purpose of this unit is to enable recording high amounts of data (e.g. raw I/Q data or baseband bits) for later post-processing. MicroSD cards provide two modes of communication to the host: 1- xxxx 2- SPI based communication. The SPI-based communication is much simpler than the first option and can provide around xxxx~Mbps. To fast prototype our design, we have implemented the SPI-based communication between the FPGA and the MicroSD card.
\subsection{Over-the-Air Update}
\subsection{Protocols}
\subsubsection{LoRa}
\subsubsection{Zigbee}
\section{System Design}\label{sec:system}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{./figs/foo.eps}
\caption{\footnotesize{{\bf \name Hardware Design.}}}
\label{fig:sdr_hardware}
\end{figure}
The \name system is designed to fulfill the vision of an SDR based IoT testbed. In particular, \name system has flexible PHY and MAC layer, is capable of updating over the air (OTA) and low power. In this section, we explain the design of the \name system and present design choices and considerations. As shown in fig.~\ref{fig:system}, the system is divided to six different functional blocks: i) PHY layer, ii) MAC layer and central control, iii) Power Management Unit (PMU), iv) storage devices, v) sensor interface, and vi) OTA update unit.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figs/block_diagram/system.eps}
\caption{\footnotesize{\name system block diagram.}}
\label{fig:system}
\end{figure}
\subsection{MAC Layer and Central Control Unit}
The MAC layer controls channel access between different transceivers on the medium. In \name design, it is implemented on an MCU. Note that, since \name is designed for IoT protocols testing and implementation, it has relaxed latency requirements. This means there is no stringent timing requirement on the communication link between the PHY and MAC layer. MCU is the central controller of the system and in addition to hosting the MAC layer it controls sensor interfaces, OTA update process and all necessary functions to increase the battery life-time including duty-cycling and power-gating of different blocks.
transforms information bits into a radio waveform,
or vice versa. PHY has both control and data flows. The control
flow is relatively simple. It sets configuration parameters.
\subsection{PHY Layer}
The job of PHY layer is to convert data to RF signal on the transmit side and vice versa on the receive side. It is composed of RF front-end and baseband processing. On the transmit chain, FPGA gets data, does encoding and modulation and feeds it to the RF front-end to transfrom it to RF signal. On the receive side, RF front-end receives RF signal and sends it FPGA for decoding and demodualtion. The combination of a RF front-end capable of exchanging I/Q samples with an FPGA provides us with a flexible PHY layer. MCU controls the settings of this layer.
\subsection{Power Management Unit (PMU)}
PMU is designed to meet the expectations of the IoT testbed which are being power-optimized and programmable. To this end, we used high-efficiency programmable switching regulators controlled by the MCU. Moreover, the system is divided to different power domains based on the voltage requirements of the blocks of the system each powered by a separate regulator. The granular power distribution design in combination with the programmable regulators enables power-gating functionality and saves power as a result.
\subsection{Storage Devices}
We provisioned two storage devices in \name design: - Flash memory - microSD card. Flash memory is used to save the programming data of the FPGA and also programming data of the MCU in the case of multi-boot design. microSD card is several Gigabyte memory used to record I/Q samples to be post-processed offline.
\subsection{Over-the-Air Update}
One of the key features of our design is the OTA functionality. OTA is enabled by adding a second dedicated RF chip talking to the MCU as backbone. In \name design, after every preset time duration, \name's backbone radio talks to the IoT testbed central access point and checks if there is an update. If the answer is yes, first, it receives the compressed programming data of the FPGA and MCU itself and saves it in the flash memory. Then, MCU reads the data from flash memory, uncompresses and saves it in the flash memory again. Then, MCU activates the FPGA programming mode and FPGA code will be updated. As the last step, the MCU activates its boot loader and updates its own code if necessary. Note that during the update process, all the blocks except the backbone radio, MCU and flash memory are power-gated to save the energy.
\subsection{Sensor Interface}
To enable testing of different application scenarios, \name is utilized with common sensor interfaces like ADC, I2C and SPI. These interfaces are handled by the MCU.
\section{System Design}\label{sec:system}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{./figs/foo.eps}
\caption{\footnotesize{{\bf \name Hardware Design.}}}
\label{fig:sdr_hardware}
\end{figure}
The \name system is designed to fulfill the vision of an SDR based IoT testbed. In particular, \name system has flexible PHY and MAC layer, is capable of updating over the air (OTA) and low power. In this section, we explain the design of the \name system and present design choices and considerations. As shown in fig.~\ref{fig:system}, the system is divided to six different functional blocks: i) PHY layer, ii) MAC layer and central control, iii) Power Management Unit (PMU), iv) storage devices, v) sensor interface, and vi) OTA update unit.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 108
|
@interface HomeViewController : BaseViewController<SinaWeiboRequestDelegate>
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,231
|
Home Facility Locator Decommissioning Sites by Location or Name Uranium Recovery Sites
Page Last Reviewed/Updated Thursday, August 23, 2018
Rio Algom – Lisbon Valley
Uranium Recovery
San Juan County, UT
UT1900481
Uranium mining and milling occurred at the Site from 1972 to 1989. Two tailings impoundments were operated during mining. The former Lisbon Valley Uranium Mill has been decommissioned and was done under NRC oversight. Rio Algom has been the only licensee and operator for the former Mill site. It is believed that the site was decommissioned using a concentration-based criteria in accordance with 40 CFR 192.12(a), 192.32(b)(2), and 192.41. However, the DWMRC is waiting for the licensee to submit a closure report to confirm this assumption. Rio Algom is in the process of revisiting the groundwater modeling to re-establish new ACLs for uranium. Seepage of tailings water from the impoundments was the source of uranium contamination in the groundwater. The existing ACLs were approved by the NRC before the State of Utah became an Agreement State.
On February 7, 2011, an enforcement action was taken against Rio Algom for exceeding the Target Action Levels for uranium in wells RL-1 and EF-8 for two consecutive sampling events. Rio Algom is in the process of revising the ACLs for uranium. The revised ACLs must be approved by the DWMRC before the licensee can return to compliance. Another option available to the DWMRC is to require the licensee to remediate the aquifer to meet the Utah GWQS for uranium (30 μg/L). This is something the DWMRC is considering if the licensee fails to submit a closure report in the near future.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,879
|
'use strict';
Object.defineProperty(exports, "__esModule", {
value: true
});
var _pangu = require('../modules/pangu');
var _pangu2 = _interopRequireDefault(_pangu);
var _extend = require('../utils/extend');
var _extend2 = _interopRequireDefault(_extend);
var _upperCaseFirst = require('../utils/upper-case-first');
var _upperCaseFirst2 = _interopRequireDefault(_upperCaseFirst);
var _expsDict = require('../stores/exps-dict');
var _expsDict2 = _interopRequireDefault(_expsDict);
function _interopRequireDefault(obj) { return obj && obj.__esModule ? obj : { default: obj }; }
var lifeExp = function lifeExp(talks) {
var options = arguments.length <= 1 || arguments[1] === undefined ? {} : arguments[1];
// Init exps
var exps = talks;
// Init options
var defaultOptions = {
isTrim: true,
isWearGlasses: false
};
options = (0, _extend2.default)({}, defaultOptions, options);
/* ==== Teaching ==== */
// Replace text according to expsDict
for (var pattern in _expsDict2.default) {
exps = exps.replace(new RegExp(pattern, 'ig'), _expsDict2.default[pattern]);
}
// TODO: Wear glasses
if (options.isWearGlasses) {}
// Add space between Chinese and English characters
exps = (0, _pangu2.default)(exps);
// Remove leading and trailing excess spaces
if (options.isTrim) {
exps.trim();
}
// Upper case the first character of each sentence
var endPunctuation = ['。', '!', '?', // Fullwidth
'. ', '! ', '? ' // Halfwidth
];
endPunctuation.forEach(function (mark) {
exps = exps.split(mark).map(_upperCaseFirst2.default).join(mark);
});
return exps;
};
exports.default = lifeExp;
module.exports = exports['default'];
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,963
|
Q: geom_boxplot not displaying correctly In the assignment I am doing, it wants me to use geom_boxplot. However, I have been unable to get the graph to display the boxplots correctly.
# Convert To Factor
census_data$CIT <- as.factor(census_data$CIT)
class(census_data$CIT)
ggplot(census_data, aes(census_data[["VALP"]], (census_data[["CIT"]])) +
geom_boxplot(color = "blue", fill = "orange") +
ggtitle("Property value by citizenship status") +
xlab(""Citizenship status") + ylab("Property value")
I am slightly concerned that the CIT may not have been converted correctly to a factor.
A: I think you have your x and y aesthetics the wrong way around. you have VALP first which is then assumed to be x and CIT second which is asssumed to be y. Given your labels I think you want them in the other order.
I always find it helps to label them explicitly, ie aes(x=.., y=...) so you don't get confused!
You also don't need to use census_data[["VALP"]] in the aes function call, since you have supplied the census_data in the data argument just saying aes(x=CIT, y=VALP) should be enough.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,447
|
{"url":"https:\/\/markthegraph.blogspot.com\/2013\/05\/budget-charts.html","text":"## Saturday, May 18\n\n### Budget charts\n\nSome quick charts from the 2013-14 Budget papers. First, the revenue forward estimates and projections look far more plausible than they have in a long time. The last time the Budget revenue estimates have looked this plausible was MYEFO in 2009.\n\nWhile there has been quite a bit said about the size of revenue write-downs since the GFC, these observations should be contextualised with an assessment of the plausibility of past estimates and projections.\n\nOn the expenditure side, I am less convinced by the forward estimates and projections. Actual expenses in 2011-12 were $377.7b. The Budget last year projected$376.3b for 2012-13 (you may recall a whole heap of 2012-13 payments were prepaid from 2011-12 to achieve this \"no growth\" result). This week's Budget papers have the estimate for expenses at $381.4b for 2012-13, up$5.1b. Notwithstanding the \\$5.1b growth in expenses this year, the out-year estimates and projections in this year's Budget papers are the same or less than they were in last year's Budget papers.\n\nSo let's look at the balance statements.\n\nMuch is made of Treasury independence and expertise in framing Budget estimates and projections. Do not be fooled by these statements. Budget papers are profoundly political documents: they frame an important narrative for any government in respect of its economic competence. For this reason, they are not left to bureaucrats alone to write. The estimates and projections in the Budget papers are based on hundreds of parameters that are the subject of intense, iterative discussions between senior officials in the Treasury and the Department of Finance and the government of the day (regardless of its hue).\n\nThese negotiations are a fine balance. The government wants to maximise its narrative without over-reaching on its immediate and longer-term credibility. This process follows the rules of repeated games under game theory. The need for a government to retain the public's confidence from one year to the next means that Budget parameters and estimates typically lie within a reasonable (market expected) range, with an occasional upside or downside bias. Cumulatively these biases are designed to benefit the narrative of the government of the day.\n\nEarlier in the week, we saw Peter Costello admit that when he was the Treasurer framing his Budget papers, he opted for the conservative estimates for revenue growth from a range presented to him by Treasury. I am unaware of any public statements on Wayne Swan's biases, but my guess is that up until now they have been on the upside for revenue growth.\n\nIt will be interesting to see how these estimates and projections are reported when the Pre-election Economic and Fiscal Outlook (PEFO) comes out on or before 23 August 2013. Unlike the Budget papers, the PEFO estimates and projections are not based on parameters negotiated between the Treasury and Finance officials and the government of the day. Indeed, it may have been the possibility of a significant PEFO contradiction that prompted the more conservative estimates of revenue growth over the out years in this year's Budget papers compared with previous years.\n\nHaving said that, I do not expect a significant contradiction between the Budget and PEFO. But I will be looking for the odd 25 basis point movement in parameters here and there that cumulatively contribute to a headline movement in the expenses and balance charts above.","date":"2020-08-04 10:36:08","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\": 0.27534615993499756, \"perplexity\": 2219.184635949357}, \"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-2020-34\/segments\/1596439735867.93\/warc\/CC-MAIN-20200804102630-20200804132630-00347.warc.gz\"}"}
| null | null |
Aon is the global leader in human resource consulting. We partner with your organisation to deliver HR strategies that help you achieve business results.
Aon with its Human Resource data and benchmarking capabilities can help you gain a holistic understanding of how your organisation compares with others in the market.
Certification courses for HR professionals and business managers in the areas of business skills, HR technical expertise and business partnership.
Aon helps design or reviews your organisation's compensation strategy to ensure it aligns with your HR and business objectives.
Sebi regulations: Who would you rather be, CEO or Chairman of the Board?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,742
|
<?php
namespace Sylius\Bundle\FixturesBundle\DataFixtures\ORM;
use Doctrine\Common\Persistence\ObjectManager;
use Sylius\Bundle\FixturesBundle\DataFixtures\DataFixture;
use Sylius\Component\Cart\SyliusCartEvents;
use Sylius\Component\Core\Model\OrderInterface;
use Sylius\Component\Core\Model\OrderItemInterface;
use Sylius\Component\Core\Model\PaymentInterface;
use Sylius\Component\Core\Model\ShipmentInterface;
use Sylius\Component\Core\SyliusCheckoutEvents;
use Sylius\Component\Order\OrderTransitions;
use Symfony\Component\EventDispatcher\GenericEvent;
class LoadOrdersData extends DataFixture
{
/**
* {@inheritdoc}
*/
public function load(ObjectManager $manager)
{
$orderRepository = $this->getOrderRepository();
$orderItemRepository = $this->getOrderItemRepository();
$channels = array(
'WEB-UK',
'WEB-DE',
'WEB-US',
'MOBILE',
);
for ($i = 1; $i <= 50; ++$i) {
/* @var $order OrderInterface */
$order = $orderRepository->createNew();
$channel = $this->getReference('Sylius.Channel.'.$this->faker->randomElement($channels));
$order->setChannel($channel);
for ($j = 0, $items = rand(3, 6); $j <= $items; ++$j) {
$variant = $this->getReference('Sylius.Variant-'.rand(1, SYLIUS_FIXTURES_TOTAL_VARIANTS - 1));
/* @var $item OrderItemInterface */
$item = $orderItemRepository->createNew();
$item->setVariant($variant);
$item->setUnitPrice($variant->getPrice());
$item->setQuantity(rand(1, 5));
$order->addItem($item);
}
$this->createShipment($order);
$order->setCurrency($this->faker->randomElement(array('EUR', 'USD', 'GBP')));
$order->setShippingAddress($this->createAddress());
$order->setBillingAddress($this->createAddress());
$order->setCreatedAt($this->faker->dateTimeBetween('1 year ago', 'now'));
$this->dispatchEvents($order);
$order->calculateTotal();
$order->complete();
$paymentState = PaymentInterface::STATE_COMPLETED;
if (rand(1, 10) < 5) {
$paymentState = PaymentInterface::STATE_NEW;
}
$order->setCustomer($this->getReference('Sylius.Customer-'.rand(1, 15)));
$this->createPayment($order, $paymentState);
$order->setCompletedAt($this->faker->dateTimeThisDecade);
$this->setReference('Sylius.Order-'.$i, $order);
$manager->persist($order);
}
$manager->flush();
}
/**
* {@inheritdoc}
*/
public function getOrder()
{
return 60;
}
/**
* @param OrderInterface $order
* @param null|string $state
*/
protected function createPayment(OrderInterface $order, $state = null)
{
/* @var $payment PaymentInterface */
$payment = $this->getPaymentRepository()->createNew();
$payment->setOrder($order);
$payment->setMethod($this->getReference('Sylius.PaymentMethod.StripeCheckout'));
$payment->setAmount($order->getTotal());
$payment->setCurrency($order->getCurrency());
$payment->setState(null === $state ? $this->getPaymentState() : $state);
$payment->setDetails($this->faker->creditCardDetails());
$order->addPayment($payment);
$this->get('event_dispatcher')->dispatch(SyliusCheckoutEvents::FINALIZE_PRE_COMPLETE, new GenericEvent($order));
}
protected function createShipment(OrderInterface $order)
{
/* @var $shipment ShipmentInterface */
$shipment = $this->getShipmentRepository()->createNew();
$shipment->setMethod($this->getReference('Sylius.ShippingMethod.UPS Ground'));
$shipment->setState($this->getShipmentState());
foreach ($order->getInventoryUnits() as $item) {
$shipment->addItem($item);
}
$order->addShipment($shipment);
}
protected function dispatchEvents($order)
{
$dispatcher = $this->get('event_dispatcher');
$dispatcher->dispatch(SyliusCartEvents::CART_CHANGE, new GenericEvent($order));
$dispatcher->dispatch(SyliusCheckoutEvents::SHIPPING_PRE_COMPLETE, new GenericEvent($order));
$this->get('sm.factory')->get($order, OrderTransitions::GRAPH)->apply(OrderTransitions::SYLIUS_CREATE);
}
protected function getPaymentState()
{
return array_rand(array_flip(array(
PaymentInterface::STATE_COMPLETED,
PaymentInterface::STATE_FAILED,
PaymentInterface::STATE_NEW,
PaymentInterface::STATE_PENDING,
PaymentInterface::STATE_PROCESSING,
PaymentInterface::STATE_VOID,
PaymentInterface::STATE_CANCELLED,
PaymentInterface::STATE_REFUNDED,
PaymentInterface::STATE_UNKNOWN,
)));
}
protected function getShipmentState()
{
return array_rand(array_flip(array(
ShipmentInterface::STATE_PENDING,
ShipmentInterface::STATE_ONHOLD,
ShipmentInterface::STATE_CHECKOUT,
ShipmentInterface::STATE_SHIPPED,
ShipmentInterface::STATE_READY,
ShipmentInterface::STATE_BACKORDERED,
ShipmentInterface::STATE_RETURNED,
ShipmentInterface::STATE_CANCELLED,
)));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,161
|
\section{Introduction}
\sloppy
The idea that networks should learn to drive themselves is gaining traction \cite{feamster2017and}, taking inspiration from self-driving cars where driving and related functionality do not require human intervention. The networking community wants to build a similar cognitive control in networks where networks are able to configure, manage, and protect themselves by interacting with the dynamic networking environment.We refer to such networks as cognitive self-organizing networks \textit{CSON}. The expected complexity and heterogeneity of CSON makes machine learning (ML) a reasonable choice for realizing this ambitious goal. Recently artificial intelligence (AI) based CSON have attained a lot of attention in industry and academia.
In 2003, Clark et al. \cite{clark2003knowledge} proposed that ML and cognitive techniques should be used for operating the network, this knowledge plane incorporation will bring many advantages in networks such as automation of network management, efficient and real-time anomaly and intrusion detection and many related tasks. Due to limited computational resources and lack of ML abilities, the idea of knowledge plane was not implemented in networks. In recent years, the field of ML, especially ``neural networks'', have evolved rapidly and we have witnessed its success in vision, speech, and language processing. This huge success motivated networking research community to utilize deep ML tools for building CSON.
Deep ML or deep learning (DL) is a branch of ML where hierarchical architectures of neural networks are used for unsupervised feature learning and these learned features are then used for classification and other related tasks. DL classifiers are function approximators that require a lot of data for generalization. Although they have outperformed all other statistical approaches on large datasets, due to generalization error, they are very vulnerable to adversarial examples. Adversarial examples are carefully crafted perturbations in the input which when ML/DL algorithms are subjected to get classified in a different class with high probability.
In this paper, we take security to encompass the securing of all of the functional areas of CSON (i.e., ISO defined functional areas often abbreviated as FCAPS: fault, configuration, accounting, performance, and security) and experiment with multiple adversarial attacks on ML/DL based malware classification systems. Our experimental results demonstrate that the current state of the art ML/DL based networking solutions do not have substantial deterrence against adversarial attacks. Specifically, our experiments utilize the highly cited malware image dataset provided by Nataraj et al. \cite{nataraj2011malware} to perform adversarial attacks on malware classifier to demonstrate that using current ML/DL techniques in conjunction with CSONs can be a potential security risk.
\textbf{Contributions}:
In this paper, we have made the following contributions:
\begin{itemize}
\item To the best of our knowledge, we have made the first attempt to show that CSON utilizing ML/DL techniques are very vulnerable to attacks based on adversarial perturbations.
\item We have argued that existing defenses to overcome adversarial perturbations are not appropriate and efficient for CSON applications. We have also highlighted that protection schemes against adversarial examples create an arms race between adversaries.
\end{itemize}
The rest of the paper is organized as follow. In the next section, we review related research studies that focus on CSON and adversarial attacks on networking applications. Section III describes our research methodology; particularly, with reference to the dataset, the ML/DL model, used dataset, and threat model assumptions, and the adversarial attacks. In Section IV, we provide the details of our experimental evaluations and the potential defense against these attacks. In Section V, we discuss the posed questions as well as some future directions and challenges. Finally, Section VI concludes our study.
\section{Related Work}
Many applications of ML/DL in networking have been proposed in the last few years highlighting the applications, opportunities, and challenges of using ML/DL in networking domain \cite{alsheikh2014machine, bartulovic2017biases, bkassiny2013survey, fadlullah2017state, hodo2017shallow,
klaine2017survey, patcha2007overview, usama2017unsupervised, zhang2018deep, wang2018machine}.
Although many ML-based solutions for networking applications have been proposed, the networking community has not yet standardized any ML-based solutions for CSONs. This arises partly from the complexity of the CSON environment that is characterized by dynamically changing network environment, data sparsity, expected tussles between control loops, high dimensionality, label data scarcity, heterogeneity, offline data processing, and many other architectural issues.
CSON are expected to resolve the challenges of optimization, configuration, healing, and coordination in the communication and data networks by incorporating AI/ML based cognitive techniques. Latif et al. \cite{latif2017artificial} highlights AI as a potential enabler for CSON. Similar ideas based on deep reinforcement learning for learning from environment and experience termed as experience-driven networking are presented in \cite{xu2018experience}. Feamster et al. \cite{feamster2017and} termed this idea of learning from network environment for measuring, analyzing, and configuring network without any human intervention as self-driving networks. Jiang et al. \cite{jiang2017unleashing} highlighted the benefits and challenges in developing an intelligent data-driven network with the ability of learning from dynamic nature of the networking environment by using exploration and exploitation processes. Koley et al. \cite{koley2016zero} proposed and provided a framework for zero-touch networking and highlighted the need for CSON using Google's infrastructure network as an example. Mestres et al. \cite{mestres2017knowledge} revisited the possibilities of embedding artificial intelligence in networking and proposed an ML/DL based knowledge plane for networking applications and this new networking paradigm was termed as knowledge defined networking.
While ML/DL applications will be a core part of CSON, recent studies demonstrated that ML/DL models are very susceptible to adversarial examples \cite{liu2018survey} \cite{akhtar2018threat}. Although most existing studies in this domain have targeted image classification applications in which high-dimensional images are perturbed in a way that fools the algorithm without being the change being conspicuous to naked human eye, these attacks also pose a significant challenge to CSON since the underlying algorithms are largely similar.
Such adversarial attacks are performed to compromise the integrity in terms of misclassification, accuracy reduction, targeted misclassification, or decision boundary evasion of the ML/DL techniques. We can divide these adversarial attacks into two broader categories based on the adversary's/attacker's knowledge.
\begin{itemize}
\item \textbf{White-box Attack}: This attack assumes that the adversary has complete knowledge about the ML/DL architecture, training data, and hyper-parameters. For adversarial attacks on CSON, we assume a white-box attack setting.
\item \textbf{Black-box Attack}: This attack assumes that the adversary/attacker has no information about the ML/DL technique and hyper-parameters. The adversary acts as a standard user who can query the ML/DL based system and gets a response. These query-response pairs are later used for crafting the adversarial examples.
\end{itemize}
Most of the adversarial attacks are white-box attacks, but white-box adversarial examples can be converted into black-box attacks by exploiting the ML/DL transferability property \cite{papernot2016transferability}.
Since these adversarial attacks on ML algorithms have not yet been applied much in the case of networks, we will initially review their applications in other domains. Szegedy et al. \cite{szegedy2013intriguing} proposed the first successful adversarial attack that has fooled the state of the art image classifiers with very high probability. Goodfellow et al. \cite{goodfellow2014explaining} proposed an adversarial sample generation method called \textit{fast gradient sign method}, where adversarial perturbation was generated by taking the sign of the gradient of the cost function with respect to the input. Kurakin et al. \cite{kurakin2016adversarial} explored the vulnerability of ML/DL techniques in the physical world and demonstrated that a small invisible tweak to the input of an ML/DL techniques can result in incorrect results. Carlini et al. \cite{carlini2017towards} proposed three attacks by exploiting the three different distance matrices $(L_0, L_2,$ and $L_\infty)$ and showed that the defensive distillation method \cite{papernot2016distillation} used to prevent against adversarial attacks does not increase the robustness of the ML/DL techniques. Papernot et al. \cite{papernot2016limitations} proposed a \textit{saliency map based attack}, where saliency map is used to find the most discriminative features of the input that are then fractionally perturbed to form an adversarial attack on the ML/DL based classifiers. In 2017, Papernot et al. \cite{papernot2017practical} proposed a black-box attack, where adversarial attack transferability \cite{papernot2016transferability} is exploited to form a successful evasion attack. Further details about adversarial attacks on different vision, language, and text processing systems can be found in \cite{yuan2017adversarial} and \cite{akhtar2018threat}.
Adversarial attacks have not yet been explored for CSON, we will cover some general networking applications. In 2013, Corona et al. \cite{corona2013adversarial} highlighted the possibilities and open research challenges of adversarial attacks on intrusion detection systems. Hu et al. \cite{hu2017generating} proposed a generative adversarial network (GAN) based black-box attack on malware examples but training a GAN on malware examples is difficult and computationally exhaustive. Grosse et al. \cite{grosse2016adversarial} proposed an adversarial perturbation attack against deep neural networks for malware classification, where a restricted amount of feature perturbations are used to fool a deep neural network with $0.85$ probability which was previously classifying malware with $97\%$ accuracy. In the next section, we provide the details of the proposed approach to perform multiple adversarial attacks on CSON.
\section{Methodology}
In this section, we describe the approach followed in designing adversarial examples to evade the ML/DL based malware classification system which we use as a proxy for the functional areas of CSON. To the best of our knowledge, no standardized deep learning based solution for malware classification in the CSON has been proposed yet. In this work, we propose a deep neural network based solution for malware classification. Before delving deep into the details of the proposed model, we describe the threat model and few related assumptions.
\subsection{Threat Model}
In the following, we outline the salient assumptions regarding the adversarial threat:
\begin{itemize}
\item The adversary may have the knowledge about the trained model which includes model architecture and hyper-parameters, but the adversary cannot make any changes to the architecture or model parameters. This is a common assumption in the adversarial machine learning domain \cite{akhtar2018threat}.
\item The adversary can only perform attacks during the testing phase, attacks on the training data (i.e., poisoning attacks) are not within the scope of this study.
\item For malware classification, we assume that similar families of malware, when represented as grayscale images exhibit similar visual and texture representations. This hypothesis was proposed and defended in \cite{nataraj2011malware}. In this work, we utilize convolutional neural networks (CNN) for malware classification because CNN is by far the best feature extractors.
\item The goal of an attack is to compromise the integrity of the ML/DL based classification techniques through a reduction in the classification accuracy with small perturbations.
\end{itemize}
\subsection{Malware Image Representation}
In this paper, we have used grayscale malware image dataset provided in \cite{nataraj2011malware}, where a malware executable is converted to a grayscale image. This approach of conversion includes both static and dynamic code analysis. The executable code is converted to binary and then represented as 8-bit unsigned vectors, these 8-bit unsigned vectors are then reshaped to a 2D array which can be visualized as a grayscale image. Figure 1 is depicting the procedure of converting malware executable to a grayscale image.
\begin{figure}
\centering
\centerline{\includegraphics[width=0.5\textwidth]{cap.png}}
\caption{Depiction of malware executable as an image}
\end{figure}
\subsection{Malware Classification Model}
We propose a CNN based malware classification architecture. Table \ref{ta1} depicts the proposed architecture. CNN is a powerful DL technique that learns spatial feature representations using convolutional filters. CNN has the capability to tolerate the distortion and spatial shifts in the input data and extract features from raw input data. CNN provides the state-of-the-art solution for network traffic feature extraction and classification \cite{lotfollahi2017deep}, motivated by these successes, we explore the use of CNN for grayscale malware image classification.
In the proposed architecture, we re-scale the input grayscale images of various sizes to $28$ pixel wide and $28$ pixel high, where pixel values are between $0$ to $255$. These input values are subjected to a two-dimensional convolutional layer with $64$ filters of receptive field $14$ pixel wide and $14$ pixel high. After that, we use a rectified linear unit (i.e., ReLU) as an activation function. The resultant activation values are then passed on to a second convolution layer with $128$ filters and $5\times5$ receptive field. Again, we use a ReLU as an activation function. Similarly, the third convolution layer follows the same procedure mentioned earlier but with $128$ filters of $1\times1$ receptive field. After the third convolution layer, the resultant activation values are flattened and passed on to a fully connected layer with softmax as an activation function producing resulting probabilities. We use a variant of the stochastic gradient descent (SGD) as an optimization function and categorical cross-entropy as a loss function to train the CNN.
\begin{table}[!ht]
\centering
\scriptsize
\caption{Proposed CNN architecture for malware classification}
\label{ta1}
\begin{tabular}{|c|}
\hline
\begin{tabular}[c]{@{}c@{}}\textbf{Input}: Malware gray scale image,\\ Size: (28*28)\end{tabular} \\ \hline
\begin{tabular}[c]{@{}c@{}}\textbf{2D Convolution Layer}\\ (Filter Size: 14*14,\\ No. of filters: 64, \\ Activation Function: ReLU)\end{tabular} \\ \hline
\begin{tabular}[c]{@{}c@{}}\textbf{2D Convolution Layer}\\ (Filter Size: 5*5,\\ No. of filters: 128,\\ Activation\\ Function: ReLU)\end{tabular} \\ \hline
\begin{tabular}[c]{@{}c@{}}\textbf{2D Convolution Layer}\\ (Filter Size: 1*1,\\ No. of filters: 128,\\ Activation\\ Function: ReLU)\end{tabular} \\ \hline
\begin{tabular}[c]{@{}c@{}}\textbf{Dense Layer} \\ (Number of neurons: 25,\\ Activation function: Softmax)\end{tabular} \\ \hline
\begin{tabular}[c]{@{}c@{}}\textbf{Output}: \\ Malware\\ Classification Probabilities\end{tabular} \\ \hline
\end{tabular}
\end{table}
\subsection{Adversarial Attacks}\label{adv}
We performed \textit{fast gradient sign method}, \textit{basic iterative method}, and \textit{Jacobian-based saliency map} attacks on malware classifiers to demonstrate that ML/DL based malware classification methods in CSON are vulnerable to adversarial examples.
\subsubsection{Fast Gradient Sign Method}
Goodfellow et al. \cite{goodfellow2014explaining} proposed a fast method of generating adversarial examples, this method is called the fast gradient sign method (FGSM). This method exploits the vulnerability of deep neural networks to adversarial perturbations. FGSM performs one step gradient update along the sign of the gradient to solve the optimization problem. Formally, the perturbation is calculated as:
\begin{equation}\label{eq1}
\eta =\epsilon \textit{sign}(\nabla_x j_\theta(x,l))
\end{equation}
In equation \ref{eq1}, $\epsilon$ represents the update step width or magnitude of the perturbation, $\eta$ is the difference between original and perturbed input, $\nabla_x$ represents the gradient with respect to each example, lastly $j_\theta(x,\textit{l})$ is the loss function used for training the neural network for original example $x$ and its corresponding label $\textit{l}$.
The generated adversarial example $x^{'}$ is calculated as:
\begin{equation}\label{eq2}
x^{'} = x + \eta
\end{equation}
FGSM is a very powerful attack because it is resilient to the regularization techniques such as dropout and norm-based regularization methods.
\subsubsection{Basic Iterative Method}
Kurakin et al. \cite{kurakin2016adversarial} proposed an element-wise basic iterative method (BIM) for adversarial falsification. It is an iterative procedure for generating adversarial example for physical world applications. They improved the success rate of the FGSM attack by including an iterative clipping method for each pixel to avoid large changes in the pixel values.The generated adversarial example is calculated via multiple iterations. The adversarial example generation procedure is given as:
\begin{equation}\label{eq3}
x_{0} = x,
\end{equation}
\begin{equation}\label{eq4}
x_{n+1} = \textit{Clip}{_x,_\xi}({{x{_n} + \epsilon \textit{sign}(\nabla_x j_\theta(x,l))}})
\end{equation}
Where $x_{n+1}$ is an adversarial example after $n+1$ iterations. The rest of the parameters are similar to the one utilized in the FGSM attack.
\subsubsection{Jacobian-based Saliency Map Attack}
Papernot et al. \cite{papernot2016limitations} proposed a new efficient method for generating adversarial examples called the Jacobian-based saliency map attack (JSMA). This attack is an iterative method for generating a saliency map to find out the most discriminative features, a Small perturbation is added to these discriminative features to fool the classifier. This attack is based on calculating the Jacobian of the forward propagating examples with respect to the input sample. The procedure of generating the saliency map of each sample is given as:
\begin{equation}\label{eq5}
J(x) = \frac{\partial f(x)}{\partial x} = [\frac{\partial f{_j}(x)}{\partial (x{_i})}]
\end{equation}
This attack achieved $97\%$ accuracy by altering only $4.2\%$ of the input features. Although this attack provides very effective adversarial examples but it is computationally very expensive \cite{papernot2016limitations}.
\section{Experimental Evaluation} \label{ee}
We evaluated the CNN based malware classifier against adversarial examples. Through our experiments, we want to answer the following questions:
\begin{itemize}
\item \textbf{Question 1:} \textit{Since ML/DL techniques are necessary to fuel the CSON, do these techniques provide the necessary robustness required to deal with adversarial perturbations?}
\item \textbf{Question 2:} \textit{How to build deterrence against adversarial attacks in CSON?}
\item \textbf{Question 3:} \textit{Do the deterrence techniques against adversarial examples create an arms race between adversaries?}
\end{itemize}
Before answering these questions, we provide the details of the dataset used for our experiments.
\subsection{Dataset}\label{data}
Nataraj et al. \cite{nataraj2011malware} provided a malware grayscale images dataset based on their novel image processing technique where malware execute-able are viewed as a grayscale image for visualizing malware families for classification purposes. We evaluated the performance of our proposed CNN architecture and adversarial attacks on malware classifiers using this dataset. The dataset consists of $9,458$ malware images divided into $25$ different malware families like Allaple.L, Allaple.A, Lolyda. AA etc. These malware families belong to major malware types such as worm, PWS, trojan, Dialer, Tdownloader, rouge, and backdoor, more details about malware types and related families in the dataset is available in \cite{nataraj2011malware}. Here we want to highlight that to keep the excutability of the malware we have limited the scope of the perturbation to the uninitialized data and zero padding portion of the malware image. We utilized $70\%$ of the data for training and $30\%$ for testing. Figure 2 depicts a sample malware image and its associated attributes.
\begin{figure
\centering
\centerline{\includegraphics[width=0.25\textwidth]{mal1.png}}
\caption{Malware image and related features in the image.}
\end{figure}
\begin{table*}[]
\centering
\tiny
\label{table2}
\caption{FGSM attack and defense results with different values of epochs and $\epsilon$}
\begin{tabular}{|c|l|c|c|c|}
\hline
\multicolumn{5}{|c|}{Fast Gradient Sign Method Attack} \\ \hline
\multicolumn{1}{|l|}{Epochs} & Epsilon & \begin{tabular}[c]{@{}c@{}}Test accuracy on\\ Legitimate Samples in (\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Test accuracy of \\ Adversarial Examples in (\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Test accuracy after\\ Adversarial training in (\%)\end{tabular} \\ \hline
\multirow{3}{*}{10} & 0.1 & 98.39 & 1.87 & 78.16 \\ \cline{2-5}
& 0.2 & 98.29 & 0.37 & 88.17 \\ \cline{2-5}
& 0.3 & 97.97 & 0.7 & 91.7 \\ \hline
\multirow{3}{*}{50} & 0.1 & 98.02 & 1.61 & 94.91 \\ \cline{2-5}
& 0.2 & 98.72 & 0.48 & 98.07 \\ \cline{2-5}
& 0.3 & 97.97 & 0.27 & 97.75 \\ \hline
\multirow{3}{*}{100} & 0.1 & 98.34 & 1.55 & 96.73 \\ \cline{2-5}
& 0.2 & 97.34 & 0.32 & 96.6 \\ \cline{2-5}
& 0.3 & 97.64 & 0.32 & 97.43 \\ \hline
\end{tabular}
\end{table*}
\begin{table*}[]
\centering
\tiny
\caption{BIM attack and defense results with different values of epochs and $\epsilon$}
\label{tab3}
\begin{tabular}{|c|l|c|c|c|}
\hline
\multicolumn{5}{|c|}{Basic Iterative Method Attack} \\ \hline
\multicolumn{1}{|l|}{Epochs} & Epsilon & \begin{tabular}[c]{@{}c@{}}Test accuracy on\\ Legitimate Samples in (\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Test accuracy of \\ Adversarial Examples in (\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Test accuracy after\\ Adversarial training in (\%)\end{tabular} \\ \hline
\multirow{3}{*}{10} & 0.1 & 97.22 & 0.90 & 61 \\ \cline{2-5}
& 0.2 & 96.95 & 0.75 & 36 \\ \cline{2-5}
& 0.3 & 97.32 & 0.91 & 35 \\ \hline
\multirow{3}{*}{50} & 0.1 & 97.91 & 0.48 & 72 \\ \cline{2-5}
& 0.2 & 98.07 & 0.27 & 38 \\ \cline{2-5}
& 0.3 & 97.38 & 0.70 & 30 \\ \hline
\multirow{3}{*}{100} & 0.1 & 97.91 & 0.70 & 77 \\ \cline{2-5}
& 0.2 & 98.81 & 0.64 & 47 \\ \cline{2-5}
& 0.3 & 97.59 & 1.02 & 31 \\ \hline
\end{tabular}
\end{table*}
\begin{table*}[]
\centering
\tiny
\caption{JSMA attack with average number of features perturbed for different values of epochs and $\gamma$}
\label{tab4}
\begin{tabular}{|c|l|c|c|c|}
\hline
\multicolumn{5}{|c|}{Jacobian-based Saliency Map Attack} \\ \hline
\multicolumn{1}{|l|}{Epochs} & Gamma & \begin{tabular}[c]{@{}c@{}}Test accuracy on\\ Legitimate Samples in (\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Test accuracy of \\ Adversarial Examples in (\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Average number of\\ Features Perturbed (\%)\end{tabular} \\ \hline
\multirow{3}{*}{10} & 0.1 & 95.93 & 9.13 & 90.5 \\ \cline{2-5}
& 0.2 & 97 & 4.8 & 95.8 \\ \cline{2-5}
& 0.3 & 96.62 & 4.8 & 95.2 \\ \hline
\multirow{3}{*}{50} & 0.1 & 97.53 & 4.83 & 94.28 \\ \cline{2-5}
& 0.2 & 97.48 & 5.0 & 91.04 \\ \cline{2-5}
& 0.3 & 96.62 & 4.8 & 95.2 \\ \hline
\multirow{3}{*}{100} & 0.1 & 98.28 & 19.67 & 75.88 \\ \cline{2-5}
& 0.2 & 98.28 & 7.83 & 88.09 \\ \cline{2-5}
& 0.3 & 97.21 & 5.0 & 90.83 \\ \hline
\end{tabular}
\end{table*}
\subsection{Results}\label{sec4}
We evaluated the performance of adversarial attacks on CSON using malware classifiers as a proxy. The dataset details are provided in section \ref{data}. Both FGSM and BIM attacks are element-wise attacks, with individual perturbation scope, non-targeted specificity and same perturbation magnitude parameter $\epsilon$. We performed both attacks using multiple values of $\epsilon$ with $10$, $50$ and $100$ epochs. Our experimental results are shown in Tables II and III. JSMA is a targeted, iterative, Euclidean distance based attack. It has two major controlling parameters; namely, \textit{maximum distortion parameter} $\gamma$ and \textit{rate of perturbation in the features} $\theta$. For this experiment, we fixed $\theta$ to be $+1$ and varied the value of $\gamma$ between $0.1$, $0.2$ and $0.3$ for $10$, $50$ and $100$ epochs. The achieved adversarial test accuracy values along with the average number of features perturbed for a successful adversarial example are reported in Table \ref{tab4}. For all aforementioned experiments, a batch size of $128$ and a learning rate of $0.001$ were used.
\subsubsection{Performance impact}
The CNN based malware classifier has a classification accuracy of $98.39\%$ when trained on legitimate examples. This accuracy is better than the best accuracy reported on the dataset in consideration. Adversarial test examples created by employing FGSM have reduced the classification accuracy from approximately $99\%$ to $1.87\%$ which is nearly $97\%$ loss in the accuracy of classification and prevention against adversarial examples. It also means that the probability of an adversary evading the malware classifier has increased from $1\%$ to $97\%$ which is very alarming. Similarly, the BIM attack reduces the test accuracy of adversarial samples to $0.9\%$ which is even worse than the FGSM attack. In case of JSMA, the classification accuracy decreased from $98.28\%$ to $7.87\%$ but it requires an $88.09\%$ of average feature perturbations to create successful adversarial examples, which is computationally very expensive. The full experimental results are summarized in Tables 2, \ref{tab3} and \ref{tab4}.
Malware classifiers are an integral part of the security architecture of CSON and we demonstrated that a very small perturbation in the test example has the potential to evade the integrity of the classifier. This performance degradation depicts the potential risks of applying ML/DL methods in the context of CSON without considering the robustness of ML/DL classifiers and building proper deterrence against adversarial examples. Without such deterrence, ML/DL models might cause more harm than good in CSON.
\subsubsection{Computational complexity}
Adversarial attacks are not just random noise/values added to the test samples. Instead, they are carefully calculated perturbations. These perturbations are based on exploiting the inherent generalization error and gradient variations in of ML/DL techniques. As the shown in Table \ref{tab4}, detecting and exploiting these errors to make effective adversarial examples is a computationally very complex and expensive process. Since JSMA works on saliency maps and forward derivatives to find the most discriminant features, it becomes computationally very expensive. Table \ref{tab4} depicts the average number of features perturbed to construct an adversarial example for each class, these values are surprisingly very high because for each example the underlying data contains $784$ features and each feature has a value greater than zero which is not the case in other standard datasets like MNIST \cite{lecun2010mnist}. This unusual property of the malware image dataset increases the search space to find the most discriminating features; thus, resulting in rapid increase in that computational complexity and poor performance of the JSMA attack.
\subsection{Adversarial Defense}
We need to identify that adversarial settings have been assumed in networks before through tools such as game theory, but unique challenges emerge and the stakes get higher when we give more control of the network to ML and algorithms in CSON \cite{manshaei2013game}. Barreno et al. \cite{barreno2006can} provided a taxonomy of defences against adversarial attacks, they have highlighted that regularization, randomization and information hiding can ensure defence against adversarial perturbation but these countermeasures are not very effective against attacks described in section \ref{adv}.
There are two major types of defenses against adversarial examples; namely, \textit{proactive} and \textit{reactive}. Proactive defenses include \textit{adversarial training} and \textit{network distillation}. Whereas reactive defenses include \textit{input reconstruction} and \textit{adversarial detection}. In this paper, we only consider proactive countermeasures against adversarial examples. More detail about reactive countermeasures against adversarial examples are explored in \cite{barth2010learning}.
\subsubsection{Adversarial Training}
One countermeasure against adversarial examples is to include adversarial examples in the training data for ML/DL techniques. Goodfellow et al. \cite{goodfellow2014explaining} proposed this idea and showed that ML/DL classifiers can be made more robust against adversarial examples by training them with adversarial examples. The purpose of including adversarial examples in the training is to regularize the ML/DL technique. This regularization helps to avoid over-fitting which in turn increases the robustness of the ML/DL technique against adversarial examples.
In this paper, we also explored adversarial training for making CNN models robust against FGSM and BIM attacks. Test accuracies before and after the adversarial training are reported in Tables II and \ref{tab3}. The results clearly show that performing adversarial training can increase the deterrence against adversarial attacks but it only provides defense against the adversarial examples on which it is trained, while other adversarial perturbations continue to pose a threat of evading the integrity of the classifier.
\subsubsection{Network Distillation}
Network distillation is another approach of forming a defense against adversarial examples. Hinton et al. \cite{hinton2015distilling} proposed the idea of distillation to improve the generalization of the deep neural networks. Papernot et al. \cite{papernot2016distillation} used the distillation process to form a defense against adversarial examples. Network distillation is a process of training a classifier such that the generation of adversarial examples becomes very difficult. This defense is based on hiding the gradients between pre-softmax layers and the softmax output, which reduces the chances of developing a gradient-based attack against deep neural networks. Since in this paper we consider white-box attacks where an adversary knows the model parameters (i.e., architecture, hyper-parameters, gradients, etc.), this defensive scheme is not applicable to our study. More information on defence schemes against adversarial examples can be found in \cite{yuan2017adversarial}.
\section{Discussions, challenges and Future Extensions}
Our experimental results clearly demonstrate that applying ML/DL techniques in CSON without taking into account adversarial perturbation threats can potentially lead to major security risks. To date, there does not exist any appropriate solution that provides deterrence against all kinds of adversarial perturbations. Our experiments answer the questions posed earlier in Section \ref{ee}. Furthermore, they provide the following insights:
\begin{itemize}
\item \textbf{Robustness of ML/DL for CSON}: In section \ref{sec4}, we have shown that CSON are very vulnerable to adversarial attacks. Sparsity, high dimensionality, unstructured nature, unique data packing scheme, large salient feature decision space of network data and less fault tolerance makes adversarial attacks more lethal for CSON as compared to other vision and language data. Given the adversarial threat, networking community has to come up with new ML/DL mechanism to ensure appropriate deterrence against adversarial examples. Robustness can be introduced by incorporating approximation and fault tolerance on top of defense techniques against adversarial threats.
\item \textbf{Deterrence against adversarial attacks in CSON}: We have performed proactive defense against adversarial attacks by training on adversarial examples. This adversarial training procedure provides deterrence against the adversarial examples it is trained on but an unknown adversarial perturbation can evade the classifier. Table II depicts that when the classifier is trained via an adversarial training procedure, it enables the malware classifier to classify FGSM based adversarial examples correctly with $97.43\%$ accuracy after $100$ epochs but the same classifier was unable to classify BIM attacks with appropriate accuracy even after $100$ epochs of adversarial training.
This shows that before incorporating ML/DL techniques in support of CSON applications like routing, intrusion detection, traffic classification, malware detection, the research community needs to figure out an appropriate defense against all adversarial perturbations. The margin of error in adversarial examples classification is very narrow in networking application when compared to computer vision problems.
Building deterrence against adversarial examples requires a method to improve generalization, this can be achieved via constraint objective function optimization, distributed denoising, and exploiting vicinal risk minimization instead of empirical losses. Apple Inc. \cite{moosavi2018divide} proposed a distributed denoising scheme for building deterrence against adversarial attacks for security-critical applications whereas Zhang et al. \cite{zhang2017mixup} proposed a method for improving the generalization of the ML/DL schemes which uses vicinal risk minimization rather than conventional empirical loss minimization. This procedure improves the robustness of ML/DL techniques against adversarial examples. Our experiments demonstrate that CSON are currently lacking the capability to provide appropriate defense against adversarial attacks on ML/DL techniques.
\item \textbf{Arms race between adversaries}: Our experiments also highlight that using ML/DL techniques in CSON can lead to an arms race situation between adversaries. Consequently, adversarial attacks and defense mechanisms will be in an arms race where attackers keep on dynamically changing the adversarial perturbations and defenders have to adapt accordingly.
\end{itemize}
ML/DL techniques will enable future CSON but before their deployments, the research community has to figure out an effective way to deal with adversarial attacks.
\subsection{Open issues}
\begin{itemize}
\item \textbf{Standardized datasets}: Progress in CSON largely depends upon learning from data obtained from the user, operating system, and application. Unfortunately, there does not exist a single standardized dataset for benchmarking ML/DL techniques for real-time networking applications. In order to ensure a proper utilization of ML/DL techniques with efficient deterrence against adversarial examples networking community has to come up with standardized datasets for security-critical applications.
\item \textbf{Learning from untapped network data}: Building deterrence in CSON against adversarial examples can be achieved by improving the generalization of ML/DL techniques. Generalization can be improved by harnessing the features from untapped networking data (network data that is recorded but not utilized in decision making) by introducing new network telemetry schemes for CSON. This can be a very promising way forward in realizing security critical CSON.
\item \textbf{New ML/DL mechanisms}: Conventional ML/DL techniques are very vulnerable to adversarial examples as shown in section \ref{sec4} and related defense schemes do not qualify for CSON applications. Developing new ML/DL schemes for unstructured networking data which are robust to adversarial threats is still an open avenue. Geometric and graph ML/DL techniques have the potential to solve this issue but have not yet been explored in this context.
\end{itemize}
\section{Conclusion}
In this paper, we evaluated the feasibility of employing ML/DL techniques to realize CSON in security critical applications and their ability to defend against adversarial examples. We demonstrated that network data is highly susceptible to adversarial attacks. We also evaluated the proactive defense mechanisms to build a defense against adversarial perturbations. Our experiments demonstrate that the application of ML/DL techniques in networking can push the limits on the state-of-the-art in CSON. However, without taking into account the threat of adversarial examples, significant security risks will be a major hindrance to the deployment of these networks.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,381
|
<?php
namespace SpBar\Bundle\BlogBundle\Model;
use Doctrine\ORM\EntityManager;
use SpBar\Bundle\BlogBundle\Entity\Tag;
class TagManager
{
protected $em;
public function __construct(EntityManager $em)
{
$this->em = $em;
}
public function createTag()
{
return new Tag();
}
public function updateTag(Tag $tag, $flush=true)
{
$this->em->persist($tag);
if($flush)
{
$this->em->flush();
}
}
public function removeTag(Tag $tag)
{
$this->em->remove($tag);
$this->em->flush();
return;
}
public function getTagById($id)
{
return $this->em->getRepository("SpBarBlogBundle:Tag")->findOneBy(array('id'=>$id));
}
public function getTagByName($name)
{
return $this->em->getRepository("SpBarBlogBundle:Tag")->findOneBy(array('name'=>$name));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,173
|
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